*** START OF THE PROJECT GUTENBERG EBOOK 74253 ***





                                  THE
                    PHILOSOPHICAL and MATHEMATICAL
                       COMMENTARIES OF PROCLUS,
                                  ON
                 THE FIRST BOOK OF EUCLID’S ELEMENTS.

                          TO WHICH ARE ADDED,
          A History of the Restoration of PLATONIC THEOLOGY,
                       BY THE LATTER PLATONISTS:
                  And a Translation from the Greek of
                    PROCLUS’S THEOLOGICAL ELEMENTS.

                            IN TWO VOLUMES.

  [Illustration]


                                VOL I.

                    LONDON, PRINTED FOR THE AUTHOR:
 And Sold by T. PAYNE and SON; B. WHITE and SON; J. ROBSON; T. CADELL;
   LEIGH and CO.; G. NICOL; R. FAULDER; and T. and J. EGERTON. 1792.

                    [Price Two Guineas in Boards.]




      _Extracts from Curiosities of Literature._ Second Edition.
                     Printed for Murray. Page 385.

Mr. _T. TAYLOR_, the Platonic Philosopher and _the modern Plethon_,
consonant to that philosophy, professes Polytheism.[1]




The Reader is requested to correct the following Errors.


   Page 4. of the Dissertation, Vol. I. line 8. for _admitted_,
   read _omitted_. Page 16. line 8. for _from_, read _form_. Page
   51. Vol. I. of the Commentaries, line 16. instead of _They are
   surely not the_, &c. read _For surely it cannot be said that
   there are_, &c. Line 17. for _but we_, read _but that we_. And
   line 19. for _is by much prior to_, read _is by a much greater
   priority_.

   Vol. II. page 18. line 26. for _and one is_, read _and one part
   is_. And line 27. for _another_, read _the other_. Page 114.
   line 13. for _The angle_, read _Let the angle_; and instead
   of _is bisected_, in the same line, read _be bisected_. Page
   411. line 2. for _is filled with intellect_, read _fills
   intellect_. And line 3. for _it also participates_, read _also
   it participates_.




                                  TO

                              THE SACRED

                                MAJESTY

                                  OF

                                TRUTH.




                               PREFACE.


The design of the present work is to bring us acquainted with
the nature and end of Mathematics in general, and of Geometry in
particular: and in the execution of this design our Author has
displayed an uncommon elegance of composition, and a most valuable
store of recondite learning. He is not content with every where
unfolding the full, and most accurate meaning of Euclid; but he
continually rises in his discourse, and leads us into the depths of the
Pythagoric and Platonic philosophy. We are surprised to find an use in
Geometry, which at present it is by no means suspected to afford. For
who would conceive that it is the genuine passage to true theology, and
the vestibule of divinity? This, indeed, is by no means the case when
it is studied for lucre, and applied to mechanical purposes; for then
the soul is neither elevated nor enlightened; but degraded and filled
with material darkness. Hence these Commentaries are alone valuable
to the liberal part of mankind, who look beyond _sense_ for
certainty; and who prefer things desirable for their own sakes, before
such as minister to the necessities of life.

The translation of this work is attended with great difficulty and
labour; not only from the sense of the philosopher, which is always
profound, and frequently obscure, but from the great incorrectness of
the Greek edition, in which, exclusive of numberless typographical
errors, entire sentences, essential to the connection, are frequently
omitted; and in one place two pages of the Latin translation are
wanting in the original, as will be shewn in our following notes.
Indeed, the Latin translation of Francis Barocius the Venetian, (Patav.
1560.) which was made from a variety of manuscripts, is inconceivably
valuable; for the diagrams, so necessary to a work of this kind, but
which are omitted in the Greek, are here inserted; and the version
is every where faithful, and sufficiently perspicuous to those who
are conversant in the ancient philosophy. Barocius justly cautions
the reader not to compare his version with the printed Greek, which
he observes is rather lacerated than printed; as indeed, without his
translation, it is impossible for any one to read the half of this
invaluable work, even though he should be as perfect in Greek as in
his native tongue. If I had not, therefore, fortunately acquired this
translation, which is at present very rare, I would have by no means
engaged in this arduous undertaking. Barocius, indeed, gives evident
proofs of his possessing the philosophical genius, by the excellence of
his translation, and his preface to the reader; and it is greatly to
be lamented that he did not adorn his version with explanatory notes,
which this profound work frequently demands, and which he was doubtless
well qualified to accomplish. This defect I have endeavoured, as far
as I was able, to supply; and at the same time have been cautious
neither to weary the reader by prolixity, nor by too much brevity to
leave him destitute of proper information. In the distribution of
the first book of this work into chapters, I have followed the order
of Barocius, because it is natural and obvious; and must beg leave
to solicit the reader’s indulgence for using the words _partible_
and _impartible_, differently from their common signification. These
words I have generally employed to express the meaning of μεριστός[2]
and αμεριστος[3] in the Greek, as I do not conceive that the words
_divisible_ and _indivisible_ always convey their full signification.
I have likewise used _quadrangle_ instead of square, and _quinquangle_
for the word pentagon. For if τρίγωνος be rendered _triangle_, why
should not τετραγώνος be rendered _quadrangle_? And, as Barocius
observes, why, for a similar reason, should not πεντάγωνος and
ἑξάγωνος be rendered _quinquangle_ and _sexangle_; and so of the rest?
Uniformity is always desirable when it can be obtained; and is no where
so necessary as in scientifical disquisitions.

It is likewise necessary to inform the reader, that though I have
always endeavoured to give the faithful meaning of my Author, yet I
have occasionally paraphrased his sense, when most obscure, and added
such elucidations of my own, as I either thought necessary to the
full comprehension of his matter; or which were naturally excited by
the fire and spirit of the Original. If it shall appear that I have
succeeded in the execution of this work, and rendered it intelligible
to the lovers of truth, I shall rejoice in my success, and consider
my labours sufficiently rewarded. The applause of the multitude I am
neither likely nor desirous to gain; but I am anxious to procure the
approbation of the discerning few, who know that the age of philosophy
is past; and who esteem the works of her ancient heroes as the most
precious treasures which have escaped the ravages of time.

       *       *       *       *       *

Time, indeed, is like a deep and rapid river; whatever is trifling and
light, is precipitately borne on its surface, and what is valuable and
weighty, sinks to its bottom. Hence, the superficial observer collects
nothing more than the rubbish, which it is forever devolving into the
abyss of oblivion; while the profound and contemplative genius explores
the depths of the stream, and accounts himself happy if he can gather
any of the pearl which its bottom contains. Thus the discoveries of
experimental philosophy, float like straws on the surface, while
the wisdom of Pythagoras and Plato lies concealed in the depths of
the river. I am well aware it will be said, that the reverse of this
similitude is true; that the modern philosophy is the pearl, and
the ancient the stubble; and that the former will be celebrated by
posterity, and increase in reputation when the latter shall scarcely
be known. But let us attentively examine the truth of this assertion,
and shut our ears to the unsubstantial echoes of popular applause. Is
it reasonable to suppose that men of such exalted abilities, as the
Pythagoric and Platonic philosophers possessed, even in the estimation
of their opponents, accompanied with the greatest advantages of birth
and fortune, and the most unwearied attention, have discovered nothing
valuable, and have left nothing behind them, but jargon and reveries?
Is it to be supposed, that in an age when philosophy was almost
adored; when it was esteemed by kings, cultivated by noblemen, and
even reverenced by the vulgar; when empire was relinquished for its
pursuit, and every danger encountered for its possession: is it to be
supposed, that nothing but delusion was the offspring of so glorious a
period, and nothing but folly the reward of such generous endurance?
Or shall we say, that the discovery of truth was reserved for the age
of experiment; and that she is alone to be apprehended in the infinite
labyrinth of particulars? That she is to be investigated with the
corporeal senses, and not with the powers of intellect; and that
the crucible, the alembic, and the air-pump, are the only mediums of
detection? If this be the case, truth is material, and may be calcined,
distilled, and rarefied, like any other corporeal substance. It is
no longer eternal and immutable, but perishable and fluctuating; the
phantastic subject of sensible inspection, and not the steady and real
object of the permanent energies of science. Shall we call this the
age of philosophy, in which talents are prostituted for sustenance,
and learning submits to the impudence of wealth? Shall we say that we
have strengthened the cause of philosophy, by demolishing her schools;
and increased her independence, by enlarging the empire of commerce?
Where shall we find the man, who is at present reverenced for the
profession of teaching speculative truth, or indeed who teaches it
at all? Or should we chance to meet with such an obsolete character,
shall we find him supported by the profession? It is a well known fact,
that men formerly lived in the highest esteem by its propagation: it
is equally as notorious, that a man at present would starve by such an
attempt. Dare we assert, that the reason of this difference must be
ascribed to the greater liberality, and more philosophical spirit of
the present age? Shall we not rather say, that the period, in which
these ancient heroes lived, was the golden age of philosophy;--a
period so different from the present, as to appear fabulous on the
comparison? For mark the distinguishing characteristics of our
inferiority. The great object of ancient philosophy, was an accurate
speculation of principles and causes: but that of the modern, is a
confused investigation of effects. And if pursuits participate of the
nature of their subjects, and causes are more noble than effects, the
ancient philosophy must undoubtedly be more elevated than the modern.
Again, the object of the Pythagorean and Platonic philosophy was to
make its possessors wise and virtuous; and to elevate them above the
common frailties and imperfections of degraded humanity; and this end
was happily accomplished in its votaries, as their lives abundantly
evince: but the object of modern philosophy, is a promotion of the
conveniencies and refinements of life, by enlarging the boundaries
of traffic; and the Mathematical Sciences are studied solely with a
view to this enlargement. The design of the ancient philosophy was
to remove the causes of wonder, by contemplating effects in their
causes: the grand object of the modern, is to increase admiration, by
attempting to investigate causes through the infinity of particular
effects. So that philosophy, as Mr. Harris justly observes, now
ends where it formerly began. For either there is no such thing as
science, or if its existence be admitted, it can never be obtained by
experimental enquiries; as these must be liable to all the inaccuracy
and imperfection of their material subjects.

In short, the philosophy of Pythagoras and Plato will be found, when
impartially considered, to contain every thing which can enlighten the
mind, improve the morals, and exalt the character of man. It is built
on the steady basis of truth, and will survive the wreck of ages. Its
foundation is deep, and its summit reaches the heavens. It is a mighty
rock, which modern systems _may_ assail, like a raging sea; but,
like stormy waves, they will only be broken about its impenetrable
sides. To war against wisdom is folly; for opposition in this case is
the destruction of its author. The moderns may, indeed, expect, because
their merit is raised by the present age, above that of the ancients,
to appear as giants in the eyes of posterity; but they will only verify
the elegant observation of the poet[4], that

    Pygmies are Pygmies still, though perch’d on Alps,
    And Pyramids are Pyramids in vales.




                                   A

                             DISSERTATION

                                ON THE

                    PLATONIC DOCTRINE OF IDEAS, &c.




                              SECTION I.


The Platonic doctrine of Ideas has been, in all ages, the derision of
the vulgar, and the admiration of the wise. Indeed, if we consider
that ideas are the most sublime objects of speculation, and that their
nature is no less bright in itself, than difficult to investigate, this
opposition in the conduct of mankind will be natural and necessary;
for, from our connection with a material nature, our intellectual eye,
previous to the irradiations of science, is as ill adapted to objects
the most splendid of all, “as the eyes of bats to the light of day[5].”
And yet (as I presume, it will appear from the following discourse),
unless the existence of these lucid beings is admitted, there can
be no such thing as science; nor, indeed, any genuine knowledge at
all. Hence, an enquiry concerning their nature and reality, is highly
proper, as an introduction to the ensuing Commentaries, in which they
are considered as the stable pillars of all truth, and the prolific
principles of the universe.

But previous to this enquiry, it is proper to observe, that Plato
was not the inventor, though he was a strenuous asserter, of ideas;
for, in the Sophista he affirms, that ideas were the discovery of men
who excelled in wisdom and piety, and who contended for an invisible
essence. Diogenes Laërtius, indeed, asserts, that Plato received
the doctrine of ideas from Epicharmus. But Epicharmus was not their
inventor, because Pythagoras, and others of still higher antiquity,
were well acquainted with ideas; so that it may be affirmed, with much
greater truth, that Plato was instructed in their nature by Philolaus
his preceptor, and the disciple of Pythagoras. For Pythagoras, after
his mysterious manner, signified ideas by numbers. But, prior to
Pythagoras, Orpheus was an asserter of ideas, and called Jupiter, or
the dimiurgus of the world, “the idea of all things.” And, according
to Syrianus, the mundane sphere, celebrated by Empedocles, is no other
than the ideal world; so that the doctrine of ideas is as ancient as
that of wisdom itself.

But to begin with our enquiry: in the first place, without universals
there can be no science; for the flowing and perishing nature of
particulars is perfectly foreign from that stability and duration which
is requisite to objects of invariable truth. Neither is it possible,
that infinite individuals can exist without the subsistence of one
cause endued with infinite power; for all multitude must necessarily
originate from one, and must resemble its cause in as great a degree of
perfection as its nature can admit; by a diffused infinity, shadowing
forth that infinite power which subsists in indivisible union. Hence,
if this be the case, and if infinite men, horses, and a multitude
of other univocals, are produced in an infinite time, an unity of
infinite power must be the source of each, according to which they are
generated in a terminated manner to infinity in the universe. Again,
all animals are transmuted from that which is in capacity (i. e. seed),
into energy. But if this be true, it is requisite there should be some
animal in the universe, subsisting in ever-vital energy, which may
call forth that which is concealed in dormant capacity, into perfect
actuality. Thirdly, the celestial orbs would not perpetually revolve
in the same spaces, and after the same manner, unless one and the same
universal number, or idea, ruled in each. So, likewise, there is a
natural number in every animal; or those of the same species, would
not always (when perfect) be distinguished with the same invariable
organs; nor would they be subject to puberty and old age, at the same
time, unless they were detained by the same measure of nature. Besides,
the participation of universals, is evident in every sensible object.
Thus, the rational nature is united with every individual man. Thus,
animal subsists in a lion and a horse, in a man and a dog. And thus
the pentad, or number five, is participated in the five fingers, and
the duad in the nostrils, eyes, hands, and feet. But since these do
not subsist without a cause, but are perfected by certain determinate
natures, it is necessary there should be an universal animal, in the
whole of nature, separate from sensibles, by means of which this
sensible animal is generated. And that there should subsist in nature
a pentad, through which the hands are always adorned with that number
of extremities; and a duad, from which the two eyes and nostrils are
derived. But if nature does not possess these numbers from herself, as
she is not the first cause of all, but derives them from another cause,
in the same manner as matter from nature, it is necessary there should
be universals and numbers prior to nature, subsisting in far greater
purity and perfection.

Again, we may demonstrate the existence of ideas as follows: if the
Deity, in fabricating the universe, operated essentially (and there
is no other way in which we can conceive him to operate), he must
fabricate the universe, an image of Himself. But, if this be the case,
he contains in himself, in the manner of an exemplar, the causes of
the universe; and these causes are no other than ideas. Besides, this
consideration is not to be omitted, that the perfect must necessarily
antecede and preside over the imperfect; unity over multitude; the
impartible over the partible; and that which is perpetually the same,
over that which admits of variation and change. From whence it may be
inferred, that things do not originate from baser natures, but that
their gradual processions end in these; and that they begin from the
most perfect, best, and most beautiful natures. But let us pursue this
reasoning more minutely, as it affords the strongest arguments for the
existence of ideas.

When the Deity fabricated the various species of animals, and bestowed
on them the different senses, it was doubtless with a view to the
benefit of their possessors, as he foresaw, that without these, the
animal could neither provide for its own support, nor defend itself
from surrounding dangers. But may we not enquire from whence this
previous perception originated? For it is not to be supposed, that he
first made animals destitute of senses, and so, being admonished by
their sudden destruction, afterwards assigned them to their nature.
Shall we say, this foreknowledge was the result of a reasoning process?
But then, we again ask, What were the principles of this ratiocination?
For if they originated from other reasonings, it is necessary, at
length, to arrive at something prior to these discursive operations,
on which they ultimately depend; since all reasoning must be founded
on indemonstrable principles. Was sense, then, or intellect the
principle of this previous perception? But, sense, in the present
instance, had not then a being, for it could not exist prior to the
animal nature: it was, therefore, intellect. But if intellect be the
repository of certain propositions, and the conclusion be science, it
must follow, that there could not then be a consultation of any thing
sensible. For the principle and the conclusion must both depend on
something intelligible. Besides, may we not ask, how such a habit of
thought arose before the existence of a sensible nature! It is absurd
in the extreme, to say from chance, and to resolve it into a sudden
volition of the Deity, is an assertion that may, indeed, satisfy vulgar
minds, but can by no means quiet the restless spirit of philosophical
investigation. Since, to suppose the cause of the universe, actuated by
sudden volitions, is to place him on a level with the vilest natures,
and subject him to the irrational impulses of the brute. Hence we
infer that the formation of animals, and by the same arguments of
the world, was not the result of any reasoning process. For, indeed,
argument and foreknowledge cannot with propriety be attributed to
the Deity; but when they are ascribed to him, we must consider it as
nothing more than an indication of his constituting particulars, in a
manner somewhat similar to the providence of a wise man, in inferior
concerns. For, in subordinate natures, whose operations cannot take
effect prior to enquiry, reason is necessary, on account of the
inferiority of that power which precedes the reasoning energy. In
like manner, foreknowledge is necessary, because a power is wanting
to its possessor, which might render him superior to its use. For
foreknowledge is directed to this end, that one particular circumstance
may take place in preference to another. But if it be requisite that
every energy in the Deity should be void of defect, and if it is not
lawful that any thing should be present with him, which is not total
and universal, it is necessary that all things should be contained in
every thing essential to the nature of the Deity. Hence, since even
futurity is with him present, there is nothing in him posterior; but
what is present in him becomes posterior, by its participation in
another. If then futurity be present with the Deity, it is necessary it
should be so present, as if foreknown in a posterior nature; that is,
in such a manner that nothing may be wanting to any being; and that is,
lastly, so that every thing may be complete.

Besides, reasoning cannot, by any means, belong to an eternal essence
like the deity; for if this be admitted, he must be forgetful of his
former operations. And if, in consequence of reasoning, he produces
more perfect natures afterwards, his works could not be perfectly
beautiful before: but if they were beautiful before, they must be
co-existent with their cause, i.e. they must be eternally beautiful,
antecedent to the reasoning energy. Again, if we suppose the supreme
intellect, the demiurgus of the world, to operate by enquiry, his
energy could not be spontaneous, and truly his own; but his essence
would be similar to that of the artificer, who does not derive his
productions from himself, but procures them as something adventitious
by learning and enquiry. But if the universe was not formed by
deliberation, it must be co-existent with its cause, and reside in
his essence; for if it be not co-existent there must have been some
particular time, in which its artificer determined on its production;
and this determination must have been the result of a reasoning
process, concluding that it would not be good to produce it before
that particular time, (from whence, by the way, we infer the eternity
of the world.) And if the universe be co-existent with its author, it
must perpetually emanate from his nature, and be dependent on it, like
the shadow on its forming substance. But in this case, its archetype
must be contained in the essence of its author; for every cause is
that primarily, which its effect is secondarily. And hence we infer,
that if the sensible universe be replete with forms of every kind, the
exemplars of those forms, must subsist in immaterial perfection, in the
artificer of the world.

If this sensible world, then, be formed according to the exemplar of
that which is intelligible; may we not say, with the great Plotinus,
that it is requisite universal animal should there primarily subsist in
perfect vital energy, containing all things in its omniform essence.
“Hence (says he[6]) the heavens are there a divine animal, replete with
ideal stars. Earth too does not there subsist solitary, but is much
more vital than this corporeal earth, for it is full of intellectual
life. The sea too is there, and all water subsisting in life, and an
ever-abiding stream. For how is it possible that any thing not vital,
can be the progeny of life itself? He, therefore, who enquires from
whence animals originate in the intelligible world, might as well
enquire from whence all life, and soul, and universal intellect, arose.
For here there is nothing indigent nor defective, but every thing is
perfect and exuberant. Here they all flow from one fountain, not as
from a certain spirit, or heat, but as if from an universal quality,
possessing and preserving in itself, all qualities; such as sweetness,
accompanied with fragrance of smell, the vigour of wine, and the
strength of all juices, bright colours, and whatever is perceived by
the taste.”

3. Such then are the arguments which the Platonic philosophy affords
in defence of ideas; the existence of which was so evident to Plato,
that, in the Sophista, he compares those who oppose the friends of
ideas to the giants of old, warring, as it were, on celestial souls,
and such as are engaged in sublime investigations. Let us now consider
to what universals these lucid beings are confined; since, according to
the Pythagoreans and Platonists, there are not ideas of all universal
conceptions. “For, in the first place (says Syrianus[7]), there are no
ideas of things evil and base, because these subsist in nature rather
by a privation and absence of ideas. And, on this account, they are
said to exist contrary to nature. Nor, secondly, of negations, for
these are destructive of the bound and limitation which is attributed
to every thing from the unifying and comprehending nature of ideas;
and hence, separation is rather the result of material infinity than
of that which is formal or ideal. Nor again, are there any ideas of
things which at different times receive a variety of conditions. For
these participate of transmutation from a moveable cause, but not from
the immoveable and stable illustration of ideas. Nor again of parts,
such as the hand, head, fingers, and the like. For the causes of
things existing entire, produce whole species and forms; not divided
about the parts of these, like the reasons of nature. But neither did
these wise men place in intellect the determinate causes of accidents
in bodies, such as sweetness and whiteness. For they considered that
natural reasons were sufficient for the production of accidents.
Nor again, of composites, as of a wise man. For since ideas are
simple, they preside over the simple essence of every thing. But the
composition and division of things is the business of our intellect;
ideas, at the same time, and that intellection which is co-ordinate
to ideas, being exempt from all these, on account of superlative
simplicity. Neither, therefore, must we establish ideas of things
generated from dissimilars, such as mules; nor of fruit produced by
engrafting from different trees. For all these have a posterior and
adventitious generation, and are not the work of nature alone, nor
of nature proceeding according to her own reasons, but, as it were,
compelled to labour contrary to her own determinations. Hence it is
manifest, that all art, which imitates nature, and alone ministers
to the use of mortal life, is separated from the cause of ideas. But
neither are the works which, depending on the purpose of the soul, are
perfected by a concourse of many causes, and which we are accustomed
to call the operations of fortune, to be conjoined to the cause of
ideas. For things which are there perfected, are eternal, and subsist
perpetually the same, free from the nature of contingent events. It
remains, therefore, that ideas must be confined to universal and
perfect essences, and to whatever confers to their natural disposition;
as for instance, to man, and every thing perfective of man, such as
wisdom and virtue. For ideas existing as the generative and energetic
causes of the perfection of every thing, distribute being to essences,
and convert them to the inexhaustible plenitude of their own omniform
natures.”

4. But let us now consider the nature of numbers; for as every form
is a number, according to the Pythagoreans[8], a speculation of this
kind must afford no small light to the arduous investigation of ideas.
Will it not, therefore, be proper, in the first place, to enquire,
with the great Plotinus[9], whether multitude is not a departure and
distance from _one_, so that infinity itself is a separation
from unity in the extreme, because it is no other than innumerable
multitude; that on this account it becomes evil; and that we contract a
similar nature when departing from intellectual unity, we are divided
by sensible multitude? For a being then properly becomes many, when no
longer able to remain collected in itself, the same, it is diffused
abroad, and thus, being dispersed, is variously extended; so that
when, by diffusion, it is absolutely deprived of unity, it becomes
perfect multitude, destitute of that universal cement, which unites
one part with another. But whenever the conciliating one is present,
then that which was scattered and diffused, becoming permanent by its
bounding power, passes into magnitude. But if any one should deny the
subsistence of unity, asserting that one is no where to be found,
which is not some particular one; and should hence affirm, that what
is called one abstractedly, is only a certain affection of the soul
towards any being; we ask, what prohibits the appellation of essence,
from being nothing more than an affection of the soul, and consequently
the existence of _being_, a delusion? For we predicate unity of
particulars with as great propriety as being. I am well aware, that
philosophers of the present day will answer, that we have an evident
proof of the reality of being, from its agitating the soul, and
becoming apparent in the phantasy: to which we reply, that in like
manner, the soul is agitated, and the imagination influenced about the
_one_. For every individual as much excites the perception of one,
as of being.

Besides, it is necessary to enquire whether we behold this passion and
conception of the soul, as one or multitude. And again, when we say
not one, we do not then possess one from the thing itself; for we say
that one is not contained in that individual. And hence we must possess
one in our own nature, and this must reside in the soul, separate
from that which is denominated some particular one. But here it may
be objected, that the one we possess is received from externals, and
is nothing more than a conception of the mind, produced by the thing
itself. For it will be said, that as multitude is nothing besides a
number of individuals, which are called many, so one is nothing besides
one thing; and is formed by thought separating that one particular from
others. To this we reply as follows:

How can it be consonant to reason to suppose that the conception of
_one_ arises from the sensation of some one particular subject?
For one particular man, who is discerned by sense, is by no means the
same with one itself, since, if this were the case, thought could never
predicate one of that which is not a man. Besides, as cogitation, on
beholding the different positions of things, affirms that this is here
or there, so when it perceives an individual, pronounces one; for that
passion is not vain, nor does it assert one of a non-entity. Nor must
we think it predicates ones, because this individual is different from
another; for when cogitation affirms such a thing is this, and not
another, it declares, in the mean time, that the other is _one_.
Likewise when it affirms that any thing is this _alone_, it then
declares, that what is alone is one: on which account, it predicates
one, prior to alone. Besides, if there be multitude, it is necessary
that one should antecede; since when it predicates _many_, it
pronounces more than one. And when it affirms that an army contains a
multitude of men, it conceives the soldiers reduced to one order.

For thought, indeed, does not permit multitude to remain perfect
multitude, destitute of the conciliating power of unity; in which
very circumstance, the subsistence of one is evinced; for acutely
and swiftly perceiving the one which results from order, it reduces
the nature of the many into one. Besides, we affirm that a house and
an army are each one, but that a house is more one than an army, on
account of the continuity of its parts. If therefore, one is contained
more in that which is continued than in that which is discrete, and
still more in what is perfectly indivisible, it is evident that _the
one_ is a certain nature, and has a real being. For it is impossible
that the more and the less should take place among things which have no
subsistence. If then it be not possible to understand any thing without
one or two, or some other number, it is by no means proper to deny
existence to that, without which we cannot comprehend the existence
or properties of any being: but it is requisite _that_ nature
should antecede all discourse, and intelligence, which is every where
necessary to their existence.

Again, if unity has no real subsistence, and is nothing more than
a name or conception of the mind, it may be destroyed without the
destruction of its subject. The unity, therefore, of a house may be
taken away, without the ruin of a house. But if a house is nothing more
than certain materials, reduced into one form, this is impossible. And,
on the contrary, the alteration of that subject, of which unity is
predicated, can make no real alteration in unity (on this hypothesis)
any more than the death of a man can affect his name. When, therefore,
a body, of which _one_ was predicated, is divided into a multitude
of parts, there is no real alteration made in the unity of the body,
because unity is nothing more than a name.

It was in consequence of this reasoning, and perceiving that unity
was participated by every being, that the Pythagoreans placed a
super-essential one at the top of the universe, intelligibly abstracted
from all beings in simplicity and excellence of nature. For they
considered, that unless there was a self-subsisting one in all things,
there could neither be universals nor particulars. Not the first,
because they are by nature _one_ and _many_. But it is requisite that
the one itself, should preside over that which is not one alone.
Nor again, the second, because they are many and one, (that is,
they participate more of multitude than unity, and their nature is
determined more by the _many_ than the _one_.) And because of things
in participation, unless an unparticipated one is added, there can
be no cause of union to beings; in the same manner as the cause of
essence to beings, is taken away by those who deny that being itself,
is the principle of all essence. For as the good itself, is the one
principle of good to the universe, and is nothing besides good; and
as a self-motive nature, which is nothing besides self-motion, is the
cause of motion to all things; so all things proceed from being itself,
and all united natures receive their union from _the one_, abstracted
from all things.

Hence (such is the absolute dominion of unity), continued quantities
would have no existence without its participation; for when they are
divided, so far as they lose unity, they change their being into some
other form. Hence, the bodies of plants or animals, which are each of
them one, when they fly from unity, and are dissipated into multitude,
immediately lose the essence they formerly possessed, and become
something else; which new state of being they likewise possess so far
as they are one. Add too, that health then flourishes in the corporeal
frame, when the body is conciliated into one; then beauty flourishes,
when the power of one connects the members into proportion and consent;
and then virtue reigns in the soul, when the soul is reduced into one
similitude with that which is divine.

5. But let us now investigate the nature of numbers. All number,
according to the Pythagoreans, originates from unity and the indefinite
duad; the first having the relation of form, and the second, that
of matter to all the orders of numbers. But they likewise divided
number into two kinds, essential and monadic. The essential number
they considered as first subsisting in the intelligible world,
together with being, and from thence distributed into all the various
gradations of forms. But the monadic, or that which is composed from
certain units, they justly considered as nothing more than the image
of essential number. And with respect to the numbers which the human
soul participates, these from its imperfect condition have a middle
subsistence; i. e. they exist in a vital, gnostic, and speculative,
but not in an operative manner. Hence, when receiving one thing with
another, we affirm, that they are two, as a dog and a man, or two men;
or when we compute more than two, as ten, and say that there is a decad
of men, this number is not essential to the two or ten individuals,
nor is it to be conceived as subsisting in sensible natures; but it
is purely quantity. But when we distribute this ten, into units,
we produce the principle of quantity, and generate a subject in
opinion[10], capable of participating the essential decad of our soul.
But when, considering man in himself, we affirm that he is a certain
number, as the duad, composed of animal and rational, we do not observe
one mode in this predication; but so far as by a discursive operation
of the soul, we numerate, we effect a particular quantum; but so far as
the subjects are two, and at the same time both one (since one fills
the essence of both, and in both unity is contained), we pronounce
another, and an essential number: and this duad is not of a posterior
origin, nor alone signifies a certain quantity, external to the
subject, but a duad subsisting in the essence of man, and containing
his nature. For here we do not produce a number by a discursive
operation, while we pursue essential natures. But when we number any
ten things, which are not connected by any conciliating unity, like
a choir, or an army, then this decad, which we predicate of the ten
particulars, subsists alone in our numerating soul, which renders
the ten individuals in opinion, a definite quantum. But in a choir,
or an army, essential number is participated exclusive of that which
subsists in our soul. And if it be enquired how number subsists in the
human soul, we must say, that the soul, by her self-moving energies,
procreates number, while she numerates, and by this energy, causes the
existence of quantity; in the same manner as in walking, we give rise
to a certain motion. Thus, monadic number, or a collection of units of
various kinds, subsists in opinion, in a manner correspondent to that
of geometrical figures; and by this means participates the essential
number of the soul. For as a triangular figure in the phantasy, is the
recipient of a triangular nature, or of triangle itself; so every three
units in opinion, receive the essential triad of the soul, and, by this
means, form a definite quantum.

In short, as in every being we may discern the resemblances of matter
and form, so in the pentad, or any other number, the five units, which
are the subject of participation, and the quantity of the number,
originate from the duad; but the form, that is the pentad itself, from
unity. For every form is an unity, which unites its subject quantity,
and connects it with its ideal species. It is, therefore, requisite
to understand, that the two principles of mathematical numbers are
resident in our souls, with which every mathematical number is
co-existent; I mean unity, comprehending in itself all the forms of
numbers, and which corresponds to unity in intellectual natures; and
the duad, endued with a generative power, of a formless nature, and of
infinite virtue; and which is called boundless, on account of its being
the image of never-failing and intelligible duality. Hence, the unity
of the soul, with a never-ceasing energy, continually distinguishes and
forms all the orderly processions of her numbers, suffers no vacuum
to intervene, and leaves no quantity formless and innumerable. Hence
too, no essential number of the soul, as for instance, the pentad,
is composed from substance and accident, as a white man; nor from
genus and difference, as man from animal and biped; nor again, from
five unities mutually touching each other, like a bundle of wood; nor
from things mixt, like water and wine, nor from things subsisting
by position, in the manner that stones compose a house; nor lastly,
does it subsist like things numerable; for it is not because they are
composed from indivisible units, that they possess any thing besides
units. For many points are indivisible, yet quantity is not produced
on this account; but because they participate of two natures, the one
corresponding to matter, and the other to form. Lastly, it is not
proper to say, that the number seven (and so of any other number), is
composed from the triad and the tetrad; for units, indeed, composed
with units, form a subject adapted to the reception of the heptad,
or the ideal and essential number seven; but the definite numerical
quantity seven, is formed from so many units, and the ideal heptad.
Hence, as the soul of the shipwright gives form to the timber, from her
inherent art; so the numerative soul, from the unity endued with the
relation of a principle which she possesses, gives form and subsistence
to all her inherent numbers. But there is this difference between the
two, that the shipwright’s art is not essential to our nature, and
requires manual operation, because it is conversant with sensible
matter; but the numerative art is essentially inherent in the soul,
and is therefore present with all men, and possesses an intellectual
matter, which it easily forms without the assistance of time. And
this, perhaps, is what deceives many, who think that the heptad is
nothing more than seven units. For the imagination of the vulgar,
unless it first perceives a thing destitute of ornament, and afterwards
the operations of the adorning artificer supervening its nature; and
lastly, beholds the thing perfect, and invested with form, cannot be
persuaded that it possesses two natures, the one formless, but the
other endued with an energetic and forming power.

And here it is necessary to observe, that though unity is the form
of all arithmetical forms, yet it is chiefly the form of the decad.
For what unity is simply to all the series of numbers, that the decad
is to the following hundreds, thousands, and millions; from whence,
according to a secondary progression, it is denominated unity. As
intellect, therefore, is the form of all things, but especially of the
soul, so unity, though it is the idea of all numbers, yet especially
of the decad. But the reason why the Pythagoreans extended ideal
numbers no farther than ten, is because this number is the ultimate
perfection of beings, containing all things in its omniform nature.
For all proportion subsists within the number ten; the arithmetical in
a natural progression of numbers from unity; the geometrical in the
numbers 1, 2, 4, and 1, 3, 9, and the harmonical in the numbers 2,
3, 6, and 3, 4, 6. And since the causes of all things are contained
in numbers, as far as to the decad[11], it is superfluous to suppose
exemplars of the following numbers.

If it should be asked in what manner we must conceive number as
subsisting in the intelligible world, we answer, with the great
Plotinus, that we must conceive it as subsisting in being itself,
with a power of impelling it to the production of multitude. “Hence
(says he, Ennead vi. lib. vi.) number is either the essence or the
energy of being, and animal itself, and intellect is number. But,
perhaps, we must call being, number united (ἀριθμὸς ηνωμένος),
but beings, number evolved, or unfolded; (ἐξεληλεγμένος ἀριθμὸς)
intellect, number moving in itself; (ἀριθμὸς ἐν ἐαυτῶ κινούμενος)
and lastly, animal, number comprehending (ἀριθμὸς περιέχων.“) It
was in consequence of this reasoning, that the Pythagoreans called
ideas numbers; because the gradual evolution of these from ineffable
unity, produced all the beautiful variety of forms. Their exalted
conceptions of numbers, likewise, originated from the same sublime
theory. Hence, [12]Pythagoras, in the sacred discourse, calls number
“_the ruler of forms and ideas_.” But [13]Philolaus, “_the
commanding and self-begotten container of the eternal duration of
mundane concerns_.” And [14]Hippasus, and all those who were
called ἀκουσματικοὶ (or such as were yet under the probation of the
quinquennial silence), “_the first exemplar of the mundane fabric,
and the judiciary instrument of its artificer_.”

6. And here I cannot but take notice, with regret, of the very
unphilosophical mistake committed by that great mathematician Dr.
Barrow[15]: I say, with regret, on account of the extraordinary
obligations I am under to his writings, for my proficiency (whatever it
may be) in mathematical learning. But respect must yield to the truth.
“Unity, says he, is not indivisible. (For how ex. gr. can 2/6 added
to 4/6 be equal to unity, if unity be indivisible and incomposed, and
represent a point) but rather only unity is properly divisible, and
numbers arise from the division of unity.” Here the Doctor evidently
confounds sensible units, which are the subjects of vulgar practical
arithmetic, with those units which are the objects of science.
Every individual sensible object, is indeed an unit, so far as it
participates the connecting and conciliating power of an immaterial
_one_: but the unity which stands at the top of speculative
arithmetic, is perfectly indivisible, or arithmetic would cease to be a
science. The truth of this is evident from Euclid’s definition: “Unity
(says he) is that according to which each of the things which are, is
called one.” But if unity be a composite, the definition is false;
since a composite, or a certain multitude, can never be the cause of
unity, but the contrary. And that this immaterial _one_ subsists
in sensible natures, has, I hope, been sufficiently proved in the
preceding part of this discourse. But the Platonic Theo[16] of Smyrna,
fully establishes the indivisibility of unity, as follows: “Unity is
terminating quantity, the principle and element of numbers, which
remains undiminished by the most immense multitude of subtractions,
and being deprived of all number, continues firm and fixt, because
it is impossible for division to proceed beyond the bound of unity.
Thus, if we separate any one corporeal substance into parts, the
_one_ again becomes _many_; and by subtracting the several
parts, we end in one part; and from this remaining part, again divided,
arises multitude; and by taking away every part, we again arrive at
_one_. So that _one_, considered as _one_, is incapable
of diminution, and perfectly indivisible. On the contrary, every
number is diminished by division, and is separated into parts less
than itself; as the number 6 into 3 and 3, or into 4 and 2, or into 5
and 1. But unity in sensible particulars, if divided, is diminished
after the manner of body, and by section is distributed into parts less
than itself: but it receives increase after the manner of number; for
instead of the one, multitude is produced. In this sense, therefore,
is unity indivisible; for nothing is divided into parts greater than
itself. But that which is cut into parts greater than the whole,
and into parts equal to the whole, is divided as number. Thus, for
instance, if any one sensible body is divided into six parts, 1, 1, 1,
1, 1, 1, these shall be equal to the whole; but by a section into 4
and 2, it is divided into parts greater than the whole, considered as
_one_; for 4 and 2 considered as numbers, exceed unity, and the
body was supposed to be one. Unity, therefore, as number is perfectly
indivisible. But unity is called by the Greek word μονάς, only, or
alone, either because it remains immoveable, and does not desert
itself, nor surpass the bounds of its nature (for it remains the same,
however multiplied into itself, through an infinite progression) or
because it is placed separate and apart from the multitude of other
numbers, it is denominated the _monad_, or _one_.”

In consequence of this very mistaken hypothesis, which opposes not
only all the wisdom of antiquity, but the sublimest truths, the Doctor
asserts, that an arithmetical cypher is the principle of numbers; and
that it is analogous to a point in geometry. Just as if a cypher, which
is nothing more than a mark expressive by its position with numbers,
of a certain quantity, had a real existence, and was productive of
number: when, at the same time, any other arbitrary character would
serve the same purposes, if applied in a similar manner. It must surely
afflict every thinking mind, to see how dreadfully the mechanical
system of philosophy, which has been so long in fashion, enslaves and
perverts the minds of its votaries; for there cannot, I think, be a
more egregious instance of its fatal tendency, than the present, in
which _nothing_ is considered as the foundation of that noble
science, arithmetic; which was deservedly placed by the ancients, in
the first rank of the mathematical disciplines. Such a foundation,
indeed, _may be_ proper to the _mechanical philosophy_, but
is very ill adapted to support the solid fabric of the arithmetical
science. But let us attend to the arguments of this most learned man,
in defence of so strange an assertion, “A cypher, or arithmetical
nothing (says he) is really the bound of every number coming between
it and the numbers next following, but not as a part. A cypher being
added to, or taken from a number, does neither increase nor diminish
it; from it is taken the beginning of computation, while itself is not
computed; and it bears a manifest relation to the principal properties
of a geometrical point.” But in what manner are we to conceive the
_nothing_ which intervenes between any two numbers, to be their
term or boundary? For Euclid defines a term to be the extremity of
any thing; implying by the extremity, something belonging to that of
which it is the bound. But how can a cypher, or _nothing_, in any
respect belong to number, or _something_? For if _nothing_
be a boundary, merely from its intervention, a point existing between
any two disjoined lines, though at the greatest distance from each,
must be their common boundary, which is evidently absurd. Besides, what
relation does it bear to a point, which is endued with a generative
power, by its flux forming the simple extension of a line, and, at
the same time, every where limiting its progression, and subsisting
in infinite capacity in its every part? Where are the real and divine
properties to be found in an arithmetical nothing, which Proclus,
in the following Commentaries, exhibits in a point? And how can
computation originate from a mere non-entity?

But a little consideration will convince us, that this Saracen, or
Indian cypher, is nothing more than an arbitrary character, invented
for the purpose of facilitating computation. For, suppose the letter
(_a_) to be placed in its stead, and to signify, when connected
with the mark for unity, ten, or ten multiplied by one; when connected
with the mark for two, ten multiplied by two, and so on. And again,
when placed twice after unity, let it express the second power of ten,
or one hundred, in this manner, _a a_; when thrice connected,
one thousand, or the third power of ten, and so on: shall we say, in
consequence of this, that (_a_) is the bound of numbers, and
the principle of arithmetic? Or, shall we not rather say, that it is
an arbitrary symbol, like any other algebraic character, having no
real connection with numbers, and depending, for its existence and
application, entirely on the will of its inventor. But this opinion is
too absurd to need any farther refutation.

7. It may here, perhaps, be expected, that I should explain how, in the
language of Syrianus[17], “divine number proceeds from the immortal
retreats of unity, until it arrives at the divine tetrad[18];” and
that I should unfold the properties of the tetractys, according to
the Pythagoreans; but an undertaking of this kind, would not only far
exceed the limits of this dissertation, but, perhaps, in the present
age, might be justly deemed, by the lovers of wisdom, a prostitution
and profanation of the most exalted truths. Enough, I hope, has been
said to excite the curiosity, and rouse the attention of the thinking
and liberal part of mankind; and those who understand what is here
briefly delivered, may apply themselves, with advantage, to Proclus on
Plato’s Theology, where they will find all the mysteries of numbers
unravelled; and to the works of the great Plotinus, who will lead them
into the penetralia of the most recondite wisdom. But, in perusing
the works of these great men, the reader must not expect to find the
sublimest truths explained in a familiar manner, and adapted, like
many modern publications, _to the meanest capacities_. For this,
indeed, is impossible to be effected. “Mankind (says Petvin[19]),
are not to be made any more truly knowing than happy by another’s
understanding.--There is no man can at once convey light in the higher
subjects, to another man’s understanding. It must come into the mind
from its own motions, within itself: and the grand art of philosophy,
_is to set the mind a-going_; and, even when we think nothing
of it, to assist it in its labour.” After which he observes, that
“the ancients never attempt to lead us into knowledge, by a continued
chain of reasoning; on the contrary, they write in such a manner,
as to force us to think for ourselves.” And, previous to this, he
remarks, “that there are certain truths acquired by a long exercise
of reason, both in _particular_, and likewise in those subjects
that are most _general_, as much, perhaps, out of the reach of the
greatest mathematician, as Sir Isaac Newton’s speculations are above
the capacity of some that are now called mathematicians.” The truth of
this observation is sufficiently evinced, in Plato’s definition of a
philosopher (in his Sophista), “The philosopher (says he) is the man
who sufficiently sees one idea every way extended through many, every
one of them lying apart; and many ideas different from one another,
externally comprehended under one.--And farther, one idea, throughout
all manys, wrapt up in one; and many ideas, every way separate or
discreet. This is to have the knowledge to discern how ideas, as they
are general, agree and disagree.” Now, he who thinks that a perception
of this kind may be acquired by barely reading an accurate discourse
on the nature of ideas, composed in intelligible terms, without, at
the same time, employing a long course of profound meditation, and
patient thought, knows but little the difficulty of the task, and
until he changes his opinion will never be the wiser. But the folly
and presumption of men, with respect to this sublime philosophy, is
really unpardonable; for there are very few who conceive that much
previous instruction is requisite to its acquisition; but almost
every man decides peremptorily on the most abstract speculations, and
reckons himself sufficient for the most profound investigations. In the
sciences and arts they are willing to proceed to perfection by gradual
advances; but they consider philosophy as easy, of instant access, and
hastily approach to her embraces with an assured confidence of success.
Though, like unhappy Ixion, through their presumption, instead of a
goddess, they grasp nothing but an empty cloud. Plato was so sensible
of this truth, that, in his seventh epistle to Dion, he expressly
affirms, that he neither has written, nor ever will write explicitly
concerning these sublime speculations; “For a thing of this kind (says
he) cannot be expressed by words, like other disciplines, but by a
lasting familiarity, and conjunction of life, with this _divine
object_, a bright light[20] on a sudden, as it were leaping from
a fire, will illuminate the soul, and there preserve and nourish its
splendor. He adds, that a publication of such concerns, is alone
useful to a few of mankind, who from some small vestiges previously
demonstrated, are sufficiently sagacious to their invention. But it
will fill others partly with a base contempt, and partly with a rash
and vain confidence, as if they had now learned some very excellent
things.” He then subjoins the following instance of the difficulty
attending such an undertaking: “There are three things (says he),
from which science must necessarily be produced; but the fourth is
science itself. And it is requisite to establish the fifth as that
which is the object of knowledge, and has a true existence. One of
these is the name of a thing; the second its definition; the third
the resemblance; the fourth science. Now take each of these, desiring
to learn what we have lately asserted, and think concerning them all,
in a similar manner. A circle is called something, whose _name_
we have just expressed. After this follows its definition, composed
from nouns and verbs. For that which every where is equally distant
from the extremes to the middle, is the definition of that which we
signify by the name of a round, and a circumference, and a circle. But
the third is the circle which may be painted, or blotted out, which
may be made by a wheel, or destroyed. None of which affections, the
circle itself, which each of these respects, suffers, as being of a
different nature. But the fourth is science, and intellect, and true
opinion about these. And this again must be established as one whole,
which neither subsists in voice, nor in corporeal figures, but in
intellect and intelligence. It is therefore manifest, that this fourth
is different from the nature itself of the circle, and again different
from the three we have previously mentioned. But among the number of
these, intellect, by its relation and similitude, proximately adheres
to the fifth, while the rest are more remote from its nature. The same
may likewise be affirmed of a straight and crooked figure, of colour,
and of the good, the beautiful, and the just. And again, of every
body, whether fashioned by the hand, or the work of nature, whether
fire or water, and the rest of this kind; likewise of every animal,
and the manners of animals; and of all actions and passions. For
unless, among these, some one, after a manner, receives that fourth,
he will never perfectly participate the science about the fifth.” He
then proceeds to shew in what respect each of the preceding four are
different from the fifth. “Every circle (says he) which by the hands
of men is either painted, or fashioned by a wheel, is plainly contrary
to our fifth. For it every where participates of the right-line. But
we must affirm, that the circle itself has neither more nor less of
any thing whatever; that is, it possesses in itself, nothing of a
contrary nature. Besides, none of these are endued with any stability
of name. For nothing hinders our applying the appellation of straight
to that which we now denominate round, and calling the straight by
the denomination of the round; nor will there be any less stability
in these, when their names are changed into the contrary. The same
reasoning is likewise true of definition, since it is composed from
nouns and verbs, which possess no stability. And in a variety of ways,
it may be proved, that no one of these four is certain and firm.” Now,
this fifth division of Plato’s entirely respects ideas, considered as
flourishing in intellect; by a conjunction with which, we acquire true
intelligence, and the perfection of human knowledge. The first three of
the preceding are obnoxious to various mutations; the fourth less; but
the last is perfectly stable and invariable. The three first are rather
conversant about the qualities of things, about the image and shadow;
the fourth raises us to the participation of truth; but the fifth to
truth itself, and permanent essence. In the first degrees almost all
are conversant; in the fourth a few; in the fifth, all the gods, but a
very small part of mankind, as it is asserted in the Timæus. The four
first may be known, indeed, without the fifth, confusedly; but from
the knowledge of the fifth they become perfectly manifest, as effects
from the knowledge of their cause. But we cannot, by any means, attain
to the apprehension of the fifth, unless we have been first accurately
conversant with the rest; for from our imperfect condition we are
compelled to rise from difference to identity, from multitude to unity,
and from shadow to substance. While we investigate the knowledge of
things, if we are alone desirous to apprehend their resemblance (which
is the case with the multitude) we shall be placed in the third degree,
and may easily acquire the object of our pursuit. But if we should
fortunately possess the true philosophical genius, which is rare in the
extreme, and aspiring to the fifth degree, should, by a happy event,
attain to its conjunction, though such a contact is clearer and more
certain than all knowledge; yet it is difficult to express it in words,
and to manifest it to others. And the reason of this is obvious: first,
because words are wanting, which exactly correspond to the essence of a
thing, since these are only the symbols of shadows. Secondly, because
we speak with those, who are alone conversant with shadows, and are on
this account derided by them, when they find that our fifth does not,
by any means, accord with material resemblances, which they consider as
the only realities.

8. And here a question very naturally presents itself for our solution,
whether the soul, while united with the body, is able to perceive
ideas, without the assistance of the phantasy, For it seems difficult
to apprehend how the soul, thus depressed and weighed down with the
terrene mass, should be able to raise herself to the supernal light
of ideas, and become united with their refulgence. The opinion of the
Peripatetics is well known, that some phantasm must always accompany
intelligence; but this is denied by the Platonists, and I think with
great reason. For the operations of intellect are not dependent on
the phantasy, though the perceptions of the latter proceed from the
energies of the former. Besides, as Plotinus beautifully observes, our
most vigorous energies are accompanied with the least animadversion;
and there is no absurdity in supposing that by increasing the force of
intellectual energy, we may speculate free from all imagination; since
the phantasms attending our conceptions, became weak in proportion
as the intellectual sight increases in vigour. On this account, the
Platonists affirm, that the moral virtues free us from the vehemence
of perturbations; but the contemplative from imagination, and the
senses. Hence too, the sciences may be called living waters; in which
the wings of the soul being dipt, her feathers, which were either
separated or broken by her lapse into body, are repaired, and restored
to a resemblance of their former perfection. For the wings are the
powers of the soul, leading to intelligibles: but the feathers are as
well the natural instincts to good and truth, as reasons inserted in
the soul; which either fall off, or are broken by her descent into
body, and conjunction with its ruinous bonds. But these are repaired
and invigorated by the sciences, which, like living streams, flowing
from the fountains of ideas, restore life and perfection to the soul.
Hence Plato, in the Phædrus, asserts that these wings of the soul
are increased by every thing which confers to supernal elevation; as
beauty, wisdom, and the like; and by a convenient metaphor, in the same
dialogue, he considers the chariot of the souls lives, her charioteer,
and the horses by which her car is drawn; and lastly, every thing which
contributes to the elevation of the soul, and her conjunction with
intellect and ideas. We may therefore conclude, that this conjunction
is possible to be effected, though it is rarely obtained; and that it
is a flight too arduous and sacred for the groveling and sordid; a
splendor too bright for the sensible eye; and a contact too ineffable
to be described by the unstable composition of words.

But I cannot conclude this section, without soliciting the reader’s
attention to a comparison of the difference between the ancient
philosophy, and that invented by Mr. Locke, and the moderns. According
to Mr. Locke’s system ideas are formed from sensible particulars, by a
kind of mechanical operation; so that truth is something by its nature,
posterior to sensation, and entirely dependent on it for existence.
According to Plato, ideas are eternal and immaterial beings, the
originals of all sensible forms, and the fountains of all evidence
and truth; so that on this system truth ranks among the first, and
not in the last of things; and would still retain its nature, though
the corporeal senses were no more. According to Mr. Locke, the soul
is a mere _rasa tabula_, an empty recipient, a mechanical blank.
According to Plato, she is an ever-written tablet, a plenitude of
forms, a vital and intellectual energy. On the former system, she is
on a level with the most degraded natures, the receptacle of material
species, and the spectator of delusion and non-entity[21]. Hence,
her energies are nothing but somnolent perceptions, and encumbered
cogitations; for all her knowledge terminates in sense, and her science
in passion. Like a man between sleeping and waking, her visions are
turbid and confused, and the phantoms of a material night, continually
glide before her drowsy eye. But on the latter system, the soul is the
connecting medium of an intelligible and sensible nature, the bright
repository of all middle forms, and the vigilant eye of all cogitative
reasons. Hence she is capable of rousing herself from the sleep of a
corporeal life, and emerging from this dark Cimmerian land, into the
regions of light and reality. At first, indeed, before she is excited
by science, she is oppressed with lethargy, and clouded with oblivion;
but in proportion as learning and enquiry stimulate her dormant powers,
she wakens from the dreams of ignorance, and opens her eye to the
irradiations of wisdom. On Mr. Locke’s system, the principles of
science and sense are the same, for the energies of both originate
from material forms, on which they are continually employed. Hence,
science is subject to the flowing and perishable nature of particulars;
and if body and its attributes were destroyed, would be nothing but
a name. But on the system of Plato, they differ as much as delusion
and reality; for here the vital, permanent, and lucid nature of ideas
is the fountain of science; and the inert, unstable, and obscure
nature of sensible objects, the source of sensation. On Mr. Locke’s
system, body may be modified into thought, and become an intelligent
creature; it may be subtilized into life, and shrink, by its exility,
into intellect. On that of Plato, body can never alter its nature by
modification, however, it may be rarefied and refined, varied by the
transposition of its parts, or tortured by the hand of experiment. In
short, the two systems may be aptly represented by the two sections of
a line, in Plato’s Republic. In the ancient, you have truth itself, and
whatever participates of the brightest evidence and reality: in the
modern, ignorance, and whatever belongs to obscurity and shadow. The
former fills the soul with intelligible light, breaks her lethargic
fetters, and elevates her to the principle of things; the latter
clouds the intellectual eye of the soul, by increasing her oblivion,
strengthens her corporeal bands, and hurries her downwards into the
dark labyrinths of matter.

Nor is it wonderful there should be so great a difference between the
two systems, and so much in favour of the ancients, if we consider the
great advantages these ancients possessed over the moderns in every
thing which contributes to the advancement of philosophy. For, in
the first place, they lived in an age when abstract investigations
were in the greatest request, and the professors of such pursuits in
the highest estimation. Besides this, they united the most exalted
abilities with the most unwearied attention and obstinate perseverance;
they devoted their whole lives to the search of truth; and relinquished
every thing which might be an obstacle to its acquisition. We may
add, likewise, the advantages of a language extremely philosophical;
and a freedom from the toil of learning any tongue but their own. Now
the reverse of all this is the portion of the moderns: for in the
present age, abstract speculations are ridiculed; and its professors
despised. The pursuit of truth is considered as perfectly consistent
with ordinary avocations, and is rather prosecuted as a relief from
the toils of business than as a thing desirable for its own sake,
and of the greatest dignity and worth. Hence, a few years desultory
application at a college, where language is one of the first objects
of attention, qualifies a modern for philosophy, raises him above
Pythagoras and Plato, and persuades him, with presumptuous confidence,
to enter the lists against these venerable heroes. And lastly, all
modern languages are barbarous with respect to the Greek; falling far
short of its harmony and energy, its copiousness and propriety. If
such then be the true state of the case, what judgment must we form
of men who, with all these disadvantages, philosophized without the
assistance of the ancients, despising their works, and being ignorant
of their contents? Shall we call it prudence or presumption, wisdom
or folly? Truth will certainly pronounce the latter; and the general
voice of posterity will confirm her decision. There are two egregious
instances in our own country of this daring presumption; I mean Bacon
and Locke. The former of these is celebrated for having destroyed the
jargon of the schoolmen, and brought experimental enquiries into
repute; and for attempting to investigate causes through the immensity
of particular effects. Hence, he fondly expected, by experiment piled
on experiment, to reach the principle of the universe; not considering
that his undertaking was as ridiculous as that of the giants of old,
who attempted to invade the heavens, by placing Ossa upon Pelion, and
Olympus upon Ossa; and ignorant that

    Heaven still, with laughter, the vain toil surveys,
    And buries madmen in the heaps they raise.

The latter of these, Mr. Locke, is applauded for having, without
assistance from the ancients, explained the nature, and exhibited
the genuine theory of human understanding. But that this applause
is false, the preceding comparison between his and the ancient
philosophy, may evince; and the variety of other self-taught systems
which, like nocturnal meteors, blaze for a while, and then vanish
in obscurity, abundantly confirms. Had these men, indeed, when they
justly derided the barbarous writings of the schoolmen, explored the
works of antiquity, penetrated the wisdom they contain, and enriched
their native language with its illustration, they had doubtless been
celebrated by the latest posterity: but, desirous of becoming masters
in philosophy by intuition, they disdained the instruction of the
ancients, and vainly attempted to soar on artificial wings to the
very summit of science. They are, however, destined, like Icarus, to
a precipitate fall; for the influence of time, which is continually
dissolving the cement of their plumes, is likewise continually
weakening their force, and will at last effect their final separation.
And thus much concerning the doctrine of ideas, and numbers, according
to Pythagoras and Plato.




                            SECTION II.[22]


But let us now consider the properties of the demonstrative syllogism,
and endeavour to unravel its intricate web; appointing Aristotle for
our guide in this arduous investigation. For an enquiry of this kind
is naturally connected with the doctrine of ideas, as it enables us to
gain a glimpse of the universals participated in mathematical forms,
and to rise to the principles of science. It brings us acquainted with
the laws which bind demonstration; and teaches us that objects of
intellect are alone the objects of science, and the sources of truth.

Previous to the acquisition of all learning and ratiocinative
discipline, it is necessary we should possess certain natural
principles of knowledge, as subservient to our future progress and
attainments. Thus, in every science there are some things which require
an immediate assent as soon as proposed; whose certainty is too evident
and illustrious to stand in need of any demonstrative proof deduced
from that particular science which, like stately pillars, they equally
support and adorn. Hence we are informed by the geometrician, that a
point is that which is destitute of all parts whatever; but we must
previously understand the meaning of the word _part_. Thus the
arithmetician defines an odd number, that which is divided according
to unequal parts; but it is necessary we should antecedently know
the meaning of the word _unequal_. Thus, too, art as well as
science operates by antecedent knowledge; and hence the architect,
the statuary, and the shipwright, learn the names and the use of their
respective implements, previous to the exercise of the materials
themselves. This is particularly evident in the discursive arts of
rhetoric and logic; thus the logician reasons by syllogism, the
rhetorician by induction, and the sophist by digressions and examples;
while each proceeds in an orderly progression from principles simple
and evident, to the most remote and complicated conclusions.

2. The antecedent knowledge of things may be divided into two parts:
the one a knowledge of their existence, or that they exist; the other
a knowledge of the terms expressive of their existence. Thus, previous
to the enquiry why iron is attracted by the magnet, it is necessary
we should learn the reality of this attraction, and the general mode
of its operation: thus too, in an enquiry concerning the nature of
motion and time, we must be previously convinced of their existence
in the nature of things. The second division of antecedent knowledge
takes place in subjects whose very existence admits of a dispute: thus
previous to a solution of the questions, Whether there are any gods
or not? Whether there is a providence or not? and the like, it is
necessary we should first understand the meaning of the terms; since
we in vain investigate the nature of any thing while we are ignorant
of the meaning of its name; although, on the contrary, we may have a
perfect conception of the meaning of some words, and yet be totally
ignorant whether the things they express have a real, or only an
imaginary existence. Thus, the meaning of the word _centaur_ is
well understood by every one; but its existence is questioned by most.

3. From hence it will easily appear, that no small difference subsists
between _learning_ and _knowledge_. He who is about to understand the
truth of any proposition, may be said to possess a previous conception
of its truth; while, on the contrary, it may happen that he who is in
the capacity of a learner, has no antecedent knowledge of the science
he is about to learn. Thus we attain to the distinct knowledge of a
thing which we formerly knew in a general way; and frequently, things
of which we were ignorant are learned and known in the same instant.

Of this kind are the things contained under some general idea, of which
we possess a previous knowledge: thus, he who already knows that the
three interior angles of every triangle are equal to two right, and
is as yet ignorant that some particular figure delineated on paper
is a triangle, is no sooner convinced from inspection of its being
a triangle, than he immediately learns and knows: he learns it is a
triangle; he knows the equality of its angles to two right ones. That
it is now a triangle he both sees and learns; but the equality of its
angles he previously knew in that general and comprehensive idea, which
embraces every particular triangle.

Indeed, a definite knowledge of this triangle requires two conditions:
the one, that it is a triangle; and the other, that it has angles
equal to two right. The first we receive from inspection; the second
is the result of a syllogistic process; an operation too refined for
the energies of sense, and alone the province of _intellect_
and _demonstration_. But demonstration without the knowledge of
that which is universal, cannot subsist; and since the proposition
is universal, that in every triangle the angles are equal to two
right; as soon as any figure is acknowledged to be a triangle, it must
necessarily possess this general property.

Hence we infer, that of the triangle delineated on paper, and
concealed, we are partly ignorant of this general property, the
equality of its angles (because we are ignorant of its existence);
and we partly understand it as included in that universal idea we
previously possessed. Hence too, it is evident that actual science
arises from a medium between absolute ignorance and perfect knowledge;
and that he who possesses the principles of demonstration, possesses in
capacity the conclusions also, however complicated and remote; and that
by an evocation of these principles from dormant power into energy, we
advance from general and abstracted knowledge to that which is sensible
and particular.

4. Two acceptations of knowledge may be admitted; the one common and
without any restriction; the other limited and peculiar. Since all
knowledge, whether arising from accidents, or supported by necessary
principles, is called science. Knowledge, properly so called, arises
from a possession of that cause from which a thing derives its
existence, and by which we infer the necessity of its existence; and
this constitutes simple and absolute science. Thus too, the definitions
of those general conceptions and suppositions, which from their
primary nature are incapable of demonstration, are called science.
But the science which treats of the method of arriving at knowledge,
is called demonstration; for every demonstration is a syllogism
producing science. Hence, if in every syllogism it is necessary that
the propositions should be the cause of the conclusion; and to know
any thing properly, a knowledge of its cause is requisite; in the
propositions of demonstration, both these conditions are required: that
they should be effective of the conclusion; and the causes of the thing
demonstrated.

Thus, from the ruins of a stately edifice, we may justly infer, that
the building was beautiful when entire; and from the smoke we may
collect the existence of the fire, though concealed: but the ruins of
the edifice are not the cause of its beauty; nor does fire originate
from smoke, but, on the contrary, smoke is the natural result of fire:
the inference, therefore, is in neither case a demonstrative one.
Again, since every cause is both prior to, and more excellent than its
effect, it is necessary that the propositions should be more peculiar,
primary, and excellent than the conclusions. And because we then know
a thing properly when we believe it to have a necessary existence,
hence it is requisite that the propositions should be true; for if
false, a false conclusion may ensue, such as, that the diameter of a
square is commensurable with its side. But if every science arises
from antecedent knowledge, demonstration must be founded on something
previous; and on this account it is requisite that the propositions
should be more known than the conclusions. The necessary properties,
then, of all demonstrative propositions, are these; that they exist
as causes, are primary, more excellent, peculiar, true, and known,
than the conclusions. Indeed, every demonstration not only consists of
principles prior to others, but of such as are eminently first; for
if the assumed propositions may be demonstrated by other assumptions,
such propositions may, indeed, appear prior to the conclusions, but
are by no means entitled to the appellation of first. But others, on
the contrary, which require no demonstration, but are of themselves
probable or manifest, are deservedly esteemed the first, the truest,
and the best. Such indemonstrable truths were called by the ancients,
axioms, from their majesty and authority; as the assumptions which
constitute the best syllogisms derive all their force and efficacy from
these.

And on this account, above all others, they merit the title of
the principles of demonstration. But here it is worth observing,
that these primary propositions are not the first in the order of
our conceptions; but first to nature, or in the nature of things.
To us, that which is first is particular, and subject to sensible
inspection; to nature, that which is universal, and far remote from
the apprehension of sense. Demonstration does not submit itself to the
measure of our ingenuity, but, with invariable rectitude, tends to
truth as its ultimate aim; and without stopping to consider what our
limited powers can attain, it alone explores and traces out the nature
of a thing, though to us unperceived and unknown.

This demonstrative syllogism differs not a little from others, by the
above property; the rest can as well educe a true conclusion from false
premises, which is frequent among the rhetoricians, as that which is
prior from that which is posterior; such as, Is every syllogism derived
from conjecture?

With respect to the rest, as we have already confessed, they may be
formed from principles that are true, but not from such as are proper
and peculiar; as if a physician should endeavour to prove an orbicular
wound the most difficult to coalesce and heal, because its figure is of
all others, the most capacious; since the demonstration of this is not
the province of the physician, but of the geometrician _alone_.

5. That proposition is called immediate, which has none superior to
itself, and which no demonstration whatever can confirm: such as these
are held together by the embraces of universals. There are some,
indeed, united from that which is sensible and particular: thus, that
the garment is white, is an immediate proposition, but not of that
kind whose principles require to be demonstrative ones; the cause
of which we shall hereafter investigate. Of immediate propositions
subservient to the purposes of demonstration, some are of such a
superior nature, that all men possess a knowledge of them without any
previous instruction; and these are called axioms, or general notions;
for without these all knowledge and enquiry is vain. Another species
of immediate propositions is position; incapable of being strengthened
by demonstration, yet not necessarily foreknown by the learner, but
received from the teacher. With respect to the genus of position, one
of its species is definition, and another hypothesis. Definition is an
oration, in which we neither speak of the existence, nor non-existence
of a thing; but alone determine its nature and essence. It is common to
every hypothesis, not to be derived from nature, but to be the entire
result of the art of the preceptor.

It likewise always affirms the existence or non-existence of its
subject: such as, that motion is, and that from nothing nothing is
produced. Those which are not so perspicuous are called postulates, or
petitions; as that a circle may be described from any centre, and with
any radius; and such as these are properly hypotheses and postulates.

6. We have now seen the privilege assigned to the principles of
demonstration:--whether or no our decision has been just, the
ensuing considerations will evince. We said that the assumptions in
demonstration were more known than the conclusions,--not indeed without
reason, since through these our knowledge and belief of the conclusion
arises. For universally, that quality which is attributed to many
different things so as to be assigned to one through the medium of
another, abounds most in that medium by which it is transmitted to the
rest.

Thus the sun, through the medium of the moon, illuminates the earth by
night; thus the father loves the preceptor through the medium of his
child. And in the first instance the moon is more lucid than any object
it enlightens: in the second, the child possesses more of the father’s
regard than his preceptor. If then we assent to the conclusions through
our belief of the principles alone, it is necessary that the principles
should be more known, and inherit a greater degree of our assent.
Hence, if it be true that the principles are more known than the
conclusions, it follows, that either our knowledge of them is derived
from demonstration, or that it is superior to any demonstrative proof;
and after this manner we must conceive of those general self-evident
notions which, on account of their indemonstrable certainty, are
deservedly placed at the top of all human science.

These propositions not only possess greater credibility than their
conclusions, they likewise inherit this property as an accession to
their dignity and importance; that no contrary propositions deserve
greater belief; for if you give no more assent to any principle than
to its contrary, neither can you give more credit to the conclusion
deduced from that principle than to its opposite. Were this the
case, the doctrine of these propositions would immediately lose its
invariable certainty.

7. There are, indeed, some who, from erroneously applying what we
have rightly determined, endeavour to take away the possibility
of demonstration. From the preceding doctrine it appears that the
principles are more aptly known than the conclusions. This is not
evident to some, who think nothing can be known by us without a
demonstrative process; and consequently believe that the most
simple principles must derive all their credit from the light of
demonstration.

But if it be necessary that all assumptions should be demonstrated by
others, and these again by others; either the enquiry must be continued
to infinity, (but infinity can never be absolved), or if, wearied by
the immense process, you at length stop, you must doubtless leave those
propositions unknown, whose demonstration was declined through the
fatigue of investigation. But how can science be derived from unknown
principles? For he who is ignorant of the principles, cannot understand
the conclusions which flow from these as their proper source, unless
from an hypothesis or supposition of their reality.

This argument of the sophists is, indeed, so far true, that he who does
not understand that which is first in the order of demonstration, must
remain ignorant of that which is last:--But in this it fails, that
all knowledge is demonstrative; since this is an assertion no less
ridiculous than to maintain that nothing can be known. For as it is
manifest that some things derive their credit and support from others,
it is equally obvious that many, by their intrinsic excellence, possess
indubitable certainty and truth; and command our immediate assent as
soon as proposed. They inherit, indeed, a higher degree of evidence
than those we assent to by the confirmation of others; and these are
the first principles of demonstration: propositions indisputable,
immediate, and perspicuous by that native lustre they always possess.
By means of these, we advance from proposition to proposition, and
from syllogism to syllogism, till we arrive at the most complicated
and important conclusions. Others, willing to decline this infinite
progression, defend the necessity of a circular or reciprocal
demonstration. But this is nothing more than to build error upon error,
in order to attain the truth; an attempt no less ridiculous than that
of the giants of old. For since, as we shall hereafter accurately
prove, demonstration ought to consist from that which is first, and
most known; and since it is impossible that the same thing should be
to itself both prior and posterior: hence we infer the absurdity of
circular demonstration; or those syllogisms in which the conclusions
are alternately substituted as principles, and the principles as
conclusions. It may, indeed, happen, that the same thing may be both
prior and posterior to the same; but not at one and the same time, nor
according to the same mode of existence. Thus, what is prior in the
order of our conceptions, is posterior in the order of nature; and what
is first in the arrangement of things, is last in the progressions of
human understanding. But demonstration always desires that _first_
which is prior in the order and constitution of nature. But the folly
of such a method will more plainly appear from considering its result:
let us suppose every _a_ is _b_, and every _b_ is _c_; hence we justly
infer, that every _a_ is _c_. In like manner, if we prove that every
_a_ is _b_, and by a circular demonstration, that every _b_ is _a_, the
consequence from the preceding is no other than that every _a_ is _a_;
and thus the conclusion terminates in that from which it first began; a
deduction equally useless and ridiculous. However, admitting that, in
the first figure, circular demonstration may be in some cases adopted,
yet this can but seldom happen from the paucity of reciprocal terms.

But that reciprocal terms are very few, is plain from hence: let
any species be assumed, as _man_; whatever is the predicate of
man, is either constitutive of his essence, or expressive of some
accident belonging to his nature. The superior genera and differences
compose his essence, among which no equal predicate can be assigned
reciprocable with man, except the ultimate differences which cannot be
otherwise than one, i. e., risibility, which mutually reciprocates
with its subject; since every man is risible, and whatever is risible
is man. Of accidents some are common, others peculiar; and the common
are far more in number than the peculiar; consequently the predicates
which reciprocate with man, are much fewer than those which do not
reciprocate.

8. It is now necessary to enumerate the questions pertaining to
demonstration; and for this purpose, we shall begin with propositions,
since from these, syllogisms are formed; and since every proposition
consists of a subject and predicate, the modes of predication must be
considered, and these are three which I call _total_, _essential_,
and _universal_; a _total_ predication takes place when that which is
affirmed or denied of one individual is affirmed or denied of every
individual comprehended under the same common species.

Thus, animal is predicated of every man, and it has this farther
property besides, that of whatever subject it is true to affirm man, it
is at the same time true to affirm animal.

Those things are said to be _essentially_ predicated; first,
when the predicate is not only total, but constitutes the essence
of the subject; instances of this kind are, animal of man; tree of
the plantain; a line of a triangle; for a triangle is that which is
contained under three right-lines. But here we must observe, that
not every total predicate is an essential one; thus, whiteness is
predicated of every swan, because it is inherent in every swan, and at
every instant of time; but because whiteness does not constitute the
essence of a swan, it is not essentially predicated; and this, first,
is one of the modes of essential predication of the greatest importance
in demonstration. The second mode is of accidents, in the definition
of which their common subject is applied: thus, a line is essentially
inherent in rectitude, because in its geometrical definition, a line
is adopted; for rectitude is no other than a measure, equally extended
between the points of a line. In the same manner, imparity is contained
in number; for what is that which is odd, but a number divided into
unequal parts? Thus, virtues are resident in the soul, because, in
their definition, either some part of the soul, or some one of its
powers, is always applied. The third mode of essential predicates
pertains to accidents which are inseparably contained in some
particular subject, so as to exclude a prior existence in any other
subject; such as colour in superficies. The fourth mode is of things
neither contained in another, nor predicated of others; and such are
all individuals, as Callias, Socrates, Plato. Causes are likewise said
to exist substantially, which operate neither from accident nor
fortune.

Thus, digging up the ground for the purposes of agriculture, may be the
cause of discovering a treasure, but it is only an accidental one. But
the death of Socrates, in despite of vigilance, is not the result of a
fortuitous cause, but of an essential one, viz. the operation of poison.

9. These posterior significations of essential predicates are added
more for the sake of ornament than use; but the two former have a
necessary existence, since they cannot but exist in the definition of
names which predicate the essence of a thing, and in subjects which are
so entirely the support of accidents, that they are always applied in
their definition. But it is a doubt with some, whether those accidents
are necessary, which cannot be defined independent of their common
subject? To this we answer, that no such accident can, from its nature,
be contained in every individual of any species; for curvature is not
contained in every line; nor imparity in every number; from whence we
infer, that neither is curvature necessarily existent in a line, nor
parity in number. The truth of this is evident from considering these
accidents abstracted from their subjects; for then we shall perceive
that a line may exist without curvature, and number without imparity.

Again, I call that an _universal_ predicate, which is predicated
of a subject totally and essentially, and considered as primarily
and inseparably inherent in that subject: for it does not follow
that a predicate, which is total, should be immediately universal;
for whiteness is affirmed of every swan, and blackness of every
crow, yet neither universally. In like manner, a substantial
predicate is not consequently an universal one; for the third mode
of essential predicates, and the two following (instanced before)
cannot be universal. Thus, colour, although inherent in superficies
essentially, is not inherent in every superficies, and consequently
not universally. Thus again, Socrates, Callias, and Plato, though they
exist essentially, are not universals, but particulars; and thus,
lastly, the drinking of poison was an essential cause of the death of
Socrates, but not an universal one, because Socrates might have died
by other means than poison. If then, we wish to render an accurate
definition of an universal predicate, we must not only say it is total
and essential, but that it is primarily present to its subject and no
other. Thus, the possession of angles equal to two right, primarily
belongs to a triangle; for this assertion is essentially predicated of
triangle, and is inherent in every triangle. This property, therefore,
is not universally in figure, because it is not the property of every
figure, not of a square, for instance; nor as universal in a scalene
triangle: for although it is contained in every scalene, and in every
equilateral, and isosceles triangle, yet it is not primarily contained
in them, but in triangle itself; because these several figures inherit
this property, not from the particular species to which they belong,
but from the common genus _triangle_. And thus much concerning
_total_, _essential_, and _universal_ predicates.

10. Concerning that which is universal, we are frequently liable to
err; often from a belief that our demonstration is universal, when it
is only particular; and frequently from supposing it particular when it
is, on the contrary, universal. There are three causes of this mistake;
the first, when we demonstrate any particular property of that which is
singular and individual, as the sun, the earth, or the world. For since
there is but one sun, one earth, and one world, when we demonstrate
that the orb of the earth possesses the middle place, or that the
heavens revolve, we do not then appear to demonstrate that which is
universal.

To this we answer: when we demonstrate an eclipse of the sun to arise
from the opposition of the moon, we do not consider the sun as one
particular luminary, but we deduce this consequence as if many other
suns existed besides the present.

Just as if there were but one species of triangles existed; for
instance, the _isosceles_; the equality of its angles at the base
would not be considered in the demonstration of the equality of all its
angles to two right ones: but its triangularity would be essential,
supposing every species of triangles but the isosceles extinct, and no
other the subject of this affection. So when we prove that the sun is
greater than the earth, our proof does not arise from considering it
as this particular sun alone, but as sun in general; and by applying
our reasoning to every sun, if thousands besides the present should
enlighten the world. This will appear still more evident, if we
consider that such conclusions must be universal, as they are the
result of an induction of particulars: thus, he who demonstrates that
an eclipse of the sun arises from the opposition of the moon between
the sun and earth, must previously collect, by induction, that when any
luminous body is placed in a right-line with any two others opaque,
the lucid body shall be prevented, in a greater or less degree, from
enlightening the last of these bodies, by the intervention of the
second; and by extending this reasoning to the sun and earth, the
syllogism will run thus:

  Every lucid body placed in a right-line with two others opaque,
  will be eclipsed in respect of the last by the intervention of
  the second;

  The sun, or every sun, is a luminous body with these conditions;

  And consequently the sun, and so every sun, will be eclipsed to
  the earth by the opposition of the moon.

Hence, in cases of this kind, we must ever remember, that we
demonstrate no property of them as singulars, but as that universal
conceived by the abstraction of the mind.

Another cause of deception arises, when many different species agree in
one ratio or analogy, yet that in which they agree is nameless. Thus
number, magnitude, and time, differ by the diversities of species;
but agree in this, that as any four comparable numbers correspond
in their proportions to each other, so that as the first is to the
second, so is the third to the fourth; or alternately, as the first
to the third; so is the second to the fourth: in a similar manner,
four magnitudes, or four times, accord in their mutual analogies and
proportions. Hence, alternate proportion may be attributed to lines
as they are lines, to numbers as they are numbers, and afterwards to
_times_ and to bodies, as the demonstration of these is usually
separate and singular; when the same property might be proved of all
these by one comprehensive demonstration, if the common name of their
genus could be obtained: but since this is wanting, and the species
are different, we are obliged to consider them separately and apart;
and as we are now speaking of that universal demonstration which is
properly one, as arising from one first subject; hence none of these
obtain an universal demonstration, because this affection of alternate
proportion is not restricted to numbers or lines, considered in
themselves, but to that common something which is supposed to embrace
all these, and is destitute of a proper name. Thus too we may happen
to be deceived, should we attempt to prove the equality of three
angles to two right, separately, of a scalene, an isosceles, and an
equilateral triangle, only with this difference, that in the latter
case the deception is not so easy as in the former; since here the
name triangle, expressive of their common genus, is assigned. A third
cause of error arises from believing that to demonstrate any property
inherent after some particular manner in the whole of a thing, is
to demonstrate that property universally inherent. Thus, geometry
proves[23] that if a right-line falling upon two right-lines makes
the outward angle with the one line a right-angle, and the inward
and opposite angle with the other a right one, those two right-lines
shall be parallel, or never meet, though infinitely extended. This
property agrees to all lines which make right-angles: but they are not
primarily equidistant on this account, since, if they do not each make
a right-angle, but the two conjointly are equal to two right, they
may still be proved equidistant. This latter demonstration, then, is
primarily and universally conceived; the other, which always supposes
the opposite angles right ones, does not conclude universally; though
it concludes totally of all lines with such conditions: the one may
be said to conclude of a greater _all_; the other of a lesser.
It is this greater _all_ which the mind embraces when it assents
to any self-evident truth; or to any of the propositions of Euclid.
But by what method may we discover whether our demonstration is of
this greater or lesser all? We answer, that general affection which
constitutes universal demonstration is always present to that subject,
which when taken away, the predicate is immediately destroyed, because
the first of all its inherent properties.

Thus, for instance, some particular sensible triangle possesses these
properties:--it consists of brass; it is scalene; it is a triangle. The
query is, by which of these we have just now enumerated, this affection
of possessing angles equal to two right is predicated of the triangle?
Take away the brass, do you by this means destroy the equality of its
angles to two right ones? Certainly not:--take away its scalenity, yet
this general affection remains: lastly, take away its triangularity,
and then you necessarily destroy the predicate; for no longer can this
property remain, if it ceases to be a triangle.

But perhaps some may object from this reasoning, such a general
affection extends to figure, superficies, and extremities, since, if
any of these are taken away, the equality of its angles to two right
can no longer remain. It is true, indeed, that by a separation of
figure, superficies, and terms, from a body, you destroy all the modes
and circumstances of its being; yet not because these are taken away,
but because the triangle, by the separation of these, is necessarily
destroyed; for if the triangle could still be preserved without figure,
superficies, and terms, though these were taken away it would still
retain angles equal to two right; but this is impossible. And if all
these remain, and triangle is taken away, this affection no longer
remains. Hence the possession of this equality of three angles to two
right, is primarily and universally inherent in triangle, since it is
not abolished by the abolition of the rest:----such as to consist of
brass; to be scalene, or the like. Neither does it derive its being
from the existence of the rest alone; as figure, superficies, terms;
since it is not every figure which possesses this property, as is
evident in such as are quadrangular, or multangular. And thus it is
preserved by the preservation of triangle, it is destroyed by its
destruction.

11. From the principles already established, it is plain that
demonstration must consist of such propositions as are universal and
necessary. That they must be universal, is evident from the preceding;
and that they must be necessary, we gather probably from hence; that in
the subversion of any demonstration we use no other arguments than the
want of necessary existence in the principles.

We collect their necessity demonstratively, thus; he who does not
know a thing by the proper cause of its existence, cannot possess
science of that thing; but he who collects a necessary conclusion from
a medium not necessary, does not know it by the proper cause of its
existence, and therefore he has no proper science concerning it. Thus,
if the necessary conclusion _c_ is _a_, be demonstrated by
the medium B, not necessary; such a medium is not the cause of the
conclusion; for since the medium does not exist necessarily, it may
be supposed not to exist; and at the time when it no longer exists,
the conclusion remains in full force; because, since necessary,
it is eternal. But an effect cannot exist without a cause of its
existence; and hence such a medium can never be the cause of such a
conclusion. Again, since in all science there are three things, with
whose preservation the duration of knowledge is connected; and these
are, first, he who possesses science; secondly, the thing known; and
thirdly, the reason by which it is known; while these endure, science
can never be blotted from the mind, but on the contrary, if science be
ever lost, it is necessary some of these three must be destroyed.

If then you infer that the science of a necessary conclusion may be
obtained from a medium not necessary, suppose this medium, since
capable of extinction, to be destroyed; then the conclusion, since
necessary, shall remain; but will be no longer the object of knowledge,
since it is supposed to be known by that medium which is now extinct.
Hence, science is lost, though none of the preceding three are
taken away; but this is absurd, and contrary to the principles we
have just established. The thing known remains; for the conclusion,
since necessary, cannot be destroyed;--he who knows still remains,
since neither dead, nor forgetful of the conclusion:--lastly, the
demonstration by which it was known, still survives in the mind; and
hence we collect, that if science be no more after the corruption
of the medium, neither was it science by that medium before its
corruption; for if science was ever obtained through such a medium,
it could not be lost while these three are preserved. The science,
therefore, of a necessary conclusion can never be obtained by a medium
which is not necessary.

12. From hence it is manifest, that demonstrations cannot emigrate from
one genus to another; or by such a translation be compared with one
another. Such as, for instance, the demonstrations of geometry with
those of arithmetic. To be convinced of this, we must rise a little
higher in our speculations, and attentively consider the properties
of demonstration: one of which is, that predicate which is always
found in the conclusion, and which affirms or denies the existence of
its subject: another is, those axioms or first principles by whose
universal embrace demonstration is fortified; and from whose original
light it derives all its lustre. The third is the subject genus, and
that nature of which the affections and essential properties are
predicated; such as magnitude and number. In these subjects we must
examine when, and in what manner a transition in demonstrations from
genus to genus may be allowed. First, it is evident, that when the
genera are altogether separate and discordant, as in arithmetic and
geometry, then the demonstrations of the one cannot be referred to
the other. Thus, it is impossible that arithmetical proofs can ever
be accommodated with propriety to the accidents of magnitudes: but
when the genera, as it were, communicate, and the one is contained
under the other, then the one may transfer the principles of the other
to its own convenience. Thus, optics unites in amicable compact with
geometry, which defines all its suppositions; such as lines that are
_right_, angles acute, lines equilateral, and the like. The same
order may be perceived between arithmetic and music: thus, the double,
sesquialter, and the like, are transferred from arithmetic, from which
they take their rise, and are applied to the measures of harmony.

Thus, medicine frequently derives its proofs from nature, because
the human body, with which it is conversant, is comprehended under
natural body. From hence it follows, that the geometrician cannot, by
any geometrical reasons demonstrate any truth, abstracted from lines,
superficies, and solids; such as, that of contraries there is the same
science; or that contraries follow each other; nor yet such as have an
existence in lines and superficies, but not an essential one, in the
sense previously explained.

Of this kind is the question, whether a right-line is the most
beautiful of lines? or whether it is more opposed to a line perfectly
orbicular, or to an arch only. For the consideration of beauty, and the
opposition of contraries, does not belong to geometry, but is alone the
province of metaphysics, or the first philosophy.

But a question here occurs, If it be requisite that the propositions
which constitute demonstration should be peculiar to the science they
establish, after what manner are we to admit in demonstration those
axioms which are conceived in the most common and general terms; such
as, if from equal things you take away equals, the remainders shall be
equal:----as likewise, of every thing that exists, either affirmation
or negation is true? The solution is this: such principles, though
common, yet when applied to any particular science for the purposes
of demonstration, must be used with a certain limitation. Thus the
geometrician applies that general principle, if from equal things, &c.
not simply, but with a restriction to magnitudes; and the arithmetician
universally to numbers.

Thus too, that other general proposition:----of every thing,
affirmation or negation is true; is subservient to every art, but not
without accommodation to the particular science it is used by. Thus
number _is_ or is _not_, and so of others. It is not then
alone sufficient in demonstration that its propositions are true,
nor that they are immediate, or such as inherit an evidence more
illustrious than the certainty of proof; but, besides all these, it
is necessary they should be made peculiar by a limitation of their
comprehensive nature to some particular subject. It is on this
account that no one esteems the quadrature of Bryso[24], a geometrical
demonstration, since he uses a principle which, although true,
is entirely common. Previous to his demonstration he supposes two
squares described, the one circumscribing the circle, which will be
consequently greater; the other inscribed, which will be consequently
less than the given circle. Hence, because the circle is a medium
between the two given squares, let a mean square be found between
them, which is easily done from the principles of geometry; this mean
square, Bryso affirms, shall be equal to the given circle. In order to
prove this, he reasons after the following manner: those things which
compared with others without any respect, are either at the same time
greater, or at the same time less, are equal among themselves: the
circle and the mean square are, at the same time, greater than the
internal, and at the same time less than the external square; therefore
they are equal among themselves. This demonstration can never produce
science, because it is built only on one common principle, which may
with equal propriety be applied to numbers in arithmetic, and to times
in natural science. It is defective, therefore, because it assumes no
principle peculiar to the nature of the circle alone, but such a one as
is common to quantity in general.

13. It is likewise evident, that if the propositions be universal,
from which the demonstrative syllogism consists, the conclusion must
necessarily be eternal. For necessary propositions are eternal; but
from things necessary and eternal, necessary and eternal truth must
arise. There is no demonstration, therefore, of corruptible natures,
nor any science absolutely, but only by accident; because it is not
founded on that which is universal. For what confirmation can there
be of a conclusion, whose subject is dissoluble, and whose predicate
is neither always, nor simply, but only partially inherent? But as
there can be no demonstration, so likewise there can be no definition
of corruptible natures; because definition is either the principle of
demonstration, or demonstration differing in the position of terms, or
it is a certain conclusion of demonstration. It is the beginning of
demonstration, when it is either assumed for an immediate proposition,
or for a term in the proposition; as if any one should prove that man
is risible, because he is a rational animal. And it alone differs
in position from demonstration, as often as the definition is such
as contains the cause of its subjects existence. As the following:
an eclipse of the sun is a concealment of its light, through the
interposition of the moon between that luminary and the earth. For
the order of this definition being a little changed, passes into a
demonstration; thus,

  The moon is subjected and opposed to the sun:

  That which is subjected and opposed, conceals:

  The moon, therefore, being subjected and opposed, conceals the
    sun.

But _that_ definition is the conclusion of demonstration, which
extends to the material cause; as in the preceding instance, the
conclusion affirming that the subjection and opposition of the moon
conceals the sun, is a definition of an eclipse including the material
cause.

Again, we have already proved that all demonstration consists of
such principles as are prior in the nature of things; and from
hence we infer, that it is the business of no science to prove its
own principles, since they can no longer be called principles if
they require confirmation from any thing prior to themselves; for,
admitting this as necessary, an infinite series of proofs must ensue.
On the contrary, if this be not necessary, but things most known
and evident are admitted, these must be constituted the principles
of science. He who possesses a knowledge of these, and applies them
as mediums of demonstration, is better skilled in science, than he
who knows only posterior or mediate propositions, and demonstrates
from posterior principles. But here a doubt arises whether the first
principles of geometry, arithmetic, music, and of other arts, can ever
be demonstrated? Or shall we allow they are capable of proof, not by
that particular science which applies them as principles or causes
of its conclusions? If so, this will be the office of some superior
science,--which can be no other than the first philosophy, to whose
charge the task is committed; and whose universal embrace circumscribes
the whole circle of science, in the same manner as arithmetic
comprehends music, or geometry optics.--This is no other than that
celebrated _wisdom_ which merits the appellation of science in
a more simple, as well as in a more eminent degree than others: not,
indeed, that all causes are within its reach, but such only as are the
principal and the best, because no cause superior to them can ever be
found. Hence the difficulty of knowing whether we possess science or
not, from the difficulty of understanding whether it is founded on
peculiar or common principles; since it is necessary that both these
should be applied in the constitution of all real knowledge and science.

[25]Again, axioms differ from postulates in this:--they demand our
assent without any previous solicitation, from the illustrious
certainty they possess. Their truth may, indeed, be denied by external
speech, but never from internal connection. He who denies that
equal things shall remain from the subtraction of equal, dissents,
as Euripides says, with his tongue, and not with his heart. But
demonstration depends not on external speech, but on intellectual and
internal conviction; and hence, axioms derive all their authority
from intrinsic approbation, and not from public proclaim. For the
prompt decisions of the tongue are frequently dissonant from the
sentiments concealed in the secret recesses of the heart. Thus the
[26]geometrician does not speculate those lines which are the objects
of corporeal sight, but such as are exhibited by mental conception,
and of which the delineations on paper, or in the dust, are no more
than imperfect copies, notes, and resemblances. Thus, when he draws
a pedal line which is not pedal, or an equilateral triangle which is
not equilateral, we must pay no regard to the designations of the pen,
but solely attend to the intellection of the mind; for the property
demonstrated of some particular line, is in the conclusion applied
to one that is universal, and this true line could be no otherwise
signified to the learner than by a material description.

The certainty of axioms is, indeed, in a measure obvious to every one.
For what more evident than that nothing exists of which it is possible,
at the same time, to affirm and deny any circumstance of being? Indeed,
so illustrious and indubitable is the light of this axiom, that in any
demonstration we are ashamed to assign it the place of an assumption.
It would almost seem prolix and superfluous, since there is nothing
more manifest and certain; and yet there are cases in which it is
necessary to rank it among assumptions. And these take place whenever
the intention is to conclude the existence of something as true, and
of its opposite as false. Thus, for instance, in the demonstration
that the world is finite, we assume this principle, and then reason as
follows:

  Bound and infinite cannot be at the same time affirmed and denied
    of any _body_:

  The world is a body:

  Therefore the world is not at the same time finite and infinite.

And in this genus of demonstration, the major proposition ought always
to assimilate with the conclusion. But the above axiom is not the
only one obvious, for the following possesses equal certainty; that
of every thing which exists, either affirmation or negation is true.
This axiom is of great use in demonstrations leading to an absurdity;
for he who demonstrates the impossibility of any opposite assertion,
necessarily establishes his own. Hence it is we affirm that the
diameter of a square is either commensurable or incommensurable with
its side; and this general principle is accommodated, and, as it were,
descends into its proper matter as often as that which it possesses of
universal is contracted to a certain genus; for, as we have previously
observed, common principles are not admitted in demonstration without
any restriction; but then only when their general nature is limited to
some particular subject, by which they become peculiar and apposite.

14. [27]Wisdom, or the first philosophy and logic, agree in not using
axioms after the same manner as other arts; but on the contrary,
they confirm and establish their certainty, though with this
difference, that the logician reasons only from probabilities, but the
metaphysician from the highest certainty and evidence. Besides, we do
not rank logic in the order of the sciences, because it is destitute
of some determinate genus or subject, as it is neither conversant about
lines, nor numbers, nor proportions. And its chief concern is about
apparent properties, and not such as are essential to a subject.

Hence, in logical disquisitions, we confidently employ interrogations,
as equally subservient to the affirmation or negation of an opinion:--a
method utterly impracticable, if we only employed those principles
which are universally acknowledged; since it is impossible of the same
thing to prove contrary properties,--as of the soul, that it is mortal
and immortal; but he who demonstrates, assumes one definite part of a
question, because his purpose is not to interrogate, but to trace out
the latent paths of truth. And hence, if any one affirms that the soul
is moved, and immediately after denies it, he is no longer a subject
worthy the exercise of our discursive and reasoning powers.

Again, it may so happen, that the same science at one time considers
_why_ a thing is, at another only explains its existence, or that
it exists, without considering the cause. Thus, the syllogism which
concludes by mediate propositions, demonstrates without assigning the
proper cause: but that which determines by immediate ones, in a great
measure explains the cause or reason of existence. Thus, he who infers
that trees do not breathe because they are not animals, reasons from a
mediate and secondary cause, because there are many animals, such as
insects, which exist without breathing: but he who infers this from
their want of lungs, demonstrates from the immediate and primary cause.

Thus, the following syllogism is a mediate one, or such as requires one
or more mediums to establish its certainty:

  Every thing that is not an animal does not breathe;

  A tree is not an animal;

  Therefore a tree does not breathe.

Here the major proposition is evidently mediate, because we are still
to seek why that which is not an animal does not breathe, which the
following immediate syllogism solves.

  Every thing that is not endued with lungs does not breathe;

  Every thing that is not an animal is not endued with lungs; ergo,

  Every thing that is not an animal does not breathe.

Again, the same science may demonstrate the existence of a thing,
or that it exists, and the cause of such existence as often as it
assigns two immediate reasons; but the one from the proper cause,
the other only from a sign. Thus, he who demonstrates the increase
of the moon, from the plenitude of her orb, infers the cause of such
increase; but on the contrary, he who collects the plenitude of her
orb from her increase, reasons only from a sign, and can alone declare
its existence. And, indeed, it often happens that the _cause_
and _sign_ reciprocate, so that as from the sign we advance to
the cause, demonstration from the cause frequently recurs to the
_sign_. Thus, from the breadth and firmness of the basis, we
collect the permanent duration of the pyramid; and from its extended
existence we infer the strength of its support. Whenever, then, the
argument originates from a sign, it gives evidence to the conclusion,
as from something more known than its cause. When it begins from the
cause, it proceeds from that which is first in the order of nature, to
that which is last, and reasons as from the proper principle of the
thing.

Sometimes the _cause_ and _sign_ do not reciprocate. Thus,
although wherever there is smoke, we infer the existence of fire; yet
we cannot infer, that wherever there is fire smoke exists. Thus, from
the palace and the picture we collect the existence of the architect
and painter; but the last may exist without the first;--the living
architect without the actual palace; and the living painter without
the energies of his art. And thus it is that the cause is illustrated
by its sign; but not always the sign by its cause.

Hence then, as all causes do not reciprocate with their effects; so
neither is it always causes and effects which do reciprocate: because
a multitude of signs, mutually inferring each other, may accompany a
certain cause. Thus, the signs which attend the causes of a fever, are
a quick pulsation of the artery, and an intense heat: and these signs
mutually assert each other; but no syllogism can be composed from
either expressing the _why_, but only simply _that_ the other
exists.

15. We now propose to consider the mode in which the two preceding
demonstrations are distributed in different sciences. When sciences
then are so related, that the one is dependent on the other, as optics
on geometry, navigation on astronomy, and music composed by the
arbitration of the ear, on that which consists in the knowledge of
mathematical proportions: in this case, the demonstration of simple
existence, or _that_ they _exist_, pertains to the science of
sensibles; but the demonstration _why_ they _exist_ to the
science which is speculative and mathematical.

Thus the mathematician speculates the causes of a certain sensible
effect, without considering its actual existence; for the contemplation
of universals excludes the knowledge of particulars; and he whose
intellectual eye is fixed on that which is general and comprehensive,
will think but little of that which is sensible and singular. Thus,
by mathematics we may learn the responsive harmony of the _last
chord_, and its consonance with the _mean_; but we cannot
perceive this concord, if unaccustomed to the practice of the musical
art. In fine, those sciences which are more of a mathematical nature,
I mean such as are more amply conversant with the inspection of
things, considering their forms abstracted from every material subject,
always demonstrate the _why_; and such is geometry in respect
of optics. Thus geometry considers only such things as are peculiar
to right-lines, independent of every sensible connection. For the
geometrician does not investigate a right-line as contained in stone or
brass; but considers it as entirely detached and unconnected with any
object of sense.

On the contrary, optics receives a right-line just as it is perceived
in a rule, or engraved in brass. And, indeed, in treating of some
particulars, natural science has the same relation to optics, as optics
to geometry. Thus, in considering the reason of the appearance of
the rainbow, the natural philosopher defines the bow to be an image
refracted from a certain cloud against the sun; but why it is endued
with such a form, and seen with such a colour, must be assigned by him
who is skilled in optics. There are, again, sciences, one of which is
not subordinate to the other, because founded on principles totally
different; yet, in some particulars they agree with the preceding.
Thus, to know that an orbicular wound is the most difficult of cure,
belongs to the physician; but to know _why_, to the geometrician.

16. Of all syllogistic figures, the first is the best adapted to
science, since the arithmetician, geometrician, and lastly all those
who demonstrate any effect from its proper cause, fabricate their
reasonings according to this figure. For the middle figure is seldom
used, because only adapted to a few occasions: and since the knowledge
of the _why_ is of all others the most important, which is alone
obtained by this figure: hence, in the pursuit of science, it is
always preferred before the rest. Besides, it is equally accommodated
to the knowledge of final causes; to which it alone tends: for it
composes definitions from words universal, and affirmative. In the
second figure, a complex negative is conceived; and in the last, a
particular one. Add to this, that mediate propositions are no other
ways reducible to immediate ones than by this figure, in which the
mediate proposition tends, by a continued series, to that which is
immediate. But the second does not conclude affirmatively, nor the
last universally; from whence it appears, that a mediate proposition
can never become immediate by these figures: not that all affirmative
propositions are immediate ones, since some negatives are of this
kind; for all propositions are equally immediate, which cannot be
confirmed by syllogism; and such are those negatives, of whose terms it
is impossible any genus can be affirmed. Thus the proposition, _no
substance is quality_, is an immediate negative of this kind, whose
terms are two of the most universal genera of things.

Again, as we have frequently affirmed that he who demonstrates, always
assumes such things as are essentially predicated; but that he who
argues dialectically or topically, not always, but generally assumes
such as are accidentally predicated, and which appear more probable and
known than such as are essentially inherent; it is proper we should
define what is meant by accidental predication; or something predicated
by means of another. Indeed, the term has a diffuse signification:
for, first, a body is said to be white by something else, because by
its superficies; and in this manner vines are white, because their
branches are white. Thus, if accident be predicated of accident, it is
by means of another; as when we say the musician is fair; for the being
a musician is an accident of man, and the being fair of the musician:
and man is the subject of each. The predicate of substance is equally
accidental, when not included in the number of things substantially
inherent; as when we affirm of any particular man that he is red, or
black. But the predication is especially accidental, as often as, by
perverting the order of nature, substance is predicated of accident; as
when we say something white is an animal: for this assertion differs
from that other, _animal is white_. In the latter, the subject
_animal_ is neither inherent in another, nor subsists by another,
but has an essential existence. In the former, what is assumed as a
subject derives its existence from that of which it is the accident.
It is only dialectically, therefore, that we can argue from predicates
as probable and known without any distinction: but in demonstration,
all that are preposterous and accidental must be carefully avoided,
excepting such accidents as being essentially in a subject, admit of an
essential predication; and some of these we have enumerated before.

17. We are now entering on a disquisition neither ignoble nor useless:
it is this, whether the number of things predicated essentially of a
subject is finite, or whether things in a continued series run on to
infinity. For instance, let us suppose some ultimate subject, which
is not the predicate of any thing besides; and let _c_ represent such
a subject, of which _b_ is the first and immediate predicate; and in
the same manner _d_ of _b_, and _e_ of _d_: the query is, Whether or
not this extraction must necessarily stop, or will admit of an immense
progression, so that _f_ may be predicated of _e_, and _g_ of _f_,
and so on infinitely; the power of the predicates, which supplies the
common identity, still remaining inexhaustible and undiminished? The
second query is this, Supposing some general subject, which we call
_a_, of such a nature as to be no longer the subject of any farther
predication, but to be itself the supreme and primary predicate; and
supposing that it is immediately inherent in _f_, and _f_ in _e_,
and _e_ in _g_, whether or not the process must stop, or extend to
infinity, and no subject be found which is not directly predicable of
another? There is a remarkable difference in the two considerations;
for, in the former we enquire whether any ultimate subject can supply
an infinite ascent of predicates; in the latter, whether any first
predicate can exist in an infinite descending series of subjects. The
third question is, supposing two extremes constituted from a first
predicate and last subject, whether it is possible an infinite number
of mediums can intervene? And this is no other than to enquire whether
demonstrations admit of an infinite progression, so that whatever is
assumed in proof of another, must be proved itself? Or whether it
is not more agreeable to truth, that there should be some immediate
propositions and ultimate terms, whose discovery may give respite to
enquiry, and stay the elaborate process of demonstration? The same
question occurs in negatives. But that some of these are immediate, the
instance lately alledged sufficiently evinces. The solution of this
enquiry is not so difficult in subjects which mutually reciprocate;
for in these, when the ultimate subject is given, no one can doubt the
existence of their primary predicate; nor when the primary predicate
is admitted, can there be any doubt of the existence of some ultimate
subject. For, in things which mutually reciprocate, whatever is
enquired of the one, is immediately questioned of the other; and
wherever there is a last subject, there must be a first predicate;
for by the conversion of the ultimate subject you effect the primary
predicate.

Previous to the discussion of the first question, it is necessary to
know that infinite intermediates cannot intervene between two finite
terms in an ascending and descending series of predications. I call the
series ascending which rises to universals; but descending, which, by
a contrary process, stops at particulars. Thus, if any one admits that
_a_ is some first predicate, and _g_ some ultimate subject,
and should contend, that between these terms there may be infinite
mediums, he contradicts himself; since he who begins from _a_ in a
descending progression, will never, by this means, arrive at _g_;
and he who departs from _g_ in an ascending series, can never
finally rise to _a_. So that the extremes can be no longer finite,
as the hypothesis admitted. Indeed, the absurdity of such a supposition
is the same as to contend that between one and ten, an infinity of
numbers may exist; which is evidently impossible, because the discrete
nature of numbers excludes their actual existence in infinitum,
between any finite limits; since they can only become infinite from
their actual existence and precedence, and not from any dormant power
or capacity they possess: for between any two given numbers there is
nothing similar to number in capacity, which can ever become number
in energy; as in quantity continuous between any two points there are
always parts in capacity, which, whenever a proper agent is at hand,
become immediately actual. In like manner, he who admits the terms
finite, but believes that the mediums are infinite, asserts what is
impossible, since these logical predications are of the same discrete
nature with numbers themselves. Thus all the predicates which can exist
between Socrates and substance, must exist actually, or not at all; for
surely between these two terms, or periods, no predicate in capacity
can ever be supposed to subsist. If it be urged, that the capacity of
receiving these predicates exists between Socrates and substance, still
we reply, it is not that kind of capacity in which these predicates can
retain the most shadowy existence; out of which they can ever be called
forth into energy, as from some latent retreat; or into which they
can finally retire, when energy is no more. And hence we conclude it
impossible that infinite mediums can exist between any finite terms.

18. It now remains that we prove, first, by probable arguments,
and then by such as are demonstrative, that the extremes in any
series of predications are finite; and that an infinite progression
is impossible, not only in substantial predicates, but in such as
are accidental. For every thing predicated of another is either
essentially or accidentally inherent; and is predicated in a natural
or preposterous order. It is predicated according to nature, when
accident is declared of substance; contrary to nature, when substance
of accident. That essential predicates are finite, appears from hence,
because a contrary hypothesis excludes the existence of definition,
by admitting that all things are contained in some superior genus,
and acknowledge some farther definition; since it is impossible that
the definitions of genus can ever be circumscribed, while there is a
continual supply of other genera, which can never be known without
definition; for thus we shall never obtain either a beginning or an
end. But to define all things is not possible, because infinity can
never be absolved by the most unwearied progression. Predictions then,
of this kind, are always circumscribed by a certain number of terms,
which prevent their infinite process, and cause all the strength of
demonstration, and all the certainty of human knowledge. The same may
be proved in accidents; for such as are predicated of substance, are
either predicated as qualities or quantities, as relatives, or as
actions and passions; as expressive of some habit, or significant of
some place; or as connected with some time. Thus we say the wood is
white, the triangle is scalene; whiteness being accidental to the
wood, and scalenity to the triangle. It is therefore certain, that
every accident is predicated of substance; and it is no less certain
that the predicates of substance are finite, since they are all
included in the ten universal genera of things.

19. We have hitherto defended the impossibility of an infinite
progression of logical predicates and subjects, in a demonstrative
process, by such arguments as are dialectical and common: it now
remains that we adopt such as are peculiar and certain. Demonstrations,
then, are derived from affections essentially inherent in a subject;
and these are either such as take place in definitions of a subject,
as multitude and quantity, are essentially predicated of number; or,
secondly, accidents which are defined from their subjects, as imparity
by number. But the predication cannot, in either case, be extended to
infinity. For it is not necessary that in the same manner that imparity
is predicated of number, something else, suppose _c_, should be
predicated of imparity; and so imparity be contained in its definition,
similar to number in the definition of imparity. For in predications
of this kind, the terms are always assumed more contracted than their
subject; and at length, by a continued procession, must terminate in an
indivisible. Thus, as imparity is more contracted than number, _c_
must be more contracted than imparity. Hence, these predications either
finally stop, for the reasons we have assigned; or because whatever is
predicated of imparity, is necessarily predicated of number; so that
one thing as number would be actually contained in the definition of an
infinity of things; and so actual infinity must ensue, which is absurd.
Lastly, whatever is said to reside in the terms, must be allowed to
reside in the subject; so number must be applied in the definition
of every affection; and an infinite number of properties will be
essentially inherent in number; and number will inherit infinite
definitions. But affections essentially resident in a subject cannot
be infinite, because it is necessary they should exist in energy.
Thus, imparity cannot exist potentially in number; nor reason in man;
nor rotundity in a circle, because wherever these subjects have an
actual being, it is necessary these essential attributes should be
actually inherent. Again, in the definitions of a subject, an infinite
process is impossible, because from such an hypothesis nothing could
ever be defined; and thus it appears that neither can demonstrations
be infinitely extended, nor every thing admit of demonstration, an
opinion we have already noticed in the beginning of this section: for
if neither universally, nor in every proposition a middle term can
be assumed, but as soon as we arrive at immediate propositions, the
labour of investigation is finished, the possibility of demonstrating
every thing can no longer be defended; since it is proved above, that
by limiting the extremes, an infinite number of mediums is necessarily
excluded.

And thus, by taking away infinity from the reasoning art, we have
given a support to science, which the most vigorous efforts of subtle
sophistry can never finally subvert. We have set bounds to that
restless spirit of enquiry which wanders uncontrouled in the mind
unenlightened by science, by every where circumscribing its progress
within the limits of that which is most particular, and most universal,
a first predicate, and an ultimate subject: and finally, by asserting
that all the evidence of human knowledge results from the lustre
of primary and immediate principles, we have held up a steady and
permanent light, ever sufficient to direct our steps through the dark
mazes of ignorance and error, into the bright paths of certainty and
truth.

20. Let us next consider whether universal demonstration is preferable
to particular, or not. And first, in favour of particulars we may
say that their evidence is more exquisite and certain than that of
universals. Thus, the knowledge, from inspection, that Callias is a
rational animal, is superior to that acquired by a reasoning process
which infers his rationality, because every man is a rational animal.
By particular demonstration a thing is known as it is, by universal
only in common. Besides, particulars possess some solidity, universals
none: and the demonstration of things which have a real existence,
is more excellent than that of things which have none. And there are
no errors more frequent than those about universals; demonstration
considering them as things entirely abstracted from singulars. On
the contrary, particulars are usurped by the sight, grasped, as it
were, by the hand, and the general subject of every sense; so that
concerning these, demonstration affirms nothing false or inconstant.
But these reasons, however plausible, are easily confused. And,
first, the term essential is more closely connected with universals
than particulars. Thus the possession of three angles equal to two
right, is an affection more essential to the triangle itself, than
to one equilateral or scalene. Add too, that in the demonstration of
universals we always infer some property of a subject from its simple
existence, or because it is such a subject. Again, many affections are
contained in singulars assumed from no particular nature, but from
that which is universal; as rationality in Socrates, which is not
inferred from his existence as Socrates, but from his existence as man.
Farther, that demonstration is the more excellent which is derived
from the better cause: but an universal cause is more extended and
excellent than a particular one; since the arduous investigation of
the _why_ in any subject is stopt by the arrival at universals.
Thus, if we desire to know why the exterior angles of a triangle are
equal to four right ones, and it is answered, because the triangle
is isosceles; we again ask, But why because isosceles? And if it be
replied, because it is a triangle, we may again enquire, But why
because a triangle? To which we finally answer, because a triangle is a
right-lined figure; and here our enquiry rests at that universal idea
which embraces every preceding particular one, and is contained in no
other more general and comprehensive than itself. Add too, that the
demonstration of particulars is almost the demonstration of infinites;
of universals, the demonstration of finites.--We add farther,
_that_ demonstration is the best, which furnishes the mind with
the most ample knowledge; and this is alone the province of universals.
Again, the principles of science become immediate only in proportion
as the demonstration becomes universal; and he who knows universals,
knows particulars in capacity: but we cannot infer, that he who has
the best knowledge of particulars, knows any thing of universals.
Lastly, that which is universal, is the province of intellect and
reason, particulars are the offspring of sense; and hence we conclude
that universal demonstration exceeds particular both in dignity and
excellence, and is first in the nature of things, although last in the
progressions of the reasoning power.

Again, That affirmative demonstration is superior to negative, appears
from hence: the affirmative does not require the assistance of the
negative; but the negative cannot exist without the affirmative;
on which account, the demonstration composed from negatives alone,
is incapable of producing real evidence and conviction. Besides,
affirmation exceeds negation both in priority and simplicity of
existence.

Again, the demonstration which concludes _directly_, is better
than that which confirms a proposition by evincing the absurdity of
its contrary. The first proceeding in a regular order, establishes,
by a natural deduction, the truth which was first advanced. The
second taking a wider circuit, yet with the same intentions produces
a conclusion quite opposite to its apparent design. The one may be
compared to the open attack of a valiant and skilful soldier, who
expects the conquest of his enemy from strength and courage alone: the
progress of the other resembles the same soldier, uniting force with
stratagem, and advancing, by an irregular march, which his foe mistakes
for a retreat, but finds the secret cause of his destruction. The first
is simple and impromiscuous, as composed from propositions alone: the
second is compound and miscellaneous, calling in hypothesis to its
assistance.

21. One science is said to be prior to, and more certain than another
in many respects;--when the one reasons from primary causes, but the
other from such as are secondary:--when the one may be ranked in the
genera of intelligibles and universals; but the other in the genera of
sensibles and particulars. And such is the relation of arithmetic to
music; of geometry to optics; and lastly, of every superior to every
subordinate science. Again, this happens when the one reasons from
simple principles, the other from such as are complex and connected;
on which account arithmetic seems to possess greater certainty than
geometry. For the principle of arithmetic is unity; but of geometry
a point; and unity is without position, with which a point is always
connected. And in this manner geometry inherits greater evidence than
astronomy; for the one considers body simply, the other as connected
with a circular motion. The science is called one which contemplates
actions belonging to one genus: the genus is one which possesses the
same first principles; and hence geometry and stereometry form one
science. On the contrary, the sciences are called different which have
different principles, such as geometry and optics; the latter of which
does not originate from the principles of the former.

Again, the same thing may admit of many demonstrations, and may be
known from many mediums: at one time from the application of such
as are congenial: at another, from those of a different order or
genus. From congenials, as when we demonstrate that the plantain is a
substance, first, by the medium of a tree, and then by the medium of a
plant, thus:

  Every tree is a substance;

  The plantain is a tree:

  Therefore the plantain is a substance. And again,

  Every plant is a substance:

  The plantain is a plant:

  Therefore the plantain is a substance.

We demonstrate, from mediums, of a following order or genus, as when we
prove man to be a substance, at one one time from his being rational,
at another from his being a biped; and these mediums, in part, mutually
contain each other.

22. Fortuitous events can never, in any science, become the subject
of demonstration; since they are neither limited by necessity, nor
admit the arrangement of syllogism. Indeed, so far from obtaining
a necessary, they do not possess a frequent existence, but every
syllogism is composed from one or other of these.

Again, science is not the business of sense, since that which is
universal is the object of perception in particulars themselves. For
the object of sight is colour in general, and not this particular
colour: the object of hearing is sound in general, and not any
particular sound; and, on this account we see or hear not only this
or that colour or sound, but likewise every other which falls under
the cognizance of these senses. Hence, if it were possible for any
one to discern by his sight, the equality of the three angles of some
particular triangle to two right, he would not by this means possess a
demonstration of the conclusion which affirms this to be the property
of every triangle; but his knowledge would extend no farther than the
triangle he inspects. Thus too, if we could perceive an eclipse of
the moon to arise from the interposition of the earth, we could not
universally conclude that this is the cause of every eclipse, but
only of the particular one we behold. For the explication of causes
extends to universals; and comprehends not only the knowledge of one
particular defect of the moon, but simply of every eclipse; since the
interposition of the earth is not so much the cause of any present
eclipse, as of all which can possibly exist in every age. Whenever,
then, the cause is universal, the knowledge of any effect deduced from
such a cause is, in every respect, superior to the evidence arising
from the perceptions of sense. It is likewise more excellent than
the apprehension which subsists independent of the proper cause; as
if any one should give absolute credit to the proposition, that the
three angles of a triangle are equal to two right, without a previous
conviction that the external angle of a triangle, is equal to the two
interior opposite ones; and without applying this last proposition as
the cause of the first. The comprehension, then, which is conjoined
with the proper cause, far exceeds the strongest evidence of sense.

But perhaps it may be said that science consists in sense, because the
science of any particular, fails from a defect of the sense by which
it is apprehended. To this we reply, that science, indeed, is not
acquired without the assistance of sense, but it does not follow from
hence, that _to perceive_ is _to know_; because the object
of science is that which is universal; but of sense, that which is
particular. Thus, if we could see light penetrating the pores of glass
(on the atomical hypothesis) the cause why it illuminates would be
manifest from sensible inspection as the means, and from the universal
apprehension of science, by which we should understand this to be
universally true.

Again, the principles of all sciences cannot be the same neither
considered as remote or proximate. Not considered as proximate, because
the principles always correspond to the demonstrated conclusions; but
these are not the same, since they are often generically different;
and consequently the propositions from which they result must be
derived from discordant genera. But propositions consist of such
things as essentially exist; and hence we infer, that the principles
of geometry are essentially distinguished from those of arithmetic,
that they cannot admit of reciprocal accommodation, so that the one may
be predicated, or become the subject of the other, and that the one
can never be subservient as a medium to the other. Again, common and
first principles are not applied in every science; such as this, that
every thing must either be affirmed or denied. Nor can any thing be
proved by their assistance alone, but as often as these are required
in demonstration, other principles more proximate and peculiar to the
given proposition, must always be adopted. Again, axioms universally
conceived, cannot be assumed in syllogism, but they must be contracted,
as it were, to some subject genus. Of this kind is that common
axiom, that as often as any four quantities are proportionable, by
permutation, or changing the order of the terms, the same ratio will
result. For the arts apply this axiom in a restricted sense; geometry,
by considering the relatives as four magnitudes, and arithmetic as four
numbers; but the natural philosopher, by adapting the comparison to
four motions, or four times. Besides, if the principles of all sciences
were the same, it is necessary they should be comprehended by some
certain number, similar to the limitation of the elements: but every
science is capable of immense increase from the many different modes
of amplification the conclusions will admit; and consequently it is
requisite to establish a correspondent number of proper principles;
for such as are common cannot be alone sufficient. Lastly, if the same
principles accord with every science, it follows, that any thing may
be demonstrated from such principles: but the certainty of geometrical
conclusions cannot be established from the principles of music; and
from hence it follows, that although the principles of every science
are not the same, they do not possess an entire diversity, nor yet an
absolute affinity of nature.

23. There is a remarkable difference between science and opinion.
Whatever is the subject of science must have a necessary existence; on
the contrary, opinion is conversant with things liable to mutation and
decay. Again, as science depends on necessary propositions for support;
so opinion on such as possess only a possibility of existence; and so
there is one mode of approbation in subjects of opinion, and another
in those of science. Hence science is distinguished from opinion by
two discriminations, the one arising from their subjects, the other
from the mode of approbation. That opinion is conversant with things
possible or contingent, we may learn from hence; contingencies cannot
belong to science, because their existence is not necessary; nor to
intellect, or that principle of science by which its terms are known;
nor to the apprehension or belief of immediate propositions, called
indemonstrable science. Hence, if every habit by which truth is known,
is either science, or intellect, or opinion, it remains that opinion
alone consists of things which are, indeed, true; but not necessary.
It is, therefore, inconstant and unstable, from the mutable nature of
its subjects. Besides, no one thinks he possesses an opinion of things
which he believes to have a necessary existence, so that they cannot be
otherwise than they are; but to such conviction he properly gives the
name of knowledge, and to its contrary the name of opinion.

Again, the same thing from the same propositions may at one time
become the subject of knowledge, at another, of opinion; and this
happens according to the different formation of the syllogism which
the propositions compose; whether reasoning from the proper cause it
explains the _why_, or only simply declares a thing exists. Hence
a doubt arises why opinions of this kind may not be called science,
since both the subjects and propositions are the same? The solution is
obvious. If it is believed that the propositions cannot be otherwise
than they are, or that they have a necessary existence, such an assent
of the mind is not opinion, but science; because things which inherit
an essential existence are the ornaments of science alone. On the
contrary, if we are convinced that the propositions are true, but
at the same time not necessary, such conviction is not science, but
opinion. Hence, it is impossible that science and opinion can be the
same, since they vary in their definition and mode of approbation, and
in a different manner demand our belief. Similar to this, although
it may happen that of the same thing a true and a false opinion may
arise, it will not therefore follow, that true and false opinions
are the same. For that which is firm and constant can never be the
same with that which is mutable and frail; and that which is always
true must be essentially different from that which may be changed into
false. By the power of habit indeed in different men, the same thing
may be comprehended by opinion and science. Thus it was opinion in
Epicurus when he said that the sun was eclipsed by the moon passing
under its orb, because he thought it might otherwise happen, and that
the moon might be interposed without obscuring the light of the sun.
It was science in Hipparchus, because he knew it as a necessary event.
But in the same mind, at the same time, and of the same thing, it is
impossible that science and opinion can exist. And thus much concerning
the difference of the two.

24. Lastly, sagacity is an acute and sudden apprehension of the medium,
or proper cause of a certain effect: as if any one, beholding the
moon, should in a moment conjecture the cause of the part opposite to
the sun being lucid, and the other parts obscure, because she derives
her splendor from the sun. Hence he is universally called acute and
sagacious, who, from the aspect or hearing of the extremes, can readily
perceive the medium which exists between them: as the term imports a
certain revolution of the conclusion into its first propositions, and,
as it were, a swift comprehension and continuation of the medium.




                             SECTION III.


In the ensuing Commentaries, the soul is considered as immaterial; and
as possessing a middle nature between intelligibles and sensibles;
but that this important assertion may not remain without proof,
the following demonstrative arguments, derived from the Platonic
philosophy, are offered to the reader’s consideration. And first, that
the soul is an unextended, and consequently immaterial essence, may
be thus proved. If, after the manner of magnitude, it consisted of
continuous parts, it would be impossible that any one part could be
sentient from the passion of another; but the soul, for instance, which
is situated in the finger, would be sensible of passion, as if detached
from soul in the other parts of the body, and existing by itself; for
the soul, from this hypothesis, would be sentient by parts, and not
considered as a whole. Besides, there must be many souls governing
each part of us, different among themselves, and endued with their
own peculiar energies. For whatever may be said of continuity, is to
no purpose, unless it conduces to unity of sensation; so that the
hypothesis which supposes that the sensations gradually arrive at the
principal part of the soul by a certain continued succession, is not
to be admitted, since it may be reasonably asked, How is the principal
part to be peculiarly distinguished? By what rule of quantity can the
parts be discerned, by what difference are they to be distinguished,
where the quantity is one, and the bulk continuous? Besides, is
the principal part alone, or are the other parts, sentient? If the
principal part alone be perceptive, the soul can then alone be sentient
when a sensible passion meets with this principal part, situated in
its peculiar seat; but if a sensible passion falls upon any other part
of the soul destitute of sense, it is impossible that such a part
should be able to transmit the same passion to the principal, or be at
all sentient. For how can that which is void of sense, receive passion,
and convey it entire to a sentient part? Besides, if passion accedes
to the principal, it either falls on one of its parts, and so either
one part alone will be sentient, and the rest without sensation, and
consequently superfluous, or there must be innumerable and dissimilar
sensations; for if the sensation of each of the parts singly, is the
same with the aggregate of them all, of what use is a multitude of
parts? But if the sensations are various, a man may say, as it were, I
am primarily sentient in this place, and secondarily in another; and
every sentient part besides the first, will be ignorant where sensation
is most powerful: or perhaps (from such an hypothesis) every part of
the soul will be equally deceived, each part thinking the passion to
arise in the place where it is situated. But if not the principal
part alone, but every part of the soul be sentient, a principal part
is superfluous for the purpose of sensation; and if the soul be
divisible like magnitude, how is it able to recognize, as belonging
to one subject, the qualities which flow, as it were, through many
senses, as through the eyes and ears? For that part of the soul which
is perceptive through the eyes, distinguishes nothing but colours;
that which energises through the ears, nothing but sounds; and that
which acts through the medium of the touch, nothing but the surfaces
of bodies: what is it then which perceives all these properties of
bodies united in one subject, or what is that which perceives any thing
as a certain one? For unless the intentions of the senses, and of
sensible objects, were collected together in one, the soul could never
be able to judge of the peculiar and different properties of bodies:
Hence it is necessary that the soul should be, as it were, a general
centre; that the several senses should be extended on all sides to
this, similar to lines verging from the circumference of a circle to
the centre; and that a power of this kind, comprehending all things;
should be truly one. For if the soul was any thing divisible, and the
intentions of the senses reached the soul, and ended in its essence
like the extremities of various lines, they must either again concur
in one and the same as a medium, or have different situations, in such
a manner that each sense may perceive different from one another:
as if, for instance, the sense of sight should perceive the form of
Socrates, and the sense of hearing recognize his voice; that essence
which pronounces the whole to be one person, that of Socrates, must
be something different from each of the senses. Hence it is necessary
that the soul should be an indivisible essence; for if she possessed
magnitude, she must be divided along with every sensible object she
perceives; so that one part of the soul would perceive a certain part
of a sensible object, and we should possess no sentient power capable
of perceiving the whole, or of pronouncing any thing one. Thus, in
the perception of a man, considered as one, how is it possible that
the soul can be divided, so that the perception of a part shall be
the same with the perception of the whole. But if we suppose the soul
divisible in all her perceptions, since it is impossible she should
be co-extended with every sentient object, in how many parts is the
division to be made? Is the soul to be distributed into the same number
of parts as the sensible object she perceives, so that every part of
the soul may perceive the same part of the object? Or shall we say that
the parts of the soul have no sensation of the parts of the object?
But this would be absurd and impossible. If every part then of the
soul perceived every part of a magnitude as a whole, since magnitude
is divisible to infinity, and since, on this hypothesis, there must
be innumerable sensations of every sensible object, there must be
innumerable images, as it were, of the same thing in our principal
part. Besides, if that which perceives is corporeal, it will not be
possible for it to perceive in any other manner than as if certain
images were impressed from a seal in wax, or in brass, or in any other
sensible substance. But if the images of sensation exist as in humid
bodies (which is most probable), they will certainly be confounded
like images in water; nor can there be any memory, the image departing
with its forming substance. And if we suppose the figures to remain
like impressions in solid bodies, either it will not be possible for
others to succeed while the former endure; and thus, sensations of
other things cannot take place; or if others succeed, the former images
must be immediately destroyed, and memory be no more. So that if we
allow it possible to remember, and, besides this, to perceive other
things, without any hindrance from former impressions, it is impossible
that the soul should be corporeal. Since the soul, therefore, is an
unextended, indivisible, and immaterial substance, it is consequently
incorruptible and immortal; for every thing capable of dissolution and
dispersion is either corporeal and composite, or exists in some subject
from which it is inseparable. And indeed, whatever may be dissolved
is corruptible, as being compounded from many. But whatever naturally
subsists in something different from itself, when it is separated
from its subject, immediately vanishes into non-entity. But the soul,
as we have proved, is incorporeal; it is likewise removed from every
subject, and naturally reverts to itself, and is therefore immortal and
incorruptible.

2. Let us now consider how, and on what account the soul is said to be
of a middle nature, and to be the receptacle of all middle energies,
both vital and gnostic. Since, then, there is a long gradation of
beings, proceeding from the first being, even to formless matter,
which is nothing more than the dark shadow of essence, it is requisite
to enquire what the properties are of the first and last beings, and
what the condition is of the middle orders; for thus we shall know
where the essence of the soul ought to be placed. The properties of
intelligible natures, therefore, are as follows: true being, eternal,
indivisible, immoveable, total, perfect, full of essence, replete with
life, free, moving all things, similitude, presiding over all things,
and at the same time separated from all; for each of these properties
appears in intelligibles, according to the processions of being.
But the properties of sensible natures, different from these by the
greatest interval, are such as, not-true-being, temporal according to
essence, partible, moveable, particular, indigent of another, always
replete with subsistence, living by participation, moved by another,
dissimilitude, and occupying place by parts. But the middle properties
of these are, not-true-being, an essence better than non-being, and
inferior to true-being, according to essence eternal, but according to
its energies extended with time, indivisible according to its divine
part, but divisible according to the various processions of reasons,
self-motive, governing things moved by another, but subordinate to
such as are immoveable, bearing before itself a particular nature,
together with its totality; (for, because it contains in itself all
reasons, it is after a manner a whole, but because it is diminished and
fallen, ends in parts, and suffers a transition of its energy, it must
be esteemed a particular nature): and again, perfecting itself, yet,
nevertheless, perfected by natures prior to its own; filling itself
with power and strength, and at the same time filled by others: living
from itself, and receiving life from others, being more divine, indeed,
than things which live only by participation, but inferior to things
primarily vital; moving other things, and itself moved by others; at
the same time similar and dissimilar; and separated, at the same time,
from last natures, and co-ordinated with them. Such then, being the
properties of the first, middle, and last orders, let us consider where
we ought to place the soul, whether in the first order, or in those
which retain the last place: but if we establish it among the first,
it must be true being, every way eternal and immoveable, and it must
consequently possess every thing which we have attributed to the first
beings; add too, that on this hypothesis we can no longer attribute
to the soul a power of self-motion, nor the discursive processions of
reasoning, nor a variety of other particulars, which manifestly belong
to the soul. But can we place it with propriety among the last of
beings? The least of all: for on this hypothesis we shall make it alone
moved by others, divisible, composite, and alone possessing perfection
from others, the opposite of which is evident in all our souls; since
they move and perfect themselves, and are led wherever they please.
Since then it is not possible to place the soul either in the first,
or last order of beings, it is requisite to assign it a middle place,
in imitation of its divine cause Rhea (according to the theologists),
who is the conciliating band of the two parents Saturn and Jupiter,
and is reported, from her prolific bosom to produce the life of the
soul. But though the soul is thus the extremity of intelligible, and
the principle of sensible natures, we must not conceive it to be such
a principle or extremity as a point in a line, for it is not in both
the natures it terminates, like a point in both the sections of a
line; but it is to be called the extremity of intelligibles, because it
appears after an intelligible essence, and the principle of sensibles,
as being abstracted from them, and the source of their motion. And
thus it will preserve to us a certain proportion, that as the natures
which are moved by others, are to those which are moved by themselves,
so are these last to immoveable natures; and hence it will obtain the
condition of a bond, on account of its peculiar mediocrity, unfolding,
indeed, united causes, but reducing the dissipated powers of sensibles
into _one_, and being contained by an immoveable and perpetually
abiding cause; but containing itself the generation, which, moved by
another, is subject to continual mutation. It is likewise intelligible,
if we regard generated natures; but generated, if we compare it with
intelligibles; and thus it exhibits in its middle nature both extremes,
imitating also, by this means, (according to the Greek theologists) its
divine cause, for it is said to be on both sides refulgent, ἀμφιφαὴς,
and to be endued with two faces ἀμφιπρόσωπος, and to receive in its
bosom the processions of intelligible natures. It is likewise said to
be replenished with intellectual life, and to be the fountain of the
ever-running streams of corporeal life, and to contain in itself the
centre of the processions of all beings. On this account it is, with
great propriety, affirmed to be generated, and at the same time without
generation. For true being, according to the Platonists, is without
generation, because it has an infinite power of being totally present
at the same time: and body is said to be generated, because it always
possesses in itself an infinite flowing power, which it cannot at
once totally receive. The soul, therefore, because it is incorporeal,
abiding in itself, has an infinite power of being, and this total
with respect to its essence, and immortal without generation; but
according to parts it may be considered in infinite production. For it
has not the same total infinity ever present, or there would be the
same infinity of the whole and part, of the perfect and imperfect,
of the contained and containing, which is impossible. But neither is
it possible that the whole of its essence should be in the act of
perpetual production, any more than that a part of it should be eternal
being, lest the part should be more worthy and better than the whole.
Hence the ὑπόϛασις, or subsistence of the soul, is at the same time
of infinite power, and is generated in infinitum; for by this means
it participates of being, and obtains the first place among generated
natures; while body alone, both with respect to its whole and parts, is
obnoxious to a perpetual generation.

3. But let us now enquire from what genera Plato composes this
nature of the soul, which contains in itself the bond of all beings;
previous to which it will be requisite to explain what these genera
are, and from whence they originate. Of the species, then, existing
in the intelligible world, or the divine intellect, which contains in
itself the causes of all posterior natures, some are most general,
extending themselves to the universality of things; but others are
more particular, like the most special species, and others subsisting
between these, expand themselves, indeed, to a multitude of things,
but not to all, according to the division, of the Elean guest in the
Sophista. For man is produced from the ideal man, and horse from the
ideal horse, in the intelligible world; but the similitude which is
found in man and horse, and other animals, is produced from likeness
itself, or the ideal similitude, as dissimilitude from unlikeness
itself; but the sameness and difference which are found in all beings,
proceed from the sameness and difference which subsist in ever-vital
energy and perfection, in the supreme intellect, or the ideal world.
Now, as among the sciences some are especially universal, so in
intelligible causes some are perfectly particular, presiding alone
over the proper and peculiar number of one species; but others extend
themselves to a multitude, such as equality, likeness, totality, (for
the whole considered as a whole is not common to all things, since
the part is not a whole); but others, again, expand themselves to all
things, as all beings participate of these, considered as beings, and
not considered as vital or animated, or possessing any other property
exclusive of the denomination of being. Because, therefore, _being_
is the first, the causes of _being_ obtain the most universal order
among genera; and these are five in number, as follows, _essence_,
_sameness_, _difference_, _motion_, and _station_. For every being
is endued with essence; is united to itself; is by itself, or its
own sameness, separated from others; proceeds from itself, and its
own state and principle, and no less appears to participate of a
certain _abiding_, in preserving its own proper species. All things,
therefore, whether intelligibles or sensibles, or subsisting between
both, depend on these genera for their existence. For without the
being of _essence_, nothing could subsist; in like manner, without
_sameness_ every whole would be dissipated, and divided from itself;
and _difference_ being taken away, all things would be one alone, and
multitude be destroyed. But without _motion_ and _station_, all things
would either be inefficacious and dead, or, losing their proper state
and stability, would end in non-entity.

4. Such then being the middle nature of the soul, Plato, with great
propriety, in the Phædrus, and in his tenth book of laws, defines it to
be number moving itself; which definition he received from Philolaus,
and Philolaus from Pythagoras. For since mathematical species have
a middle subsistence, as is proved in the following Commentaries,
they are of all things most accommodated to the nature of the soul.
Hence Plato, in imitation of Pythagoras, sometimes explains the
soul by number, as in the present instance; and sometimes by figure
and magnitude, as in the Timæus; while he considers in the soul the
intersection of lines, and a twofold circle. For since mathematical
forms are separated from the flux and inconstancy of matter, they
participate of a certain, exact, sure, and exquisite condition, by
means of which they eminently confer to the elevation of our ingenuity,
and the explication of latent concerns; and, on this account, as they
pertain to numbers, we may say, preserving the analogy, that there are
five orders of numbers, _the divine_, _the essential_, _the
animative_, _the natural_, and _the mathematic_. The first
of these is uniform, the second immoveable, the third self-motive, the
fourth moved by another, and the last the image of the others, and
their external measure. The divine number is considered eminently in
the deity, as in the principle of all things; the essential belongs to
intellect through ideas, and is called _essence_, _unity_,
and _the first being_; the animative number belongs to the soul,
through the medium of her inherent reasons; the natural to physical
concerns, through the seeds of nature; and lastly, the mathematical
belongs to opinion, as it is nothing more than the image of essential
number, formed by the energies of the rational soul. The soul,
therefore, is number, not limited by quantity, and mathematical,
but animative; it is number, not indeed numbering, but numbered,
generating and converted into itself. Hence too, because harmony arises
from number, the soul is called harmony; not, indeed, a harmony of
the parts of the body, nor the harmonic quantity which subsists in
sound or in voice; but a harmony arising from its essential numbers,
placed in its inherent reasons, and in the genera which constitute its
nature. It is this harmony which produces, as from its proper cause,
the harmony of the corporeal parts, the rhythm of motions, and the
melody of voices and sounds. It is this which produces that delight
in the soul from sensible harmony, which sufficiently indicates it to
be something familiar and domestic to her nature. From hence it may
be inferred, that Plato is not inconsistent with himself when, in the
Phædo, he denies, and in the Timæus affirms, the soul to be harmony;
for he denies that it is a harmony of a definite quantity, or such as
arises from the parts of the body; but he asserts it to be a harmony
in the manner already explained. It may likewise be inferred, that
Plato is ignorantly accused by a many, for affirming that the soul is
harmony, or number; for they only regard vulgar mathematical number,
and sensible harmony; while Plato, far more elevated, discourses of
intelligible numbers, and ideal harmony, subsisting in immaterial
energy and perfection.

5. And here it is necessary to consider what number, in a particular
manner belongs to the soul; for various numbers, differently
considered, accord with her self-motive nature. In the first place,
union and unity may be considered in the soul, as in her proper degree
she participates of divine unity; and likewise with relation to her
totality, for she is one certain whole. And because _a whole_ may
be considered in a triple respect, one before the parts, another rising
from the aggregate of parts, and a third subsisting in the single
parts; the soul is a whole in each of these respects. Thus she is a
whole prior to the parts, while she is considered as divisible into
them, in an incorporeal manner; she is a whole rising from parts, while
assuming the parts in the first place, we consider how her nature is
fabricated from their conjunction; and she is a whole in the single
parts, since she is total in the whole, and in every part. Besides,
the duad belongs to the soul, because she contains in her nature bound
and infinite, sameness and difference; and lastly, a conversion to
intelligibles and sensibles. And, indeed, the duad conjoined with
unity, very properly accords with the soul; for to intellect above
soul, unity particularly belongs; to body beneath soul, the infinite
alone; and to soul situated in the middle, duality properly agrees,
being, as it were, infinite, connected with unity. Again, the ternary
number is attributed to the soul, as well on account of her beginning,
middle, and end, as because she abides in herself, proceeds to
inferiors, and returns to supernal natures. Besides, she flows from
_the one_, recedes from him, and is reflected into the one when
she acquires her proper perfection. Lastly, as Proclus observes, the
nature of the soul is divided into essence, power, and energy; so that
she may be said to rejoice in the ternary number, and to be replete
with its perfection. But the quaternary number belongs to the soul
so far as she is connected with matter, which is tempered with four
qualities, and four elements; and she is endued with four principal
faculties, nutrition, sensation, local motion, and intellection. But to
omit other numbers, and their conformity with the soul, the quinary,
and septenary numbers are especially attributed to the soul. The
quinary, because the soul is composed from the five genera of things,
we have previously explained; and because five particulars merit a
principal consideration in the soul; first, her essence; secondly,
the harmony of her reasons; thirdly, the species arising from the
concord of her parts or reasons; fourthly, her virtue; and, lastly, her
energies: and on this account, Proclus observes, the consideration
of the soul ought to receive a quintuple distribution. Besides, as
the soul consists from a divisible and indivisible nature; so the
quinary is composed from the first even and the first odd number.
Lastly, as the soul is the connecting medium of the universe; so the
quinary obtains the middle situation in universal number, that is in
the decad. But the septenary number belongs to the soul, because, as
Plato shews, in his Timæus, all harmonical reasons are contained in
the seven numbers, 1, 2, 3, 4, 9, 8, 27; and from these the soul is
composed. Hence Proclus elegantly observes, that the septenary number
is dedicated to Apollo, the parent of all harmony; because in one, two,
and four, from which the septenary results, the first bisdiapason is
found. Besides, the writers on harmony affirm that all the difference
of voices proceeds as far as to the seventh degree. So that Plato uses,
with great propriety, the septenary number for the composition of the
soul. Again, in these numbers of the soul, every medium is found in
a convenient proportion; in these the geometric medium is contained,
corresponding to the right institution of laws; of which Plato, in his
Republic, says, that by this cities are properly governed: there is
found too, an harmonical medium, which is the similitude of justice:
and lastly, we may discover an arithmetical medium, which is called
the illustrious symbol of peace. After this manner, therefore, Plato,
with a wonderful fecundity of significations, affirms that the soul is
composed from numbers. He likewise considers the figures she contains,
I mean the circle and triangle; because as the soul is the first
nature which verges to body, so these are the first of all figures, as
well rectilinear as curvilinear. Besides, an orbiculation agrees to
the soul, through intellect; but progression and rectitude according
to her own proper nature; and on this account she contains both a
straight and circular figure. I omit other correspondencies of a circle
and triangle with the soul, as they are exhibited in the following
Commentaries; and particularly by Proclus, in the third book of his
Commentaries on the Timæus: it is sufficient to the design of this
Dissertation, just to have mentioned this analogy, that the nature of
the numbers and figures may appear, which are considered by Plato in
the composition of the soul.

6. Let us now pass from contemplating the nature of the soul, to
a survey of its various gradations of knowledge, and the means
by which it acquires the illuminations of science; as this is a
speculation perfectly essential to a full comprehension of the ensuing
Commentaries. According to Plato, then, in the sixth and seventh
books of his Republic[28], there are four degrees of the internal
cognitions of our soul; imagination, or assimilation; faith, cogitation
(διάνοια); and lastly, science or wisdom. The two first degrees
conjoined constitute opinion; but the two last equally joined produce
intelligence in its large acceptation. I say in its large acceptation,
because the word intelligence is considered by the Platonists in a
triple respect. First, as it rises from opinion and science; as Plato
asserts, in the seventh book of his Republic. Secondly, as it passes
into the same with science; for thus, in the end of the sixth book,
he considers intelligence and science as the same. Lastly, as it is
distinguished from science, and intelligible from that which is the
object of cognition: thus, science regards the essential reasons
of the soul; but intelligence elevates us to ideas, and this is
intelligence in its proper acceptation. The distinction of these four
degrees, especially depends on the distinction of things with which
the soul is conversant. For these four degrees of things are, the
image of a sensible object, the sensible object itself, the image of
an intelligible object, and the intelligible itself. Imagination or
assimilation regards the image of the sensible object; which image is
nothing more than the shadow or resemblance appearing in water, or
other lucid and polished bodies. Faith is conversant with sensible
objects; and these are animals, plants, and every thing subject to
the energies of sense. From the junction of imagination and faith,
opinion is produced. Cogitation is conversant with the image of an
intelligible object; which is nothing more than a certain universal,
collected from sensibles, related to the reasons of things existing
in the soul, and constituted by their assistance; but not elevated to
ideas, and resolved into their lucid nature. Those who are conversant
with this image of an intelligible object, use hypotheses; which do
not elevate us to principles, which are reasons and ideas, but bring
us down to subordinate objects. Lastly, science, considered as the
same with intelligence, is conversant with that which is intelligible,
or the essence of things; and of this kind are reasons pertaining to
the soul, and ideas to intellect. Hence, as _intelligible_ is to
_sensible_ in splendor and truth, and form to its image; such is
the relation of the superior to the inferior degrees of cognition.
And as these four degrees may be resolved into five, by separating
intelligence from science; so they are reduced by Plato into two
principal degrees; so far as the two first are conversant about
generation; but the two last about essence. And thus much for the first
particular proposed.

It now remains that we investigate the mode in which science is
produced in the soul, according to the doctrine of Plato. For this
purpose, I think it will be necessary to consider the rational soul,
in whose nature all reasons corresponding to ideas, have been inserted
from eternity, received into the human body, as into the plain of
oblivion; bordering on the river of negligence, that is placed near to
the flux of humours; which producing in its nature various kinds of
perturbations, are the causes of its self-oblivion and neglect. Hence
the soul, thus constituted, as it were, sleeping and intoxicated;
sleeping before it is roused; intoxicated before it is purified; begins
from external sensible objects to be moved and excited, and to seek
with avidity, the knowledge congenial to its nature. The soul now,
enquiring after truth and the sciences, is first conversant with the
external images of things, in which the glimmering light of similitude
to truth, presents itself to the view; then it distinguishes these
images among themselves by its reasoning power; and if they agree
in any particular, collects them into one. Thus, being employed in
separating into many, things united, and reducing many into one, it
advances from shade to substance, and is elevated from similitude
to truth itself; and thus apprehends the essence of a thing free
from every foreign or contrary quality, shining in _reason_ and
_idea_. Afterwards, from this contact, especially salutary to
itself, it experiences an ineffable joy, as from a return into its
proper nature, and best disposition; and so great is its exultation,
that it neglects and despises the shadows which it formerly pursued.
Then the soul truly knows, that while a man regards corporeal natures,
he is employed in resemblances; and that though he may esteem himself
knowing in many things, he knows nothing in reality; but is then alone
elevated to the sublime degree of science, when he arrives at ideas.
Hence it appears, that there are four subordinate dispositions of
mankind in order to science. For, in the first place, children, as
new guests of this world, are ignorant of every thing, without being
conscious of their ignorance. But as they advance in years, they are
employed in the shadows and images of natural concerns; and not being
yet converted to the essences of things, though they possess no real
knowledge, yet they conceive themselves to abound in knowledge. But
in the third place, being elevated to supernal natures, and judging
these to be alone true, they affirm themselves to be ignorant of all
they formerly imagined themselves to know; in which degree Socrates
professed to find himself, when he said, _this one thing I know,
that I know nothing_; an ignorance preferable to all the knowledge
gained by the most unwearied experimental enquiries. Indeed, this order
is indicated by Socrates himself, in the Phædo, when he says, that on
his first acquaintance with natural concerns, he thought he abounded
in knowledge, but as he advanced in these pursuits, he perceived that
he knew nothing. Lastly, when, through the piercing sight of the soul,
men are elevated to ideas, they become illustrated with the splendors
of true science, and pass into the regions of perfect reality. And
hence it appears how true science, which is the same with intelligence
and wisdom, produces piety and religion: for science elevates us to
intellect and divinity; and copulates the soul with natures of the
greatest purity and perfection; so that an union of this kind cannot
take place without piety, sanctity, and religion; as dissimilars can
never be blended in amicable conjunction.




                              SECTION IV.


We are informed by Proclus, in the ensuing Commentaries, that the end
of geometry, and, indeed, of mathematics in general, is to be referred
to the energies of intellect; and that it is degraded when made
subservient to the common utilities of a mere animal life. But as the
very opposite to this is the prevailing opinion of the present age,
let us examine the truth of this doctrine, and attend to the arguments
which the Platonic philosophy affords in its defence. For if we can
prove that this assertion of Proclus is supported by the strongest
evidence, we shall vindicate the dignity of true geometry, restore it
to its ancient esteem in the minds of the liberal, and shew how much it
is perverted by applying it to contrary purposes.

In order to this, I shall endeavour to prove the following position,
that things valuable for their own sakes, are preferable to such as
refer to something else. Now, this may be demonstrated, by considering
that every natural production was made with reference to some end, as
is evident from an induction of particulars; and if this be the case,
it may be safely inferred, that every thing exists for the sake of the
end. But that for the sake of which any being subsists is the best
of all; and the end, according to nature, is that which is perfected
the last of all, from the birth of any being. Hence the human body
receives its end or perfection first, but the soul last. And hence the
soul is posterior to the body, in the accomplishment of its nature;
and its ultimate perfection is wisdom. It is on this account that old
age alone pursues and desires the goods of prudence and wisdom. Hence,
wisdom is a certain end to us according to nature; and to be wise,
is the extreme or final cause for which we were produced. It was,
therefore, beautifully said by Pythagoras, that man was constituted
by divinity, that he might know, and contemplate. If then wisdom be
the end of our nature, to be endued with wisdom must be the best of
all. So that other things are to be performed for the sake of the good
which this contains. But to enquire in every science something besides
this, and to require that it should be useful, is alone the employment
of one ignorant of the great difference between the most illustrious
goods, and things necessary. For they differ, indeed, widely; since
things are to be called necessaries, which are the objects of desire
for the sake of others, and without which it is impossible to live. But
those concerns alone are properly good, which are loved by themselves,
though nothing else should fall to the lot of their possessor; for one
thing is not to be desired for the sake of another infinitely, but it
is requisite to stop at some limited object of desire, of which it
would be ridiculous to require any utility abstracted from itself.
But you will ask, What is the emolument of contemplative wisdom, what
the good it confers on its possessor? What if we should say (for such
is the truth of the case) that it transports us by intellect and
cogitation, to regions similar to the fortunate islands; for utility
and necessity are strangers to those happy and liberal realms. And
if this be admitted, ought we not to blush, that having it in our
power to become inhabitants of the fortunate islands, we neglect the
pursuit, through a sordid enquiry after what is useful and profitable
according to vulgar estimation? The rewards of science, therefore, are
not to be reprehended, nor is it a trifling good which results from
its acquisition. Besides, as men travel to the mountain Olympus for
its spectacle alone, preferring a view of its lofty summit to much
wealth; and as many other spectacles are desired for their own sakes,
and valued beyond gold, in like manner the speculation of the universe
is to be prized above every thing which appears useful to the purposes
of life: for it is surely shameful that we should eagerly frequent
the theatre, and the race, for the sake of the delight afforded to
our corporeal sight, and should look for no farther utility in these
than the pleasure they produce; and yet should be so sordidly stupid
as to think that the nature of things, and truth itself are not to be
speculated without some farther reward than the sincere delight their
contemplation affords.

It is on this account that the apprehension of truth is compared to
corporeal vision; for the sight is the most liberal of all the senses,
as is confirmed by the general testimony of mankind. Hence, the sight
of the sun and moon, and the glorious spectacle of the stars is desired
by the most illiterate as well as the most knowing, for the delight
such visions afford; while, on the contrary, the desires of the other
senses are for the most part directed to something farther than the
mere objects of their energy. Thus, even the sense of hearing, which
is the next in dignity to the sight, is not always desirable for its
own sake; for light is the general object of sight, and sound that of
hearing; but it is evident that light is more universally desired than
sound, since all light, when not excessive, is always pleasing, but
this is by no means the case with every kind of sound. Hence it is,
that all contemplation is so delightful, and this in proportion as it
becomes abstracted from sensible objects; for the most beautiful forms
do not produce genuine delight, until they are strongly represented in
the phantasy, as is evident in the passion of love; since the fairest
face then alone causes love when it presents itself clearly to the
inward eye of thought, in the mirror of imagination, accompanied with
living elegance, and a resistless energy of form.

Indeed, so liberal and so exalted an employment is contemplation,
that Plotinus, with his usual profundity, proves that the universe
subsists for its sake; that all the productions of nature originate
from this; and that even actions themselves are undertaken with a
view to the enjoyment of after-speculation. May we not, therefore,
say that the sportsman follows the chace for the sake of a subsequent
review of his favourite pursuit? That the glutton for this rejoices in
the meal; and even the miser in his wealth? And that conversation is
alone sollicited, that it may recal past images to the soul? In short,
contemplation is the first spring of action, and its only end; since we
are first incited to any external object by speculating its image in
the phantasy: and our subsequent conduct tends, without ceasing, to the
energy of reflection; for destroy prior and posterior contemplation,
and action is no more.

Now if this be the case, and if geometry is a speculative science (I
mean the geometry of the ancients), it is both desirable for its own
sake, and for still higher contemplations, the visions of intellect,
to which it is ultimately subservient. For, when studied with this
view, it opens the eye of the soul to spectacles of perfect reality,
and purifies it from the darkness of material oblivion. Away then,
ye sordid vulgar, who are perpetually demanding the utility of
abstract speculations, and who are impatient to bring down and debase
the noblest energies, to the most groveling purposes; ignorant of
that mighty principle of action, which influences every part of the
universe, and through which even division and discord tend as much as
possible to union and consent; ignorant that from the depravity of your
nature, and the blindness of your inward eye, you are incapable of
speculating the substance of reality, and are therefore eagerly gazing
on its shadow: and lastly, unconscious that this is the point about
which you are continually making excentric revolutions, mistaking the
circumference for the centre, motion for rest, and a departure from
good for a tendency to felicity.

It was for the sake of this most exalted and liberal contemplation that
Heraclitus yielded his right of succession to a throne, to his brother;
and that Anaxagoras neglected his patrimony, esteeming one drop of
genuine wisdom preferable to whole tuns of riches. Led by a desire of
this, as by some guiding star, Pythagoras travelled into Egypt, and
cheerfully encountered the greatest difficulties, and maintained the
most obstinate perseverance, until at length he happily penetrated
the depths of Egyptian wisdom, and brought into Greece a treasury of
truth for future speculation. But these were happy days; this was the
period destined to the reign of _true philosophy_, and to the
advancement of the human soul to the greatest perfection its union with
this terrene body can admit. For in our times, the voice of wisdom is
no longer heard in the silence of sacred solitude; but _folly_
usurping her place, has filled every quarter with the barbarous and
deafning clamours of despicable sectaries; while the brutal hand of
commerce has blinded the liberal eye of divine contemplation. For
unfortunately, the circle of time, as it produces continual variations,
at length reverses the objects of pursuit; and hence, that which was
once deservedly first, becomes at length, by a degraded revolution, the
last in the general esteem.

2. If geometry, therefore, be both valuable for its own sake, and
for its subserviency to the most exalted contemplations, there can
be no doubt but that the great perfection to which this science was
brought by the Greeks, was entirely owing to their deep conviction
of this important truth. Euclid, we are informed by Proclus, in
this work, was of the Platonic sect; and Archimedes is reported, by
Plutarch, in his Life of Marcellus, to have possessed such elevated
sentiments of the intrinsic dignity of geometry, that he considered
it perverted and degraded, when subservient to mechanical operations;
though, at the request of king Hiero, he fabricated such admirable
engines for the defence of Syracuse. From this source alone, the great
accuracy and elegance of their demonstrations was derived, which have
been so deservedly applauded by the greatest modern mathematicians,
and the warmest advocates for the farrago of algebraic calculation.
Algebra, indeed, or as it is called, _specious analysis_, is the
modern substitute for the perfect method adopted by the ancients in
geometrical demonstrations; and this solely, because it is capable of
being applied with greater facility to the common purposes of life.
Hence, hypotheses have been eagerly admitted in geometry, which the
ancients would have blushed to own: I mean the multiplications and
divisions of lines and spaces as if they were numbers, and considering
geometry and arithmetic as sciences perfectly the same. But we have
fortunately the testimony of the first mathematicians among the moderns
against the unlawfulness of this ungeometrical invasion. And to begin
with the great sir Isaac Newton, in his Universal Arithmetic[29]:
“Equations (says he) are expressions of arithmetical computation, and
properly have no place in geometry, except so far as quantities truly
geometrical (that is, lines, surfaces, solids, and proportions), may
be said to be some equal to others. _Multiplications, divisions,
and such sort of computations, are newly received into geometry, and
that unwarily, and contrary to the first design of this science._
For whoever considers the construction of problems by a right-line and
a circle, found out by the first geometricians, will easily perceive
that geometry was invented that we might expeditiously avoid, by
drawing lines, the _tediousness of computation_. _Therefore,
these two sciences ought not to be confounded._ The ancients so
industriously distinguished them from one another, that they never
introduced arithmetical terms into geometry. _And the moderns, by
confounding both, have lost the simplicity in which all the elegancy
of geometry consists._” And in another part[30] of the same work he
observes, that “_the modern geometers are too fond of the speculation
of equations_.” To this very high authority we may add that of Dr.
Halley, in the preface to his translation of Apollonius de Sectione
Rationis; for which work he conceived so great an esteem, that he was
at the pains to learn Arabic in order to accomplish its translation
into Latin[31]: “This method, says he, (of Apollonius) contends with
specious algebra in facility, but far excels it in evidence and
elegance of demonstrations; as will be abundantly manifest if any one
compares this doctrine of Apollonius de Sectione Rationis, with the
algebraic analysis of the same problem, which the most illustrious
Wallis exhibits in the second volume of his mathematical works, cap.
liv. p. 220.” And in the conclusion of his preface, he observes[32],
“that it is one thing to give the resolution of a problem some how
or other, which may be accomplished by various ways, but another to
effect this by the most elegant method; by an analysis the shortest,
and at the same time perspicuous; by a synthesis elegant, and by
no means operose.” And Dr. Barrow, notwithstanding he was so great
an advocate for the identity of arithmetic and geometry, expressly
asserts[33], that _algebra is no science_. To these authorities
we may add Simson and Lawson, who, sensible of the superior skill
of the ancients, both in analysis and synthesis, have made laudable
attempts to restore the _Greek geometry_ to its pristine purity
and perfection.

Again, the greatest men of the present times have been of opinion, that
algebra was not unknown to the ancients; and if this be true, their
silence respecting it is a sufficient proof of their disapprobation.
Indeed, if we consider it when applied to geometry, as an art alone
subservient to the facility of practice, as conveying no evidence, and
possessing no elegance of demonstration, we shall not wonder at its
being unnoticed by the ancients, with whom practice was ever considered
as subservient to speculation; and in whose writings elegance of theory
and accuracy of reasoning are found perpetually united.

3. But the lives of the first cultivators of this science (I mean the
Egyptian priests) as well as of the Pythagoreans and Platonists, by
whom it afterwards received such improvements, sufficiently evince
that this science advanced to perfection from an intellectual theory
as its source, and from being referred to contemplation as its end;
and this will be evident, by attending to the following history of the
Egyptian priests, as preserved to us by Porphyry, in his excellent
work on abstinence[34]; a translation of which will not, I presume, be
unacceptable to the philosophical reader, “Chæremon, the Stoic (says
he) explaining the rites of the Egyptian priests, who, he says, are
accounted philosophers by the Egyptians, relates, that they choose a
place best adapted to the study and performance of sacred rites; so
that a desire of contemplation is excited by only frequenting those
recesses which are dedicated to their use, and which procure safety
to the priests, on account of that reverence of the divinity, whose
sacred mysteries they perform; so that all possible honour is paid to
these philosophers, in the same manner as to some sacred animals. But
he says they live entirely solitary, except at particular times, when
they mix with others in such assemblies as are usually held, and in
public feasts; and that on all other occasions they are scarcely to be
approached. For he who desires to converse with them must first purify
himself, and abstain from a multitude of things after the manner of
these Egyptian priests. He adds, that these men, renouncing every other
occupation, and all human affairs, give themselves entirety, through
the whole of life, to the contemplation of divine concerns, and to
enquiring into the divine will: by the latter of these employments
procuring to themselves honour, security, and the estimation of
piety; by contemplation, tracing out the latent paths of science; and
by both these occupations united, accustoming themselves to manners
truly occult, and _worthy of antiquity_. For to dwell always on
divine knowledge, and be disposed for divine inspiration, removes a
man beyond all immoderate desires, calms the passions of the soul, and
raises her intellectual eye to the perception of that which is real
and true. But they studied tenuity of aliment, and frugality in their
apparel, and cultivated temperance and patience, together with justice
and equity, in all their concerns. Indeed, a solitary life rendered
them perfectly venerable; for during that period which they call the
time of purification, they scarcely mixed with the associates of their
own order, or saw any one of them, except him who was conversant with
them in that exercise of purity, on account of necessary uses. But
they by no means concerned themselves with those who were unemployed
in the business of purification. The remaining part of their time they
conversed familiarly with those similar to themselves; but they lived
separate and apart from those who were estranged from their ceremonies
and manner of living. He adds, they are always seen employed among the
resemblances of the gods, either carrying their images, or preceding
them in their accustomed processions, and disposing them with gravity
of deportment, and in a graceful order. In all which operations
they did not indicate any pride of disposition; but exhibited some
particular natural reason. But their gravity was conspicuous from their
habit; for when they walked, their pace was equable, and their aspect
so perfectly steady, that they refrained from winking whenever they
pleased. Their risibility too, extended no farther than to a smile.
But their hands were always contained within their garments; and as
there were many orders of priests, every one carried about him some
remarkable symbol of the order he was allotted in sacred concerns.
Their sustenance too was slender and simple; and with respect to wine,
some of them entirely refrained from it; and others drank it very
sparingly, affirming that it hurt the nerves, was an impediment to the
invention of things, and an incentive to venereal desires. They also
abstained from many other things, never using bread in exercises of
purity; and if they ate it at other times, it was first cut in pieces,
and mingled with hyssop. But they abstained, for the most part, from
oil, and when they used it mixt with olives, it was only in small
quantities, and as much as was sufficient to mitigate the taste of the
herbs.

In the mean time, it was not lawful for any one to taste of the
aliment, whether solid or fluid, which was brought into Egypt from
foreign parts. They likewise abstained from the fish which Egypt
produced; and from all quadrupeds having solid or many fissured hoofs;
from such as were without horns; and from all carniverous birds: but
many of them abstained entirely from animal food. And at those times
when they all rendered themselves pure, they did not even eat an egg.
But when the time drew near in which they were to celebrate some sacred
rites, or festival, they employed many days in previous preparation,
some of them setting apart forty-two days, others a greater length
of time than this; and others again a shorter; but never less than
seven days; abstaining, during this period, from all animals, and
from all leguminous and oily nutriment, but especially from venereal
congress. Every day, they washed themselves three times in cold water;
after rising from bed, before dinner, and when they betook themselves
to rest. And if they happened to be polluted in their sleep, they
immediately purified their bodies in a bath. They made cold water
too subservient to the purposes of purification at other times, but
not so often as the bath. Their beds were composed from the branches
of palm, which they called βαίς, _bais_. A piece of wood, of a
semi-cylindrical form, and well planed, served them for a pillow.
But through the whole of life, they were exercised in the endurance
of hunger and thirst, and accustomed to a paucity and simplicity of
nutriment.

But as a testimony of their temperance, though they neither used the
exercise of walking nor riding, yet they lived free from disease, and
were moderately strong. For, indeed, they endured great labour in
their sacred ceremonies, and performed many services exceeding the
common strength of men. They divided the night between observations of
the celestial bodies, and offices of purity; but the day was destined
by them to the cultivation of the divinities, whom they worshiped
with hymns each day three or four times; in the morning and evening,
when the sun is at his meridian, and when he is setting. But the rest
of their time they were occupied in arithmetical and geometrical
speculations, always laborious and inventing, and continually employed
in the investigation of things. In winter nights also, they were
diligent in the same employments, and were ever vigilant to literary
studies; since they were not solicitous about external concerns,
and were freed from the base dominion of intemperate desires. Their
unwearied and assiduous labour, therefore, is an argument of their
great patience; and their continence is sufficiently indicated by their
privation of desire. Besides this, it was esteemed very impious to sail
from Egypt, as they were particularly careful in abstaining from the
manners and luxuries of foreign nations; so that to leave Egypt was
alone lawful to those who were compelled to it by state necessities.
But they discoursed much concerning a retention of their native
manners; and if any priest was judged to have transgressed the laws
in the least particular, he was expelled from the college. Besides,
the true method of philosophizing was preserved in Commentaries and
Diaries, by the prophets and ministers of sacred concerns: but the
remaining multitude of priests, _Pastophori_, or priests of Isis
and Osiris, governors of temples, and servants of the gods, studied
purity, yet not so exactly, nor with so great continence as those we
have mentioned. And thus much is related of the Egyptians, by a man who
is equally a lover of truth, and of accurate diligence, and who is
deeply skilled in the Stoic philosophy.”

4. But the lives of the Pythagoreans and Platonists, who carried this
divine science to its ultimate perfection, no less eminently evince
the truth of our position. For, as Porphyry informs us, in the same
invaluable treatise[35], “some of the ancient Pythagoreans, and wise
men, inhabited the most desert places; and others retired into temples,
from which the multitude and every tumult were expelled. But Plato was
willing to fix his academy in a place not only solitary, and remote
from the city, but, as they report, insalubrious. Others, again, have
not spared their eyes, through a desire of more perfectly enjoying that
blissful contemplation, from which they wished never to be separated.”
After this, he presents us with a description from Plato[36] of
those intellectual men, by whom the world has been enlightened with
the sublimest wisdom and truth: “For it was not falsly, or in vain
(says he), that a certain philosopher, speaking of contemplative men,
affirms, that such as these are ignorant, from their early youth,
of the way which leads to the forum, or in what place the court or
senate-house is situated, or any public council of the state. They
neither see nor hear the laws, whether decreed or promulgated, or
written; and with respect to the factions and contentions of their
companions for magistracy, for assemblies and splendid entertainments,
luxurious eating and minstrels, they do not even think of these as in a
dream. Such an one knows no more of the evil which has happened to some
one of his ancestors, whether male or female, or any thing belonging
to them, than how many pitchers of water are contained in the sea. Nor
does he abstain from things of this nature for the sake of acquiring
fame; but in reality, his body alone abides in the city, and wanders
about from place to place, but his intellect esteeming all these as
of small importance, or rather as non-entities, he despises them,
and, according to Pindar, “from these on every side he soars:” by no
means applying himself to things which are near him, and to sensible
concerns.”

If such then were the lives of the men who brought this contemplative
science to its present perfection, and who are to this day our masters
in geometry; if such were the exalted sentiments they entertained
of its dignity and worth, what greater proof can we require of its
being valuable for its own sake, and as subservient to the energies
of intellect? We have ample evidence too, of its being degraded
when brought down to the common purposes of life, in the example
of those who, with this view, have disguised it with the dark and
sordid involutions of algebraic calculation; for it was solely to
facilitate practice, that this barbarous invasion has been admitted by
the moderns. Let me then be permitted to persuade the few who study
geometry in its ancient purity, and who consider the ruins of Grecian
literature on this, as well as on every other science, the models
of perfection, to enter with avidity on the study of the ensuing
Commentaries, and endeavour to fathom the depth of our profound and
elegant philosopher: for by this means they may happily obtain the end
of all true science, the purification of the soul; and be able to draw
the light of perfect wisdom, from the undecaying and inexhaustible
fountain of good.

But if it should be asked in what these energies of intellect
consist, to which all science ultimately refers? I answer, in the
contemplation of true being, or those ideal and divine forms, with
which the intelligible world is replete. Now this great end is not
to be accomplished without previous discipline, a long exercise of
the reasoning power, and a continued series of philosophic endurance.
For this end, when attained, is no other than the enjoyment of that
felicity congenial to the soul previous to her immersion in body. But,
for the further information of the liberal reader on this important
subject, the following paraphrases from Porphyry and Proclus are
subjoined; the former instructing us in the various purifications
necessary to this end; and the latter exhibiting the gradations by
which we may rise to the speculation of reality, and (leaving all
_multitude_ behind) ascend to the divinely solitary principle of
things, the ineffable _One_.

5. “In the first place, then (says Porphyry[37]) my reasons are not
addressed to those who are occupied in illiberal arts, nor to those
engaged in corporeal exercises, neither to soldiers nor sailors,
neither to rhetoricians nor to those who have undertaken the duties of
an active life. But I write to the man continually employed in thinking
what he is, from whence he comes, and whither he ought to tend: and
who, with respect to every thing pertaining to food, and other offices
of life, is entirely changed from those who propose to themselves a
different manner of living; for to a man of this kind alone is my
present discourse addressed. Indeed, in this common state of existence,
one and the same mode of persuasion cannot be addressed to the sleeper,
who, if it was possible, would conciliate to himself perpetual sleep,
and who, for this purpose, seeks on every side for soporiferous
incentives, as to him who studies continually to drive away sleep,
and to dispose every thing about him to vigilance and intellectual
activity. But to the former, it is necessary to advise intoxication,
surfeiting, and satiety, and to recommend a dark house; and, as the
poets say, _a bed luxurious, broad, and soft_. Such a one should
chuse whatever tends to produce stupor, and give birth to indolence
and oblivion, whether consisting of odours, ointments, or medicaments
which are accustomed to be eat or drank. But it is necessary that the
intellectual man should use sober drink, unmixed with the lethargic
fumes of wine; nutriment slender, and almost approaching to fasting;
a lucid house, receiving a subtle air and wind; that he should be
continually agitated with cares and griefs; and lastly, that he
prepares for himself a small and hard bed, while thus employed in
purifying his soul from the stains contracted by corporeal involution.
But whether we are born for this exalted purpose, I mean for vigilant
intellectual energies, allowing as small a part of our life as possible
to sleep; (since we do not exist in a place where souls perpetually
vigilant abide), or whether we are destined to a contrary purpose,
I mean, to sleep and oblivion, would be foreign from our design to
explain; and would require a longer demonstration than the limits of
our work will admit.

But whoever once cautiously surmizes the delusions of our life in the
present world, and the inchantments of this material house in which we
are employed, and who perceives himself naturally adapted to vigilant
energies; lastly, who apprehends the soporiferous nature of the place
in which he acts, to such a one we would prescribe a diet congruous
to his suspicion of this fallacious abode, and to the knowledge he
possesses of himself; in the mean time, advising him to bid a long
farewel to the sleeper, stretched on his couch, as on the lap of
oblivion. Nevertheless, we should be careful lest, as those who behold
the bleer-eyed, contract a similar defect, and as we gape when present
with those who are gaping, so we should be filled with drowsiness
and sleep, when the place in which we reside is cold, and adapted to
fill the eyes with watery humours, from its abounding with marshes and
vapours, which incline their inhabitants to heaviness and sleep. If
then, legislators had composed the laws with a view to the utility of
the state, and had referred these to a contemplative and intellectual
life as their end, we ought to submit to their institutions, and
acquiesce in the diet they have prescribed for our subsistence. But if
they, only regarding that life which is according to nature, and is
called of the middle kind, ordain such things as the vulgar admit, who
only estimate good and evil as they respect the body, why should any
one, adducing these laws, weary himself in endeavouring to subvert a
life which is far more excellent than every law written and composed
for the sake of the vulgar, and which follows a law not written, but
divinely delivered? For such is the truth of the case.

_That_ contemplation which procures us felicity, is not a mass
of discourses, and a multitude of disciplines; or, as some may
think, consisting from hence; nor does it receive any increase from
a quantity of words. For if this was the case, nothing could hinder
those from being happy, who comprehend all disciplines, and _are
accurately skilled in a variety of languages_. But the whole
circle of the sciences cannot by any means accomplish this blissful
contemplation, nor even those disciplines which are conversant with
true and substantial being, unless there is also a conformation of
our nature and life to this divine end. For since there are, as they
say, three ends of living, if we regard the particular objects to
which mankind tend, the end with us is to follow the contemplation of
_true being_, promoting, as much as possible, by an acquisition
of this kind, an intimate union of the contemplating individual with
the object of contemplation. For, in nothing else besides _true
being_, is it possible for the soul to return to its pristine
felicity; nor can this be effected by any other conjunction. But
intellect is _true being itself_: so that the proper end is to
live according to intellect. And on this account, exoteric discourses
and disciplines, retarding the purgation of the soul, are far from
filling up the measure of our felicity. If then, felicity was defined
by the comprehension of words or sciences, they who do not pay a proper
attention to the kind and quantity of their food, nor to any thing
else pertaining to their present existence, might obtain this end: but
since it is requisite to change our life, and to be pure both in speech
and action, let us consider what discourses and what works may render
us partakers of this most necessary means of acquiring substantial
felicity.

Are, then, those things which separate us from sensible objects,
and from the affections which they excite, and which lead to a life
intellectual, and void of imagination and passion, are these the
means we are in pursuit of? So that every thing contrary is foreign
from our purpose, and worthy to be rejected? And in such proportion
as it draws us aside from intellect? Indeed, I think it is consonant
to truth, that we should eagerly contend where intellect leads; for
in this material abode, we are similar to those who enter or depart
from a foreign region, not only in casting aside our native manners
and customs, but from the long use of a strange country, we are imbued
with affections, manners, and laws foreign from our natural and true
region, and with a strong propensity to these unnatural habits. Such
an one, therefore, should not only think earnestly of the way, however
long and laborious, by which he may return to his own, but that he may
meet with a more favourable reception from his proper kindred, should
also meditate by what means he may divest himself of every thing alien
from his true country, which he has contracted; and in what manner he
may best recal to his memory, those habits and dispositions without
which he cannot be admitted by his own, and which, from long disuse,
have departed from his soul. In like manner, it is requisite, if we
wish to return to such things as are truly our own, and proper to man
considered as a rational soul, to lay aside whatever we have associated
to ourselves from a mortal nature, together with all that propensity
to material connections, by which the soul is allured, and descends
into the obscure regions of sense; but to be mindful of that blessed
and eternal essence intellect, our true father, and hastening our
return to the contemplation of the uncoloured light of _good_, to
take especial care of these two things; one, that we divest ourselves
(as of foreign garments) of every thing mortal and material; the
other how we may return with safety, since thus, ascending to our
native land, we are different from ourselves before we descended into
mortality. For we were formerly intellectual natures; and even now we
are essences purified from every stain contracted by sense, and from
that part which is destitute of reason: but we are complicated with
sensible connections, on account of our impotence and infirmity, which
is the cause that we cannot always be conversant with intellectual
concerns; but with mundane affairs we can be present with frequency
and ease; for all our energetic powers are stupified and clouded
with oblivion, through body and sense; the soul not remaining in an
intellectual state; (as the earth when badly affected, though good
fruit is deposited in its bosom, produces nothing but weeds); and this,
through the improbity of the soul, which does not, indeed, destroy
its essence, while it acquires brutality; but by such an accession it
becomes complicated with a perishing nature, is bound in the dark folds
of matter, and is drawn aside from its proper state, into one that is
foreign and base.

So that it is highly requisite to study, if we are solicitous of
returning to our pristine state of felicity, how to depart from sense
and imagination, and her attendant brutality, and from those passions
which are raised by her phantastic eye, as much as the necessity of our
nature will permit. For the intellect must be accurately composed; and
it is proper it should obtain a peace and tranquility free from the
contentions of that part which is destitute of reason, that we may not
only hear with attention concerning intellect and intelligible objects,
but to the utmost of our ability, may enjoy their contemplation; and
thus, being reduced into an incorporeal nature, may truly lead an
intellectual life, and not in a false delusive manner, like these
who are at the same time entangled with corporeal concerns. We must,
therefore, divest ourselves of the various garments of mortality by
which our vigour is impeded; as well this visible and fleshly garment,
as that more interior one with which we are invested contiguous to
the skin. We must enter the place of contest naked, and without the
incumbrance of dress, striving for the most glorious of all prizes, the
Olympiad of the soul. But the first requisite, and without which it is
not lawful to contend, is, that we strip off our garments. And since
our vestments are some of them exterior, and some interior, so with
respect to the denudation of the soul, one process is by things more
open, another by such as are more occult. For instance, not to eat, or
not to accept what is offered, is among things obvious and open; but
not to desire is more obscure; so that it is here requisite not only
to abstain from things improper in deeds, but likewise in desire. For
what does it profit to abstain in actions from what is base, in the
mean time adhering to the causes which produce such actions, as if
bound in indissoluble chains?

But this receding from material affections is brought about partly
by force, and partly by persuasion; and by the assistance of reason
the affections languish, and are, as it were, buried in oblivion, or
in a certain philosophical death; which is, indeed, the best mode
of desertion, without oppressing the terrene bandage from which the
soul departs. For in things which are the objects of sense, a violent
devulsion cannot take place without either a laceration of some part,
or at least a vestige of separation. But vice steals in upon the
soul through continual negligence: and carelessness is produced by
not sufficiently attending to intelligible objects; the affections
in the mean time being excited by the drowsy perceptions of sense,
among which must be also reckoned the sensations arising from food.
We must therefore abstain, not less than from other things, from such
food as usually excites the passions of our soul. Let us then in this
particular enquire a little farther.

There are two fountains, whose noxious streams detain the soul in
matter, and with which, as if saturated with lethargic potions, she
forgets her own proper speculations: I mean pleasure and grief, the
artificer of which is sense and its perceptions, together with the
operations attendant on the senses, imaginations, opinions, and memory.
The passions, roused by the energies of these, and the irrational part,
now fattened with noxious nutriment, draw down the soul, and avert
her inclinations from her native love of true being. It is requisite,
therefore, that we revolt from these to the utmost of our ability.
But true defections can alone take place by avoiding the passions and
rash motions produced by the senses. But, sensation respects whatever
moves the sight, or the hearing, or the taste, or the smell. And sense
is, as it were, the metropolis of that foreign colony of passions which
reside in the soul, and which must be expelled by him who wishes, while
connected with body, to become an inhabitant of the royal regions of
intellect. Let us then enquire how much fuel of the passions enters
into us through each of the senses; and this either when we behold the
spectacles of horses in the race, and the labours of the athletic, or
the contests of those who twist and bend their bodies in leaping, or
when we survey beautiful women. For all these insnare us, unconscious
of the danger, and subject to their dominion the irrational appetite,
by proffered inchantments of every kind.

For by all such inchantments the soul, as if driven into fury, compels
the compound man to leap rashly, and without reason, and full of the
brutal nature to bellow and exclaim. In the mean time, the perturbation
appearing from without, being inflamed by the internal, which was
first of all roused by sense. But the vehement motions excited by the
hearing, arise from certain noises and sounds, from base discourse,
and mixed assemblies; so that some, exiled from reason, behave as if
struck mad; and others, enervated by effeminate softness, agitate
themselves by a multitude of trifling gesticulations. And who is
ignorant how much the soul is fattened, and infested with material
grossness, by the ointments and perfumes which commend lovers to each
other? But why is it necessary to speak of the passions originating
from the taste: in this respect especially, binding the soul in a
double band; one of which is thickened by the passions excited by the
taste; the other becomes strong and powerful by the different bodies
which we receive in food. For as a certain physician observed, those
are not the only poisons which are prepared by the medical art, but
such things as we daily receive for food, as well liquid as solid, are
to be reckoned among this number; and much greater danger arises to
our life from these, than to our bodies from poisons. But the touch
does all but transmute the soul into body, and excites in it, as in a
dissonant body, certain broken and enervated sounds. The remembrance,
imagination, and cogitation of all these raise a collected swarm of
passions, i. e. of fear, desire, anger, love, emulation, cares, and
griefs, they fill the soul with perturbations of this kind, cloud its
intellectual eye with oblivion, and bury its divine light in material
darkness.

On which account it is a great undertaking to be purified from all this
rout of pollutions; and to bestow much labour in meditating day and
night, what measures we shall adopt to be freed from these bonds, and
this because we are complicated with sense, from a certain necessity.
From whence, as much as our ability will permit, we ought to recede
from those places in which we may (perhaps unwillingly), meet with
this hostile rout; and it is requisite we should be solicitous not to
engage in combat with these dangerous foes, lest, through too great a
confidence of victory and success, instead of vigorous contention, we
produce only unskilfulness and indolence.”

And in the conclusion of the first book, he adds, “For, indeed, if
it be lawful to speak freely, and without fear, we can by no other
means obtain the true end of a contemplative, intellectual life, but
by adhering to the Deity (if I may be allowed the expression), as if
fastened by a nail, at the same time being torn away and separated
from body and corporeal delights; having procured safety from our
deeds, and not from the mere attention to words. But if friendship
is not to be conciliated with a divinity, who is only the governor
of some particular region, with any kind of food, or by the use of
animal nutriment, much less can a gross diet effect an union with
that God who is exalted above all things, and who is superior to a
nature simply incorporeal; but after every mode of purgation, and the
greatest chastity of body, and purity of soul, we shall scarcely be
thought worthy to obtain the vision of his ineffable beauty; though
this is sometimes permitted to him whose soul is well disposed, and
who has passed through life with the greatest sanctity and purity of
manners. So that, by how much the Father of all exceeds every nature
in simplicity, purity, and self-sufficiency, as being infinitely
remote from all suspicion of material contagion, by so much the more
ought he who approaches to the Deity, to be entirely pure and holy,
first in his body, and afterwards in the most secret recesses of his
soul; having distributed a purgation adapted to every part, and being
completely invested with purity, as with a transparent garment, fit
for the intimate reception of divine illumination.” Thus far Porphyry,
whose excellent sentiments on this subject are a lasting monument of
the elevation and purity of soul which the Platonic philosophy affords;
and at the same time sufficiently prove the arrogance and ignorance of
those who depreciate the wisdom of the ancients, and consider their
greatest philosophers as involved in mental darkness and delusion.
But presumption of this kind is continually increased by indolence,
and strengthened by interest; and it is common to find _scribblers
of every kind_, laughing at Plato and his philosophy, who are too
mean for criticism, and even too insignificant for contempt. Let us,
therefore, leave such in their native inanity, and listen to the
instructions of the divinely elegant Proclus, by which we may ascend to
the contemplation of true being, and the ineffable principle of things.

6. [38]“Pythagoras and Plato command us to fly from the multitude,
that we may pursue the most simple truth, and apply ourselves wholly
to the contemplation of real being. From the multitude of exterior
people drawing us aside in various ways, and deceiving us by fallacious
appearances. But much more to shun the multitude of interior people;
for this much more distracts and deceives. We must, therefore, fly from
the various multitude of affections, the obscure informations of sense,
the shadowy objects of imagination, and the dusky light of opinion. For
every multitude of this kind is so different in itself, that its parts
are contrary to one another; from whence it is necessary to betake
ourselves to the sciences, in which multitude has no contrariety.
For though affections are contrary to affections, one perception
of sense to another, imaginations to imaginations, and opinions to
opinions, yet no one science is found contrary to another. In this
multitude, therefore, of propositions and notions, we may collect
into one the number of sciences binding them in one according bond.
For they are so remote from contrariety to each other, that notion is
subservient to notion, and inferior sciences minister to superior,
depending on them for their origin. Above all, it is here necessary,
from many sciences which pre-suppose one, to betake ourselves to one
science itself, no longer supposing another, and in an orderly series
to refer them all to this original one. But after science, and its
study, it will be necessary to lay aside compositions, divisions, and
multiform discourses, and from thence to ascend to intellectual life,
to its simple vision, and intimate perception. For science is not the
summit of knowledge, but beyond it is intellect; not that intellect
only which is separated from soul, but the illustration infused from
thence into the soul, which Aristotle affirms to be the intellect by
which we acknowledge the principles of science; and Timæus says, that
this exists in no place but the soul. Ascending, therefore, to this
intellect, we must contemplate together with it intelligible essence,
by indivisible and simple perceptions, speculating the simple genera
of beings. But after venerable intellect itself, it will be proper to
contemplate that summit of the soul, by which we are one, and under
whose influence our multitude is united. For as by our intellect
we touch the divine intellect; so by our unity, and as it were the
_flower_ of our essence, it will be lawful to touch that first
one, from whom all subordinate unities proceed. And by this our
_one_, we are especially conjoined with divinity. For similitude
may be every where comprehended by that which is similar; the objects
of knowledge by science; things intelligible by intellect; and the most
unifying measures of being, by the unity of the soul. But this unity
and its energy is the summit of our actions; for by this we become
divine, when, flying from all multitude, we retire into the depths of
our unity, and, being collected into one, uniformly energize. Thus far
we admonish to shun the multitude, by steps proceeding from the order
of knowledge: in the next place, we shall proceed in the same design
by the series of knowable objects. Fly then every sensible species,
for they are heaped together, are divisible, and perfectly mutable,
and incapable of affording sincere and genuine knowledge. From these
dark informations, therefore, betake yourself to incorporeal essence;
since every sensible object possesses adventitious unity, is by itself
scattered and confused, and full of formless infinity. Hence its good
is divisible, and adventitious, distant and separated from itself, and
residing in a foreign seat. When you have ascended thither, and are
placed among incorporeal beings, you will behold above the fluctuating
empire of bodies, the sublime animal order, self-moving, spontaneously
energizing in itself, and from itself possessing its own essence, yet
multiplied, and anticipating in itself a certain apparition or image
of the essence divisible about the unstable order of bodies. You will
there perceive many habitudes of reasons, various proportions, and
according bonds. Likewise the whole and parts, vivid circles, and
a multiform variety of powers; together with a perfection of souls
not-eternal, not subsisting, together as a whole, but, unfolded by
time, gradually departing from their integrity, and conversant with
continual circulations. For such is the nature of the soul.

But after the multitude belonging to souls, betake yourself to
intellect, and the intellectual kingdoms, that you may possess the
unity of things. There remain in contemplation of a nature ever abiding
in eternity, of life ever flourishing, intelligence ever vigilant, to
which no perfection of being is wanting, and which does not desire
the chariot of time, for the full energy of its essence. When you
have beheld natures of this exalted kind, and have seen by how great
an interval they are superior to souls; in the next place enquire
whether any multitude is there, and if intellect, since it is one, is
also universal; and again, since it is uniform, if not also multiform:
for you will find it subsists after this manner. When, therefore, you
have intimately beheld this intellectual multitude, though profoundly
indivisible and united, transport yourself again to another principle,
and having considered, as in a more exalted rank, the _unities_
of intellectual essences, in the last place proceed to unity perfectly
separate and free from all things. And when advanced thus far, lay
aside all multitude, and you will at length arrive at the ineffable
fountain of _good_. And since it appears, from these various
gradations, that the soul then properly obtains perfection, when she
flies from all external and internal multitude, and the boundless
variety of the universe, we may likewise conclude from hence, that
our souls do not alone collect their knowledge from the obscure
objects of sense, nor from things particular and divisible discover a
perfect whole, and a perfect one, but draw forth science from their
inmost recesses, and produce accuracy and perfection from whatever
in appearances is inaccurate and imperfect. For it is not proper to
suppose that things false and obscure, should be the principal sources
of knowledge to the soul; and that things discordant among themselves,
which require the reasonings and arguments of the soul, and which are
ambiguous and confused, should precede science which is immutable; nor
that things variously changed, should generate reasons abiding in one;
nor that indeterminate beings should exist as the causes of determinate
intelligence. It is not, therefore, fit to receive the truth of
eternal entities from boundless multitude; nor from sensible objects
the judgment of universals; nor from things destitute of reason,
accurate discrimination of that which is good: but it is proper that
the soul, retiring into her immortal essence, should there scrutinize
the _good_ and the _true_, and the immutable reasons of all
things: for the essence of the soul is full of these, though they are
clouded by oblivion. The soul, therefore, beholding exteriors, enquires
after truth, in the mean time possessing it in the depths of her
essence, and deserting herself, explores the good in the dark regions
of matter. Hence, every one in the pursuit of reality ought to begin
with the knowledge of himself. For, if we constantly extend our views
among the multitude of men, we shall never discern the one species man,
obscured by the multitude, and distracted by the division and discord,
and the various mutations of those who participate the species. But
if we turn our eye inwards, there, remote from perturbation, we shall
behold one reason and nature of men; since multitude is an impediment
to the conversion of the soul into herself. For here variety darkens
unity, difference obscures identity, and dissimilitude clouds
similitude; since species are confused in the folds of matter; and
every where that which is excellent is mixed with the base.” Thus far
Proclus; and thus much for our intended Dissertation.




                                 LIFE
                                  AND
                             COMMENTARIES
                                  OF
                               PROCLUS.




                                  THE
                           LIFE OF PROCLUS,
                            BY MARINUS[39];
                                  OR,
                         CONCERNING FELICITY.


When I consider the magnitude of mind, and dignity of character
belonging to Proclus, a philosopher of our time, and attend to those
requisites, and that power of composition which those ought to possess
who undertake a description of his life; and lastly, when I regard
my own poverty of diction, I am inclined to believe it more proper
to refrain from such an undertaking, not to leap over the fosse
(according to the proverb), and to decline a discourse involved in so
much difficulty and danger. But my scruples are something diminished
when I consider, on the other hand, that even in temples, those who
approach to the altars do not all sacrifice alike; but that some are
solicitously employed in preparing bulls, goats, and other things of a
similar kind, as not unworthy the beneficence of the Gods to whom those
altars belong: likewise that they compose hymns, some of which are more
elegant in verse, but others in prose; while some, who are destitute
of all such gifts, and sacrifice with nothing more perhaps than a cake
and a small quantity of bread, with frankincense, and who finish their
invocations with a short address to the particular divinity they adore,
are not less heard than others. While I thus think with myself, I am
afraid, according to Ibycus[40], lest I should not offend against the
Gods (for these are his words) but against a wise man, and thus obtain
the praise of men.

For I do not think it lawful, that I who was one of his familiars,
should be silent concerning his life; and should not, according to my
utmost ability, relate such particulars concerning him as are true, and
which perhaps ought to be published in preference to others. And indeed
by such a neglect I shall not perhaps obtain the esteem and honour
of mankind, who will not entirely ascribe my conduct to the desire
of avoiding ostentation, but will suppose I avoided such a design
from indolence, or some, more dreadful disease of the soul. Incited,
therefore, by all these considerations, I have taken upon me to relate
some illustrious particulars of this philosopher, since they are almost
infinite, and may be depended on for their undoubted reality.

I shall begin therefore not according to the usual manner of writers,
who are accustomed to distribute their discourse into chapters; but
I consider that the felicity of this blessed man ought, with the
greatest propriety, to be placed as the foundation of this treatise.
For I regard him as the most happy of those men who were celebrated
in former ages; I do not say happy only from the felicity of wisdom,
though he possessed this in the highest degree of all men; nor because
he abundantly enjoyed the goods of an animal life; nor again on account
of his fortune, though this belonged to him in a most eminent degree,
for he was supplied with a great abundance of all such things as are
called external goods: but I call him happy, because his felicity
was perfect, complete in all parts, and composed from each of the
preceding particulars. Having then in the first place distributed[41]
virtues according to their kinds into natural, moral, and political,
and also into those of a sublimer rank, which are wholly conversant
with purification and contemplation, and are therefore called Cathartic
and Theoretic, and also such as are denominated Theurgic, by which we
acquire a similitude with some particular divinity; but omitting such
as are superior to these, as beyond the reach of man, we shall begin
from such as are more natural, and which are first in the progressions
of the human soul, though not first in the nature of things.

This blessed man, then, whose praise is the subject of this treatise,
naturally possessed, from the hour of his birth, all those physical
virtues which fall to the lot of mankind; the traces of which were
manifest in the latest period of his life, and appeared to surround and
invest his body after the manner of a tenacious shell. In the first
place, he was endued with a singular perfection of sensation, which
they denominate corporeal prudence; and this was particularly evident
in the nobler senses of seeing and hearing, which are indeed given by
the gods to men for the purpose of philosophizing, and for the greater
convenience of the animal life; and which remained entire to this
divine man through the whole of his life. Secondly, he possessed a
strength of body which was not affected by cold, and which was neither
weakened nor disturbed by any vicious or negligent diet, nor by any
endurance of labours, though it was exhausted day and night, while he
was employed in prayer, in perusing the works of others, in writing
books himself, and in conversing with his familiars; all which he
performed with such expedition, that he appeared to study but one thing
alone. But a power of this kind may with propriety be called fortitude
of body, from the singular strength employed in such exertions. The
third corporeal virtue with which he was endued was beauty, which, when
compared with temperance, the authors of these appellations have very
properly considered as possessing a similitude of nature. For as we
consider temperance as consisting in a certain symphony and consent of
the powers of the soul, so corporeal beauty is understood to consist
in a certain agreement of the organical parts. He was indeed of a most
pleasing aspect, not only because he was endued with this excellent
proportion of body, but because the flourishing condition of his soul
beamed through his corporeal frame like a living light, with splendors
too wonderful for language to explain. And indeed he was so beautiful
that no painter could accurately describe his resemblance; and all the
pictures of him which were circulated, although very beautiful, were
far short of the true beauty of the original. But the fourth corporeal
virtue which he possessed was health, which they affirm corresponds
to justice in the soul; and that this is a certain justice in the
disposition of the corporeal parts, as the other in those of the soul.
For justice is nothing more than a certain habit, containing the parts
of the soul in their proper duty. Hence, that is called health by
physicians, which conciliates the jarring elements of the body into
union and consent; and which Proclus possessed in such perfection, that
he affirmed he was not ill above twice or thrice, in the course of so
long a life as seventy-five years. But a sufficient proof of this is
evident from hence, that, in his last illness, he was entirely ignorant
what the disorders were which invaded his body, on account of the great
rarity of their incursions.

Such then were the corporeal goods which Proclus possessed, and which
may be called the forerunners, and as it were messengers, of those
forms into which we have distributed perfect virtue. But the first
powers and progeny of his soul, which he naturally possessed, previous
to instruction, and those parts of virtue with which he was adorned,
and which Plato reckons the elements of a philosophic nature[42], must
excite the wonder of any one who considers their excellent quality. For
he was remarkable for his memory and ingenuity; he was of a disposition
magnificent, gentle, and friendly; and a companion, as it were, of
truth, justice, fortitude and temperance; and his love of truth was
so great, that he never admitted any prudent dissimulation, but hated
falsehood vehemently. Indeed it is necessary that he who prosecutes
truth, with so much earnestness and sincerity, should be extremely
desirous of it from his infancy, since truth is the source of every
good, both to gods and men. But that he despised corporeal pleasures,
and was an eminent lover of temperance, is sufficiently evident from
his great propensity to disciplines, and his desire of every kind
of studies; for dispositions of this kind never suffer beastly and
illiberal pleasure to dwell it the mind, but are able to excite in the
soul, from her own internal operations, sincere pleasure and delight.
But it is impossible to say how foreign he was from avarice, so that
when a boy he despised the wealth of his parents, though very rich, on
account of his incredible love towards philosophy. Hence he was far
removed from illiberality, and from the care of lesser concerns, as he
was most studious of the universe, and of every thing divine and human.
But from such a disposition of the rational soul, having acquired true
magnanimity, he considered human life as of no account, and, unlike the
multitude, viewed nothing dreadful in death. So that he by no means
feared all that rout of molestations which appear terrible to others,
and this in consequence of that natural affection which it is proper to
call by no other name than that of Fortitude alone. But, from all these
virtues, I think it must be evident to those who have not experienced
his best of dispositions, that he loved equity from a boy; that he was
just and mild, and by no means difficult or unjust in his associations
or contracts. To us indeed he certainly appeared modest and elegant,
neither avaritious nor illiberal, neither arrogant nor timid.

But will it not be superfluous to mention the goodness and fertility of
his ingenuity? Especially among those who know and who have heard, that
he was full of the most beautiful disciplines, and who are acquainted
with the multitude he produced and published to the world, so that
he alone seemed to have drank nothing of the cup of oblivion, as he
was endued with a power of memory which was never disturbed, and that
which belongs to the oblivious, never happened to him. Besides, he
never neglected fresh acquisitions, as if possessing a sufficiency of
disciplines; and as one who is merely delighted with their study. But
he was most remote from a nature rustic and horrid, and averse from the
Muses, and particularly propense to more cultivated endowments: for on
account of his singular urbanity and festivity (without transgressing
the bounds of true honesty) in his common associations, sacred feasts,
and other actions, he allured and charmed his familiars, and always
dismissed them more cheerful and pleased.

His mother, therefore, Marcella, lawfully united to his father
Patricius, both of the Lycian nation, and excelling in birth and
virtue, produced our philosopher, thus endued from the beginning with
all these, and other gifts of nature. And[43] Minerva, the tutelar
goddess of Byzantium, received him when born, and took care of him as a
midwife, she being the cause of his birth in that city: but afterwards
she provided for his well-being, when he was numbered among boys and
young men; for she appeared once to him in a dream, and exhorted him
to the study of philosophy, from whence arose his great propensity
to this goddess, as he particularly performed her sacred rites, and
cultivated with a greater fury (as I may say) her institutions. Lastly,
his parents brought him, when born, into their native country Zanthus,
consecrated to Apollo: and I cannot but think that this country
happened to him by a certain divine providence; as it was requisite
that he, who was to be the prince of all sciences, should be educated
under the presiding deity of the Muses. Here, being instituted in the
most elegant manners, he pursued moral virtues, and was accustomed
to right conduct, and to a declination of its contrary, that which
is base.

But at that time the love of the gods, who had attended him from his
nativity, manifestly appeared; for being once detained by some disease
of body, and it appearing very difficult, and scarcely possible to
cure him, there stood at his bed a youth of a more than ordinary
appearance, so that even previous to the declaration of his name,
he might be considered as[44] Telesphorus or Apollo: but the god
proclaiming who he was, and pronouncing his name, touched the head of
Proclus (for he stood reclining his head on Proclus’ pillow) and having
immediately restored him to health, vanished from his sight. And such
was the[45] divine vision, and the divine benevolence at that time
exhibited to our youth.

But having, for a short space of time, in Lycia, applied himself
to grammar, he went to Alexandria in Egypt, bringing with him very
singular moral virtues, by which he excited towards himself the love
of the masters resident in that place. Hence Leonas the rhetorician,
who derived (as I think) his lineage from Isaurus, and was illustrious
among many of that profession, who were then at Alexandria, not only
made him a partaker of his studies, but thought him worthy to become
his domestic, and ordered that he should be supplied with food,
together with his wife and children, no otherwise than if he had been
his[46] true son. He likewise took care to procure him the notice of
the principal men in Egypt, who being wonderfully delighted with the
acumen of the youth’s ingenuity, and with the elegance and integrity
of his morals, reckoned him among their greatest friends. But he was
also instructed by Orion the grammarian, whose ancestors discharged
the sacerdotal office among the Egyptians, and who had made such a
progress in the knowledge of his art, that he composed elaborate books
on this subject, which he left not without advantage to posterity. He
also went to the schools of the Roman preceptors, and made a great
progress in that language; for he was at first led to the study of his
father’s profession, in which he was illustrious, his employment being
the study of law in the royal city. But when it appeared how vehemently
the young man was delighted with the study of rhetoric, as he had not
yet touched the writings of the philosophers, he both acquired great
glory from his acquisitions, and became the admiration of his fellow
pupils and masters, on account of the elegance of his discourse, and
his celerity in perceiving; and from his exhibiting more the habit and
industry of the master, than that of the scholar.

But while he yet frequented the rhetorical school, the sophist
Leonas, made him the companion of his journey to Byzantium: which he
undertook for the purpose of gratifying his friend Theodorus, who was
at that time præfect of Alexandria, and who was a man both polite and
magnificent, and a lover of philosophy. But Proclus, though a youth,
followed his master more cheerfully in this journey, lest he should
be compelled to interrupt his studies. However, that I may speak
more truly, a certain good fortune brought him back to the source of
his nativity. For, on his return, his tutelar goddess exhorted him
to philosophy, and to visit the Athenian schools. But having first
returned to Alexandria, and bid farewel to rhetoric, and the other arts
which he had formerly studied, he gave himself up to the discourses of
the philosophers then resident at Alexandria. But he frequented[47]
Olympiodorus, the most illustrious of philosophers, for the sake of
imbibing the doctrine of Aristotle; and in mathematical disciplines
gave himself to Hero[48], a religious man, and one who was eminently
skilful in the proper methods of institution. But these men were so
delighted with the manners of the youth, that Olympiodorus wished
him to espouse his daughter, whom he had taken care to instruct in
philosophy, and Hero committed to him all his religion, and made
him his constant companion. But having, on a certain time, heard
Olympiodorus, a man who was endued with a great power of speaking,
and on account of the celerity of his speech, and the gravity of
his subjects, was understood by very few of his auditors, as he was
departing with the dismissed multitude, he repeated to his companions
all that was said, and almost verbatim, though the discourse was
copious; as Ulpianus Gazæus, one of his fellow-disciples informed
me, who also consumed not the least part of his life in the study of
philosophy. But he likewise learned, with great facility, the writings
of Aristotle, pertaining to rational philosophy[49], though the bare
reading of them is difficult to those who are engaged in the attempt.

Having therefore, at Alexandria, applied himself to these masters,
and enjoyed their confidence in such instruction as they were able to
afford, when upon reading together with them a certain author, they
appeared to him not to interpret the mind of the philosopher as they
ought; conceiving a contempt for these schools, and at the same time
being mindful of the exhortation which had been divinely sent to him
at Byzantium, he went to Athens, attended by the presiding deities of
eloquence and philosophy, and by beneficent dæmons. For that he might
preserve the genuine and entire succession[50] of Plato, he was brought
by the gods to the guardian city of philosophy, as the circumstances
which happened on his first entrance into the city, and all the divine
excitations manifestly evince: for they openly presaged, that this
gift was sent from the father Apollo, and was a future suffrage of his
succession confirmed by divine events. For when his vessel drove to
the Pyræum, and it was told to the citizens, Nicolaus, who afterwards
flourished in the rhetorical art, but at that time studied under the
masters of Athens, descended to the shore as if to an acquaintance,
received him for his guest as if he had been a citizen, and brought
him to the city; for Nicolaus was also a Lycian. But Proclus, who
perceived himself weary from his journey, sat down at the temple of
Socrates, though he did not yet know, nor had heard that Socrates was
worshipped in that place, and requested Nicolaus that he would stay
there for a short time, and, if possible, procure him some water,
as he said that he was exceeding thirsty. Immediately Nicolaus,
from that very consecrated place, brought him some water; for the
fountain belonging to the statue of Socrates was not far distant. But
while he was drinking, Nicolaus, for the first time, considering the
circumstance: This is an omen, says he, because you have sate in the
temple of Socrates, and have there first drank the Attic water. Then
Proclus rising, and having paid due reverence to the place, proceeded
to the city. But when he came to the tower, the porter who was present
at his entrance, and was about to close the gates with bars, said to
him, (that I may repeat the words of the man,) “Certainly unless you
had come, I should have closed the gates.” And what omen could be more
manifest, or could less require the interpretations of[51] Polletes or
Melampodes, or such like diviners?

But Proclus, despising the schools of the rhetoricians, though they
very much desired his association, as if he had come for that very
purpose, met with the prince of philosophers Syrianus[52], the son of
Philoxenus. At that time too, Lachares was in the same company, a man
much conversant in philosophy, and the companion of Syrianus in his
study, but in eloquence he was in as great admiration as Homer in the
poetic art. He then was, as I have said, present at the same time. But
it was now the evening twilight, and while they were engaged in mutual
converse, the sun sate, and the moon made her first appearance after
her change: wherefore, having saluted the stranger, they endeavoured to
dismiss him, as being a young man, from their company, that they might
adore the goddess apart. But he not having proceeded far, beheld also
the moon appearing from the same house, and laying aside his sandals,
in their presence saluted the goddess. Here Lachares, admiring the
confidence of the youth, said, speaking to Syrianus, “This is what
Plato[53] divinely affirms of great geniuses; that they either produce
great good, or its contrary.” And such, that I may mention a few out
of many, were the signatures of divine original, which happened to our
philosopher, on his first arrival at Athens.

But Syrianus brought him to the great Plutarch[54], the son of Nestor,
who, when he saw the young man, not yet twenty years old, and heard
of his love and desire of a philosophic life, he was much delighted,
and immediately made him a partaker of his philosophic study, though
his age almost forbade such an attempt; for he was then very old. He
therefore read to Proclus his commentary on Aristotle’s books on the
soul, and on the Phædo of Plato: and this great man exhorted him to
commit to writing what he heard, employing the ambition of the youth
as an instrument, by telling him, that if he completed those scholia,
they would be reported as the commentaries of Proclus. And as he
loved the youth very much on account of his inclination to the best
studies, he called him his son, and caused him to reside with him as a
domestic. But after he saw his temperance, with regard to animal food,
he exhorted him not to abstain from animals entirely, but to use them
so far as was necessary to the vital energies of the corporeal part. He
likewise gave the same advice to Syrianus, concerning the youth’s diet.
But he answered the old man, as that divine head (Proclus) informed
us: “Suffer him with that frugality to learn what I wish, and then, if
he pleases, he may die.” Such was the care of his masters respecting
him, in every concern. But the old man lived about two years after the
arrival of Proclus; and dying, commended the youth to his successor
Syrianus, as also his grandson Archiadas. But Syrianus, when he had
received Proclus as his pupil, not only much assisted him in learning,
but made him his domestic as to other concerns, and the companion of
his philosophic life, having found him such an auditor and successor
as he had for a long time sought for; and one who was adapted for the
reception of a multitude of disciplines, and divine dogmata.

In a shorter space, therefore, than two years, he read, together
with Syrianus, all the works of Aristotle, i. e. his logic, ethics,
politics, physics, and theological science. But being sufficiently
instructed in these, as in certain[55] proteleia, and small mysteries,
Syrianus led him to the sacred discipline of Plato, and this by an
orderly progression, and not[56] according to the oracle, with a
transcendent foot. And he was careful that he might survey with him
true mysteries, with the eyes of his soul, free from material darkness,
and with a speculation of intellect refined and pure. Hence Proclus
was employed night and day in vigilant energies, and in writing
compendiously what he had heard, employing his own judgment in the
selection and order. And in consequence of this unwearied assiduity,
he made so great a progress in a short time, that by then he was
twenty-eight years of age, he composed a multitude of works; and among
the rest his very learned and elegant commentaries on the Timæus. But
from an institution of this kind, his manners also received a greater
ornament, since as he advanced in science he accumulated virtue.

In the original κατὰ τὸ λόγιον, which I wonder Fabricius should
translate, _quod aiunt_, as it is usual with the Platonists, to
cite the Zoroastrian oracles exactly in these words, instances of which
may be found in Proclus on Plato’s theology; and the very words prove
themselves to be a part of an oracle, when attentively considered.]

But he likewise pursued the civil virtues from Aristotle’s political
writings, and Plato’s books concerning laws and a republic. However,
lest he should be conversant only in the contemplation of these, and
should attempt nothing actual, since he was hindered from engaging
in public affairs himself, from his being employed in more important
concerns, he exhorted Archiadas, who was a religious man, to a
political life, at the same time instructing him in its nature and
virtues, and explaining to him the methods proper to be observed in
the discharge of its duties. And in the same manner as they incite
those who run in the race, so he exhorted him to preside in common
over the whole of his city, and to confer benefits privately upon each
according to all the virtues, but particularly in conformity with the
laws of justice. But he excited in reality the emulation of himself in
Archiadas, when he exhibited to him his own liberality and magnificence
with respect to riches, by bestowing gifts at one time on his friends,
and at another time on his kindred, guests, and citizens; proving
himself by this means not in the least affected with avarice, and
conferring not a little to the good of public concerns. But when dying,
he bequeathed that part of his possessions which he did not leave to
Archiadas, to his own country, and to Athens. But Archiadas became so
great a lover of truth, as well from the company of his own associates
as from the friendship of Proclus, that he is never mentioned by
our acquaintance, without being at the same time celebrated as the
religious Archiadas.

But the philosopher also employed himself in civil consultations
among public assemblies, where things pertaining to the republic
were discussed; giving the most prudent counsels, conferring with
governors concerning equity, and not only exhorting them to an
impartial distribution of justice, but in a manner compelling them
by philosophical authority. For he had a certain public care of the
morals of princes, and not only instructed them in the art of temperate
government by his discourse, but also by his own example through the
whole of his life; since he was, as it were, the exemplar of temperance
to the rest. But he gave a specimen of civil fortitude perfectly
Herculean: for since at that time there was, as it were, a sea of
troubles upon him, and mighty waves of stormy employments were roused
by adverse winds against his upright life, he conducted himself, though
in danger, with gravity, and an unshaken constancy. And when he was
once very much molested by the improbity of some violent men, which
was both pernicious and dangerous to himself, he undertook a journey
into Asia, which contributed greatly to his own advantage: for as he
was not unskilful in the more ancient rites of that place, which he
yet preserved, a divine power afforded him this occasion of departure.
Hence, as he well knew the whole of their concerns, he taught them
more accurately in things pertaining to the gods, if they happened to
have neglected any thing through a long interval of time. And while
he was engaged in all these employments, and lived in a correspondent
manner, he was so concealed from the multitude that he even excelled
the Pythagoreans, who keep with unshaken constancy this precept of
their founder, λάθε βιώσας, _live concealed_. But having passed
a year only in the parts of Lydia, he returned to Athens, under the
providential protection of the presiding goddess of philosophy. And
thus fortitude was perfected in our philosopher, first by nature,
then by custom, and afterwards by science and the consideration of
causes. Besides this, he exhibited in another manner his politic habit
practically, by composing letters for noblemen; and by this means
procuring good to entire cities. But of this I have a sufficient
testimony from those on whom they were bestowed, as well Athenians as
Argives, and others of different nations.

But he likewise much promoted and increased literary studies, demanding
of princes rewards for the preceptors, according to their several
deserts. Nor did he undertake this rashly, nor with any interested
views, but he compelled them (as he considered it a matter of great
moment) to be diligent in their profession, interrogating and
discoursing with them respecting every particular: for he was a judge
sufficiently instructed in the employments of them all. And if he ever
found any one negligent in his profession, he sharply reproved him;
so that he appeared very vehement and ambitious, because he was both
willing and able to give a just determination on every subject: and
he was indeed a lover of glory. But this was not a fault in him, as
in most, because it alone regarded virtue and goodness. And, perhaps,
without an energy of this kind, nothing great and excellent would ever
subsist in the human mind. But he was in this respect vehement: this I
will not deny. Yet, at the same time, he was gentle; for he was easily
pleased, and demonstrated in a moment that his anger was as pliable
as wax. For, almost at the same time, he was (as I may say) wholly
transported in reprehension, and with a desire of becoming subservient
to their interest, and that he might intercede with princes in their
names; being moved with a certain natural conjunction of soul, and, as
it were, sympathy of grief.

And here, I very opportunely recollect a peculiar example of his
natural sympathy of soul with others: nor do I think the like was
ever related of any other man. For, notwithstanding he was unmarried,
and had no children (because he was not desirous of such connections,
but remained free from them all, though many noble and rich alliances
were offered him), yet his care of all his familiars and friends,
and of their wives and children, was as great as if he had been
some common father, and the cause of their birth; for he bestowed a
singular attention on the life of each. And whenever any one of them
was detained by any disease, he first earnestly supplicated the gods
on their behalf with sacrifices and hymns; afterwards he gave a prompt
attendance on the sick person himself, convened the physicians, and
urged them to make an immediate trial, if they knew of any thing in
their art advantageous to the condition of the diseased; and sometimes
he produced some singular advice himself, among the physicians; and
thus delivered many from imminent dangers. And the greatness of this
blessed man’s humanity towards his servants, may be understood by
those who desire it, from his will. But of all his familiars, he loved
Archiadas and his kindred the most; because, in the first place, their
succession was derived from the genus of Plutarch the philosopher;
and afterwards on account of that[57] Pythagoric friendship which he
maintained with Archiadas, as he was both the companion and preceptor
of his studies. And this other kind of friendship, differing from
the two already mentioned, appears to have been the most firm and
excellent, among these illustrious men. For Archiadas desired nothing,
which was not also the wish of Proclus; and on the contrary, the
desires of Proclus were the constant wishes of Archiadas.

But having now brought the political virtues, which are inferior to
the true ones, to an end, and terminating them in[58] friendship, as
their proper bound, we shall now pass to the Cathartic differing from
the politic virtues. Indeed, the employment of these last consists in
purifying the soul, that so being liberated from the body as much as
they are able to effect, it may regard human concerns, and possess a
certain similitude with divinity; which is the soul’s best and most
exalted end. Yet they do not all liberate after the same manner,
but some more, and others less. Since there are certain political
purgations which adorn their possessors, even while connected with
body, and reduce them to a better condition; bringing under the
dominion of reason, anger and desire, and entirely destroying passion
and every false opinion: but the Cathartic virtues, which are superior
to these, separate entirely from this truly leaden weight of body, and
procure an easy flight from mundane concerns. And in these, indeed, our
philosopher was studiously employed during the whole of his life, which
was devoted to philosophy; since he both taught by his discourses what
they were, and after what manner they were preparatory to felicity,
and in a particular manner conformed his life to their institutions;
performing every thing which could contribute to the separation of
his soul, using both night and day prayers, lustrations, and other
purifications, as well according to the Orphic as the Chaldaic
institutions: and every month he descended, with great diligence, to
the sea; and this sometimes twice or thrice. But he was exercised in
these, not only in the vigour of his age, but also towards the close of
his life; and these customs he observed perpetually, as if they were
certain invariable statutes.

But he used meat and drink, and other necessary pleasures, only so
far as was necessary to avoid the molestations of disease; for he was
in these by much the most frugal, and particularly loved abstinence
from animal food. And if at any time he was invited to eat it more
vehemently, he was so cautious in its use, that he ate it merely after
the manner of a taster. But he purified himself every month by the
sacred rites, in honour of the mother of the gods, celebrated by the
Romans, and prior to them by the Phrygians: he likewise more diligently
observed the unfortunate days of the Egyptians than they themselves;
and, besides this, fasted, on certain days in a peculiar manner, on
account of the lunar appearances[59]. He likewise instituted a fast
on the last day of the month, not having supped the day before. But
in what a splendid manner, and with what piety, he celebrated the new
moon, and properly observed, with sacrifices, the more illustrious
feasts of almost all nations, according to the manner of each country;
and how from these he did not, according to the custom of many, take
occasion of becoming idle and intemperate, but employed himself in
continual prayers, hymns, and the like, his hymns sufficiently evince,
which not only celebrate the divinities of the Greeks, but likewise
Marna Gazæus, Esculapius Leonteuchus, Ascalonites, and Theandrites,
another god much venerated by the Arabians; together with Isis,
worshipped by the Philians; and lastly, all the rest which were the
subjects of his devotion. For this sentiment was very familiar to
this most religious man, that it was proper a philosopher should not
be careful in the observance of the rites and institutions of one
particular city, nor of certain nations only, but that he should be the
general priest of the universe. And thus was he pure and holy, so far
as pertains to the virtue of temperance.

But he declined, as much as possible, pain: and if it ever happened
to him, he bore it with gentleness, and diminished it with this view,
that his best part might not at the same time be affected with its
molestations. And the fortitude of his soul in this respect, was
sufficiently evinced in his last illness; for when, at that time,
he was oppressed and tormented with the most excruciating pains, he
endeavoured to the utmost to mitigate and expel their afflictive
invasions. Hence, on such occasions, he often commanded us to repeat
certain hymns, which when repeated procured him a remission and
cessation of pain. And what is more wonderful, he remembered what he
heard of these, though forgetful of almost all human concerns, from the
dissolution of his corporeal part continually increasing. For when we
began to repeat, he supplied what was unfinished of the hymns, together
with many of the Orphic verses; for it was these we were then reciting.
Nor was he only thus constant in enduring corporeal evils, but much
more so in external unfortunate events, and such as appeared to happen
contrary to expectation. So that he would say, concerning particulars
of this kind, _So it is, such things are usual_; which seemed to
me, or rather at that time appeared to be, worthy of remembrance, and
an evident argument of our philosopher’s magnanimity. But besides this,
he restrained anger as much as possible, so that it might either remain
free from all excitation, or that at least reason might not consent to
its indulgence, but the irrational faculty alone, contrary to his will,
might be moderately and lightly excited. And with respect to venereal
concerns, he used them in the natural way, but so as that he might not
proceed beyond a very moderate and light phantasy, in their indulgence.

And thus the soul of this blessed man, having collected itself from
all parts, and retiring into the depths of its essence, departed
after a manner from body; while it yet appeared to be contained in
its dark receptacle. For he possessed a prudence, not like that of a
civil nature, which is conversant in the administration of fluctuating
particulars, but prudence itself, by itself sincere, which is engaged
in contemplating, and converting itself into itself, without any longer
consenting to a corporeal nature. He likewise possessed a temperance
free from evil; and which is not even moderately influenced by
perturbations, but is abstracted from all affections. And lastly, he
acquired a fortitude, which does not fear a departure from body. But
reason and intellect having obtained in him a perfect dominion, and the
inferior powers of his soul no longer opposing themselves to purifying
justice, his whole life was adorned with the divine irradiations of
genuine virtue.

Our philosopher, therefore, having most happily absolved this form of
virtues, advancing now, as it were, by the highest and most mystical
step, he ascended to the greatest and most consummate or telestic
virtues; employing for this purpose, the felicity of his nature, and
a sciential institution. Hence, being now purified, and the victor of
his nativity, and despising the vain Thyrsus-bearers, and boasters of
wisdom, he happily penetrated into her profound recesses, and enjoyed
the contemplation of the truly blessed spectacles she contains. No
longer requiring prolix dissertations, or demonstrations, for the
purpose of collecting the science of these, but, with a simple vision
and energy of intellect, beholding the exemplar of the divine mind,
he obtained a virtue which cannot with sufficient propriety be called
prudence, but is more properly denominated wisdom, or something,
if possible, still more venerable and divine. But the philosopher
energizing, according to this virtue, easily comprehended all the
theology of the Greeks and Barbarians, and whatever is shadowed over
by the figments of fables, and placed it in a clear light, for the use
of those who are willing and able to pursue its latent signification.
But having interpreted divinely every thing of this kind, and shewing
the symphony between them all; at the same time, investigating all the
writings of the ancients, whatever he found in them of genuine wisdom,
and approved by general consent, this he judiciously applied to use;
but if he found any thing of a different and dissonant nature, this
he entirely rejected, as vicious and false. And whatever he met with
contrary to wisdom, though endued with a friendly appearance, this he
vigorously subverted by a diligent examination. Nor did he employ less
force and perspicuity in his association with other men. For he was a
man laborious to a miracle; as he often, in one day absolved five, and
sometimes more lectures; and writ besides, many verses, often to the
number of seven hundred. Besides this, he went to other philosophers,
and frequented their company; and again celebrated with them an
evening association, ceasing from the labour of writing. And all
these employments he executed in such a manner, as not to neglect his
nocturnal and vigilant piety to the gods, and assiduously supplicating
the sun when rising, when at his meridian, and when he sets.

But he was the parent of many dogmata, which were never known before,
both in physics, and in intellectual and more divine concerns. For he
first taught, that there was a kind of souls[60], endued with the power
of contemplating many forms at once, which he placed, not without great
probability, between intellect, collectively, and as it were with one
intuition comprehending every thing, and souls, which are alone able to
direct their vision to one particular form. And those who are willing
to peruse his works will meet with a great variety of dogmata, peculiar
to him alone; the relation of which I shall omit, lest I should give a
too great extent to my discourse. But he who evolves his writings, will
easily perceive that all we have above related of him is most true, and
much more if he happens to have known him, to have seen his face, and
to have heard him interpreting in the most excellent dissertations, and
delivering the Platonic and Socratic dogmata in his yearly schools.
Nor did he seem destitute of divine inspiration; for he produced from
his wise mouth, words similar to the most white and thick falling
snow[61]; so that his eyes emitted a bright radiance, and the rest of
his countenance was resplendent with a divine light. Hence, when on a
certain time, one Rufinus, a man of a great name in the republic, who
was studious of truth, and in other respects worthy of veneration, came
to him when he was teaching and interpreting, he perceived that the
head of Proclus was surrounded with a light; and when the philosopher
had finished his interpretation, Rufinus rising, adored him; and
offered to give a public testimony, by oath, of the divine vision which
he had observed. And much gold was offered to him, by the same Rufinus,
on his return from Asia, having escaped the danger of the war. But
Proclus likewise rejected this gift, and was by no means willing to
receive it.

But that we may return to our first design, having now discoursed
concerning the contemplative wisdom of the philosopher, though
in a manner but little suited to its dignity, it remains that we
now speak of the justice pertaining to this kind of virtues. For
this, unlike that of which we spoke before, is not conversant in
distribution, or proportion; but must be equally removed from the kind
of self-energizing justice, by which all things are alone directed to
the rational soul. For to that, concerning which we now treat, it is
alone proper to refer every energy to intellect and the deity, which
our philosopher performed in the most exalted manner. For he scarcely
rested from his diurnal labours, or refreshed his body with sleep,
and perhaps even then was not free from meditation and contemplation.
This is certain, that having very speedily roused himself from sleep,
as from a certain torpor of the soul, he aspired after the morning,
the time of prayer; and lest the greater part of the night, should
glide from him without advantage, as he was lying alone in his bed, he
either composed hymns, or examined and fortified those dogmata which
afterwards, in the day time, he committed to writing.

After a similar manner he pursued that temperance which has an affinity
with these virtues, and which consists in a conversion of the soul to
intellect, so as not to suffer itself to be touched, nor moved with any
other concerns. Lastly, he joined fortitude in alliance with these, by
a certain perfect method, zealously aspiring after that liberty which
is ignorant of all passion, and which he perceived was natural to the
divine object of his contemplation. And thus, through the whole of his
conduct, he did not lead the life of a man merely good, to which, as
Plotinus says, the political virtues may lead, but leaving this far
behind him, he endeavoured to change it for one much more perfect and
divine, the life of the gods themselves; since, to become similar to
these, and not to virtuous men, was the great object of his contention.

And thus he had rendered virtues of this kind familiar to himself,
while he frequented the philosopher Syrianus, and evolved and studied
the commentaries of the ancients. But he received from the mouth of
his preceptor certain small seeds, as it were, of the Orphic and
Chaldaic theology; because he was prevented from hearing the complete
interpretation of his master on the Orphic verses. For Syrianus left to
the choice of Proclus and one Domninus[62], a philosopher of the Syrian
nation, and who afterwards succeeded Syrianus, the exposition of the
Orphic writings, or the oracles. But they were by no means unanimous
in their choice; for Domninus preferred the interpretation of the
Orphic verses, and Proclus that of the oracles. But our philosopher did
not perfect his undertaking, because the death of the great Syrianus
happened not long after. Having therefore, as I have said, received
the outlines from the mouth of his master, he applied himself with
the greatest diligence to the written commentaries of Syrianus upon
Orpheus; and being assiduously nourished with the copious lucubrations
of Porphyry and Jamblichus on the oracles, and similar writings of
the Chaldeans, he arrived, as much as is possible to man, to the top
of those highest virtues, which the divine Jamblichus was accustomed
to call after a truly divine manner, theurgic. He laboured therefore,
not without exquisite judgment, in collecting the expositions of
philosophers prior to his time; and contracted into one, other
Chaldaic hypotheses, and the most excellent of the commentaries on
the divine oracles, completing this great work in the space of five
years; concerning which this divine vision appeared to him in his
sleep. For he saw the great Plutarch approach to him, affirming that
he should live so many years as he had composed tetrads or quaternions
on the oracles. Afterwards, having collected the number of these, he
found they amounted to seventy. But that this was a divine dream,
was sufficiently evinced by the last part of his life; for though
he lived seventy-five years, as we have mentioned above, yet he had
not the perfect use of his powers, in the five last. But his body,
though naturally of the best constitution, being debilitated by that
hard and scarcely tolerable kind of diet which he used, and by so
many frequent labours and fastings, began to languish exactly at his
seventieth year, so that he then became much more remiss in all his
energies than before. Yet, even at this period, and thus affected, he
composed orations and hymns: he also writ some things, and conversed
with his friends; but his ancient vigour was wanting in each. Hence
the memory of the dream excited his wonder, and he every where said,
that he had only lived seventy years. But while he laboured under this
infirmity of body, a certain youth, named Hegias, rendered him more
alert in the business of interpretation. This young man, who already
exhibited from his tender years such egregious signs of all the virtues
of his ancestors, was one of that golden chain of philosophers, who
had formerly appeared to men; and adhered most diligently to Proclus
delivering the Platonic and other theologies. But Proclus, at this
advanced period, was not moderately rejoiced in communicating with
the young man, his own writings, after he understood that he made
cubital advances in every kind of disciplines. And thus we have briefly
discoursed concerning the apparatus of the philosopher in the Chaldean
oracles.

But I, on a certain time, having read with him the Orphic verses, and
heard, among his interpretations, not only the recondite theology
which is to be found in Jamblichus and Syrianus, but also, among many
other divine men, I requested the philosopher that he would not leave
these divine verses also without his explanation: but his answer was,
That he had often thought of writing commentaries on Orpheus, but that
he had been strongly prohibited in more than one dream. For Syrianus
appearing to him in his sleep, had deterred him with threats from the
design. Having therefore employed other machines, I intreated that at
least he would mark what he principally approved of in the books of
his master; which when this best of men had performed, in consequence
of my persuasions, and had noted some things in the front of each of
Syrianus’s commentaries, we obtained a collection of all these, and
by this means scholia, and commentaries of no small bulk; though to
accomplish this on the whole of that divine poetry, and on all the
Orphic rhapsiodies, was not the intention of Proclus.

But since, as we have said, his incredible study of these concerns,
procured him a greater and more perfect degree of theurgic virtue,
hence he no longer remained in the contemplative order, nor contented
with either of the twofold properties in divine concerns, exercised
intellect and thought alone on the more excellent objects of
speculation: but he was careful to obtain another kind of virtues more
divine than the former, and separated from the politic mode; for he
used the Chaldean assemblies and conferences, and their divine and
ineffable concealments. And having comprehended these, he learned the
manner of pronouncing and expressing them, with their remaining use,
from Asclepigenia, the daughter of Plutarch: for she alone, at that
time, preserved the knowledge of the great Orgies, and of the whole
theurgic discipline, delivered to her by her father, who received it
from Nestor. Besides this, our philosopher also being purified in an
orderly manner in the Chaldean lustrations, was an inspector of the
lucid hecatic phasmai (or visions) of which he himself makes mention in
one of his commentaries. But by moving a certain hecatic sphærula[63],
he very opportunely brought down showers of rain, and freed Athens
from an unseasonable heat. Besides this, by certain phylacteria, or
charms, he stopt an earthquake, and had thoroughly tried the energies
of a divining tripod, having learned from certain verses concerning its
defect. For when he was in his fortieth year, he thought in a dream,
that he repeated to himself, the following verses:

    High above æther there with radiance bright,
    A pure immortal splendor wings its flight;
    Whose beams divine with vivid force aspire,
    And leap resounding from a fount of fire.

And in the beginning of his forty-second year, he appeared to himself
to pronounce these verses, with a loud voice:

    Lo! on my soul a sacred fire descends,
    Whose vivid pow’r the intellect extends;
    From whence far-beaming through dull bodies night,
    It soars to æther deck’d with starry light;
    And with soft murmurs through the azure round,
    The lucid regions of the gods resound.

Besides, he clearly perceived that he belonged to the Mercurial series;
and was persuaded from a dream that he possessed the soul of[64]
Nicomachus the Pythagorean.

Indeed, if I were willing to be prolix, I could relate many theurgical
operations of this blessed man; from the infinite number of which,
I will only relate one in this place of a very wonderful nature.
Asclepigenia, the daughter of Archiades and Plutarch, but the wife of
Theagenes, from whom I have received many benefits, while she was yet a
girl, and was nourished by her parents, fell into a disease, grievous,
and incurable by the physicians. And Archiades, who had reposed in her
alone the hope of his race, was deeply afflicted with her condition.
But the physicians despairing of her recovery, the father fled to his
last anchor, Proclus the philosopher, which he was accustomed to do in
affairs of the greatest moment, considering him as his best preserver,
and earnestly intreated him to intercede by his prayers with the god
for his daughter, whose condition was well known to our philosopher.
Proclus therefore, taking with him Pericles[65] the Lydian, a man
well deserving the name of a philosopher, went to the temple of
Esculapius[66], that he might pray to the god for the health of the
sick virgin: for the city at that time happily possessed this divinity,
and rejoiced in the temple of the saviour god, which was as yet free
from the destruction of the Christians. As soon, therefore, as Proclus
had prayed after the ancient manner, the girl immediately perceived a
great change and alleviation of her disease; for the preserving god had
easily restored her to health. The sacred rites being finished, Proclus
went to Asclepigenia, and found her delivered from the molestations of
disease, and in a healthy condition. But this affair, with many others,
was performed privately, so that no traces of it remained to succeeding
investigators; and the house in which he resided greatly assisted him
in this design. For besides his other fortunes, he enjoyed a most
convenient dwelling, which his father Syrianus and his grand-father
Plutarch (for so he did not hesitate to call him) had once inhabited;
and this was situated near the temple of Esculapius, together with
that of the celebrated Sophocles, and of Bacchus, which is next to the
theatre, and looks towards, or is at least seen from, the lofty towers
of Minerva.

But how much Proclus was loved by the philosophic goddess is
abundantly evinced by his philosophic life, which he chose through
her persuasions, and that with the great success we have hitherto
described. But she clearly demonstrated her affection to Proclus, by
the following circumstance. When her image, which had been so long
dedicated in the parthenon, or temple, was taken away by those[67] who,
without any hesitation, moved out of their places things the most holy,
and which ought to be immoveable, there appeared to the philosopher
in a dream, a woman of a graceful form, who admonished him to build a
temple with great expedition, for, says she, _it pleases Minerva, the
presiding deity of philosophy, to dwell with you_. And how familiar
he was with Esculapius, besides what we have mentioned above, may be
evinced from his perceiving the presence of the same god in his last
illness. For while he was between sleeping and waking, he perceived a
dragon creeping on his head, which vision occasioned a remission of
his disease, and a mitigation of his pain; so that it is probable he
would have been restored to perfect health, if his desire and vehement
expectation of death had not prevented his recovery, or his no longer
bestowing a diligent attention on his body.

But he also related the following circumstances, (worthy of being
remembered) but not without tears, through the sympathy of his mind.
For, when a young man, he was afraid lest he should be infested
with the gout, which was the disease of his father, and which loves
to descend from parents to their children. Nor was he afraid, as
it seems, without reason; for prior to that which we have already
related concerning him, he was tormented with pains of this kind, when
another extraordinary circumstance happened to this blessed man. In
consequence of a certain person’s prescription, whom he had consulted,
he applied a plaster to his tormented foot, which an unexpected bird
flew away with as he was lying on his bed. And this was certainly a
divine and salutary symbol to Proclus, and might have persuaded him
not to be afraid of this calamity in future. But he, as I have said,
was notwithstanding afflicted with the dread of this disease; he
therefore supplicated the healing god concerning this, and intreated
him to afford him a more evident token of his will: and after this he
saw in his sleep (it is bold indeed to conceive such a circumstance in
the mind, but we must dare, nevertheless, nor dread to bring truth
to light) Esculapius approaching to him, and accurately contemplating
his leg, nor disdaining, through his great philanthropy to embrace his
knee. Hence, in consequence of this vision, Proclus was, through the
whole of his life, free from the apprehensions of this disease, and was
never more tortured with this kind of pains.

Again, the god who is worshipped by the Adrotteni, did not less
conspicuously evince his great familiarity with this friend of the
gods; for, more than once, the god benevolently presented himself to
the sight of Proclus, frequenting his temple. And when he was doubtful,
and wished to know what god or gods resided and were worshipped in
that place, because the inhabitants were of different opinions in
this affair, some believing that the temple of Esculapius was there,
persuaded of this by various signs, as voices are certainly said to
be heard in that place, and a table is consecrated to the gods, and
oracles are given concerning the recovery of health, and those who
approach are miraculously delivered from the most grievous dangers
of life. Others, on the contrary, think that the Dioscuri reside in
that place, because there have been those who saw in the way tending
towards Adrotta, two young men of the most beautiful form, and riding
with great celerity on horseback; at the same time declaring, that
they hastened to the temple. They add besides, that the countenance of
these was indeed human, but that they immediately gave evidence of a
more divine presence; for when the men were arrived at the temple, the
above mentioned youths appeared to them, making no enquiry concerning
the affair, and occupied in the sacred concerns; but presently after,
withdrew themselves from their sight. As the philosopher, therefore,
was in doubt concerning these divinities, and did not discredit the
relations, having requested the gods who inhabited that place, that
they would condescend to manifest who they were; a god appeared to him
in a dream, and clearly spoke to him as follows: _What! Hast thou not
heard of Jamblichus teaching concerning, and celebrating those two,
Machaon and Podalirius?_ And besides this, the god condescended to
afford so great an instance of benevolence to this blessed man, that
he stood after the manner of those who bestow encomiums on others in
the theatres, and with a clear voice, and composed habit, extending his
right hand, did not hesitate to exclaim, (that I may relate the words
of the god,) _Proclus the ornament of the city_. And what could
possibly be a greater testimony of this blessed man’s friendship with
divinity? But he, indeed, in consequence of a most remarkable sympathy,
by which he was united with the gods, could never refrain from tears
when he mentioned this affair to us, and related the divine encomium of
himself.

Indeed, if I were willing to pursue every particular in like manner,
and to relate his familiarity with the Hermetic Pan, together with
the great benevolence and manifold assistance which that divinity
condescended to afford Proclus at Athens, and of that perfectly
singular kind of felicity which he obtained from the mother of the
gods, and in which he was accustomed especially to triumph and rejoice;
I should perhaps seem to many readers, to be rash in my assertions,
and to others, the author of things perfectly incredible. For many and
great were the daily instances of this goddess’s benevolence towards
him, in words and actions, which are both innumerable and unheard
of, and concerning which, I do not at present retain an accurate
remembrance. But if any one is desirous of knowing how great he was in
these, he must evolve his book concerning the mother of the gods, from
which he will understand, that it was not without divine assistance
he produced all the theology of that goddess into light; as likewise,
whatever is acted or affirmed concerning the same in the fables of
Athis, which he has explained after a philosophical manner, that
vulgar ears might afterwards cease to be disturbed, on perceiving the
lamentations and other obscure ceremonies with which her mysteries are
celebrated.

After having, therefore, run through the theurgic virtue of Proclus,
and its energies, and the happy circumstances which befell him,
respecting its operations, and having shewn, that he did not less excel
in every kind of virtue, and that he was a man such as mortals have
not beheld for a long period of time, it remains that we now bring our
discourse concerning him to a conclusion: for we are not now at the
beginning of our narration, nor does the half of the whole remain to
us, according to the proverb, but the whole itself is now perfect and
complete. Since having begun from the felicity of the philosopher,
and proceeded in its exemplification, our discourse now returns to
it again. For we have explained the great goods, and providential
exertions, which were granted to this most excellent man from the
gods, and have shewn their prompt attention to his prayers, the divine
visions which he enjoyed, and the help and solicitude which the gods
testified towards him. We have likewise explained his prosperous fate,
and propitious fortune, his country, parents, the good habit of his
body, his masters, friends, and other external advantages; each of
which we have shewn to have been far greater and more splendid in
him than in other men; and have likewise diligently enumerated such
things as cannot be reckoned amongst outward allurements, but entirely
depended on his will, such as are the upright and illustrious deeds of
his soul, according to universal virtue. And thus we have summarily
demonstrated that his soul arrived in reality at the summit of the most
consummate virtue, and was happily established in a perfect life, by
human and divine goods of every kind.

But that the lovers of more elegant studies way be able to conjecture
from the position of the stars under which he was born, that the
condition of his life, was by no means among the last or middling
classes, but ranked among the highest and most happy orders, we have
thought fit to expose in this place, the following scheme of his
nativity[68].

            ☉      16°    26’    ♒︎
            ☽      17     29     ♊︎
            ♄      24     23     ♉︎
            ♃      24     41     ♉︎
            ♂       29     50     ♐︎
            ♀       --     23     ♓︎
            ☿       4     42     ♒︎
       Horos.        8     19     ♈︎
    Mid.    }
    Heaven. }        4     42     ♑︎
            ☋ or the head of the dragon.
                    24     33     ♏︎
    The new moon preceding his birth.
                     8     51     ♒︎

But Proclus departed from this corporeal life, in the one hundred and
twenty-fourth year after the government of Julian, on the seventeenth
day of the Attic Munichion, or the April of the Romans. Nicagoras,
the junior, being at that time the Athenian archon. The dead body was
taken care of according to the Athenian rites, as he himself while
living had directed. And if that diligent attention which is due to
the deceased, was ever paid by any one, it was certainly bestowed by
this most blessed man. For he did not neglect any particular of the
accustomed ceremonies, but every year, on certain days, visited the
sepulchres of the Athenian heroes and philosophers, and besides this
sacrificed himself, and not through the medium of another, to the
manes of his friends and familiars. And having thus exhibited to each
a proper reverence and honour, proceeding into the Academy, he there,
in a certain place, pleased one by one the manes of his ancestors and
kindred; and shortly after, in another part of the same Academy,
he supplicated in common the souls of all philosophers. And these
ceremonies being ended, this most excellent man, having chosen a third
place, performed sacred rites to all the souls of deceased mankind. The
dead body of Proclus, therefore, being disposed of according to his
appointment, as we have said, was carried by his friends and buried
in the eastern part of the suburbs[69], near to Lycabetus, where also
the body of his preceptor Syrianus reposes, who, when he was alive,
had requested this of our philosopher, and, in consequence of this,
had taken care to procure a twofold receptacle in his sepulchre. But
when this most holy man was consulted how he would wish his funeral to
be conducted, lest there should be any thing indecent, or without a
proper decorum, he desired flutes, with which he was threatened in a
dream, and nothing besides. An epigram, consisting of four verses, is
inscribed on his tomb, which he composed himself, as follows:

    I Proclus, here the debt of nature paid,
    (My country Lycia) in the dust am laid;
    Great Syrianus form’d my early youth,
    And left me his successor in _the truth_.
    One common tomb, our earthly part contains,
    One place our kindred souls,--th’ ætherial plains.

Nor were prognostics wanting in the year prior to his decease, such
as an eclipse[70] of the sun, so great that night was produced in the
day-time: for the darkness was so profound, that the stars became
visible. And this happened when the sun was in Capricorn, in the
eastern centre. Besides, the writers of Ephemeride observe that there
will be another eclipse of the sun, at the conclusion of the next year.
But such like affections of the heavenly bodies are said to portend
events on the earth: to us indeed, the eclipse perspicuously signified
a privation and, as it were, defection of the illustrious luminary of
philosophy.

And thus much may suffice for the life of the philosopher. It now
remains for those who are willing to undertake the labour, to write
concerning his disciples and friends. For it appears that many resorted
to him from all parts; among which some were only his auditors, but
others firm adherents to his doctrine; and, on account of philosophy,
his familiars. I could likewise wish that some one, who is more
laborious than myself, would give a particular account of his writings.
For I have alone delivered these particulars of his life, that I might
satisfy my conscience, and might testify that I religiously reverence
the soul of Proclus, and his good dæmon. I shall, therefore, add
nothing concerning the philosopher’s writings, except that I have
always heard him prefer his commentaries on the Timæus of Plato beyond
all his works. He likewise vehemently approved of his commentaries on
the Theætetus. And he used frequently to say, that if he was endued
with despotic power, he would alone preserve, of all the writings of
the ancients, the Oracles and the Timæus. _For_, said he, _I
would abolish all the rest, and remove them from the present age,
because it happens that many are offended, who undertake to read them
rashly, and without proper institution_.




             Concerning the Published Writings of PROCLUS.


Marinus having neglected to give us an account of the writings of
Proclus, I thought it would not be unacceptable to the reader, to
present him with a catalogue of his works which are still extant;
and the most perfect relation I am able of such as are unfortunately
lost. In the execution of this design, I shall follow, for the most
part, the accurate Fabricius in the order, and critical account of his
works; but shall not hesitate to dissent from him in deciding on their
philosophical merit: for it is very rare that philology and philosophy
are united in the same person, and coalesce in amicable conjunction.

       *       *       *       *       *

_1. Four Hymns._ In the preceding life we are informed, by
Marinus, that Proclus composed many hymns, in which the divinities,
both of Greeks and Barbarians, were celebrated; but unfortunately there
are only four preserved, the first of which is to the Sun, the second
and third to Venus (which may be found in the Florilegium of Stobus, p.
249. edit. Grotii,) and the fourth to the Muses. They are collected by
Fabricius, in the eighth volume of his Bibliotheca Græca; and are most
happy specimens of philosophical poetry. Indeed, they bear most evident
marks of a mind full of divine light, and agitated with the fury of
the Muses; and possess all that elegance of composition for which the
writings of Proclus are so remarkable. So that it is very strange
Gyraldus should ascribe them to a Hierophant of Laodicæa, of that name;
since, as Fabricius observes, Suidas mentions other writings of this
priest, but does not speak of him as the author of any hymns. And if he
had, these hymns breathe too much of the spirit and manner of Proclus,
to be the production of any other.

_2. Two Books concerning the useful Parts of Learning, contained
in Grammar._ Three books on this subject are mentioned by Suidas,
and four by Photius; but there are alone extant fragments of two,
in the Bibliotheca of Photius; which were published separate by
Andrew Schottus, at the end of the syntax of Apollonius Alexandrinus,
Francof. 1590. Hanov. 1615. quarto. It appears from these extracts,
that Proclus, in this work, not only treats of every kind of verse, but
also of the more celebrated poets. And hence Fabricius conjectures,
that the short life of Homer, which Leo Allatius published under the
name of Proclus, was taken from the first book of this Chrestomathia.

_3. Eighteen Arguments against the Christians._ The learned Cave,
was egregiously mistaken, in supposing that this work, as well as the
answer to it by Philoponus, is lost; not considering, probably, that
these arguments in defence of the world’s eternity are (as Fabricius
well observes) directly against one of the Christian dogmata, the
creation of the world. These arguments (except the first, which is
lost,) are preserved in the answer of Philoponus; the Greek edition of
which was published at Venice, in 1535, folio; and in Latin from the
version of Joh. Mahatius, at Lyons, 1557. folio. Simplicius, in his
commentaries on the first book of Aristotle de Cœlo, and elsewhere,
opposes this work of Philoponus; at the same time, suppressing
his name. The arguments are, in my opinion, extremely subtle, and
unanswerable; and Philoponus, in his refutation, every where discovers,
that he has by no means fathomed the depth of our philosopher.

_4. A Commentary on the Timæus of Plato, in five Books._ A
most admirable work, composed in the flower of his age, when he was
twenty-eight years old, according to Marinus. This invaluable treasure,
may be said to contain all the wisdom of the highest antiquity:
for Proclus every where cites the most celebrated interpreters on
the Timæus, such as Adrastus the Peripatetic; Ælianus and Albinus,
Platonists; Aristocles, Clearchus, Crantor, Dercyllides; and especially
Jamblichus; likewise Longinus, whom he often refutes; Numenius the
Pythagorean, and Origen, (different from Adamantius,) together with
Plotinus, Porphyry the Platonic, Ptolemy, Severus, and Theodorus
Asinæus; but he always prefers the explications of his master Syrianus,
before all the rest. Add too, that he every where conciliates the
Orphic writings and the Chaldean oracles with the theology of Plato.
And with respect to Orpheus in particular, we are indebted to these
sublime commentaries for the greatest part of the fragments of that
divine poet. He is perpetually on the wing, and rises gradually in
elevated conceptions, until his mind, like Homer’s discord, reaches the
heavens. His style is at the same time nervous and diffuse, accurate
and elegant. We are astonished with the magnificence of his metaphors,
delighted with the copious variety of his matter, and filled with
a divine light, by the sacred truths he unfolds. This great work,
however, unfortunately, scarcely explains a third part of the Timæus;
from whence it is probable, as Fabricius well observes, that several
books have been lost through the injury of time. It was published in
Greek, at Basil, in the year 1556; and is full of mistakes, as is
generally the case with the Basil editions of books, so that a more
correct edition is greatly to be desired, though at present not much
to be expected. On concluding my account of this inestimable work, I
find my indignation roused by the following words of Dr. Cudworth,
in his Intellectual System, p. 306. “Proclus (says he) had some
peculiar fancies and whims of his own, and was indeed a confounder
of the Platonic theology, and a mingler of much unintelligible stuff
with it.” I must confess, (and I am neither afraid, nor ashamed of
the declaration,) that I never found any thing in Proclus, but what
by patient thought, accompanied with a sincere and vehement thirst
after truth, I have been able to fathom. Had Dr. Cudworth been endued
with these requisites, he would doubtless have had equal success;
but without them, the sublimest truths will certainly appear to be
_unintelligible stuff_. Besides this consideration is not to be
omitted, that a modern priest makes a bad philosopher.

_5. On Plato’s Republic._ These commentaries, or rather fragments
of commentaries, are extant in Greek, at the end of Proclus on the
Timæus. Suidas mentions four books of Proclus on Plato’s politics; and
some of Proclus’s dissertations on these books, were found (according
to Fabricius) in the library of Lucas Holstenius. The chief design of
this work seems to be the unveiling the theological mysteries concealed
under the fables of Homer, and other divine poets; which Proclus has
accomplished (in my opinion) in a most wonderful manner. That Homer,
indeed, every where abounds with Egyptian learning, is obvious to
every one; but few are acquainted with the profound wisdom which his
fables conceal. The latent meaning of most of these is unfolded in
the present invaluable, though imperfect work; and he whose mind is
sufficiently enlightened by the ancient philosophy, to comprehend the
beauty of these illustrations, will receive an additional delight from
the study of Homer, which it is impossible to express. An epitome of
this work was published in Latin, by the learned Gesner, 1542. 8vo.
under the following title: Apologiæ quædam pro Homero, et Arte Poetica,
Fabularumque aliquot Enarrationes ex commentariis Procli Lycii Diadochi
philosophi Platonici in libros Platonis de Rep. in quibus plurimæ de
Diis Fabulæ non juxta grammaticorum vulgus historicè, physicè aut
ethicè tractantur, sed Theologicis, ut Gentiles loquntur, ex prima
Philosophia rationibus explanantur.

_6. On the first Alcibiades of Plato._ Ficinus translated parts
of this work into Latin, and published them under the title of Procli
de Anima ac dæmone, de Sacrificio et Magia, Venice 1497. and 1516.
fol. by Aldus; and in a more simple form at Lyons. Fabricius informs
us, that the manuscript commentary of Proclus in Greek, but scarcely
explaining the half part of the Alcibiades, is to be found in various
libraries of France, England, and Italy. Also at Lyons, among the books
of Isaac Vossius; and at Hamburgh in the Johannean library. From the
specimen given of this work by Ficinus, it appears, like all Proclus’s
philosophical writings, to be an invaluable treasury of wisdom; and
nothing certainly, reflects greater disgrace on a nation than suffering
such monuments of ancient learning and wisdom to lie concealed in
colleges, covered with dust, and never consulted.

_7. Six Books on Plato’s Theology._ A most divine work, in which
the philosopher collects into a system the theology dispersed in the
writings of Plato, and establishes it by invincible demonstrations. He
deduces, in a beautiful and connected series, all the divine orders,
from the retreats of ineffable unity; every where connects them by
proper mediums, and, after leading us through the long gradation of
principles, brings us back again to the original from whence they
flowed, and to which they constantly tend. The whole is uncommonly
profound and abstruse; and it was not before the third reading, that
I could fathom the depth it contains. Fabricius observes, “that it
is a subtle and learned work, but from which, you will sooner learn
the opinion of Syrianus and Proclus, concerning the deity and divine
concerns, than that of Plato. He adds, that it is usual with the
Platonists, even from Plotinus, to unite to the doctrine of Plato, a
thousand dogmata, foreign from his philosophy, as if Plato, though he
did not perceive after this manner, ought certainly so to perceive.”
When men mistake their abilities, they always act absurdly, and often
dangerously. As a laborious and accurate critic on philological
matters, Fabricius merits the highest commendation such attainments
can deserve; but when he leaves the beaten road in which nature
designed him to walk, and attempts the tractless paths of philosophy,
he perpetually stumbles, and often falls on the ground. The wings of
philology, like those of the swallow, were never destined for a lofty
flight:--it must be the eagle wing of genius, which can alone soar to
the sun of philosophy. The Greek and Latin edition of this valuable
work, was published at Hamburgh, by Æmilius Portus, 1618, folio.

_8. Theological Institutions_; or, as it may be called, _the
Elements of Theology_. This admirable work contains two hundred and
ten propositions, disposed in a scientific order, and supported by the
firmest demonstrations. They begin from super-essential unity, and
proceed gradually through all the beautiful and wonderful progressions
of divine causes, ending in the self-moving energies of soul. They
possess all the accuracy of Euclid, and all the subtilty and sublimity
necessary to a knowledge of the most profound theology; and may be
considered as bearing the same relation to the Pythagoric and Platonic
wisdom, as Euclid’s Elements, to the most abstruse geometry. Patricius,
the first Latin translator of this divine work, seems to have been very
sensible of the truth of this observation: for he every where carefully
distinguishes the propositions from their demonstrations; and adds
the word corollary to such consequencies as merit that appellation.
His edition was published at Ferraria, 1583. quarto, under the title
of Theological Elements. The Greek and Latin edition, is subjoined to
Proclus’s six books on Plato’s Theology, Hamburgh 1618. folio.

_9. Two Books concerning Motion._ This useful work, collected, as
Fabricius observes, from the third and following books of Aristotle’s
physics, was published in Greek at Basil, 1531, and with the Latin
version of one Justus Velsius, a physician, Basil, 1545. octavo. It
was likewise translated by Patricius, and is annexed to his version of
the Theological Institutions.

_10. An Hypotyposis, or Information concerning Astronomical
Hypotheses._ This work, which Fabricius observes is a compendium of
Ptolemy’s Almagest, was published in Greek, at Basil, 1540. quarto;
and in Latin by George Valla, folio, 1541. A part of this work, which
treats of the use of the astrolabe, Fabricius informs us, is extant in
manuscript, in various libraries. The same accurate critic likewise
observes, that a small treatise, inscribed Uranodromus, is extant,
under the name of Proclus, in some libraries, as in that of Vindobona,
and of Oxford, among the Barrocian volumes. The comprehensive variety
of Proclus’s genius equally demands our admiration and applause.

_11. A small Treatise concerning the Sphere, or Celestial
Circles._ This little work is an accurate and elegant introduction
to astronomy; and is almost wholly taken from the Isagoge of Geminus
Rhodius, on the phænomena. The best editions are the Greek and Latin
one published at Paris in 1553, quarto; and that of Bainbridge,
professor of astronomy at Oxford, London 1620. quarto.

_12. A Paraphrase in four Books, on the Quadripartite of Ptolemy._
This elegant work must, I should imagine, be an invaluable treasure to
the lovers of astrology. It was first published in Greek by Melancthon;
and afterwards in Greek and Latin by Leo Allatius, at Lyons Bat. 1654.
octavo.

_13. Four Books, on the first Book of Euclid’s Elements._ For an
account of this work, see the introduction, and the following sheets,
in which it speaks for itself, in an English dress.

_14. A Commentary on Hesiod’s Works and Days._ This work contains
a valuable moral explanation of this great poet’s meaning; and
Fabricius justly observes, that he is often assaulted without occasion,
by the petulant jeers of that vain man Joh. Tzetzes. The best edition
of this work is that of Daniel Heinsius, Lugd. Bat. 1603. quarto.

_15._ Fabricius informs us, that in some manuscripts, as in the
Vindobonensian and Barrocian, a small treatise is usually ascribed
to Proclus, entitled _Epistolic Characters_; and is prefixed to
the Epistles of Phalaris and Brutis, and published under the name of
Libanius, in Greek, with the version of Casp. Stibilinus, Commelin.
1597, octavo. But it is doubtful whether Proclus is the genuine author
of this work: from the title, I should suppose the contrary. And thus
much for an account of those writings of Proclus which have escaped
the ravages of time, and have been fortunately exposed to public
inspection: it now remains that we relate such inestimable works of
this philosopher, as are yet preserved in shameful concealment; or are
utterly lost in the ruins of antiquity.


            Concerning the Unpublished Writings of PROCLUS.

16. _On the Alcibiades of Plato._ See num. 6.

17. _On Plato’s Politics._ See num. 5.

18. _On Plato’s Parmenides._ A commentary, in seven books; the
last of which was not completed by Proclus, but by Damascius. From
occasional fragments, which have been published of this commentary, it
appears to be a most divine work; and indeed it cannot be otherwise, if
we consider it as the production of one of the greatest philosophers,
on the most sublime and profound of all Plato’s Dialogues. It is
dedicated to Asclepiodotus, a physician and philosopher, and is
not only extant in Greek MS. in the library of the German emperor,
according to Lambecius, lib. vii. p. 41. but also in Latin, from the
unpublished version of one Antonius Hermannus Gogava, as the same
Lambecius informs us, p. 41. Four books of this work are extant in
Greek, in the Bodleian library at Oxford; and it is much to be lamented
that Thomson did not publish these, instead of his trifling edition of
the Parmenides. Fabricius likewise informs us, that Livius Galantes
mentions his having found six of these books in some of the Italian
libraries. They are also extant in the Medicæan library of the great
Etruscan commander.

_19. On the Cratylus of Plato._ We have already observed, in the
dissertation on the Orphic theology, p. 105. what a great treasure of
ancient mythology, must be contained in this work; but there is little
hope of its ever emerging from the obscurity of public libraries. It
is extant in Greek, not only in the Italian libraries, but also among
the manuscript books of Isaac Vossius.

_20. Uranodromus._ See above, num. 10.

_21. Ten Doubts concerning Providence, in one Book._ Philoponus
mentions this work, in his second book against Proclus on the eternity
of the world; and a Latin version of it is extant by one William de
Morbeka, in the Johannean library of Hamburgh. Extracts from this
translation are preserved by Fabricius, in his Greek Library; and they
are in every respect worthy of the genius of Proclus.

_22. Concerning Providence and Fate, and that which is in our Power,
one Book._ This work is dedicated to one Theodorus, a mechanist; and
is extant in the Latin translation of the same Morbeka, in the Greek
Library of Fabricius. The translation is for the most part barbarous,
but is, however, sufficiently legible to discover that it is a most
valuable treatise, replete with the usual elegance, subtilty, and
sublimity of our philosopher.

_23. Concerning the Hypostasis, or Subsistence of Evil._ This
book is extant in Latin, in the Johannean library; and fragments of
it are preserved by Fabricius, in his Greek library. It is to be
regretted, that Fabricius did not preserve the whole in that excellent
philological work.


               Concerning the Lost Writings of PROCLUS.

_24. On the Speech of Diotima, in Plato’s Banquet, concerning the
Subsistence of the Beautiful._ Fabricius informs us, that this
work is distributed into many books; and Holstenius observes, that it
is mentioned in a certain scholium of the Medicean copy of Proclus’s
commentaries on Plato’s politics; but it is unfortunately no where
extant.

_25. On the Philebus of Plato_; as may be inferred from the
narration of Damascius in Photius, p. 550; and Suidas in Marinus.
For Damascius relates, that Marinus having composed a commentary on
this dialogue, on shewing it to Isidorus for his approbation, that
philosopher observed, _that those of his master were sufficient_;
which words Fabricius, with great propriety, applies to the
commentaries of Proclus on the Philebus.

_26. On the Theætetus of Plato._ This work is praised by Marinus,
in the last chapter of the preceding life; and no doubt with great
propriety: for this abstruse and sublime dialogue would naturally call
forth all the divine fire and elegance of our philosopher.

_27. Commentaries on the Enneads of Plotinus._ This work is
mentioned by Gyraldus, in his second dialogue on ancient poets; by
Ficinus on Plotinus; by Philip Labbeus, in his account of MS. books,
p. 286; and in the notes of Bullialdus to Theo of Smyrna, p. 224. But
also in a certain note prefixed to an ancient manuscript of Jamblichus,
on the Egyptian mysteries, to this effect: “The philosopher Proclus,
commenting on the Enneads of the great Plotinus, says, that it is the
divine Jamblichus who answers the epistle of Porphyry.” This note is
in Greek, in the original, and is (in my opinion) of itself sufficient
to prove that such a work was once extant, though now unfortunately
lost. How much the want of these commentaries is to be regretted, must
be deeply felt by every lover of the Platonic philosophy. For the
unequalled profundity, and divine mysteries, contained in the writings
of Plotinus, could never be more happily illustrated than by the
irradiations of such a genius as Proclus.

_28. Lectures on Aristotle’s Book_ Περὶ Ερμηνείας, _or concerning
Interpretation_. This work, it seems, was never published; but Ammonius
Hermeas, the disciple of Proclus, has inserted in his valuable
commentary on this book all that he could retain in his memory of
Proclus’s lectures.

_29. Hymns_, not a few, see num. 1.

_30. Chrestomathea._ See num. 3.

_31. On the Mother of the Gods_, one book, mentioned by Marinus,
in the preceding Life.

_32. On the Theology of Orpheus._ This work is mentioned by
Marinus, in the preceding Life, and by Suidas; and its loss must be
particularly regretted by all the lovers of recondite theology.

_33. Ten Books, on the Chaldean Oracles._ This most valuable work
is mentioned by Marinus, in the preceding Life, and by Proclus himself
on Plato’s Politics, p. 359. It was doubtless not extant at the time
when Psellus and Pletho undertook the illustration of a few of these
oracles: at least the inconsiderable merit of their commentaries,
strongly favours this supposition.

_34. A Commentary on the whole of Homer._ Suidas. A specimen
of the great value of this work may be seen in our philosopher’s
commentaries on Plato’s republic. The works of Homer are not only the
great fountain of poetry, but likewise of philosophy; and are no less
admirable for inspiring the fury of the Muses than for containing the
mysteries of the most recondite theology.

_35. Concerning the Gods, according to Homer._ Had this work been
preserved, we should doubtless have been furnished with a defence of
the heathen religion, which would have silenced the ignorant clamours
of its opponents.

_36. The Symphony or Concord of Orpheus, Pythagoras, and Plato._
Suidas. Proclus, in his published writings, is every where studious of
reconciling the doctrines of these great men, and is always successful
in this undertaking. Indeed, the same divine genius seems to have
irradiated and inspired these wonderful heroes, but in different
ways: in Orpheus it was accompanied with the fire of the Muses; in
Pythagoras it shone through the mysterious veil of numbers; and in
Plato, combining the preceding modes, it was seen enshrined in awful
majesty of thought, clothed with the graces of poetical diction, and
resplendent with ineffable light.

_37. Two Books on the Theurgic Discipline._ Suidas. How much
Proclus excelled in this art, may be seen in the preceding Life.

_38. Concerning the Oppositions of Aristotle to Plato’s Timæus._
This work is mentioned by Proclus in the 3d book of his commentary
on the Timæus, p. 226. and seems to have escaped the notice of the
accurate Fabricius. Aristotle may, no doubt, in many particulars
be reconciled with Plato; but it is also certain, that in some he
is perfectly dissonant. And thus much for the Life and Writings of
Proclus.




                             COMMENTARIES

                                  OF

                               PROCLUS.




                                BOOK I.




                               CHAP. I.

          _On the Middle Nature of the Mathematical Essence._


It is necessary that the mathematical essence should neither be
separated from the first nor last genera of things, nor from that
which obtains a simplicity of essence; but that it should obtain
a middle situation between substances destitute of parts, simple,
incomposite and indivisible, and such as are subject to partition, and
are terminated in manifold compositions and various divisions. For
since that which subsists in its inherent reasons remains perpetually
the same, is firm and durable, and cannot be confuted, it evidently
declares it is superior to the forms existing in matter. But that
power of progression which apprehends, and which besides uses the
dimensions of subjects, and prepares different conclusions from
different principles, gives it an order inferior to that nature which
is allotted an indivisible essence, perfectly constituted in itself.
Hence (as it appears to me)[71] Plato also divides the knowledge
of things which are, into first, middle, and last substances. And
to indivisible natures, indeed, he attributes an intelligence,
which, in a collective manner, and by a certain simple power, divides
the objects of intellectual perception; so that being divested of
matter, and endued with the greatest purity, it apprehends things
themselves, by a certain unifying perception, and excels the other
kinds of knowledge. But to divisible essences, and such as are
allotted the lowest nature, and to all sensible beings, he attributes
opinion, which obtains an obscure and imperfect truth. But to middle
essences (and such are mathematical forms), and to things inferior
to an indivisible and superior to a divisible nature, he attributes
cogitation. For this, indeed, is inferior to intellect, and the supreme
science dialectic; but is more perfect than opinion, and more certain
and pure. For it advances by a discursive procession, expands the
indivisibility of intellect, and unfolds that which was involved in
the unity of intellectual apprehension: but it collects things which
are divided, and brings them back to mind. Hence, as knowledges differ
among themselves, so the objects of knowledge are distinguished by
nature. So that intelligible essences having an uniform subsistence,
evidently excel all others. But sensibles are entirely excelled by
primary essences: and mathematical natures, and whatever falls under
cogitation, are allotted a middle order: for they are excelled by the
division of intelligibles; but because destitute of matter, they are
superior to sensible natures; and by a certain simple power, they are
excelled by the first; but by a certain reason are more exalted than
the last. Hence they possess notions of an intellectual essence, which
are more manifest than sensibles, but which are, at the same time,
only the images of an intellectual nature; and they imitate divisibly
the indivisible, and, in a multiform manner, the uniform exemplars
of things. And, that I may sum up the whole in a few words, they are
placed in the vestibules or entrances of primary forms, and disclose
their indivisible and prolific subsistence collected into one, but
they do not yet excel the division and composition of reasons, and an
essence accommodated to the obscurity of images; nor are they capable
of passing beyond the various notions of the soul, endued with a
discursive power, and of adhering to intellections perfectly simple,
and purified from all material imperfection. After this manner then, is
the middle nature of mathematical genera and forms to be understood; as
filling up the medium between essences entirely indivisible, and such
as are divisible about matter.




                               CHAP. II.

 _Concerning the common Principles of Beings, and of the Mathematical
                   Essence,[72] bound and infinite._


But it is necessary that, considering the principles of the whole
mathematical essence, we should return to those general principles,
which pervade through and produce all things from themselves, I mean
_bound_ and _infinite_. For from these two after that cause of _one_,
which can neither be explained, nor entirely comprehended, every other
thing, as well as the nature of the mathematical disciplines, is
constituted. In the former, indeed, producing all things collectively
and separately; but in these proceeding in a convenient measure, and
receiving a progression in a becoming order; and in some, subsisting
among primary, but in others among middle, and in others again among
posterior natures. For intelligible genera, by their simplicity of
power, are the first participants of _bound_ and _infinite_: because,
on account of their union and identity, and their firm and stable
existence, they are perfected by bound: but on account of their
division into multitude, their copious power of generation, and their
divine diversity and progression, they obtain the nature of infinite.
But mathematical genera originate, indeed, from bound and infinite,
yet not from primary, intelligible, and occult principles only; but
also from those principles which proceed from the first to a secondary
order, and which are sufficient to produce the middle ornaments of
beings, and the variety which is alternately found in their natures.
Hence, in these also, the reasons and proportions advance to infinity,
but are restrained and confined by that which is the cause of bound.
For number rising from the retreats of unity, receives an incessant
increase, but that which is received as it stops in its progression,
is always finite. Magnitude also suffers an infinite division, yet
all the parts which are divided are bounded, and the particles of the
whole exist finite in energy. So that without the being of infinity,
all magnitudes would be commensurable, and no one would be found but
what might either be explained by words, or comprehended by reason (in
which indeed geometrical subjects appear to differ from such as are
arithmetical;) and numbers would be very little able to evince the
prolific power of unity, and all the multiplex and super-particular
proportions which they contain. For every number changes its
proportion, looking back upon, and diligently enquiring after unity,
and a reason prior to itself. But _bound_ being taken away, the
commensurability and communication of reasons, and one and the same
perpetual essence of forms, together with equality, and whatever
regards a better co-ordination, would never appear in mathematical
anticipations: nor would there be any science of these; nor any firm
and certain comprehensions. Hence then, as all other genera of beings
require these two principles, so likewise the mathematical essences.
But such things as are last in the order of beings, which subsist in
matter, and are formed by the plastic hand of nature, are manifestly
seen to enjoy these two principles essentially. Infinite as the subject
seat of their forms; but bound as that which invests them with reasons,
figures, and forms. And hence it is manifest that mathematical essences
have the same pre-existent principles with all the other genera of
beings.




                              CHAP. III.

     _What the common Theorems are of the Mathematical Essences._


But as we have contemplated the common principles of things, which are
diffused through all the mathematical genera, after the same manner
we must consider those common and simple theorems, originating from
one science, which contains all mathematical knowledge in one. And we
must investigate how they are capable of according with all numbers,
magnitudes and motions. But of this kind are all considerations
respecting proportions, compositions, divisions, conversions, and
alternate changes: also the speculation of every kind of reasons,
multiplex, super-particular, super-partient, and the opposite to these:
together with the common and universal considerations respecting equal
and unequal, not as conversant in figures, or numbers, or motions,
but so far as each of these possesses a common nature essentially,
and affords a more simple knowledge of itself. But beauty and order
are also common to all the mathematical disciplines, together with
a passage from things more known, to such as are sought for, and
a transition from these to those which are called resolutions and
compositions. Besides, a similitude and dissimilitude of reasons are
by no means absent from the mathematical genera: for we call some
figures similar, and others dissimilar; and the same with respect
to numbers. And again, all the considerations which regard powers,
agree in like manner to all the mathematical disciplines, as well the
powers themselves, as things subject to their dominion: which, indeed,
Socrates, in the Republic, dedicates to the Muses, speaking things
arduous and sublime, because he had embraced things common to all
mathematical reasons, in terminated limits, and had determined them in
given numbers, in which the measures both of abundance and sterility
appear.




                               CHAP. IV.

  _How these Common Properties subsist, and by what Science they are
                             considered._


But it is requisite to believe, that these common properties do not
primarily subsist in many and divided forms, nor originate from things
many and the last: but we ought to place them as things preceding
in a certain simplicity and excellence. For the knowledge of these
antecedes many knowledges, and supplies them with principles; and
the multitude of sciences subsist about this, and are referred to
it as their source. Thus the geometrician affirms, that when four
magnitudes are proportional, they shall be alternately proportional;
and he demonstrates this from principles peculiar to his science, and
which the arithmetician never uses. In like manner, the arithmetician
affirms, that when four numbers are proportional, they shall be so
alternately: and this he evinces from the proper principles of his
science. For who is he that knows alternate ratio considered by
itself, whether it subsists in magnitudes or in numbers? And the
division of composite magnitudes or numbers, and in like manner, the
composition of such as are divided? For surely it cannot be said that
there are sciences and cognitions of things divisible: but that we
have no science of things destitute of matter, and which are assigned
a more intellectual contemplation; for the knowledge of these is by
a much greater priority science, and from these the common reasons
of many sciences are derived. And there is a gradual ascent in
cognitions from things more particular to more universal, till we
revert to the science of that which _is_, considered _as it
is_, abstracted from all secondary properties. For this sublime
science does not think it suitable to its dignity, to contemplate
the common properties which are essentially inherent in numbers, and
are common to all quantities; but it contemplates the one, and firm
essence of all the things which _are_. Hence, it is the most
capacious of all sciences, and from this all the rest assume their
own peculiar principles. For the superior sciences always afford the
first suppositions of demonstrations to such as are subordinate. But
that which is the most perfect of all the sciences, distributes from
itself principles to all the rest, to some indeed, such as are more
universal, but to others, such as are more particular. Hence, Socrates,
in the Theætetus, mingling the jocose with the serious, compares the
sciences which reside in us to doves: but he says they fly away, some
in flocks, but others separate from one another. For such, indeed,
as are more common and more capacious, comprehend in themselves many
such as are more particular: but such as being distributed into forms,
touch things subject to knowledge, are distant from one another, and
can by no means be copulated together, since they are excited by
different primary principles. One science, therefore, precedes all
sciences and disciplines, since it knows the common properties which
pervade through all the genera of beings, and supplies principles to
all the mathematical sciences. And thus far our doctrine concerning
dialectic[73] is terminated.




                               CHAP. V.

 _What the Instrument is, which judges of the Mathematical Genera and
                               Species._


Let us now consider what that instrument is[74], adapted to the
judgment of mathematical concerns; and let us appoint Plato as our
guide in this affair, who, in his Republic, divides cognitions
separately from such things as are the objects of knowledge; and
distributes cognitions in conjunction with things subject to knowledge.
For of the things which are, some he ranks among intelligibles, and
others among sensibles. And of intelligibles, some are again pure
intelligibles, and others subject to cogitation. And of sensibles,
some are purely sensibles, but others conjectural. To intelligibles,
indeed, which are the first of the four genera, he assigns an
intelligible knowledge; but to those which are subject to cogitation,
he attributes thought: to sensibles, faith; but to conjecturals, a
conjectural or assimilatory power. And he shews, that the assimilatory
power has the same proportion to sense as thought to intelligence. For
the conjectural power knows the spectres of sensible forms, while they
are beheld in water and other bodies, which perspicuously represent
their image: since, by their situation in water, they are after a
manner, allotted the last seat in the gradations of forms, and truly
become the resemblances of resemblances. In like manner, thought
beholds the images of intelligibles in a degraded state, fallen from
primary simple and indivisible forms, into multitude and division.
Hence, a knowledge of this kind, depends on other more ancient
hypotheses; but intelligence arrives at that principle which is no
longer supposed. If then, mathematical concerns are neither allotted an
essence separate from all division and variety, nor that nature which
is apprehended by sense, which is obnoxious to many mutations, and is
in every proportion divisible, it must be manifest to every one, that
they are essentially subject to cogitation: but cogitation presides
over these as an instrument adapted to judgment, in the same manner as
sense to sensibles, and the assimilatory power to conjecturals. From
whence, indeed, Socrates determines that the knowledge of these is more
obscure than the first science, but is more evident than the impulsive
apprehension of opinion. For in this the mathematical sciences are
inferior to intelligence, because they contemplate that which is
evolved, and is endued with a power of progression; but they are
superior to opinion, by that stability of reasons which they contain,
and which cannot be confuted. And they originate from supposition,
through a diminution of the first science; but they contain forms
independent of matter, from their possessing a knowledge more perfect
than that of sensibles. We have therefore determined an instrument
adapted to the judgment of all mathematical concerns, i. e. cogitation,
according to the mind of Plato; which places itself indeed above
opinion, but is excelled by intelligence.




                               CHAP. VI.

   _Concerning the Essence of Mathematical Genera and Species[75]._


It now remains, that we consider what subsistence or essence ought
to be assigned to mathematical genera and species? Whether we must
deduce their origin and subsistence from sensible objects, or from
abstraction, or from a collection of such things as are dispersed by
parts into one common definition; or must allow them an existence
prior to that of sensibles, as Plato affirms, and as the progression
of universal being demonstrates? First then, if we affirm that
mathematical species are composed from sensibles; whilst the soul
from material triangles or circles, forms in herself the trigonic, or
circular species, by a kind of secondary generation; I would ask from
whence is derived the great certainty and accuracy of definitions?
For it must either proceed from sensibles, or from the soul herself.
But from sensibles is impossible, for these, in a continual flow of
generation and decay, do not for a moment retain an exact sameness
of being; and consequently fall far short of the exactness contained
in the definitions themselves. It must therefore proceed from the
soul, which, by her immaterial nature, procures perfection from the
imperfect, accurate subtilty from that which is neither accurate nor
subtle, and rekindles the light of ideas from the obscure and unreal
objects of sense.

For where shall we find, amongst sensible objects, an indivisible
nature, such as that of a point, or a line without the dimension
of breadth, or a superficies without depth, or the ever constant
proportion of sides, and exact rectitude of angles? For my part, I
cannot see where, since all divisible natures are thus mixed and
confused together, nothing sincere, nothing free from its contrary,
but things every where yielding to separation, as well such as are
removed by distance of place, as those which are united together.
How then shall we obtain this durable essence for these immoveable
natures from the ever fluctuating forms of sense? For whatever derives
its existence from moveable beings, must of necessity be mutable and
frail. And how shall we gain this perfect accuracy for the stable
species, from the inaccurate and imperfect? For whatever is the cause
of a conception, always immutable, is itself much more stable than its
effect. We must therefore admit the soul to be the generator of these
mathematical species and reasons. But if she contains them in herself,
as first exemplars, she gives them an essential being, so that the
generations are nothing else than propagations of species, which had
a prior subsistence in herself: and thus we shall speak agreeably to
the sentiments of Plato, and discover the true essence of mathematical
entities. But if the soul, though she neither possesses nor received
the mathematical reasons prior to the energies of sense, yet fabricates
this admirable immaterial building, and generates this fair series of
speculations; how can she discern whether her productions are stable
and constant, or things which the winds may dissipate, and phantoms
rather than realities? What standard can she apply as the measure of
their truth? Or how, since she is destitute of their essence, can she
generate such a variety of reasons? For from such an hypothesis, we
make their subsistence fortuitous, not tending to any scientific bound.
Mathematical species are therefore the genuine offspring of the soul:
nor does she derive from sensible objects the definitions she frames,
but rather the first are propagated from the second; they are the
energies of soul, which, as it were, pregnant with forms, delivers her
immaterial progeny into the dark and fluctuating regions of matter, as
evidences of the permanent duration of her species.

Again, if we collect mathematical reasons from externals, why
are not demonstrations composed from sensibles, better than the
demonstrations of universal and simple species? For we say, in order
to the investigation of any thing sought, that the principles and
propositions, should be allied to the conclusions. If then, particulars
are the causes of universals, and sensibles the sources of reasoning,
why does the boundary of demonstration always refer to that which is
more universal, and not to that which is partial and particular? And
how can we prove that the essence of intelligibles is more allied to
demonstration than the essence of sensibles? For thus they speak[76]:
his knowledge is not legitimate, who demonstrates that the isosceles,
the equilateral, or the scalene triangle, have angles equal to two
right; but he possesses science, properly so called, who demonstrates
this of every triangle simply, or of triangle itself. And again,
that universals, for the purpose of demonstration, are superior to
particulars; that demonstrations concern things more universal; but
that the principles from which demonstrations are composed, have a
priority of existence, and a precedency in nature to singulars, and
are the causes of the propositions they prove. It is very remote,
therefore, from the nature of Apodictical sciences, that from converse
with things of posterior origin, and from the dark perceptions of
sense, they should falsely collect their indubitable propositions.
I add farther, that they who affirm this, make the soul of a baser
nature than the material species themselves. For if matter derives from
nature beings essential, and participating a high degree of entity and
evidence; but the soul, by a posterior energy, receives these from
sensible objects, and fashions in herself resemblances and images
of posterior origin, contemplating vile essences, and abstracting
from matter, the forms inseparable from its nature; do they not make
the soul more obscure and indigent than matter itself? For matter
is the receptacle of forms materialized, as the soul is of species
immaterialized. But in this case, matter would be the place of primary
beings, and the soul of such as are secondary and subordinate: matter
and its forms obtaining the lead in being, and existing as the sources
of the subsistence of immaterial forms. Lastly, the material forms
would have an essential existence, the others only an intentional
denomination. How then can the soul, which is the first participant of
intellect, and an intellective essence, and which derives from thence
consummate knowledge, and a plenitude of life, become the receptacle
of the most obscure species, the lowest in the order of things, and
participating the most imperfect existence. But this opinion, which has
been sufficiently exploded by others, needs no farther confutation.

If then, mathematical species do not subsist by material abstraction,
nor by a collection of those common properties inherent in individuals;
nor are at all, in their origin, posterior to sensibles, nor derived
in any manner from them: it is necessary that the soul should either
deduce them from herself, or from intellect; or lastly, from herself
and intellect united. But if from herself alone, Whence do the
images of intellectual species arise; whence do they derive their
middle nature, linking, as it were, the divisible and indivisible
essence together; if they do not participate the fullness of entity
from primary essences? Lastly, how, upon this hypothesis, are the
first exemplars, paradigms, or ideas, which subsist in intellect,
the principles of universals? But if they are derived from intellect
alone into the soul, how can the soul remain self-operative, and
self-motive, if her inherent reasons flow from an external source, and
are regulated by its operations? And in what respect does the soul
differ from matter, which is all things in mere dormant capacity,
but generates nothing appertaining to material species? It remains,
therefore, that the soul deduces these species from herself, and
intellect; and that she is the absolute consummation of the forms which
originate from intellectual exemplars, but which are allotted from
themselves a transition to permanent being. The soul, therefore, is by
no means to be compared to a smooth tablet, void of all reasons; but
she is an ever-written tablet, herself inscribing the characters in
herself, of which she derives an eternal plenitude from intellect.
For soul is a certain subordinate intellect, revolving round an
intellect prior to herself, formed to its image, and participating its
divine irradiations. If then, this superior intellect is all things
intellectually, soul will all things animally; if the first exists as
the exemplar, soul will be as its image; if as contracted and united
in itself, soul as divisible and expanded. And this is what Plato
understood, when in his Timæus, he composes the soul of the world
from all things, dividing her according to harmonical reasons, and
analogies; assigning to her the first principles effective of figures,
I mean the right and circular line, and giving an intellectual motion
to her inherent circles. All mathematical species, therefore, have a
primary subsistence in the soul: so that, before sensible numbers,
there are to be found in her inmost recesses, self-moving numbers;
vital figures, prior to the apparent, ideal proportions of harmony
previous to concordant sounds; and invisible orbs, prior to the bodies
which revolve in a circle. So that soul is the prolific abundance of
all these, and is another ornament producing herself, and produced
from a proper principle, filling herself with life, and at the same
time filled from the demiurgus of the universe, is an incorporeal
and indistant manner. When, therefore, she produces and unfolds
her latent reasons, she then detects every science and virtue. The
essence of soul then consists in these species, nor must we suppose
her inherent numbers to be a multitude of units, nor her archytipal
ideas of divisible forms to be corporeal: but we must conceive all
these as subsisting ever vitally, and intellectually, as the exemplars
of apparent numbers, figures, reasons and motions. And here we must
follow the doctrine of Timæus, who derives the origin, and consummates
the fabric of the soul, from mathematical forms, and reposes in her
nature the causes of every thing which exists. For the seven bounding
terms[77], comprehending the principles of all numbers, lines,
planes and solids, pre-exist in soul according to cause. And again,
the principles of figures are placed in her essence, according to a
demiurgical power. And lastly, the first of all motions, which embraces
every other motion in its comprehensive ambit, is co-existent with
soul. For the principle of every thing which is moved is a circle, and
the circular motion. The mathematical reasons, therefore, which fully
consummate the soul, are essential, and self-moving: and the soul, by
her cogitative power, diffusing, propagating, and evolving these, from
her profound recesses, constitutes all the fair variety of mathematical
sciences. Nor will she ever cease to generate, and waken into energy,
succeeding species, while she divests her indivisible reasons of
their intellectual simplicity. For she previously received all
things, after a primary manner; and according to her infinite power,
from pre-existent principles, deduces a beautiful series of various
speculations.




                              CHAP. VII.

   _What the Employments and Powers are of the Mathematical Science,
        and how far they extend themselves in their Energies._


But, after contemplating the essence of mathematical forms, it is
necessary we should recur to that one master-science of these, which
we have shewn is prior to a multitude of others, and that we should
contemplate what its employment is, what are its powers, and how far
it advances in its energies. The employment, therefore, of the whole
mathematical science, possessing, as we have before said, the power
of cogitation, must not be placed so high as that of intelligence;
which is firmly seated in its own stable essence, is perfect, is
contained by itself, and in itself continually verges. Nor must it be
situated so low as that of opinion and sense, since these cognitions
dwell upon external concerns, energize upon them, and do not possess
the causes of the objects of their knowledge. But the mathematical
science, receives its commencement, indeed, extrinsically from[78]
recollection, but ends in the most intimate reasons, residing in the
depths of the soul; and is excited, indeed, from things posterior,
but arrives by gradual advances at the principal essence of forms.
Nor is its energy immoveable, like that of intelligence, nor is it
affected with local motion and alteration, like sense, but it revolves
with a vital energy, and runs through the ornament of incorporeal
reasons, sometimes advancing from principles to such things as are
perfected by principles, but at other times yielding in a retrograde
progression from conclusions to their forming principles: and sometimes
proceeding from things previously known, to such as are the subject
of investigation: but at other times, from things placed in the
question, to such as precede in cognition. Besides, it does not excel
all inquisition, as if it were perfect from itself, like intellect,
nor is it perfected from others, like sense, but it proceeds by
enquiry to invention, and ascends from the imperfect to perfection.
But it likewise possesses twofold powers, one kind of these deducing
principles into multitude, and generating the different paths of
contemplation: but the other endued with a power of collecting many
transitions into proper suppositions. For since it proposes to itself
as principles, as well unity, and multitude, as bound and infinite,
and such things as are subject to its comprehension, are allotted a
middle order, between forms indivisible and every way divisible; with
great propriety (I think) the gnostic powers of the whole science of
these are essentially twofold. One species indeed, hastens to union,
and contracts the expansion of multitude: but the other possesses
a power of distinguishing things simple into such as are various,
more universals into more particulars, and reasons digested in their
principle, into things secondary and multifariously multiplied from
these principles. For rising higher from its commencement it penetrates
even to such things as are the perfections of sensible concerns,
is joined with nature, and demonstrates many things together with
natural science. Since ascending from inferiors, it accedes in a
certain respect proximate to intellectual knowledge, and touches the
contemplation of things primary and divine. And hence, in the limits
which flow from its essence, it produces the whole mechanic, optic,
and catoptric speculation, together with many other sciences which are
inwoven and entangled with sensible concerns, and which operate through
their assistance. Besides, in its ascensions from corporeal natures,
it derives intelligences indivisible and destitute of matter: and with
these it perfects its divisible apprehensions, those cognitions which
subsist in progressions, and its own genera and forms: it likewise
indicates the truth respecting the gods themselves, and in its peculiar
treatises exhibits a contemplation of the things which _are_. And
thus much concerning the employment and powers of the Mathematical
Science.




                              CHAP. VIII.

         _Concerning the Utility of the Mathematical Science._


But let us now consider the utility of this Science, which extends
itself from the most principal to the last cognitions. Timæus,
therefore, calls the knowledge of the mathematical disciplines the
path of erudition, because, indeed, it has the same proportion to
universal science, and the first philosophy, which learning has to
virtue. For this last frames our soul to a perfect life, by the
possession of worthy manners; but the former prepares our cogitation,
and the divine eye of our soul to an elevation from the obscurity of
sensible information. Hence, Socrates in the Republic, says, “That
the eye of the soul, which is darkened and buried by other studies,
can by the mathematical disciplines alone be invigorated, and again
excited to the contemplation of that which _is_, and transferred
from resemblances to real beings, from an obscure light to that
light which has the power of intelligence, and from a cave, and
those bonds which exist in it as the authors of generation, and from
material impediments be able to rise to an incorporeal and indivisible
essence. For the beauty and order of mathematical reasons, and the
firmness and stability of the contemplations they afford, conjoins
us with intelligible objects, and perfectly determines us in their
essences; which perpetually remain the same, ever shining with divine
beauty, and preserving a mutual order without end. But Socrates, in
the Phædrus, delivers to us three characters who are elevated from
sense, because they fill up and accomplish the primary life of the
soul, i. e. the philosopher, the lover, and the musician. But the
beginning and path of elevation to the lover, is a progression from
apparent beauty, using as excitations the middle forms of beautiful
objects. But to the musician, who is allotted the third seat, the
way consists in a transition from sensible to invisible harmonies,
and to the reasons existing in these. So that to the one, sight is
the instrument of reminiscence, and to the other, hearing. But to
him who is by nature a philosopher, from whence and by what means is
reminiscence the prelude of intellectual knowledge, and an excitation
to that which truly is, and to truth itself? For this character also,
on account of its imperfection, requires a proper principle: for it is
allotted a natural virtue, an imperfect eye, and a degraded manner.
It must therefore be excited from itself; and he who is of such a
nature, rejoices in that which _is_. But to the philosopher,
says Plotinus, the mathematical disciplines must be exhibited, that
they may accustom him to an incorporeal nature, and that afterwards
using these as figures, he may be led to dialectic reasons, and to
the contemplation of all the things which _are_. And thus it is
manifest, from hence, that the mathematics are of the greatest utility
to philosophy. But it is requisite that we should be more explicit, and
mention the several particulars to which they conduce, and evince that
they prepare the intellectual apprehensions of theology. For whatever
to imperfect natures appears difficult and arduous in obtaining the
true knowledge of the gods, the mathematical reasons render, by their
images, credible, manifest, and certain. Thus, in numbers, they
indicate the significations of super-essential properties, but they
evince the powers of intellectual figures, in those figures which
fall under cogitation. Hence it is, that Plato, by mathematical forms
teaches us many and admirable sentences concerning the gods, and the
philosophy of the Pythagoreans, using these as veils, conceals from
vulgar inspection the discipline of divine sentences. For such is the
whole of the _Sacred and Divine Discourse_[79], that of Philolaus
in his Bacchics, and the universal method of the Pythagoric narration
concerning the Gods. But it especially refers to the contemplation of
nature, since it discloses the order of those reasons by which the
universe is fabricated, and that proportion which binds, as Timæus
says, whatever the world contains, in union and consent; besides, it
conciliates in amity things mutually opposing each other, and gives
convenience and consent to things mutually disagreeing, and exhibits
to our view simple and primary elements, from which the universe is
composed, on every side comprehended by commensurability and equality,
because it receives convenient figures in its proportions, and numbers
proper to every production, and finds out their revolutions and
renovations, by which we are enabled to reason concerning the best
origin, and the contrary dissolution of particulars. In consequence
of this, as it appears to me, Timæus discloses the contemplation
concerning the nature of the universe, by mathematical names, adorns
the origin of the elements with numbers and figures, referring to
these their powers, passions, and energies; and esteeming as well
the acuteness as the obtuseness of angles, the levity of sides, or
contrary powers, and their multitude and paucity to be the cause of the
all-various mutation of the elements. But why may we not say, that it
profits much, and in an admirable manner, to that philosophy which is
called Politic, as well by measuring the times of actions as affording
the various revolutions of the universe, and numbers convenient to
things rising into being; I mean the assimilating, and authors of
dissimilitude, the prolific too and the perfect, and the contraries
to these; together with orderly and elegant ministers of life, and
inelegance; and finally, such numbers as procure fertility and
sterility. Which, indeed, the speech of the Muses in the Republic[80]
evinces, placing the universal Geometric Number as the author of better
and more debased generations, and as the cause of the indissoluble
perseverance of good manners, and of the mutation of the best Republics
into such as are remote from reason, and are given to affections. For
it is sufficiently evident, that it belongs to the whole mathematical
discipline to deliver the science of this number which is called
geometrical, and not to one particular science, such as arithmetic, or
geometry: since the reasons or proportions of abundance and sterility,
permeate through all the mathematical disciplines. Again, it is the
means of our institution in moral philosophy which it brings to its
ultimate perfection, and gives order and an elegant life to our
manners. Besides this, it delivers to us figures, and modulations
and motions convenient to virtue, by which the Athenian guest wishes
those to be instituted and perfected, who are destined to pursue
moral virtue from their early youth. Add too, that it places before
our view the reasons of virtues, in one manner, indeed, in numbers,
in another in figures, but differently in musical symphonies; and
lastly, it indicates the excess and defect of vices, by which we are
enabled to moderate and adorn our manners. Hence it is, that Socrates,
in the Gorgias, accusing Calicles of an inordinate and intemperate
life, says to him, “You neglect geometry and geometric equality:” but,
in the Republic, he finds out the proportion of tyrannic pleasure to
a royal interval, according to a plane and solid generation. But we
shall learn what great utility is derived to other sciences and arts
from the mathematical science, when we consider that it adds order and
perfection to contemplative arts; I mean rhetoric, and all such as
consist in discourse. But it proposes to the poetic arts, the reasons
of poems in the place of an example, because it presides over the
measures existing in these. But to the active arts it determines action
and motion, by its own abiding and immoveable forms. For all arts, as
Socrates says, in the Philebus, require arithmetic, mensuration, and
statics, either in all, or in some of their operations. But all these
are contained in the discourses of the mathematical science, and are
terminated according to their diversity. For from this science the
divisions of numbers, and the variety of dimensions, and the difference
of weights are known. The utility, therefore, of the whole mathematical
science to philosophy itself, and to other sciences and arts, may be
from hence known to intelligent hearers.”




                               CHAP. IX.

 _A Solution of an Objection raised by some against the Utility of the
                        Mathematical Sciences._


But some, who are prone to contradiction through those who wish to
subvert geometry, endeavour to destroy the dignity of this science.
One part, indeed, depriving it of ornament and good, because it
does not discourse on these. But another part[81] affirming that
sensible experiments are more useful than the universal objects of its
speculation; I mean, that Geodesia (for instance,) or the mensuration
of the earth, is preferable to geometry, and vulgar arithmetic to
that arithmetic which is conversant with theorems alone: and that
nautical astrology is more useful than that which teaches universally,
abstracted from any application to sensible concerns. For we are not,
say they, made rich by our knowledge of riches, but by using them;
nor are we happy by the merely understanding felicity, but by living
happily. Hence we must confess that those mathematical sciences,
which are conversant with cognition, do not profit human life, and
confer to action, but those only which are engaged in exercise. For
those who are ignorant of the reasons of things, but are exercised
in particular and sensible experiments, are in every respect more
excellent, for the purposes of human life, than those who are employed
in contemplation alone. Against objections then, of this kind, we
shall reply, by shewing the beauty of the mathematical disciplines
from those arguments by which Aristotle endeavours to persuade us. We
must therefore confess that there are three things which especially
cause beauty, both in bodies and souls; I mean, order, convenience, and
determination. Since corporeal baseness, indeed, arises from material
inordination, deformity, and inconvenience, and from the dominion
of the indefinite in the composite body. But the baseness of the
soul originates from its irrationality, and inordinate motion, and
from its being in a state of discord with reason, and not receiving
from thence its proper limitation. Hence, beauty exists even in
contraries, by means of order, convenience and determination. But we
may behold these in a more eminent degree in the mathematical science;
order, indeed, in the perpetual exhibition of things posterior and
more various, from such as are primary and more simple; for things
subsequent are always annexed to their precedents, the latter ranking
as principles, and the former as the first suppositions of things
consequent: but convenience is evinced in the mutual consonance of
things demonstrated, and in the relation of all of them to intellect,
since intellect is the common measure of all science, from which it
receives its principles, and to which it converts the learner: but
determination is perceived in its perpetually abiding and immoveable
reasons, for the objects of its knowledge are not, at times, subject
to variation, like those of opinion and sense, but present themselves
for ever the same, and are bounded by intellectual forms. If such
then, are the principal requisites of beauty, it is evident, that
in these sciences that illustrious ornament and gracefulness is
found. For how is it possible this should not be the case with a
science receiving a supernal illumination from intellect, to which
it continually advances, hastening to transfer us from the obscure
light of sensible information? With respect to the second objection,
we think it proper to judge of its utility, without regarding the
conveniencies and necessities of human life. For otherwise, we must
confess that contemplative virtue is also useless, which separates
itself from human concerns, which it is very little desirous to look
down upon and understand. Indeed Socrates, in the Theætetus, affirming
this concerning noblemen endued with the prophetic power, says, “that
it withdraws them from all regard to human life, and raises their
thoughts, properly liberated, from all necessity and use, to the very
summit of all true being.” The mathematical science, therefore, must
be considered as desirable for its own sake, and for the contemplation
it affords, and not on account of the utility it administers to human
concerns. But if it is necessary to refer the utility it produces to
something different from itself, it must be referred to intellectual
knowledge. For it leads us to this, and prepares the eye of the
soul for the knowledge of universals, removing and obliterating the
impediments arising from the senses, and from corporeal involution.
As therefore we call the whole of purgative virtue useful, or the
contrary, not regarding the use of the sensible life, but of that
which is contemplative, so indeed it is requisite to refer the end
of mathematics to intellect, and universal wisdom. Hence its energy
is worthy our study, both on its own account, and on account of an
intellectual life. But it appears, as Aristotle[82] says, that this
science is desirable of itself to its votaries, because though no
reward is proposed to its enquirers, yet the mathematical contemplation
receives, in a small time, an abundant increase. Besides, this is
farther evident from hence, that all men are willingly employed in
its pursuit, and wish to dwell on its speculations, omitting every
other concern; even those who have, with their lips, as it were, but
just touched its utility. And hence it follows, that they who despise
the knowledge of the mathematical disciplines, have very little
tasted of the pleasures they contain. The mathematics, therefore,
are not to be despised because their speculative parts do not
immediately confer to human utility, (for the ultimate limits of its
progressions, and whatever operates with matter, consider a use of
this kind;) but on the contrary we should admire its immateriality,
and the good which it contains, considered by itself alone. For when
mankind were entirely disengaged from the care of necessary concerns,
they converted themselves to the investigation of the mathematical
disciplines; and this, indeed, with the greatest propriety. Since
affairs familiar to human life in its most imperfect state, and which
are immediately connected with its origin, first of all employed the
studies of mankind: but, in the second place, those concerns succeeded
which separate the soul from generation, and restore its memory of
that which _IS_. After this manner, then, we are engaged in
necessaries, before things honourable for their own sakes, on account
of their intrinsic dignity and worth; and in things related to sense,
before such as are apprehended by the nobler energies of mind. For
every origin and life of the soul which is converted into herself, is
naturally adapted to proceed from the imperfect to the perfect. And
thus much against those who despise the mathematical science.”




                               CHAP. X.

  _A Solution of another Objection of certain Platonists, against the
                Utility of the Mathematical Sciences._


But, perhaps, some of our own family will here rise up against us,
and, proposing Plato as a witness, will endeavour to provoke ruder
understandings into a contemptuous disregard of the mathematical
disciplines. For they will say, that this philosopher entirely
excludes (in his Republic) the mathematical knowledge from the choir
of the sciences, and that he accuses it as being ignorant of its own
principles, that its very principle is to itself unknown, and its ends
and mediums composed from things of which it is ignorant. To these
objections they may likewise add whatever other reproaches are there
urged by Socrates against this contemplation. In answer then, to the
objections of our friends, we shall recall into their memory, that
Plato himself perspicuously asserts the mathematical science to be the
purgation of the soul, and that it is endued with a power of leading
it on high; because, like the Homeric Minerva, it removes the darkness
of a sensible nature from the intellectual light of thought, which
is better worth saving than ten thousand corporeal eyes, and which
not only participates of a mercurial gift, (preserving us from the
incantations and delusions of this material abode, which is similar
to the fascinating realms of Circe,) but also of the more divine arts
of Minerva. He likewise every where calls it by the name of science,
and asserts that it is the cause of the greatest felicity to those
who are exercised in its contemplation. But I will briefly explain
why, in the Republic he takes from it the surname of science: for my
present discourse is addressed to the learned. Plato, indeed, in most
places, calls all the knowledge (as I may say) of universals by the
name of science, opposing it in a division to sense which apprehends
only particulars, whether such a mode of cognition is accomplished
by art or experience. And in this sense, as it appears to me in
the _Civil_ Dialogue, and in the Sophista, he seems to use the
name of science; placing likewise the illustrious Sophistic science,
which Socrates in the Gorgias, says, is a certain experience: also,
the adulatory, and many others, which are experiences, but not true
sciences. But again, dividing this knowledge of universals into
that which knows causes, and into that which understands without a
cause, he thinks that the one should be called science, but the other
experience. And hence, to arts he sometimes attributes the name of
science, but to experience never. For how (says he in the Banquet)
can a thing which possesses no reason be science? All knowledge,
therefore, which contains the reason and cause of the things known,
is a certain science. Again, therefore, he divides this science which
is endued with a power from the cause of knowing, by the peculiarity
of its subjects, and he places one, conjectural of things divisible;
but the other of such as subsist by themselves, and are ever knowable
after the same manner. And according to this division he separates from
science, medicine, and every faculty which is conversant with material
concerns. But he calls mathematical knowledge, and whatever possesses
a power of contemplating eternal objects, by the name of science.
Lastly, dividing this science, which we distinguished from arts, he
considers one part as void of supposition; but the other as flowing
from supposition. And that the one which is void of supposition, has
a power of knowing universals: that it rises to good, and the supreme
cause of all; and that it considers good as the end of its elevation:
but that the other, which previously fabricates for itself definite
and determinate principles, from which it evinces things consequent to
such principles, does not tend to the principle, but to the conclusion.
And hence he asserts, that mathematical knowledge, because it makes
use of supposition, falls short of that science which is without
supposition, and is perfect. For there is one true science, by means
of which we are disposed to know all the things _which are_, and
from which also principles emerge to all sciences; to some, indeed,
constituted more proximately, but to others more remotely. We must
not say, therefore, that Plato expels mathematical knowledge from the
number of the sciences, but that he asserts it to be the second from
that one science, which possesses the supreme seat of all: nor must we
affirm, that he accuses it as ignorant of its own principles, but that
receiving these from the master science dialectic, and possessing them
without any demonstration, it demonstrates from these its consequent
propositions. For, indeed, he sometimes allows the soul, which is
constituted from mathematical reasons, to be the principle of motion:
and sometimes he affirms, that it receives its motion from genera
which are subject to intelligence. And these variations accord among
themselves. For to such things as are moved by another, the soul is a
certain cause of motion, but it is not the cause of every motion. After
the same manner, the mathematical science is indeed the second from the
first of all sciences, and, with reference to it, imperfect: but it
is, nevertheless, a science, not as being free from supposition, but
as knowing the peculiar reasons resident in the soul, and as bringing
the causes of conclusions, and containing the reason of such things as
are subject to its knowledge. And thus much for the opinion of Plato
respecting mathematics.




                               CHAP. XI.


But let us now consider what are the things which may be required of
a mathematician, and how any one may rightly judge concerning his
distinguishing peculiarities. For[83] Aristotle indeed, says, that
he who is simply learned in all disciplines, is adapted to judge of
all: but that he who is alone skilled in the mathematical sciences,
can alone determine concerning the magnitude of reasons inherent in
these. It is requisite, therefore, that we should previously assume
the terms of judging, and that we should know, in the first place,
in what things it is proper to demonstrate generally, and in what to
regard the peculiarities of singulars. For many of the same properties
reside in things differing in species, as two right angles in all
triangles: but many have indeed the same predicament, yet differ
in their individuals in a common species, as similitude in figures
and numbers. But one demonstration is not to be sought for by the
mathematician in these, for the principles of figures and numbers are
not the same, but differ in their subject genus. And if the essential
accident is one, the demonstration will also be one[84]: for the
possession of two right angles is the same in all triangles, and that
general something to which this pertains is the same in all, I mean
triangle, and a triangular reason. In the same manner, likewise, the
possession of external angles to four right ones, not only pertains to
triangles, but also to all right-lined figures; and the demonstration,
so far as they are right-lined, agrees in all. For every reason brings
with it, at the same time, a certain property and passion, of which all
participate through that reason, whether triangular, or rectilinear, or
universally figure. But the second limit by which a mathematician is
to be judged, is, if he demonstrates according to his subject-matter,
and renders necessary reasons, and such as cannot be confuted, but are
at the same time neither probable, nor replenished with a similitude
of truth. For, says Aristotle, it is just the same to require
demonstrations from a rhetorician, and to assent to a mathematician
disputing probably; since every one, endued with science and art, ought
to render reasons adapted to the subjects of his investigation. In
like manner also, Plato in the Timæus, requires credible reasons of
the natural philosopher, as one who is employed in the resemblances
of truth: but of him who discourses concerning intelligibles, and a
stable essence, he demands reasons which can neither be confuted nor
moved. For subjects every where cause a difference in sciences and
arts, since, if some of them are immoveable, others are conversant
with motion; and some are more simple, but others more composite;
and some are intelligibles, but others sensibles. Hence we must
not require the same certainty from every part of the mathematical
science. For if one part, after a manner, borders upon sensibles, but
another part is the knowledge of intelligible subjects, they cannot
both be equally certain, but one must inherit a higher degree of
evidence than the other. And hence it is, that we call arithmetic more
certain than the science of harmony. Nor must we think it just that
mathematics and other sciences should use the same demonstrations;
for their subjects afford them no small variety. In the third place,
we must affirm, that he who rightly judges mathematical reasons, must
consider sameness and difference, what subsists by itself, and what is
accidental, what proportion is, and every consideration of a similar
kind. For almost all errors of this sort happen to those who think
they demonstrate mathematically, when at the same time they by no
means demonstrate, since they either demonstrate the same thing as if
different in each species, or that which is different as if it were
the same: or when they regard that which is accidental, as if it were
an essential property; or that which subsists by itself, as if it were
accidental. For instance, when they endeavour to demonstrate that the
circumference of a circle is more beautiful than a right line, or an
equilateral than an isosceles triangle. For the determination of these
does not belong to the mathematician, but to the first philosopher
alone. Lastly, in the fourth place, we must affirm, that since the
mathematical science obtains a middle situation between intelligibles
and sensibles, and exhibits in itself many images of divine concerns,
and many exemplars of natural reasons, we may behold in it three kinds
of demonstration[85], one approaching nearer to intellect, the second
more accommodated to cogitation, and the third bordering on opinion.
For it is requisite that demonstrations should differ according to the
varieties of problems, and receive a division correspondent to the
genera of beings, since the mathematical science is connected with all
these, and adapts its reasons to the universality of things. And thus
much for a discussion of the subject proposed.




                              CHAP. XII.

 _What and how many the Species of the whole Mathematical Science are,
            according to the Opinion of the Pythagoreans._


But after these considerations, it is requisite to determine concerning
the parts of the mathematical science, what, and how many they are.
For it is just, after speculating its whole and entire genus, to
consider the differences of its more particular sciences, according
to their species. The Pythagoreans[86], therefore, thought that the
whole mathematical science should receive a fourfold distribution,
attributing one of its parts to the how-many, but the other to the
how-much; and they assigned to each of these parts a twofold division.
For they said, that discrete quantity, or the how-many, either subsists
by itself, or must be considered with relation to some other; but that
continued quantity, or the how-much, is either stable or in motion.
Hence they affirmed, that arithmetic contemplates that discrete
quantity which subsists by itself, but music that which is related
to another; and that geometry considers continued quantity so far as
it is immoveable; but spherics contemplates continued quantity as
moving from itself, in consequence of its union with a self-motive
nature. They affirmed besides, that these two sciences, discrete and
continued quantity, did not consider either magnitude or multitude
absolutely, but that alone which in each of these is definite from the
participation of bound. For sciences alone speculate the definite,
rejecting as vain the comprehension of infinite quantity. But when
these wise men assigned this distribution, we must not suppose they
understood that discrete quantity which is found in sensible natures,
nor that continued quantity which subsists about the fluctuating
order of bodies. For, I think, the contemplation of these pertains
to the natural and not to the mathematical science. But because the
demiurgus of the universe, employed the union, division, and identity
of general natures, together with difference, station, and motion, for
the purpose of completing the essence of the soul, and composed it from
these genera, as Timæus informs us, we must affirm, that cogitation,
abiding according to its diversity, its division of reasons, and its
multitude, and understanding itself to be both one and many, proposes
indeed to itself, and produces numbers, together with an arithmetical
knowledge of these: but it provides for itself music according to an
union of its multitude, and a communication and junction with itself;
and hence it is that arithmetic excels music in antiquity; since,
according to the narration of Plato, the demiurgus first divided the
soul, and afterwards collected it in harmonical proportions. Again,
thought establishing its energy according to the stability which it
contains, draws from its inmost retreats geometry, together with one
essential figure, and the demiurgical principles of all figures[87]:
but, according to its inherent motion, it produces the spherical
science. For it is moved also by circles, but abides perpetually the
same from the causes of circles. Hence, likewise, geometry precedes
spherics, in the same manner as station is prior to motion. But because
cogitation itself produces these sciences, not by looking back upon
its convolution of forms, endued with an infinite power, but upon the
inclosure of _bound_ according to its definite genera; hence they
say, that the mathematical sciences take away infinite from multitude
and magnitude, and are only conversant about finite quantity. Indeed,
intellect has placed in cogitation all the principles both of multitude
and magnitude. For since it wholly consists, with reference to itself,
of similar parts, and is one and indivisible, and again divisible,
educing the ornament of forms, it participates of _bound_ and
_infinite_, from intelligible essences themselves. But it
understands, indeed, from its participation of bound, and generates
vital energies, and various reasons from the nature of infinite.
The intellections, therefore, of thought, constitute these sciences
according to the _bound_ which they contain, and not according to
an infinity of life; since they bring with them an image of intellect,
but not of life. Such then is the opinion of the Pythagoreans, and the
division of the four mathematical sciences.




                              CHAP. XIII.

      _Another Division of the Mathematical Science, according to
                               Geminus._


Again, some think (among whom is Geminus) that the mathematical science
is to be divided in a different manner from the preceding. And they
consider that one of its parts is conversant with intelligibles only,
but the other with sensibles, upon which it borders; denominating as
intelligibles whatever inspections the soul rouses into energy by
herself, when separating herself from material forms. And of that which
is conversant with intelligibles they establish two, by far the first
and most principal parts, arithmetic and geometry: but of that which
unfolds its office and employment in sensibles, they appoint six parts,
mechanics, astrology, optics, geodæsia, canonics, and logistics, or
the art of reckoning. But they do not think that the military art, or
tactics, should be called any one part of mathematics, according to
the opinion of some[88]; but they consider it as using at one time
the art of reckoning, as in the numbering of legions; but at another
time geodæsia, as in dividing and measuring the spaces filled by a
field of camps. As, say they, neither the art of writing, nor the
art of healing, are any part of mathematics, though frequently both
the historian and physician use mathematical theorems. This is the
case with historians indeed, when relating the situation of climates,
or collecting the magnitudes and dimensions of cities, or their
compass and circuit: but with physicians, when elucidating by ways of
this kind, many things in their art. For Hippocrates himself shews
the utility derived to medicine from astrology, and almost all who
speak of opportune times and places. By the same reason he also, who
accommodates his work to tactics, uses indeed mathematical theorems,
yet is not on this account a mathematician, although he is sometimes
willing that a numerous camp should exhibit a very small multitude,
and forms his army according to a circular figure; but sometimes in a
quadrangular, quinquangular, or some other multangular figure, when
he desires it to appear numerous. But since these are the species of
the whole mathematical science, geometry is again divided into the
contemplation of planes, and the dimension of solids, which is called
stereometry. For there is not any peculiar treatise about points and
lines, because no figure can be produced from these without planes
or solids. For geometry treats of nothing else in every one of its
parts, than that it may constitute either planes or solids: or that
when constituted, it may compare and divide them among themselves.
In like manner, arithmetic is distributed into the contemplation of
linear, plane, and solid numbers. For it considers the species of
numbers separate from sensible connections, proceeding from unity,
and the origin of plane numbers; I mean of the similar, dissimilar,
and solid, even to the third increase. But geodæsia, and the art
of reckoning, are divided similarly to arithmetic and geometry, as
they do not discourse concerning intelligible numbers or figures,
but of such as are sensible alone. For neither is it the office of
geodesia to measure the cylinder or the cone, but material masses
as if they were cones, and wells as if they were cylinders. Neither
does it accomplish this purpose by intelligible right lines, but by
such as are sensible, sometimes indeed by a more certain means, as
by the solar rays: but at other times by grosser ones, as by a line
and perpendicular. In like manner, the reckoner does not survey the
passions of numbers by themselves, but as they are resident in sensible
objects. From whence he also imposes a name upon these derived from
the things which he reckons, calling them μηλίαι, & φιαλίται. Besides
this, he does not, admit of any least, like the arithmetician, who
receives that minimum, as a genus of relation. For some one man is
considered by him as the measure of the whole multitude of men, as
unity also is the common measure of all numbers. Again, optics and
canonics are produced from geometry and arithmetic. And optics uses
the visual rays which are constituted by the rays of the eyes, as
lines and angles. But it is divided into that which is properly called
optics (because it renders the cause of these appearances, which are
accustomed to present themselves to us different from their reality, on
account of the different situations and distances of visible objects,
as the coincidence of parallel lines, or the appearance of quadrangles
as if they were circles); and into universal catoptrics, which is
conversant about various and manifold refractions, and is connected
with imaginative or conjectural knowledge: as also into that which
is called sciography[89], or the delineation of shadows, which shews
how appearances in images may seem neither inelegant nor deformed, on
account of the distances and altitudes of the things designed. But
canonics (music) or the regular art, considers the apparent reasons
of harmonies, finding out the sections of rules, every where using
the assistance of sense, and, as Plato says, seeming to prefer the
testimony of the ears to intellect itself. But to the parts we have
hitherto enumerated, mechanics must he added, as it is a certain part
of the whole science, and of the knowledge of sensible objects, and
of things united with matter. But under this exists the art effective
of instruments, which is called (ὀργανoποιητικὴ) I mean of those
instruments proper for the purposes of war: such, indeed, as Archimedes
is reported to have constructed, resisting the besiegers of sea and
land; and that which is effective of miracles, and which is called
(θαυματοποιητικὴ.) One part of this constructs with the greatest
artifice pneumatic engines, such as Ctesibius and Heron fabricated: but
another operates with weights, the motion of which is reckoned to be
the cause of inequilibrity; but their station of equilibrity, as Timæus
also has determined: and again, another part imitates animate foldings
and motions by strings and ropes. Again, under mechanics is placed
the knowledge of equilibriums, and of such instruments as are called
centroponderants: also (σφαιροποιία) or the art effective of spheres,
imitating the celestial revolutions, such as Archimedes fabricated;
and lastly, every thing endued with a power of moving matter. But the
last of all is astrology, which treats of the mundane motions, of the
magnitudes of the celestial bodies, their figures and illuminations,
their distances from the earth, and every thing of this kind; assuming
many things indeed to itself from sense, but communicating much with
the natural speculation. One part of this is gnomonics, which is
exercised in settling the dimension of hororary gnomons: but the other
is metheoroscopics, which finds out the differences of elevations, and
the distances of the stars, and also teaches many other and various
astrological theorems. The third part is dioptrics, which ascertains by
dioptric instruments of this kind the distances of the sun and moon,
and of the five other stars. And such is the account of the parts of
the mathematical science, delivered by the ancients, and transmitted to
our memory by the informing hand of time.




                              CHAP. XIV.

   _How Dialectic is the Top of the Mathematical Sciences, and what
              their Conjunction is, according to Plato._


Let us again consider after what manner Plato, in his Republic, calls
dialectic the top of the mathematical disciplines; and what their
conjunction is, according to the tradition of the author of the
Epinomis[90]. And in order to this we must assert, that as intellect
is superior to cogitation, supplying it with supernal principles,
and from itself giving perfection to cogitation; in the same manner
dialectic also, being the purest part of philosophy, excels in
simplicity the mathematical disciplines, to which it is proximate,
and with which it is conjoined. Indeed it embraces the complete
circle of these sciences, to which it elevates from itself various
energies, endued with a power of causing perfection, judgment, and
intelligence. And these energies consist in resolving, dividing,
defining, and demonstrating; by which mathematics itself, receiving
assistance and perfection, invents some things by resolution, but
others by composition: and some things it explains by division, others
by definition: but collects other subjects of its investigation by
demonstration; accommodating, indeed, these ways to its subjects, but
using each of them for the purpose of beholding its middle enquiries.
From whence indeed, both the resolutions, definitions, divisions, and
demonstrations which it contains, are peculiar, and adapted to its
nature, and revolve according to the mode of mathematical cognition.
Not undeservedly, therefore, is dialectic the vertex as it were,
and summit of mathematics. Since it perfects all which mathematics
contains of intelligence; renders its certainty free from reprehension,
preserves the stability of its immovable essence, and refers what it
contains destitute of matter and pure to the simplicity of intellect,
and a nature separated from material connections. Besides, it
distinguishes the first principles of these sciences, by definitions:
exhibits the separations of genera and forms contained under the
genera themselves: and besides this, teaches the compositions, which,
from principles, produce things consequent to principles: and the
resolutions which rise and mount up to things first, and to principles
themselves. But with respect to what remains, proportion itself is not
to be considered (as Eratosthenes thought it was) as the conjunction
of the mathematical disciplines. Since proportion is said to be, and
indeed is one of those things common to the mathematics. But in short,
many other things besides proportion regard all the mathematical
disciplines, which are essentially inherent in the common nature of
the mathematics. But as it appears to me, we should say, that there
is one proximate conjunction of these, and of the whole mathematical
science, which especially embraces in itself, in a more simple manner,
the principles of all sciences; which considers their community and
difference; teaches whatever is found in these the same; together
with what things are inherent in a many, and what in a few. So
that to those who aptly learn there is a reversion from many other
sciences to this alone[91]. But, dialectic is a conjunction of the
mathematical disciplines superior to the preceding; which Plato, as I
have already observed, calls in his Republic their vertex: for, indeed,
it perfects the whole of mathematics, brings it back to intellect by
its powers, shews it to be a true science, and causes it to be certain
and obnoxious to no reproof. But, intellect obtains the third order
between these conjunctions, which comprehends in itself uniformly
all the dialectic powers, contracts their variety by its simplicity,
their partition by its indivisible knowledge, and their multitude
by its occult union. Hence, intellect itself congregates indeed the
involutions and deviations of the dialectic paths, into an intelligible
essence, but it collects supernally all the progression of mathematical
discourses: and it is the best end both of the elevating power of the
soul, and of the energy consisting in cognition. And such are the
sentiments declared by me on the present enquiry.




                               CHAP. XV.

            _From whence the Name Mathematics originated._


Again, from whence shall we say this name of mathematics, and
mathematical disciplines, was assigned by the ancients, and what
apt reason can we render of its position? Indeed, it appears to me,
that such an appellation of a science which respects cogitative
reasons, was not, like most names, invented by indifferent persons,
but (as the truth of the case is, and according to report) by the
Pythagoreans alone. And this, when they perceived, that whatever is
called mathesis or discipline, is nothing more than reminiscence;
which does not approach the soul extrinsically, like the images which
rising from sensible objects are formed in the phantasy: nor is it
adventitious and foreign, like the knowledge consisting in opinion,
but it is excited, indeed, from apparent objects, and is perfected
within, by thought intimately converted to itself. And when they
likewise perceived that though reminiscence might be shewn from many
particulars, yet it was evinced in a more eminent manner (as Plato
also says[92]) from the mathematical disciplines. For if any one, says
he, is led into the descriptions, he will there easily prove that
discipline is reminiscence. From whence Socrates also, in the Meno,
shews by this method of arguing, that learning is nothing else than the
soul’s recollection of her inherent reasons. And this, because that
which recollects, is alone the cogitative part of the soul; but this
perfects her essence in the reasons of the mathematical disciplines,
the sciences of which she previously received into herself, though she
does not always energize on their fair variety. Indeed, she contains
them all essentially and occultly; but she produces each of them when
she is freed from the impediments originating from sense. For sense
unites her with divisible objects: the phantasy fills her with forming
motions, and appetite bends her to an indulgent and luxurious life.
But every thing divisible is an obstacle to our self-conversion. And
whatever invests with form, disturbs and offends that knowledge which
is destitute of form. And whatever is obnoxious to perturbations is
an impediment to that energy which is unimpaired by affections. When,
therefore, we have moved all these from the cogitative power, then
shall we be able to understand by thought itself, the reasons which
thought contains: then shall we become scientific in energy; and
unfold our essential knowledge. But whilst we are captive and bound,
and winking with the eye of the soul, we cannot by any means attain
to a perfection convenient to our nature. Such then is mathesis or
discipline: a reminiscence of the eternal reasons contained in the
soul. And the mathematical or disciplinative science is on this account
particularly denominated that knowledge which especially confers to our
reminiscence of these essential reasons. Hence, the business and office
of this science[93], is apparent from its name. For its duty is to
move the inherent knowledge of the soul; to awaken its intelligence;
to purify its cogitation; to call forth its essential forms from their
dormant retreats; to remove that oblivion and ignorance, which are
congenial with our birth; and to dissolve the bonds arising from our
union with an irrational nature. It plainly leads us to a similitude
of that divinity who presides over this science, who manifests
intellectual gifts, and fills the universe with divine reasons; who
elevates souls to intellect, wakens them as from a profound sleep,
converts them by enquiry to themselves; and by a certain obstetric art,
and invention of pure intellect, brings them to a blessed life. To whom
indeed, dedicating the present work, we here conclude our contemplation
of the mathematical science.




                               BOOK II.




                               CHAP. I.

   _What Part Geometry is of Mathematics, and what the Matter is of
                          which it consists._


In the preceding discourses we have considered those common properties
which respect the whole of the mathematical science; and this we have
done agreeable to the doctrine of Plato; at the same time collecting
such particulars as pertain to our present design. But consequent to
this it is requisite that we should discourse on geometry itself, and
on the proposed institution of the elements, for the sake of which we
have undertaken the whole of the present work. That geometry then, is
a part of the whole of mathematics, and that it obtains the second
place after arithmetic, since it is perfected and bounded by this,
(for whatever in geometry may be expressed and known, is determined
by arithmetical reasons) has been asserted by the ancients, and
requires no long discussion in the present enquiry. But we also may
be able to relate our opinion on this particular, if we consider what
place, and what essence its subject matter[94] is allotted among the
universality of things. For from a proper survey of this, the power
of the science which knows this subject matter, the utility arising
from it, and the good acquired by its learners, will immediately
appear. Indeed, some one may doubt in what genus of things he ought
to place geometrical matter, so as not to deviate from the truth it
contains. For if the figures concerning which geometry discourses,
exist in sensible natures, and cannot be separated from the dark
receptacle of matter; how can we assert that geometry frees us from
sensible objects, that it brings us to an incorporeal essence, that it
accustoms us to an inspection of intelligibles, and prepares us for
intellectual energy? Where shall we ever survey among sensible objects
a point without parts, or a line destitute of breadth, or a superficies
without profundity, or the equality of lines from the centre to the
circumference; or the multangles, and all the figures of many bases,
concerning which geometry informs us? Lastly, after what manner can the
reasons of such a science remain free from all possible confutation;
since, indeed, sensible forms and figures are susceptive of the more
and the less, are all moveable and mutable, and are full of material
variety; among which equality subsists mixt and confused with its
contrary inequality, and into which things without parts have proceeded
into partition, and interval, darkened with the shades of matter, and
lost in its infinite folds? But if the subjects of geometry are removed
from matter, are pure forms, and are separated from sensible objects:
they will be all of them, without doubt, void of parts, incorporeal,
and destitute of magnitude. For extension, tumor, and interval,
approach to forms, on account of the material receptacle in which they
are involved, and which receives things destitute of parts, distributed
into parts; things void of dimension, extended into dimension; and
immoveable natures accompanied with motion. How then, if this is the
case, shall we cut a right line, triangle, and circle? How can we speak
of the diversities of angles, and the increments and decrements of
triangular and quadrangular figures? Or how exhibit the contacts of
circles or right lines? For all these evince that the geometric matter
consists of parts, and does not reside among indivisible reasons.
Such then are the doubts concerning the matter of geometry, to which
we may add, that Plato considers the forms of geometry as placed in
cogitation; and grants, that we advance from sensibles to forms of
this kind, and that we rise from sensibles to intellect, though (as
we have previously observed) the reasons subsisting in cogitation
are indivisible, are separated by no interval, and subsist according
to the peculiarity of the soul. But if reasons are to be rendered
agreeable to things themselves, and to the doctrine of Plato, the
following division must be adopted. [95]Every universal, and one
thing containing many, is either naturally disposed to be thought of
in particulars, or to appear such, because it possesses its existence
in these; is inseparable from them; is disposed and distributed in
them; and together with these is either moved, or firmly and immoveably
abides. Or it is adapted to subsist prior to many, and to possess a
power of generating multitude, affording to many things images from
itself, being furnished with a nature destitute of parts, from the
essences which it participates, and raising various participations to
secondary natures: or it is disposed to be formed by thought, from the
many, to possess a generating existence, and to reside in the last
place in the many. For, according to these three modes of subsistence,
we shall find, I think, that some subsist before the many, others in
the many, and others from the relation and predication which they
possess to these. But, that I may absolve all in one word, universal
forms being threefold, we shall consider the differences of that form
which many participate, which exists in many, and fills particular
natures according to its subject matter. Besides this, establishing
a twofold order of participants, one subsisting in sensible objects,
but the other in the phantasy, (since matter is twofold; one indeed,
of things united with sense, but the other of such as fall under the
inspection of phantasy, as Aristotle asserts, in a certain place[96])
we must allow that the universal, which is distributed in the many,
is likewise twofold. The one, indeed, sensible, as being that which
sensible objects participate; but the other imaginative, as that
which subsists in the _many_ of the phantasy. For the phantasy,
on account of its forming motion, and because it subsists with, and
in body, always receives impressions which are both divided and
figured. So that whatever is known by it, is allotted a correspondent
existence: on which account, Aristotle[97] does not hesitate to call
it passive intellect. But if it is intellect, why is it not impassive,
and destitute of matter? And if it operates with passion, how can it
with propriety be called intellect? For impassivity, indeed, properly
belongs to intellect and an intelligent nature: but passivity is very
remote from such an essence. But (unless I am deceived) Aristotle
being willing to explain its middle nature between cognitions the most
primary, and such as are the last, calls it at the same time intellect,
because similar to primary cognitions, and passive from that alliance
which it possesses with such as are posterior. For first cognitions are
indeed destitute of figures and forms; comprehending in themselves,
intelligible natures, energizing about themselves, united with the
objects of knowledge, and free from all extrinsical impression and
passion. But last cognitions exercise themselves through the medium
of instruments, are rather passions than energies, admit extrinsical
knowledge, and move themselves together with their various subjects.
For such (says Plato) are the sensations which arise from violent
passions. But the phantasy, obtaining a middle centre in the order of
cognitions, is excited, indeed, by itself, and produces that which
falls under cogitation: but because it is not separate from body,
it deduces into partition, interval, and figure, the objects of its
knowledge, from the indivisibility of an intellectual life. Hence,
whatever it knows, is a certain impression and form of intelligence.
For it understands the circle, together with its interval, void,
indeed, of external matter, but possessing intelligible matter. On
this account, like sensible matter, it does not contain one circle
only: for we behold in its receptacle, distance, together with the
more and the less, and a multitude of circles and triangles. If then
an universal nature is distributed in sensible circles, since each of
these completes a circular figure, and they are all mutually similar,
subsisting in one reason, but differing in magnitudes or subjects:
in like manner, there is a common something in the circles, which
subsist in the receptacle of the phantasy, of which all its circles
participate, and according to which they all possess the same form;
but in the phantasy they possess but one difference only, that of
magnitude. For when you imagine many circles about the same centre,
they all of them exist in one immaterial subject and life, which is
inseparable from a simple body, which, by the possession of interval,
exceeds an essence destitute of parts; but they differ in magnitude
and parvitude, and because they are contained and contain. Hence,
that universal is twofold, which is understood as subsisting in the
many: one, indeed, in sensible forms; but the other in such as are
imaginative. And the reason of a circular and triangular figure, and of
figure universal, is twofold. The one subsisting in intelligible, but
the other in sensible matter. But prior to these is the reason which
resides in cogitation, and that which is seated in nature herself.
The former being the author of imaginative circles, and of the one
form which they contain; but the other, of such as are sensible.
For there are circles existing in the heavens, and universally
those produced by nature, the reason of which does not fall under a
cogitative distribution. For in incorporeal causes, things possessing
interval, are distinguished by no intervals: such as are endued with
parts, subsist without parts: and magnitudes without the diffusion of
magnitude, as on the contrary in corporeal causes, things without parts
subsist divisibly, and such as are void of magnitude with the extension
of magnitude. Hence, the circle resident in cogitation, is one, simple
and free from interval: and magnitude itself is there destitute of
magnitude; and figure expressed by no figure: for such are reasons
separate from matter. But the circle subsisting in the phantasy, is
divisible, figured, endued with interval, not one only, but one and
many, nor form alone, but distributed with form. And the circle, in
sensible objects, is composite, distant with magnitude, diminished by
a certain reason, full of ineptitude, and very remote from the purity
of immaterial natures. We must therefore say, that geometry, when it
asserts any thing of circle and diameter, and of the passions and
affections which regard the circle; as of contacts, divisions, and the
like: neither teaches nor discourses concerning sensible forms, (since
it endeavours to separate us from these), nor yet concerning the form
resident in cogitation, (for here the circle is one, but geometry
discourses of many, proposing something of each, and contemplating the
same of all: and here it is indivisible, but the geometric circle is
divisible); but we must confess, that it considers universal itself;
yet as distributed in imaginative circles. And that it beholds, indeed,
one circle[98]: and by the medium of another, contemplates the circle
resident in the depths of cogitation: but by another, different from
the preceding, fabricates the fair variety of its demonstrations.
For since cogitation is endued with reasons, but cannot behold them
contractedly, separated from material figure; it distributes and
removes them, and draws them forth seated in the shadowy bosom of the
phantasy, and placed in the vestibules of primary forms; revolving
in it, or together with it, the knowledge of these: loving, indeed,
a separation from sensibles, but finding imaginative matter proper
for the reception of its universal forms. Hence, its intellection
does not subsist without the phantasy. And the compositions and
divisions of figures are imaginative; and their knowledge is the way
which leads us to that essence pursued by cogitation: but cogitation
itself, does not yet arrive at this stable essence, while it looks
abroad to externals, contemplates its internal forms according to
these, uses the impressions of reasons, and is moved from itself to
external and material forms. But if it should ever be able to return
to itself, when it has contracted intervals and impressions, and
beholds multitude without impression, and subsisting uniformly; then
it will excellently perceive geometrical reasons, void of division and
interval, essential and vital, of which there is a copious variety.
And this energy will be the best end of the geometric study; and truly
the employment of a Mercurial gift, bringing it back as from a certain
Calypso, and her detaining charms, to a more intellectual knowledge;
and freeing it from those _forming_ apprehensions with which the
mirror of the phantasy is replete. Indeed, it is requisite that a
true geometrician should be employed in this meditation, and should
establish, as his proper end, the excitation and transition from the
phantasy to cogitation alone; and that he should accomplish this by
separating himself from intervals, and the passive intellect to that
energy which cogitation contains. For by this means he will perceive
all things without an interval, the circle and diameter without a part,
the polygons in the circle, all in all, and yet every one separate and
apart. Since, on this account, we exhibit also in the phantasy, both
circles inscribed in polygons, and polygons in circles; imitating the
alternate exhibition of reasons destitute of parts. Hence, therefore,
we describe the constitutions, the origin, divisions, positions, and
applications of figures: because we use the phantasy, and distances
of this kind proceeding from its material nature; since form itself
is immoveable, without generation, indivisible, and free from every
subject. But whatever form contains occultly, and in an indistant
manner, is produced into the phantasy subsisting with intervals,
divisibly and expanded. And that which, indeed, produces the forms of
geometric speculation, is cogitation: but that from which they are
produced, is the form resident in cogitation: and that in which the
produced figure resides is what is called the passive intellect. Which
folds itself about the impartibility of true intellect, separates from
itself the power of pure intelligence free from interval; conforms
itself according to all formless species, and becomes perfectly
every thing from which cogitation itself, and our indivisible reason
consists. And thus much concerning the geometric matter, as we are
not ignorant of whatever Porphyry the Philosopher has observed in his
miscellanies, and whatever many of the Platonists describe. But we
think that the present discussions are more agreeable to geometric
dissertations, and to Plato himself, who subjects to geometry the
objects of cogitation. For these mutually agree among themselves;
because the causes, indeed, of geometrical forms, by which cogitation
produces demonstrations, pre-exist in demonstration itself: but the
particular figures which are divided and compounded, are situated in
the receptacle of the phantasy.




                               CHAP. II.

                  _What kind of Science Geometry is._


But let us now speak of that science which possesses a power of
contemplating the universal forms participated by imaginative matter.
Geometry, therefore, is endued with the knowledge of magnitudes and
figures, and of the terms and reasons subsisting in these; together
with the passions, various positions and motions which are contingent
about these. For it proceeds, indeed, from an impartible point, but
descends even to solids, and finds out their multiform diversities.
And again, runs back from things more composite, to things more
simple, and to the principles of these: since it uses compositions
and resolutions, always beginning from suppositions, and assuming its
principles from a previous science; but employing all the dialectic
ways. In principles, by the divisions of forms from their genera, and
by defining its orations. But in things posterior to principles, by
demonstrations and resolutions. As likewise, it exhibits things more
various, proceeding from such as are more simple, and returning to
them again. Besides this, it separately discourses of its subjects;
separately of its axioms; from which it rises to demonstrations;
and separately of essential accidents, which it shews likewise are
resident in its subjects. For every science has, indeed, a genus, about
which it is conversant, and whose passions it proposes to consider:
and besides this, principles, which it uses in demonstrations; and
essential accidents. Axioms, indeed, are common to all sciences
(though each employs them in its peculiar subject matter), but genus
and essential accident vary according to the sciential variety. The
subjects of geometry are therefore, indeed, triangles, quadrangles,
circles, and universally figures and magnitudes, and the boundaries of
these. But its essential accidents are divisions, ratios, contacts,
equalities, applications, excesses, defects, and the like. But its
petitions and axioms, by which it demonstrates every particular are,
this, to draw a right line from any point to any point; and that,
if from equals you take away equals, the remainders will be equal;
together with the petitions and axioms consequent to these. Hence,
not every problem nor thing sought is geometrical, but such only as
flow from geometric principles. And he who is reproved and convicted
from these, is convinced as a geometrician. But whoever is convinced
from principles different from these, is not a geometrician, but is
foreign from the geometric contemplation. But the objects of the
non-geometric investigation, are of two kinds. For the thing sought
for, is either from entirely different principles, as we say that a
musical enquiry is foreign from geometry, because it emanates from
other suppositions, and not from the principles of geometry: or it is
such as uses, indeed, geometrical principles, but at the same time
perversely, as if any one should say, that parallels coincide. And on
this account, geometry also exhibits to us instruments of judging, by
which we may know what things are consequent to its principles, and
what those are which fall from the truth of its principles: for some
things attend geometrical, but others arithmetical principles. And why
should we speak of others, since they are far distant from these? For
one science is more certain than another (as Aristotle says[99]) that,
indeed, which emanates from more simple suppositions, than that which
uses more various principles; and that which tells the _why_,
than that which knows only the simple existence of a thing; and that
which is conversant about intelligibles, than that which touches and
is employed about sensibles. And according to these definitions of
certainty, arithmetic is, indeed, more certain than geometry, since its
principles excel by their simplicity. For unity is void of position,
with which a point is endued. And a point, indeed, when it receives
position, is the principle of geometry: but unity, of arithmetic. But
geometry is more certain than spherics; and arithmetic, than music.
For these render universally the causes of those theorems, which are
contained under them. Again, geometry is more certain than mechanics,
optics, and catoptrics. Because these discourse only on sensible
objects. The principles, therefore, of geometry and arithmetic, differ,
indeed, from the principles of other sciences; but the hypotheses of
these two, alternately differ and agree according to the difference we
have already described. Hence, also, with respect to the theorems which
are demonstrated in these sciences, some are, indeed, common to them,
but others peculiar. For the theorem which says, _every proportion
may be expressed_, alone belongs to arithmetic; but by no means
to geometry: since this last science contains things which cannot be
expressed[100]. That theorem also, which affirms, that _the gnomons
of quadrangles are terminated according to the least_[101], is the
property of arithmetic: for in geometry, a minimum cannot be given.
But those things are peculiar to geometry, which are conversant about
positions; for numbers have no position: which respect contacts; for
contact is found in continued quantities: and which are conversant
about ineffable proportions; for where division proceeds to infinity,
there also that which is ineffable is found[102]. But things common to
both these sciences, are such as respect divisions, which Euclid treats
of in the second book; except that proposition which divides a right
line into extreme and mean proportion[103]. Again, of these common
theorems, some, indeed, are transferred from geometry into arithmetic;
but others, on the contrary, from arithmetic into geometry: and others
similarly accord with both, which are derived into them from the whole
mathematical science. For the permutation, indeed, conversions,
compositions, and divisions of ratios are, after this manner, common
to both. But such things as are commensurable, arithmetic first
beholds; but afterwards geometry, imitating arithmetic. From whence,
also, it determines such things to be commensurables of this kind,
which have the same mutual ratio to one another, as number to number;
because commensurability principally subsists in numbers. For where
number is, there also that which is commensurable is found; and where
commensurable is, there also number. Lastly, geometry first inspects
triangles and quadrangles: but, arithmetic, receiving these from
geometry, considers them according to proportion. For in numbers,
figures reside in a causal manner. Being excited, therefore, from
effects, we pass to their causes, which are contained in numbers. And
at one time, we indifferently behold the same accidents, as when every
polygon is resolved by us into triangles[104]: but, at another time,
we are content with what is nearest to the truth, as when we find in
geometry one quadrangle the double of another, but not finding this
in numbers, we say that one square is double of another, except by a
deficience of unity. As for instance, the square from 7, is double the
square from 5, wanting one. But we have produced our discussion to this
length, for the purpose of evincing the communion and difference in the
principles of these two sciences. Since it belongs to a geometrician
to survey from what common principles common theorems are divided;
and from what principles such as are peculiar proceed; and thus to
distinguish between the geometrical, and non-geometrical, referring
each of them to different sciences.




                              CHAP. III.

  _From whence the whole of Geometry originated, how far it proceeds,
                  and in what its Utility consists._


But, beginning still higher, let us contemplate the whole of geometry,
from whence it originated, and how far it proceeds in its energies:
for thus we shall properly perceive the ornament which it contains.
Indeed, it is necessary to understand that it is extended through the
universality of things: that it accommodates its animadversions[105]
to all beings; and contains in itself the forms of all things: that,
according to its supreme part, and which is endued with the highest
power of intelligence, it surveys true beings; and teaches by images
the properties of divine ornaments, and the powers of intellectual
forms: for it contains the reasons of these also in its peculiar
contemplations. And it exhibits what figures are convenient to the
god, to primary essences, and to the natures of souls. But, according
to its middle cognitions, it evolves cogitative reasons; explains and
beholds the variety which they contain; exhibits their existence, and
inherent passions; as also, their communities and diversities. From
which, indeed, it comprehends, in terminated bounds, the imaginative
formations of figures, and reduces them to the essential substance
of reasons. But, according to the third propagations of cogitative
intelligence, it considers nature, and delivers the manner in which
the forms of sensible elements, and the powers which they contain, are
previously received according to cause, in the reasons themselves. For
it possesses, indeed, the images of universal intelligible genera;
but the exemplars of such as are sensible: and completes its own
essence, according to such things as are subject to cogitation. And
through these, as through proper mediums, it ascends and descends to
those universals which truly are, and to sensible forms which are in a
state of perpetual formation. But always geometrically philosophising
concerning the things which are, it comprehends in all the proportions
of virtues, the images of intellectual, animal, and natural concerns.
And it delivers, in an orderly manner, all the ornaments of republics:
and exhibits in itself their various mutations. Such then are its
energies arising from a certain immaterial power of cognition: but
when it touches upon matter, it produces from itself a multitude of
sciences; such as geodæsia, mechanics, and perspective: by which it
procures the greatest benefit to the life of mortals. For it constructs
by these sciences, war-instruments, and the bulwarks of cities; and
makes known the circuits of mountains, and the situations of places.
Lastly, it instructs us in measures: at one time of the diversified
ways of the earth; and at another, of the restless paths of the deep.
Add too, that it constructs balances and scales, by which it renders
to cities a sure equality according to the invariable standard of
number. Likewise, it clearly expresses, by images, the order of the
whole orb of the earth; and by these, manifests many things incredible
to mankind, and renders them credible to all. Such, indeed, as Hiero
of Syracuse is reported to have said of Archimedes[106], when he had
fabricated a ship furnished with three sails, which he had prepared to
send to Ptolemy king of Egypt. For when all the Syracusians together,
were unable to draw this ship, Archimedes enabled Hiero to draw it
himself, without any assistance from others. But he, being astonished,
said, From this day, Archimedes shall be believed in whatever he shall
affirm. They also report, that Gelo said the same, when Archimedes
discovered the weight of the several materials from which his crown was
composed, without dissolving their union. And such are the narrations
which many of the ancients have delivered to our memory, who were
willing to speak in praise of the mathematics: and, on this account, we
have placed before the reader, for the present, a few out of the many,
as not foreign from our design of exhibiting the knowledge and utility
of geometry.




                               CHAP. IV.

            _On the Origin of Geometry, and its Inventors._


But let us now explain the origin of geometry, as existing in the
present age of the world. For the demoniacal Aristotle[107] observes,
that the same opinions often subsist among men, according to certain
orderly revolutions of the world: and that sciences did not receive
their first constitution in our times, nor in those periods which are
known to us from historical tradition, but have appeared and vanished
again in other revolutions of the universe; nor is it possible to say
how often this has happened in past ages, and will again take place
in the future circulations of time. But, because the origin of arts
and sciences is to be considered according to the present revolution
of the universe, we must affirm, in conformity with the most general
tradition, that geometry was first invented by the Egyptians, deriving
its origin from the mensuration of their fields: since this, indeed,
was necessary to them, on account of the inundation of the Nile washing
away the boundaries of land belonging to each. Nor ought it to seem
wonderful that the invention of this as well as of other sciences,
should receive its commencement from convenience and opportunity.
Since whatever is carried in the circle of generation, proceeds
from the imperfect to the perfect. A transition, therefore, is not
undeservedly made from sense to consideration, and from this to the
nobler energies of intellect[108]. Hence, as the certain knowledge
of numbers received its origin among the Phœnicians, on account of
merchandise and commerce, so geometry was found out among the Egyptians
from the distribution of land. When Thales, therefore, first went into
Egypt, he transferred this knowledge from thence into Greece: and
he invented many things himself, and communicated to his successors
the principles of many. Some of which were, indeed, more universal,
but others extended to sensibles. After him Ameristus, the brother
of Stesichorus the poet, is celebrated as one who touched upon, and
tasted the study of geometry, and who is mentioned by Hippias the
Elean, as restoring the glory of geometry. But after these, Pythagoras
changed that philosophy, which is conversant about geometry itself,
into the form of a liberal doctrine, considering its principles in a
more exalted manner; and investigating its theorems immaterially and
intellectually; who likewise invented a treatise of such things as
cannot be explained[109] in geometry, and discovered the constitution
of the mundane figures. After him, Anaxagoras the Clazomenian
succeeded, who undertook many things pertaining to geometry. And
Oenopides the Chian, was somewhat junior to Anaxagoras, and whom Plato
mentions in his Rivals, as one who obtained mathematical glory. To
these, succeeded Hippocrates, the Chian, who invented the quadrature
of the lunula[110], and Theodorus the Cyrenean, both of them eminent
in geometrical knowledge. For the first of these, Hippocrates composed
geometrical elements: but Plato, who was posterior to these, caused
as well geometry itself, as the other mathematical disciplines, to
receive a remarkable addition, on account of the great study he
bestowed in their investigation. This he himself manifests, and his
books, replete with mathematical discourses, evince: to which we
may add, that he every where excites whatever in them is wonderful,
and extends to philosophy. But in his time also lived Leodamas the
Thasian, Archytas the Tarantine, and Theætetus the Athenian; by whom
theorems were increased, and advanced to a more skilful constitution.
But Neoclides was junior to Leodamas, and his disciple was Leon; who
added many things to those thought of by former geometricians. So that
Leon also constructed elements more accurate, both on account of their
multitude, and on account of the use which they exhibit: and besides
this, he discovered a method of determining when a problem, whose
investigation is sought for, is possible, and when it is impossible.
But Eudoxus the Cnidian, who was somewhat junior to Leon, and the
companion of Plato, first of all rendered the multitude of those
theorems which are called universals more abundant; and to three
proportions added three others; and things relative to a section,
which received their commencement from Plato, he diffused into a
richer multitude, employing also resolutions in the prosecution of
these. Again, Amyclas the Heracleotean, one of Plato’s familiars, and
Menæchmus, the disciple, indeed, of Eudoxus, but conversant with Plato,
and his brother Dinostratus, rendered the whole of geometry as yet
more perfect. But Theudius, the Magnian, appears to have excelled, as
well in mathematical disciplines, as in the rest of philosophy. For he
constructed elements egregiously, and rendered many particulars more
universal. Besides, Cyzicinus the Athenian, flourished at the same
period, and became illustrious in other mathematical disciplines, but
especially in geometry. These, therefore, resorted by turns to the
Academy, and employed themselves in proposing common questions. But
Hermotimus, the Colophonian, rendered more abundant what was formerly
published by Eudoxus and Theætetus, and invented a multitude of
elements, and wrote concerning some geometrical places. But Philippus
the Mendæan[111], a disciple of Plato, and by him inflamed in the
mathematical disciplines, both composed questions, according to the
institutions of Plato, and proposed as the object of his enquiry
whatever he thought conduced to the Platonic philosophy. And thus far
historians produce the perfection of this science. But Euclid was not
much junior to these, who collected elements, and constructed many
of those things which were invented by Eudoxus; and perfected many
which were discovered by Theætetus. Besides, he reduced to invincible
demonstrations, such things as were exhibited by others with a weaker
arm. But he lived in the times of the first Ptolemy: for Archimedes
mentions Euclid, in his first book, and also in others. Besides, they
relate that Euclid was asked by Ptolemy, whether there was any shorter
way to the attainment of geometry than by his elementary institution,
and that he answered, there was no other royal path which led to
geometry. Euclid, therefore, was junior to the familiars of Plato, but
more ancient than Eratosthenes and Archimedes (for these lived at one
and the same time, according to the tradition of Eratosthenes) but he
was of the Platonic sect, and familiar with its philosophy: and from
hence he appointed the constitution of those figures which are called
Platonic[112], as the end of his elementary institutions.




                               CHAP. V.

             _What Mathematical Volumes Euclid composed._


There are, therefore, many other mathematical volumes of this man,
full of admirable diligence, and skilful consideration: for such are
his Optics[113], and Catoptrics: and such also, are his elementary
institutions, which conduce to the attainment of music[114]; and his
book concerning divisions[115]. But his geometrical institution of the
Elements is especially admirable, on account of the order and election
of those theorems and problems, which are distributed through the
Elements. For he does not assume all which might be said, but that
only which could be delivered in an elementary order. Besides this, he
exhibits modes of syllogisms of every kind; some, indeed, receiving
credibility from causes, but others proceeding from certain signs; but
all of them invincible and sure, and accommodated to science. But,
besides these, he employs all the dialectic ways, dividing, indeed, in
the inventions of forms; but defining in essential reasons: and again,
demonstrating in the progressions from principles to things sought,
but resolving in the reversions from things sought to principles.
Besides this, we may view in his geometrical elements, the various
species of conversions, as well of such as are simple as of such as
are more composite. And again, what wholes may be converted with
wholes: what wholes with parts; and on the other hand, what as parts
with parts[116]. Besides this, we must say, that in the continuation
of inventions, the dispositions and order of things preceding and
following, and in the power with which he treats every particular,
he is not deceived, as if falling from science, and carried to its
contrary, falsehood and ignorance. But because we may imagine many
things as adhering to truth, and which are consequent to principles
producing science, which nevertheless tend to that error which flows
from the principles, and which deceives ruder minds, he has also
delivered methods of the perspicacious prudence belonging to these.
From the possession of which, we may exercise those in the invention of
fallacies, who undertake this inspection, and may preserve ourselves
from all deception. And this book, by which he procures us this
preparation, is inscribed ψευδαρίος, or, concerning fallacies[117].
Because he enumerates in order their various modes, and in each
exercises our cogitation with various theorems. And he compares truth
with falsehood, and adapts the confutation of deception to experience
itself. This book, therefore, contains a purgative and exercising
power. But the institution of his elementary, skilful contemplation of
geometrical concerns, possesses an invincible and perfect narration.




                               CHAP. VI.

                 _Concerning the Purport of Geometry._


But, perhaps, some one may enquire in what the design of this treatise
consists? To this I answer, that its design is to be distinguished as
well according to the objects of enquiry, as according to the learner.
And, indeed, regarding the subject, we must affirm, that all the
discourse of geometry is concerning the mundane figures. Because it
begins from such things as are simple, but ends in the variety of their
constitution. And, indeed, it constitutes each of them separately,
but at the same time delivers their inscriptions in a sphere, and the
proportions which they contain. On which account some have thought,
that the design of each of the books is to be referred to the world;
and they have delivered to our memory, the utility which they afford
us in the contemplation of the universe. But distinguishing the
design with respect to the learner, we must affirm, that its purpose
is the institution of elements; and the perfection of the learners
cogitative powers in universal geometry. For beginning from these,
we are enabled to understand the other parts of this science, and to
comprehend the variety which they contain. And, indeed, without these,
the discipline of the rest, is to us impossible and incomprehensible.
For such theorems as are most principal and simple, and are most allied
to first suppositions, are here collected in a becoming order. And the
demonstrations of other mathematicians, use these as most known, and
advance from these in their most complicated progressions. For thus
Archimedes, in what he has writ concerning the sphere and cylinder, and
Apollonius, and the rest of mathematicians, use, as evident principles,
the things exhibited in this treatise. Its purpose, therefore, is the
institution of learners in the whole geometric science, and to deliver
the determinate constitutions of the mundane figures.




                              CHAP. VII.

  _From whence the Name of Elementary Institution originated, and why
             Euclid is called the Institutor of Elements._


But what gave rise to the name of elementary institution, and of
element itself, from which elementary institution was derived?
To this we shall reply, by observing, that of theorems some are
usually called elements, but others elementary, and others again are
determined beyond the power of these. Hence, an element is that whose
consideration passes to the science of other things, and from which we
derive a solution of the doubts incident to the particular science we
investigate. For as there are certain first principles of speech, most
simple and indivisible, which we denominate elements, and from which
all discourse is composed; so there are certain principal theorems of
the whole of geometry, denominated elements, which have the respect of
principles to the following theorems; which regard all the subsequent
propositions, and afford the demonstrations of many accidents essential
to the subjects of geometric speculation. But things elementary
are such as extend themselves to a multitude of propositions, and
possess a certain simplicity and sweetness, yet are not of the same
dignity with elements; because their contemplation is not common to
all the science to which they belong, as is the case in the following
theorem, that in triangles, perpendiculars, drawn from their angles
to their sides, coincide in one point[118]. Lastly, whatever neither
possesses a knowledge extended into multitude, nor exhibits any thing
skilful and elegant, falls beyond the elementary power. Again, an
element, as Menæchmus says, may have a twofold definition. For that
which confirms, is an element of that which is confirmed; as the first
proposition of Euclid with respect to the second, and the fourth with
regard to the fifth. And thus, indeed, many things may be mutually
called elements one of another; for they are mutually confirmed.
Thus, because the external angles of right-lined figures, are equal
to four right angles, the multitude of internal ones equal to right
angles; and, on the contrary, that from this is exhibited[119].
Besides, an element is otherwise called that into which, because it
is more simple, a composite is dissolved. But it must be observed,
that every element cannot be called the element of every thing: but
such as are more principal are the elements of such as are constituted
in the reason of the thing effected; as petitions are the elements
of theorems. And, according to this signification of an element,
Euclid’s elements are constructed. Some, indeed, of that geometry
which is conversant about planes; but others of stereometry. In the
same manner, likewise, in arithmetic and astronomy, many have composed
elementary institutions. But it is difficult, in each science, to chuse
and conveniently ordain elements, from which all the peculiarities of
that science originate, and into which they may be resolved. And among
those who have undertaken this employment, some have been able to
collect more, but others fewer elements. And some, indeed, have used
shorter demonstrations; but others have extended their treatise to an
infinite length. And some have omitted the method by an impossibility;
but others that by proportion; and others, again, have attempted
preparations against arguments destroying principles. So that many
methods of elementary institution have been invented by particular
writers on this subject. But it is requisite that this treatise should
entirely remove every thing superfluous, because it is an impediment to
science. But every thing should be chosen, which contains and concludes
the thing proposed; for this is most convenient and useful in science.
The greatest care, likewise, should be paid to clearness and brevity;
for the contraries to these, disturb our cogitation. Lastly, it should
vindicate to itself, the universal comprehension of theorems, in their
proper bounds: for such things as divide learning into particular
fragments, produce an incomprehensible knowledge. But in all these
modes, any one may easily find, that the elementary institution of
Euclid excels the institutions of others. For its utility, indeed,
especially confers to the contemplation of primary figures: but the
transition from things more simple to such as are more various, and
also that perception, which from axioms possesses the beginning of
knowledge, produces clearness, and an orderly tradition: and the
migration from first and principal theorems to the objects of enquiry,
effects the universality of demonstration. For whatever he seems to
omit, may either be known by the same ways, as the construction of a
scalene and isosceles triangle[120]: or because they are difficult,
and capable of infinite variety, they are far remote from the election
of elements, such as the doctrine of perturbate proportions, which
Apollonius has copiously handled: or, lastly, because they may be
easily constructed from the things delivered, as from causes, such as
many species of angles and lines. For these, indeed, were omitted by
Euclid, and are largely discoursed of by others, and are known from
simple propositions. And thus much concerning the universal elementary
institution of geometry.




                              CHAP. VIII.

           _Concerning the Order of Geometrical Discourses._


But let us now explain the universal order of the discourses contained
in geometry. Because then, we assert that this science consists from
hypothesis[121], and demonstrates its consequent propositions from
definite principles (for one science only, I mean the first philosophy,
is without supposition, but all the rest assume their principles
from this) it is necessary that he who constructs the geometrical
institution of elements, should separately deliver the principles of
the science, and separately the conclusions which flow from those
principles; and that he should render no reason concerning the nature
or truth of the principles, but should confirm by reasons, the things
consequent to these geometric principles. For no science demonstrates
its own principles, nor discourses concerning them; but procures to
itself a belief of their reality, and they become more evident to the
particular science to which they belong than the things derived from
them as their source. And these, indeed, science knows by themselves;
but their consequents, through the medium of these. For thus, also, the
natural philosopher propagates his reasons from a definite principle,
supposing the existence of motion. Thus too, the physician, and he
who is skilled in any of the other sciences and arts. For if any one
mingles principles, and things flowing from principles into one and
the same, he disturbs the whole order of knowledge, and conglutinates
things which can never mutually agree; since a principle, and its
emanating consequent, are naturally distinct from each other. In the
first place, therefore (as I have said), principles in the geometric
institution are to be distinguished from their consequents, which is
performed by Euclid in each of his books; who, before every treatise,
exhibits the common principles of this science; and afterwards divides
these common principles into hypotheses, petitions, and axioms. For all
these mutually differ; nor is an axiom, petition, and hypothesis the
same, according to the demoniacal Aristotle; but when that which is
assumed in the order of a principle, is indeed known to the learner,
and credible by itself, it is an axiom: such as, that things equal
to the same, are mutually equal to each other. But when any one,
hearing another speak concerning that of which he has no self-evident
knowledge, gives this assent to its assumption, this is hypothesis. For
that a circle is a figure of such a particular kind, we presume (not
according to any common conception) without any preceding doctrine. But
when, again, that which is asserted was neither known, nor admitted by
the learner, yet is assumed, then (says he) we call it petition; as
the assumption that all right angles are equal. But the truth of this
is evinced by those who study to treat of some petition, as of that
which cannot by itself be admitted by any one. And thus, according to
the doctrine of Aristotle[122], are axiom, petition, and supposition
distinguished. But oftentimes, some denominate all these hypotheses,
in the same manner as the Stoics call every simple enunciation an
axiom. So that, according to their opinion, hypotheses also will
be axioms; but, according to the opinion of others, axioms will be
called suppositions. Again, such things as flow from principles are
divided into problems and theorems. The first, indeed, containing the
origin, sections, ablations, or additions of figures, and all the
affections with which they are conversant; but the other exhibiting
the accidents essential to each figure. For, as things effective of
science, participate of contemplation, in the same manner things
contemplative previously assume problems in the place of operations.
But formerly some of the ancient mathematicians thought that all
geometrical propositions should be called theorems, as the followers of
Speusippus and Amphinomus, believing, that to contemplative sciences,
the appellation of theorems is more proper than that of problems;
especially since they discourse concerning eternal and immutable
objects. For origin does not subsist among things eternal: on which
account, problems cannot have any place in these sciences; since they
enunciate origin, and the production of that which formerly had no
existence, as the construction of an equilateral triangle, or the
description of a square on a given right line, or the position of a
right line at a given point. It is better, therefore (say they), to
assert that all propositions are of the speculative kind; but that we
perceive their origin, not by production, but by knowledge, receiving
things eternal as if they were generated; and on this account we ought
to conceive all those theorematically, but not problematically. But
others, on the contrary, think that all should be called problems;
as those mathematicians who have followed Menæchmus. But that the
office of problems is twofold, sometimes, indeed, to procure the thing
sought; but at other times when they have received the determinate
object of enquiry, to see, either what it is, or of what kind it is,
or what affection it possesses, or what its relation is to another.
And, indeed, the assertions of each are right; for the followers of
Speusippus well perceive. Since the problems of geometry are not of
the same kind, with such as are mechanical. For these are sensibles,
and are endued with origin, and mutation of every kind. And, on the
other hand, those who follow Menæchmus do not dissent from truth: since
the inventions of theorems cannot by any means take place without an
approach into matter; I mean intelligible matter. Reasons, therefore,
proceeding into this, and giving form to its formless nature, are not
undeservedly said to be assimilated to generations. For we say that the
motion of our cogitation, and the production of its inherent reasons,
is the origin of the figures situated in the phantasy, and of the
affections with which they are conversant: for there constructions and
sections, positions and applications, additions and ablations, exist:
but every thing resident in cogitation, subsists without origin and
mutation. There are, therefore, both geometrical problems and theorems.
But, because contemplation abounds in geometry, as production in
mechanics, all problems participate of contemplation; but every thing
contemplative is not problematical. For demonstrations are entirely
the work of contemplation; but every thing in geometry posterior to
the principles, is assumed by demonstration. Hence, a theorem is more
common: but all theorems do not require problems; for there are some
which possess from themselves the demonstration of the thing sought.
But others, distinguishing a theorem from a problem, say, that indeed
every problem receives whatever is predicated of its matter, together
with its own opposite: but that every theorem receives, indeed, its
symptom predicate, but not its opposite. But I call the matter of
these, that genus which is the subject of enquiry; as for instance,
a triangle, quadrangle, or a circle: but the symptom predicate, that
which is denominated an essential accident, as equality, or section,
or position, or some other affection of this kind. When, therefore,
any one proposes to inscribe an equilateral triangle in a circle, he
proposes a problem: for it is possible to inscribe one that is not
equilateral. But when any one asserts that the angles at the base
of an isosceles triangle are equal, we must affirm that he proposes
a theorem; for it is not possible that the angles at the base of an
isosceles triangle should be unequal to each other. On which account,
if any one forming problematically, should say that he wishes to
inscribe a right angle in a semi-circle, he must be considered as
ignorant of geometry; since every angle in a semi-circle is necessarily
a right one. Hence, propositions which have an universal symptom,
attending the whole matter, must be called theorems; but those in
which the symptom is not universal, and does not attend its subject,
must be considered as problems. As to bisect a given terminated right
line, or to cut it into equal parts: for it is possible to cut it
into unequal parts. To bisect every rectilinear angle, or divide it
into equal parts; for a division may be given into unequal parts. On
a given right line to describe a quadrangle; for a figure that is not
quadrangular may be described. And, in short, all of this kind belong
to the problematical order. But the followers of Zenodotus, who was
familiar with the doctrine of Oenopides, but the disciple of Andron,
distinguish a theorem from a problem, so far as a theorem enquires what
the symptom is which is predicated of the matter it contains; but a
problem enquires what that is, the existence of which is granted. From
whence the followers of Possidonius define a theorem a proposition,
by which it is enquired whether a thing exists or not; but a problem,
a proposition, in which it is enquired what a thing is, or the manner
of its existence. And they say that we ought to form the contemplating
proposition by enunciating, as that every triangle has two sides
greater than the remaining one, and that the angles at the base of
every isosceles triangle are equal: but we must form the problematical
proposition, as if enquiring whether a triangle is to be constructed
upon this right line. For there is a difference, say they, absolutely
and indefinitely, to enquire whether the thing proposed is from a given
point to erect a right line at right angles to a given line, and to
behold what the perpendicular is. And thus, from what has been said, it
is manifest there is some difference between a problem and a theorem.
But that the elementary institution of Euclid, also, consists partly of
problems, and partly of theorems, will be manifest from considering the
several propositions. Since, in the conclusion of his demonstrations,
he sometimes adds (which was to be shewn) sometimes (which was to be
done) the latter sentence being the mark or symbol of problems, and
the former of theorems. For although, as we have said, demonstration
takes place in problems, yet it is often for the sake of generation;
for we assume demonstration in order to shew, that what was commanded
is accomplished: but sometimes it is worthy by itself, since the nature
of the thing sought after may be brought into the midst. But you will
find Euclid sometimes combining theorems with problems, and using them
alternately, as in the first book; but sometimes abounding with the one
and not the other. For the fourth book is wholly problematical; but the
fifth is entirely composed from theorems. And thus much concerning the
order of geometrical propositions.




                               CHAP. IX.

    _Concerning the Design of the first Book,--its Division,--and a
                  previous Admonition to the Reader._


But, after these considerations, when we have determined the design
of the first book, and have exhibited its division, we shall enter
upon the treatise of the definitions. The design, then, of this book,
is to deliver the principles of the contemplation of right lines. For
though a circle, and its consideration, is more excellent than the
essence and knowledge of right lines, yet the doctrine concerning these
is more adapted to us, who are hastening to transfer our cogitation
from more imperfect and sensible natures, to such as are intelligible.
For, indeed, right lined figures are proper to sensibles, but a
circle to intelligibles. Because that which is simple, uniform, and
definite, is proper to the nature of the things which are: but that
which is various, and which increases indefinitely from the number
of its containing sides, regards the fluctuating essence of sensible
particulars. Hence, in this book, the first and most principal of right
lined figures are delivered; I mean the triangle and parallelogram.
For in these, as under their proper genus, the causes of the elements
are contained: viz. the isosceles and scalene, and those which are
formed from these, the equilateral triangle, and the quadrangle,
from which the four figures of the elements are composed. We shall
find, therefore, as well the origin of the equilateral triangle as
of the quadrangle; of the last, indeed, upon, but of the first from
a given right line. [[123]An equilateral triangle, therefore, is the
proximate cause of the three elements, fire, air, and water: but a
quadrangle is annexed to earth.] And lastly, the design of the first
book is adapted to the whole treatise, and confers to the universal
knowledge of the mundane elements. Besides, it instructs learners in
the science concerning right-lined figures; since it rightly invents,
and accurately collects, the first principles of these.

But this book is divided into three greatest parts, of which the first
declares the origin and properties of triangles, as well according
to angles, as also according to sides. Besides, it makes mutual
comparisons of these, and beholds every one by itself. For receiving
one triangle, sometimes it considers the angles from the sides; but
sometimes the sides from the angles: and this according to equality
and inequality. And supposing two triangles, it discovers the same
property again, by various methods. But the second part combines the
contemplation of parallelograms, describing their properties and
generations. And the third part shews the communication of triangles
and parallelograms, both in symptoms and mutual comparisons. For it
shews that triangles and parallelograms constituted on the same and on
equal bases, are affected with the same passions; and by complication,
when both stand upon one base: and again, after what manner a
parallelogram may be made equal to a triangle; and lastly, concerning
the proportion which in right angled triangles, the square made from
the side subtending, has to the squares containing the right angle. And
such is the division of the first Book.

But, previous to our enquiry into each of these parts, we think it
requisite to admonish the reader, that he must not require of us,
those small assumptions, and cases, and whatever else there may be of
that kind, which has been divulged by our predecessors. For we are
satiated with these, and shall, therefore, but rarely adopt them in our
discourse. But whatever has a more difficult contemplation, and regards
universal philosophy, of this we shall make a particular relation:
imitating the Pythagoreans, with whom this ænigma was common, “a[124]
figure and a step: but not a figure and three oboli,” shewing by this,
that it is requisite to pursue that philosophy which ascends every
theorem by a step, and raises the soul on high; but does not suffer it
to remain among sensibles, to fill up the use attendant on mortals,
and, consulting for this, to neglect the elevation which rises from
hence to an intelligible essence.




                             DEFINITIONS.




                             DEFINITION I.

                  A POINT is that which has no PARTS.


That geometry, according to the transition which takes place from
things more composite to such as are more simple, runs from body, which
is diffused into distance by three dimensions, to a superficies by
which it is bounded; but from superficies to a line, the boundary of
superficies; and from a line to a point destitute of all dimension,
has been often said, and is perfectly manifest. But because these
terms, in many places, on account of their simplicity, appear to be
more excellent than the nature of composites; but in many, as when they
subsist in things which they terminate, they are similar to accidents,
it is necessary to determine in what genera of beings each of these
may be beheld[125]. I say then, that such things as are destitute of
matter, which subsist in separate reasons, and in those forms which
are placed under themselves, are always allotted a subsistence of
more simple essences, superior to the subsistence of such as are more
composite. On this account, both in intellect, and in the ornaments,
as well of the middle kind as among those peculiar to the soul, and
in natures themselves, the terms which proximately vivify bodies,
excel according to essence the things which are terminated; and are
more impartible, more uniform, and more primary than these. For in
immaterial forms, unity is more perfect than multitude; that which is
impartible, than that which is endued with unbounded progression; and
that which terminates, than that which receives bound from another.
But such things as are indigent of matter, and abide in others, and
degenerate from the perfection of their essence, which are scattered
about subjects, and have an unnatural union, are allotted more
composite reasons, prior to such as are more simple. Hence, things
which appear in the phantasy invested with form, and the matter of
the figures which the phantasy contains, and whatever in sensibles is
generated by nature, have, in a preceding order, the reasons of the
things terminated; but the reasons which terminate, in a following
and adventitious rank[126].] For lest that which is distributed into
three dimensions, should be extended into infinite magnitude, either
according to intelligence or sense, it was every way terminated by
superficies. And lest a plane superficies should conceal itself in an
infinite progression, a line approaching opposed its diffusion, and
gave bound to its indefinite extension. And, in like manner, a point
limited the progressions of a line; composite natures deriving their
subsistence from such as are simple. For this also is again manifest,
that in separate forms the reasons of terms subsist in themselves,
but not in those which are terminated; and abiding such as they are
in reality, possess a power of constituting secondary natures. But,
in inseparable forms they give themselves up to things which are
terminated, reside in them, become, as it were, their parts, and
are replenished with baser natures. On which account, that which is
impartible is there endued with a partible essence, and that which is
void of latitude is diffused into breadth. And terms are no longer
able to preserve their simplicity and purity. For since they abide in
another, they necessarily change their own nature into the matter of
their containing subject. Matter, indeed, disturbs the perfection of
these, and causes the reason of a plane to become a profound plane;
but obscuring the one dimension of a line, causes it to be every way
partible; and gives corporeity to the indivisibility of a point, and
separates it together with the natures which it terminates. For all
these reasons falling into matter, the one kind from cogitation into
intelligible matter, but the other from nature into that which is
sensible, are replenished with their containing subjects; and depart
from their own simplicity, into foreign compositions and intervals. But
here a doubt arises how all these, existing in intellect and soul in
an impartible manner, and without any dimension, are distributed into
matter, some indeed, principally, but others on account of its nature?
Shall we say that there is a certain order in immaterial forms, so
that some are allotted the first, some the middle, and others the last
place; and that of forms some are more uniform, but that others are
more multiplied; and that some have their powers collected together,
but others tending into interval; and that some, again, border upon
bound, but that others are proximate to infinity? For though all
participate of these two principles, yet some originate from bound,
but others from infinity, of which they more largely participate.
Hence, a point is entirely impartible, since it subsists according
to bound, yet it occultly contains an infinite power, by which it
produces every interval, and the progression of all intervals, unfolds
its infinite power. But body, and the reason of body, participates
more of an infinite nature; on which account it is among the number of
things terminated by another, and divisible in infinitum, according
to all dimensions. But the mediums between these, according to the
distance of the extremes, are either among the number of things which
have an abundance of bound; or among such as have an affluence of
infinity: on which account they both terminate and are terminated.
For, indeed, so far as they consist from bound, they are able to
terminate others; but so far as they participate of infinity, they are
indigent of termination from others, Hence, since a point is also a
bound, it preserves its proper power in participation: but since it
likewise contains infinity occultly, and is compelled to be every where
present with the natures which it terminates, it resides with them
infinitely. And, because among immaterial forms there was a certain
infinite power capable of producing things distant from each other by
intervals, a point is present with its participants in capacity. For
infinity in intelligibles is the primary cause and prolific power of
the universe; but in material natures it is imperfect, and is alone
all things in dormant capacity. And in short, those forms which, on
account of their simplicity and impartibility, hold a superior rank
among principles, preserve, indeed, (in conformity to their nature,)
their own property in their participations, but become worse than
more composite reasons. For matter is able to participate these more
clearly, and to be prepared for their reception, rather than that of
the most simple causes of beings. On which account, the vestigies of
separate principles descend into matter; but the participations of
those in a second and third order, become more conspicuous. Hence,
matter participates more of the cause of body, than of a plane; and of
this more than the form of a line; and of this still more than that of
a point, which contains all these, and is the boundary of them all.
For the reason of a point presides over this whole series, unites
and contains all partible natures, terminates their progressions,
produces them all by its infinite power, and comprehends them in its
indivisible bound. On which account also, in the images of immaterial
forms, some are the boundaries of others; but a point is the limit
of them all. But that we must not think with the Stoics, that these
boundaries of bodies alone subsist from cogitation; but that there are
certain natures of this kind among beings, which previously contain
the demiurgical reasons of things, we shall be enabled to remember, if
we regard the whole world, the convolutions of its parts, the centres
of those convolutions, and the axes which penetrate through the whole
of these revolving circles. For the centres subsist in energy, since
they contain the spheres, preserve them in their proper state, unite
their intervals, and bind and establish to themselves the powers which
they possess. But the axes themselves being in an immoveable position,
evolve the spheres, give them a circular motion, and a revolution round
their own abiding nature. And the poles of the spheres, which both
terminate the axes, and bind in themselves the other convolutions, do
they not perspicuously evince, that points are endued with demiurgical
and capacious powers, that they are perfective of every thing distant
by intervals, and are the sources of union, and an unceasing motion?
From whence, indeed, Plato[127] also says, that they have an adamantine
subsistence; shewing by this, the immutable, eternal, and stable
power of their essence, ever preserving itself in the same uniform
mode of existence. He adds too, that the whole spindle of the Fates,
is turned about these, and leaps round their coercive union. But other
more recondite and abstruse discourses affirm, that the demiurgus
presides over the world, seated in the poles, and, by his divine love,
converting the universe to himself. But the Pythagoreans thought
that the pole should be called the Seal of Rhea[128]; because the
zoogonic, or vivific goddess, pours through these into the universe,
an inexplicable and efficacious power. And the centre they called the
prison of Jupiter; because, since Jupiter has placed a demiurgical
guard in the bosom of the world, he has firmly established it in the
midst. For, indeed, the centre abiding, the universe possesses its
immoveable ornament, and unceasing convolution: and the gods who
preside over the poles, obtain a power collective of divisible natures,
and unific of such as are multiplied: and those who are allotted the
government of the axes, restrain and eternally evolve their perpetual
convolutions. And, if it is lawful to offer our own opinion on this
subject, the centres and poles of all the spheres are the symbols of
the conciliating gods, shadowing forth their imperceptible and unifying
composition. But the axes express the coherencies of the universal
ornaments; and are endued with a power of comprehending the mundane
integrities and periods, in the same manner as their presiding deities,
of such as are intellectual. But the spheres themselves are images
of the gods, called perfectors of works, copulating the principle
with the end, and excelling all figures in simplicity, similitude,
and perfection. But we have been thus prolix, that we might evince
the power of impartibles, and of the terms which the world contains,
and that so far as they bear an image of primary and most principal
causes, they are allotted the most excellent order in the universe.
For centres and poles are not of the same kind with things which
are terminated; but they subsist in energy, and possess an essence,
and perfect power, which pervades through all partible natures. But
many beholding those terms which imperfectly subsist in terminated
essences, consider them as endued with a slender subsistence; and
some indeed say, that they are alone separated from sensibles by
thought; but others, that they have an essence no where but in our
thoughts. However, since the forms of all these are found both in
the nature of intellect, in the ornaments of soul, in the nature of
things, and in inferior bodies, let us consider how, according to the
order they contain, they subsist in the genera of beings. And indeed,
all of them pre-exist in intellect, but in an impartible and uniform
manner: so that they all subsist according to one form, the reason of
a point, which exists occultly and impartibly. But they all subsist
in soul according to the form of a line: on which account Timæus also
composes the soul from right and circular lines: for every circle is
a line alone[129]. But they all subsist in natures, according to the
reason of a plane; and on this account, Plato commands us to manifest
those natural reasons, which are endued with a power of constituting
bodies by a plane. And the resolution of bodies into planes leads us
to the proximate cause of appearances. Lastly, they all subsist in
bodies, but in a corporeal manner; since all forms have their being
in these, according to the partible nature of bodies. Hence, all of
them appear every where, and each according to its proper order; and
diversity arises from pre-dominating power. The point, indeed, is every
where impartible, and when that which is divisible into parts, excels
according to the diminution of beings, it vindicates to itself, an
illustrious subsistence of partible natures. And sometimes the point is
entirely superior, according to the excellence of cause; but sometimes
it is connected with divisibles, and sometimes it is allotted in them
an adventitious existence; and, as if swallowed up by the partition of
the lowest natures, loses its own proper impartibility. As, therefore,
with respect to the monad, one[130] is the mother of number, but the
other is as matter spread under, and the receptacle of numbers; and
each of them a principle, (yet neither of them is number), but in a
different respect: in the same manner a point also, is partly the
parent and author of magnitudes; but is partly a principle in another
respect, and not according to a generative cause. But is a point, then,
the only impartible? Or may we affirm this of the now in time, and of
unity in numbers? Shall we not say, that to the philosopher, indeed,
discoursing concerning the universality of things, it is proper to
behold every thing, however falling under distribution; but that to
him who is endued with the science of particulars, who produces his
contemplation from certain definite principles, and runs back even to
these, but very little scrutinizes the progressions of beings, it is
requisite to attempt, consider, and treat concerning that impartible
nature alone, which regards his first principles; and to behold that
simplicity which presides over all the particular subjects of his
knowledge? In consequence of this reasoning, therefore, a point alone,
according to the geometric matter, is destitute of partition; but unity
according to that which is arithmetical. And the reason of a point,
however in some other respects it may be imperfect, yet is perfect in
the present science. For, indeed, the physician also says, that the
elements of bodies are fire and water, and things similar to these; and
as far as to these the resolution of bodies proceeds. But the natural
philosopher passes on to more simple elements; and the one defines an
element simple as to sense, but the other simple as to reason; and both
of them properly as to their peculiar science. We must not, therefore,
think that the definition of a point is faulty, nor determine it as
imperfect; for so far as pertains to the geometric matter, and its
principles, it is sufficiently delivered. This alone, indeed, is
wanting to its completion, that the definition does not clearly say,
_that which is impartible with me is a point; and my principle, and
that which I contain as most simple, is nothing else than this_.
And after this manner it is proper to hear the geometrician addressing
us. Euclid, therefore, from a negation of parts, declares to us a
principle, leading to the theory of its whole subject nature. For
negative discourses are proper to principles, as Parmenides teaches
us, who delivers the doctrine concerning the first and last cause,
by negations alone. Since every principle consists of an essence
different from its flowing consequents; and the negations of these
exhibit to us the property of their source. For that it is, indeed, the
cause of these, yet at the same time has nothing in common with these,
becomes perspicuous from a doctrine of this kind. But here a doubt may
arise, how, since the phantasy receives all things invested with forms,
and in a partible manner, the geometrician beholds in it the point
destitute of parts? For it is not because they are reasons existing in
cogitation, but the phantasy receives the resemblances of intellectual
and divine forms according to its own proper nature, exhibiting in its
shadowy bosom the forms of formless natures, and clothing with figure
things entirely free from the affections of figure. To this ambiguity
we must say, that the species of imaginative motion is neither alone
partible, nor impartible; but that it proceeds from the impartible to
the partible, and from the formless nature to that which is expressed
by form. For if it was partible alone, it could not preserve in itself
many impressions of forms, since the subsequent would obscure the
pre-existent figures: for no body can contain at once, and according
to the same situation, a multitude of figures; but the former will
be blotted out by the succession of the latter. But if it was alone
impartible, it would not be inferior to cogitation, and to soul, which
surveys all things in an impartible manner. Hence, it is necessary that
it should indeed begin from an impartible according to its motion, and
from thence draw forth the folded and scattered form of every thing
falling under cogitation, and penetrating to its shadowy receptacle:
but, that it should at length end in form, figure, and interval. And
if it be allotted a nature of this kind, it will, after a certain
manner, contain an impartible essence: and a point, according to this,
must be said to have its principal subsistence: for the form of a
line is contracted in the phantasy according to this. Hence, because
it possesses a twofold power, impartible and partible, it will indeed
contain a point in an impartible, and intervals in a partible manner.
But as the Pythagoreans define a point to be unity having position, let
us consider what they mean. That numbers, indeed, are more immaterial
and more pure than magnitudes, and that the principle of numbers is
more simple than the principle of magnitudes, is manifest to every one:
but when they say that a point is unity endued with position, they
appear to me to evince that unity and number subsist in opinion: I mean
monadic number[131]. On which account, every number, as the pentad and
the heptad, is one in every soul, and not many; and they are destitute
of figure and adventitious form. But a point openly presents itself in
the phantasy, subsists, as it were, in place, and is material according
to intelligible matter. Unity, therefore, has no position, so far as it
is immaterial, and free from all interval and place: but a point has
position, so far as it appears seated in the bosom of the phantasy,
and has a material subsistence. But unity is still more simple than a
point, on account of the community of principles. Since a point exceeds
unity according to position; but appositions in incorporeals produce
diminutions of those natures, by which the appositions are received.




                            DEFINITION II.

                  A Line is a Length without Breadth.


A Line obtains the second place in the Definitions, as it is by far
the first and most simple interval, which the geometrician calls a
length, adding also without breadth; since a line, in respect of a
superficies, ranks as a principle. For he defines a point, as it is the
principle of all magnitudes, by negation alone; but a line, as well by
affirmation as by negation. Hence it is a length, and by this exceeds
the impartibility of a point; but it is without breadth, because it
is separated from other dimensions. For, indeed, every thing which
is void of breadth, is also destitute of bulk, but the contrary is
not true, that every thing void of bulk is also destitute of breadth.
Since, therefore, he has removed breadth from a line, he has also
removed at the same time bulk. On which account he does not add, that
a line also has no thickness, because this property is consequent
to the notion of being without breadth. But it is defined by others
in various ways: for some call it the flux of a point, but others a
magnitude contained by one interval. And this definition, indeed; is
perfect, and sufficiently explains the essence of a line; but that
which calls it the flux of a point, appears to manifest its nature from
its producing cause; and does not express every line, but alone that
which is immaterial. For this is produced by a point, which though
impartible itself, is the cause of being to partible natures. But the
flux of a point, shews its progression and prolific power, approaching
to every interval, receiving no detriment, perpetually abiding the
same, and affording essence to all partible magnitudes. However, these
observations are known, and manifest to every one. But we shall recall
into our memory, discourses more Pythagorical, which determine a point
as analogous to unity, a line to the duad, a superficies to the triad,
and body to the tetrad. [[132]Yet when we compare those which receive
interval together, we shall find a line monadic; but a superficies
dyadic, and a solid body triadic.] From whence also, Aristotle[133]
says; that body is perfected by the ternary number. And, indeed, this
is not wonderful, that a point, on account of its impartibility,
should be assimilated to unity; but that things subsequent to a point,
should subsist according to numbers proceeding from unity, and should
preserve the same proportion to a point, as numbers to unity; and that
every one should participate of its proximate superior, and have the
same proportion to its kindred, and following degree, as the superior
to this, which is the immediate consequent. [[134]For example, that
a line has the order of the duad with respect to the point, but of
unity to a superficies; and that this last has the relation of a triad
to the point, but of the duad to a solid.] And on this account, body
is tetradic, with respect to a point, but triadic as to a line. Each
order, therefore, has its proportion; but the order of the Pythagoreans
is the more principal, which receives its commencement from an exalted
source, and follows the nature of beings. For a point is indeed
twofold; since it either subsists by itself, or in a line; in which
last respect also, since as a boundary it is alone and one, neither
having a whole nor parts, it imitates the supreme nature of beings.
On which account too, it was placed in a correspondent proportion to
unity. [135]For as the oracle says, _Unity is there first, where the
paternal unity abides_. But a line is the first endued with parts
and a whole, and it is monadic because it is distant by one interval
only; and dyadic on account of its progression: for if it be infinite,
it participates of the indefinite duad; but if finite, it requires two
terms, from whence and to what place; since, on account of these it
imitates totality, and is allotted an order among totals. For unity,
according to the oracle, is extended[136], and generates two; and
this produces a progression into longitude, together with that which
is distant extendedly, and with one interval, and the matter of the
duad. But superficies, since it is both a triad and duad, as also the
receptacle of the primary figures, and that which receives the first
form and species, is in a certain respect similar to the triadic
nature, which first terminates beings; and to the duad, by which they
are divided and dispersed. But a solid, since it has a triple distance,
and is distinguished by the tetrad, which is endued with a power of
comprehending all reasons, is reduced to that order in which the
distinction of corporeal ornaments appears; as also the division of the
universe into three parts, together with the tetradic property, which
is generative and female. And these observations, indeed, might be more
largely discussed, but for the present, must be omitted. Again, the
discourse of the Pythagoreans, not undeservedly, calls a line, which is
the second in order, and is constituted according to the first motion
from an impartible nature, dyadic. And that a point is posterior to
unity, a line to the duad, and a superficies to the triad, Parmenides
himself shews, by first of all taking away multitude from one by
negation, and afterwards the whole. Because, if multitude is before
that which is a whole, number also will be prior to that which is
continuous, and the duad to the line, and unity to the point: since the
epithet _not many_, belongs to unity which generates multitude,
but to the point, the term _not a whole_, is proper, because it
produces a whole; for this is said to have no part. And these things
are affirmed of a line, while we more accurately contemplate its
nature. But we should also admit the followers of Apollonius, who say,
that we obtain a notion of a line, when we are ordered to measure the
lengths alone, either of ways or walls; for we do not then subjoin
either breadth or bulk, but only make one distance the object of our
consideration. In the same manner we perceive superficies, when we
measure fields; and a solid, when we take the dimensions of wells. For
then, collecting all the distances together, we say, that the space of
the well is so much, according to length, breadth, and depth. But a
line may become the object of our sensation, if we behold the divisions
of lucid places from those which are dark, and survey the moon when
dichotomized: for this medium has no distance with respect to latitude;
but is endued with longitude, which is extended together with the light
and shadow.




                            DEFINITION III.

               But the Extremities of a Line are Points.


Every composite receives its bound from that which is simple, and every
thing partible from that which is impartible; and the images of these
openly present themselves in mathematical principles. For when it is
said that a line is terminated by points, it seems manifestly to make
it of itself infinite, because, on account of its proper progression,
it has no extremity. As, therefore, the duad is terminated by unity,
and reduces its own intolerable boldness under bound, when it is
restrained in its comprehensive embrace: so a line also is limited by
the points which it contains. For, since it is similar to the duad,
it participates of a point having the relation of unity, according to
the nature of the duad. Indeed, in imaginative, as well as in sensible
forms, the points themselves terminate the lines in which they reside.
But in immaterial forms, the reason of the impartible point pre-exists
separate and apart; but when proceeding from thence by far the first of
all, by determining itself with interval, moving itself, and flowing
in infinite progression, and imitating the indefinite duad, it is
restrained indeed, by its proper principle, is united by its power,
and on every side seized by its coercive bound. Hence it is, at the
same time, both infinite and finite: infinite, indeed, according to its
progression; but finite according to its participation of a terminating
cause. So that, when it approaches to this cause, it is detained in its
comprehension, and is terminated according to its union. Hence too, in
the images of incorporeal forms, a point is said to terminate a line,
by occupying its beginning and end. Bound, therefore, in immaterials,
is separated from that which is bounded: but here it is twofold; for
it subsists in that which is terminated. And this affords a wonderful
symptom, that forms; indeed, abiding in themselves, precede their
participants according to cause; but when giving themselves up to their
subordinate natures, subsist according to their diversified properties:
since they are multiplied and distributed together with these, and
receive the division of their subjects. Besides, this also must be
previously received concerning a line, that our geometrician uses it in
a threefold acceptation. As terminated on both sides, and finite; as
in the problem[137] which says, Upon a given terminated right line to
construct an equilateral triangle. And as partly infinite and partly
finite; as in the problem which commands us from three right lines,
which are equal to three given right lines, to construct a triangle;
for in the construction of the problem, he says, Let there be placed a
certain right line, on one part finite, but on the other part infinite.
And again, a line is received by Euclid as on both sides infinite; as
in the problem which says, Upon a given infinite right line, from a
given point, which is not in that line, to let fall a perpendicular.
But, besides this, the following doubts, since they are worthy of
solution, must not be omitted. How are points called the extremities
of a line? and of what line, since they can neither be the bounds of
one that is infinite, nor of every finite? For there is a certain line,
which is both finite, and has not points for its extremities. And such
is a circular line, which returns into itself, and is not bounded by
points, like a right line. And such also is the ellipsis, or line like
a shield. Is it therefore requisite to behold a line, considered as a
line? for we must receive a certain circumference, which is terminated
by points, and a part of the elliptic line; having, in like manner,
its extremities bounded by points. But every circular and elliptic
line, assumes to itself another certain property, by which it is not
line alone, but is also endued with a power of perfecting figure[138].
Lines, themselves, therefore, have their extremities terminated by
points; but those which are effective of such like figures, return into
themselves. And, indeed, if you conceive them to be described, you
will also find how they are bounded by points; but if you receive them
already described, and connect the end with the beginning, you can no
longer behold their extremes.




                            DEFINITION IV.

      A Right Line, is that which is equally situated between its
                          _bounding_ Points.


Plato establishing two most simple and principal species of lines,
the right and the circular, composes all the rest from the mixture of
these; I mean such as are called curve lines, some of which are formed
from planes; but others subsist about solids; and whatever species
of curve lines are produced by the sections of solids. And it seems,
indeed, that a point (if it be lawful so to speak) bears an image of
the one itself, according to Plato: for unity has no part, as he also
shews in the Parmenides. But, because after unity itself there are
three hypostases, or substances, bound, infinite, and that which is
mixed from these, the species of lines, angles, and figures, which
subsist in the nature of things originate from thence. And, indeed, a
circumference and a circular angle, and a circle among plane figures,
and a sphere among solids, are analogous to _bound_. But a right
line corresponds to _infinity_, according to all these; for it
properly belongs to all, if it is beheld as existing in each. But that
which is mixed in all these, is analogous to the mixt which subsists
among intelligibles. For lines are mixed, as those which are called
spirals. And angles, as the semi-circular and cornicular[139]. And
plane figures, as segments and apsides; but solids, as cones and
cylinders, and others of that kind. Bound, therefore, infinite, and
that which is mixed, are participated by all these. But Aristotle[140]
likewise assents to Plato; for every species of lines, says he, is
either right or circular, or mixed from these two. From whence also
there are three motions, one according to a right line; the other
circular; and the third mixed. But some oppose this division, and
say that there are not two simple lines alone, but that there is a
certain third line given, i. e. a helix or spiral, which is described
about a cylinder[141], when, whilst a right line is moved round the
superficies of the cylinder, a point in the line is carried along with
an equal celerity. For by this means, a helix, or circumvolute line,
is produced, which adapts all the parts of itself to all, according to
a similitude of parts, as Apollonius shews in his book concerning the
Cochlea; which passion, among all spirals, agrees to this alone. For
the parts of a plane helix are dissimilar among themselves; as also of
those which are described about a cone and sphere. But the cylindric
spiral alone, consists of similar parts in the same manner as a right
and circular line. Are there, then, three simple lines, and not two
only? To which doubt we reply, that a helix of this kind is, indeed, of
similar parts, as Apollonius teaches, but is by no means simple; since
among natural productions, gold and silver are composed of similar
parts, but are not simple bodies. But the generation of the cylindric
helix evinces that its mixture is from things simple; for it originates
while a right line is circularly moved round the axis of the cylinder,
a point at the same time flowing along in the right line. Two simple
motions, therefore, compose its nature; and, on this account, it is
among the number of mixt lines, and not among such as are simple: for
that which is composed from dissimilars is not simple, but mixt. Hence,
Geminus, with great propriety, when he admits that some simple lines
may be produced from many motions, does not grant that every such line
is mixt; but that alone, which arises from dissimilar motions. For if
you conceive a square, and two motions which are performed with an
equal celerity, one according to the length, but the other according
to the breadth, a right line or the diameter will be produced; but
the right line will not, on this account, be mixed: for no other line
precedes it, formed by a simple motion, as we asserted of the cylindric
helix. Nor yet, if you suppose a right line, moving in a right angle,
and by a bisection to describe a circle[142], is the circular line, on
this account, produced with mixture: for the extremities of that which
is moved after this manner, since they are equally moved, will describe
a right line; and the bisection, since it is unequally devolved, will
delineate a circle; but the other points will describe an ellipsis. On
which account, the generation of a circular line is the consequence
of that inequality of lation arising from the bisection; because a
right line was supposed to be moved in a right angle, but not in a
natural manner. And thus much concerning the generation of lines. But
it seems, that of the two simple lines, the right and the circular,
the right line is the more simple; for in this, dissimilitude cannot
be conceived, even in opinion. But in the circular line, the concave
and the convex, indicate dissimilitude. And a right line, indeed, does
not infer a circumference according to thought; but a circumference
brings with it a right line, though not according to its generation,
yet with respect to its centre. But what if it should be said that a
circumference requires a right line to its construction! For if either
extreme of a right line remains fixt, but the other is moved, it will
doubtless describe a circle, whose centre will be the abiding extreme
of the right line. Shall we say that the generator of the circle is
the point which is carried about the abiding point, but not the right
line itself? For the line only determines the distance, but the point
composes the circular line, while it is moved in a circular manner:
but of this enough. Again, a circumference appears to be proximate to
bound, and to have the same proportion to other lines, as bound to the
universality of things. For it is finite, and is alone among simple
lines perfective of figure. But a right line is proximate to infinity;
for its capacity of infinite extension never fails: and as all the
rest are produced from bound and infinite, in the same manner from the
circular and right line, every mixt genus of lines is composed, as
well of planes as of those which consist in solid bodies. And on this
account, the soul also[143] previously assumed into herself the right
and circular according to her essence, that she might moderate all
the co-ordination of infinite, and all the nature of bound, which the
world contains. By a right line, indeed, constituting the progression
of these principles into the universe; but by a circular line, their
return to their original source: and by the one, producing all things
into multitude; but by the other, collecting them into one. And not
only the soul, but he also who produced the soul, and endued her
with these powers, contains in himself both these primary causes.
For when he previously assumed the beginning, middle, and end of all
things, he terminated right lines (says Plato[144]), by a circular
progression according to nature. And proceeding to all things by
provident energies, and returning to himself, he established himself,
says Timæus, after his own peculiar manner. But a right line is the
mark or symbol of a providence, indeclinable, incapable of perversion,
immaculate, never-failing, omnipotent, and present to all beings, and
to every part of the universe. But a circumference, and that which
environs, is the symbol of an energy retiring into union with itself,
and which rules over all things according to one intellectual bound.
When, therefore, the demiurgus of the universe had established in
himself these two principles, the right and the circular line, and had
given them dominion, he produced from himself two unities; the one,
indeed, energizing according to the circular line, and being effective
of intellectual essences; but the other according to the right line,
and affording an origin to sensible natures. But because the soul is
allotted a middle situation between intellectuals and sensibles, so
far, indeed, as she adheres to an intellectual nature, she energizes
according to the circle; but so far as she presides over sensibles, she
provides for their welfare according to the right line: and thus much
concerning the similitude of these forms to the universality of things.
But Euclid, indeed, has properly delivered the present definition of
a line; by which he shews that a right line alone occupies a space
equal to that which is situated between its points: for as much as is
the distance of one point from another, so great is the magnitude of
the lines terminated by the points. And this is the meaning of being
equally situated between its extremes. For if you take two points in a
circumference, or in any other certain line, the space of line which
is included between these, exceeds their distance from each other;
and every line, besides a right one, appears to suffer this property.
Hence, according to a common conception, the vulgar also say, that he
who walks by a right line, performs only a necessary journey: but that
they necessarily wander much, who do not proceed in a right line. But
Plato thus defines it; a right line is that whose middle parts darken
its extremes. For this passion necessarily attends things which have
a direct position; but it is not necessary that things situated in the
circumference of a circle, or in another interval, should be endued
with this property. Hence, the astrologers also say, that the sun then
suffers an eclipse when that luminary, the moon, and our eye are in one
right line; for it is then darkened through the middle position of the
moon between us and its orb. And perhaps, the passion of a right line
will evince, that in the things which are, according to processions
emanating from causes, the mediums are endued with a power of dividing
the distance of the extremes, and their mutual communication with each
other. As also, according to regressions, such things as are distant
from the extremes, are converted by mediums to their primary causes.
But Archimedes defines a right line the least of things having the
same bounds. For since, according to Euclid, a right line is equally
situated between its points, it is on this account, the least of things
having the same bounds: for if a less line could be given, it would
not lie equally between its extremes: but all the other definitions of
a right line, fall into the same conclusions; as for instance, that
it is constituted in its extremities, and that one part of it is not
in its subject plane, but another, in one more sublime: and that all
its parts similarly agree to all: and that its extremes abiding, it
also abides. Lastly, that it does not perfect figure, with one line
similar in species to itself: for all these definitions express the
property of a right line, which it possesses from the simplicity of its
essence, and from its having one progression the shortest of all from
one extremity to another. And thus much concerning the definitions of a
right line. But again, Geminus divides a line first into an incomposite
and composite; calling a composite, that which is refracted, and forms
an angle; but all the rest of them, he denominates incomposites.
Afterwards, he divides a composite line into that which produces
figure, and that which may be infinitely extended. And he calls that
which produces figure, a circular line, and the line of a shield[145],
and that which is similar to an ivy leaf[146]; but that which is not
effective of figure, the section of a rectangular and obtuse angular
cone, the line similar to a shell[147], the right line, and all of
that kind. And again, after another manner, of the incomposite line,
one sort is simple, but the other mixt. And of the simple, one produces
figure, as the circular; but the other is indefinite, as the right
line. But of the mixt, one subsists in planes, but the other in solids.
And of that which is in planes, one coincides in itself, as the figure
of the ivy leaf, which is called the cissoid; but the other may be
produced in infinitum, as the helix. But of that which is in solids,
one may be considered in the sections of solids; but the other as
consisting about the solids themselves. For the helix, indeed, which
is described about a sphere or a cone, consists about solids; but
conic, or spirical sections are generated from a particular section of
solids. But, with respect to these sections, the conic were invented by
Mænechmus, which also Eratosthenes relating, says,

    “Nor in a cone Mænechmian ternaries divide.”

But the spiric by Perseus, who composed an epigram on their invention,
to this purpose, “When Perseus had invented three spiral lines in five
sections, he sacrificed to the gods on the occasion.” And the three
sections of a cone, are the parabola, hyperbola, and ellipsis: but of
spiral sections, one kind is twisted and involved, like the fetlock of
a horse; but another is dilated in the middle, and deficient in each
extremity: and another which is oblong, has less space in the middle,
but is dilated on each side. But the multitude of the other mixt lines
is infinite. For there is an innumerable multitude of solid figures,
from which there are constituted multiform sections. For a right line,
while it is circularly moved[148], does not make a certain determinate
superficies, nor yet conical, nor conchoidal lines, nor circumferences
themselves. Hence, if these solids are multifariously cut, they will
exhibit various species of lines. Lastly, of those lines which consist
about solids, some are of similar parts, as the helixes about a
cylinder; but others of dissimilar parts, as all the rest. From these
divisions, therefore, we may collect, that there are only three lines
of similar parts, the right, the circular, and the cylindric helix. The
two simple ones, indeed, existing in a plane, but the one mixt, about
a solid. And this Geminus evidently demonstrates, when he shews, that
if two right lines are extended from one point, to a line of similar
parts, so as to make equal angles upon that line, they shall be equal
to each other. And the demonstrations of this may be received by the
studious, from his volumes; since in these he delivers the origin of
spiral, conchoidal, and cissoidal lines. But we have barely related
the names and divisions of these lines, for the purpose of exciting
the ingenious to their investigation; as we think, that an accurate
enquiry after the method of detecting the properties of each, would be
superfluous in the present undertaking: since the geometrician only
unfolds to us in this work, simple and primary lines, i.e. the right
line, in the present definition; but a circular line, in the tradition
of a circle. For he then says, that the line terminating the circle,
is the circumference. But he makes no mention of mixt lines, though
he was well acquainted with mixt angles, I mean, the semi-circular
and cornicular: as also with plane mixt figures, i.e. segments and
sectors; and with solids, viz. cones and cylinders. Of each of the
rest, therefore, he delivers three species; but of lines only two, i.
e. the right and circular: for he thought it requisite in discourses
concerning things simple, to assume simple species; and all the rest
are more composite than lines. Hence, in imitation of the geometrician,
we also shall terminate their explanation with simple lines.




                             DEFINITION V.

       A SUPERFICIES is that which has only Length and Breadth.


After a point and a line, a superficies is placed, which is distant
by a twofold interval, length and breadth. But this also remaining
destitute of thickness or bulk, possesses a nature more simple than
body, which is distant by a triple dimension. On which account the
geometrician adds to the two intervals the particle _only_,
because the third interval does not exist in superficies. And this is
equivalent to a negation of bulk, as here also he shews the excellency
of superficies compared to a solid with respect to simplicity, by
negation, or by an addition equivalent to negation: but the diminution
which it possesses, if compared with the preceding terms, by the
affirmations themselves. But others define a superficies to be the
boundary of body, which is almost affirming the same as the definition
of Euclid; since that which terminates is exceeded in one dimension,
by that which is terminated. And others, a magnitude different by
two intervals. Lastly, others declaring the same affection, form its
assignation in a somewhat different manner. But they say we have a
knowledge of superficies when we measure fields, and distinguish their
extremities according to length and breadth; but that we receive a
certain sensation of it, when we behold shadows. For as they are
without bulk, because they cannot penetrate into the interior part of
the earth, they have only length and breadth. But the Pythagoreans
say, that it is assimilated to the triad; because the ternary is by
far the first cause to all the figures; which a superficies contains.
For a circle, which is the principle of orbicular figures, occultly
possesses the ternary, by its centre, interval, and circumference. But
a triangle, which ranks as the first among all right-lined figures, on
every side evinces that it is enclosed by the triad, and receives its
form from its perfect nature.




                            DEFINITION VI.

              The Extremities of a SUPERFICIES are Lines.


From these also, as images, we may understand, that things more simple
procure bound and an end to every one of their proximate natures: for
soul perfects and determines the operations of nature; and nature
the motion of bodies. And prior to these, intellect measures the
convolutions of soul; and unity the life of intellect; for that is
the measure of all. Just as in these also, a solid is terminated by a
superficies; but a superficies by a line; and a line by a point; for
that is the boundary of them all. Hence, the line existing uniformly
in immaterial forms and impartible reasons, terminates and restrains
the various motion of a superficies in its progression, and proximately
unites its infinity. But in the images of these, when that which
bounds supervenes that which is bounded, it causes, by this means,
its limitation and bound. But if it should be enquired how lines are
the extremities of every superficies, since they are not the extremes
of every finite figure; for the superficies of a sphere is terminated
indeed, yet not by lines, but by itself? In answer to this, we must
say, that by receiving a superficies so far as it is distant by a
twofold interval, we shall find it terminated by lines according to
length and breadth. But if we behold a spherical superficies, we
must receive it as that which is endued with figure; which possesses
another quality, and conjoins the end with the beginning; and loses
its two extremities in the comprehensive embraces of one: and this one
extremity subsists in capacity only, and not in energy.




                            DEFINITION VII.

     A Plane SUPERFICIES is that which is equally situated between
                          its bounding Lines.


It was not agreeable to the ancient philosophers to establish a plane
species of superficies; but they considered superficies in general,
as the representative of magnitude, which is distant by a twofold
interval. For thus the divine Plato[149] says, that geometry is
contemplative of planes, opposing it in division to stereometry, as
if a plane and a superficies were the same. And this was likewise the
opinion of the demoniacal Aristotle[150]. But Euclid and his followers
consider superficies as a genus, but a plane as its species, in the
same manner as rectitude of a line. And on this account he defines a
plane separate from a superficies, after the similitude of a right
line. For he defines this last as equal to the space, placed between
its points. And in like manner, he says, that two right lines being
given, a plane superficies occupies a place equal to the space situated
between those two lines. For this is equally situated between its
lines; and others also explaining the same boundary, assert that it
is constituted in its extremities. But others define it as that to
all the parts of which a right line may be adapted[151]. But perhaps
others will say, that it is the shortest of superficies, having the
same boundaries; and that its middle parts darken its extremities; and
that all the definitions of a right line may be transferred into a
plane superficies, by only changing the genus: since a right, circular,
and mixt line, commencing from lines, arrive even at solids, as we
have asserted above; for they are proportionally, both in superficies
and solids. Hence also, Parmenides says, that every figure is either
right, or circular, or mixt. But if you wish to consider the right in
superficies, take a plane, to which a right line agrees in various
ways; but if a circular receive a spherical superficies; and if a
mixt, a conic or cylindric, or some one of that genus. But it is
requisite (says Geminus) since a line, and also a superficies is
called mixt, to know the measure of mixture, because it is various.
For mixture in lines, is neither by composition, nor by temperament
only: since, indeed, a helix is mixed, yet one part of it is not
straight, and another part circular, like those things which are
mixed by composition: nor if a helix is cut after any manner, does
it exhibit an image of things simple, such as those which are mixed
through temperament; but in these the extremes are, at the same time,
corrupted and confused. Hence, Theodorus the mathematician, does not
rightly perceive, in thinking that this mixture is in lines. But
mixture in superficies, is neither by composition, nor by confusion;
but subsists rather by a certain temperament. For conceiving a circle
in a subject plane, and a point on high, and producing a right line
from the point to the circumference of the circle, the revolution of
this line will produce a conical superficies which is mixt. And we
again resolve it into its simple elements, by a parallel section: for
by drawing a section between the vertex and the base, which shall
cut the plane of the generative right line, we effect a circular
line. But the idea of lines, shews that the mode of mixture is not by
temperament; for neither does it send us back to the simple nature of
elements: on the contrary, when superficies are cut, they immediately
exhibit to us their producing lines. The mode of mixture, therefore, is
not the same in lines and superficies. But as among lines there were
some simple, that is, the right and circular, of which the vulgar also
possess an anticipated knowledge without any previous instruction;
but the species of mixt lines require a more artificial apprehension:
so among superficies, we possess an innate notion of those which are
especially elementary, the plane and spherical; but science and its
reason investigates the variety of those which are composed through
mixture. But this is an admirable property of superficies, that their
mixture in generation is oftentimes produced from a circular line;
and this also happens to a spiral superficies. For this is understood
by the revolution of a circle remaining erect, and turning itself
about the same point which is not its centre. And on this account, a
spiral also is threefold; for its centre is either in a circumference,
or within, or external to a circumference. If the centre is in
the circumference, a continued spiral is produced: if within the
circumference, an intangled one; if without, a divided one. And there
are three spiral sections corresponding to these three differences.
But every spiral line is mixt, although the motion from which it is
produced is one and circular. And mixt superficies are produced as
well from simple lines, (as we have said,) while they are moved with a
motion of this kind, as from mixt lines. Since, therefore, there are
three conic lines, they produce four mixt superficies, which they call
conoids. For a rectangular conoid, is produced from the revolution of
the parabola about its axis: but that which is formed by the ellipsis,
is called a spheroid; and is the revolution is made about the greater
axis, it is an oblong; but if about the lesser a broad spheroid.
Lastly, an obtuse-angled conoid is generated from the revolution of the
hyperbola. But it is requisite to know, that sometimes we arrive at the
knowledge of superficies from lines, and sometimes the contrary; for
from conical and spiral superficies, we apprehend conical and spiral
lines. Besides, this also must be previously received concerning the
difference of lines and superficies, that there are three lines of
similar parts (as we have already observed), but only two superficies,
the plane and the spherical. For this is not true of the cylindric,
since all parts of the cylindric superficies cannot agree to all. And
thus much concerning the differences of superficies, one of which
the geometrician having chosen (I mean the plane), this also he has
defined; and in this, as a subject, he contemplates figures, and their
attendant passions: for his discourse is more copious in this than in
other superficies: since, indeed, we may understand right lines, and
circles, and helixes in a plane; also the sections of circles and right
lines, contacts, and applications, and the constructions of angles
of every kind. But in other superficies, all these cannot be beheld.
For how in one that is spherical, can we apprehend a right line, or a
right-lined angle? How, lastly, in a conic or cylindric superficies,
can we behold sections of circles or right lines? Not undeservedly,
therefore, does he both define this superficies, and discuss his
geometrical concerns, by exhibiting every thing in this as in a
subject; for from hence he calls the present treatise plane. And, after
this manner, it is requisite to understand that which is plane, as
projected and constituted before the eyes: but cogitation as describing
all things in this, the phantasy corresponding to a plane mirror, and
the reasons resident in cogitation as dropping their images[152] into
its shadowy receptacle.




                           DEFINITION VIII.

   [153]A PLANE ANGLE, is the inclination of two Lines to each
   other in a Plane, which meet together, but are not in the same
   direction.

Some of the ancient philosophers, placing an angle in the predicament
of relation, have said, that it is the mutual inclination of lines or
planes to each other. But others, including this in quality, as well
as rectitude and obliquity, say, that it is a certain passion of a
superficies or a solid. And others, referring it to quantity, confess
that it is superficies or a solid. For the angle which subsists in
superficies is divided by a line; but that which is in solids, by a
superficies. But (say they) that which is divided by these, is no other
than magnitude, and this is not linear, since a line is divided by a
point; and therefore it follows that it must be either a superficies
or a solid. But if it is magnitude, and all finite magnitudes of the
same kind have a mutual proportion; all angles of the same kind, i.
e. which subsist in superficies, will have a mutual proportion. And
hence, the cornicular will be proportionable to a right-lined angle.
But things which have a mutual proportion, may, by multiplication,
exceed each other; and therefore it may be possible for the cornicular
to exceed a right-lined angle, which, it is well known, is impossible,
since it is shewn to be less than every right-lined angle. But if
it is quality alone, like heat and cold, how is it divisible into
equal parts? For equality, inequality, and divisibility, are not less
resident in angles than in magnitudes; but they are, in like manner,
essential. But if the things in which these are essentially inherent,
are quantities, and not qualities, it is manifest that angles also are
not qualities. Since the more and the less are the proper passions
of quality[154], but not equal and unequal. On this hypothesis,
therefore, angles ought not to be called unequal, and this greater,
but the other less; but they ought to be denominated dissimilars, and
one more an angle, but the other less. But that these appellations
are foreign from the essence of mathematical concerns, is obvious to
every one: for every angle receives the same definition, nor is this
more an angle, but that less. Thirdly, if an angle is inclination, and
belongs to the category of relation, it must follow, that from the
existence of one inclination, there will also be one angle, and not
more than one. For if it is nothing else than the relation of lines
or planes, how is it possible there can be one relation of lines or
planes, but many angles? If, therefore, we conceive a cone cut by a
triangle from the vertex to the base, we shall behold one inclination
of the triangular lines in the semicone to the vertex; but two distinct
angles: one of which is plane, I mean that of the triangle; but the
other subsists in the mixt superficies of the cone, and both are
comprehended by the two triangular lines. The relation, therefore,
of these, do not make the angle. Again, if is necessary to call an
angle either quality or quantity, or relation; for figures, indeed,
are qualities, but their mutual proportions belong to relation. It is
necessary, therefore, that an angle should be reduced under one of
these three genera. Such doubts, then, arising concerning an angle,
and Euclid calling it inclination, but Apollonius the collection of
a superficies, or a solid in one point, under a refracted line or
superficies (for he seems to define every angle universally), we
shall affirm, agreeable to the sentiments of our preceptor Syrianus,
that an angle is of itself none of the aforesaid; but is constituted
from the concurrence of them all. And that, on this account, a doubt
arises among those who regard one category alone. But this is not
peculiar to an angle, but is likewise the property of a triangle. For
this, too, participates of quantity, and is called equal and unequal;
because it has to quantity the proportion of matter. But quality also,
is present with this, in consequence of its figure (since triangles
are called as well similar as equal); but it possesses this from one
category, and that from another. Hence, an angle is perfectly indigent
of quantity, the subject of magnitude. But it is also indigent of
quality, by which it possesses, as it were, its proper form and figure,
Lastly, it is indigent of the relation of lines terminating, or of
superficies comprehending its form. So that an angle consists from
all these, yet is not any one of them in particular. And it is indeed
divisible, and capable of receiving equality and inequality, according
to the quantity which it contains. But it is not compelled to admit
the proportion of magnitudes of the same kind, since it has also a
peculiar quantity, by which angles are also incapable of a comparison
with each other. Nor can one inclination perfect one angle: since the
quantity also, which is placed between the inclined lines, completes
its essence. If then we regard these distinctions, we shall dissolve
all absurdities, and discover that the property of an angle is not the
collection of a superficies or solid, according to Apollonius (since
these also complete its essence,) but that it is nothing else than
a superficies itself, collected into one point, and comprehended by
inclined lines, or by one line inclined to itself: and that a solid
angle is the collection of superficies mutually inclined to each
other. Hence, we shall find that a formed quantum, constituted in a
certain relation, supplies its perfect definition. And thus much we
have thought requisite to assert concerning the substance of angles,
previously contemplating the common essence of every triangle, before
we divide it into species. But since there are three opinions of an
angle, Eudemus the Peripatetic, who composed a book concerning an
angle, affirms that it is quality. For, considering the origin of an
angle, he says that it is nothing else than the fraction of lines:
because, if rectitude is quality, fraction also will be quality. And
hence, since its generation is in quality, an angle will be entirely
quality. But Euclid, and those who call it inclination, place it in the
category of relation. But they call it quantity, who say that it is the
first interval under a point, that is immediately subsisting after a
point. In the number of which is Plutarch, who constrains Apollonius
also into the same opinion. For it is requisite (says he) there should
be some first interval, under the inclination of containing lines
or superficies. But since the interval, which is under a point, is
continuous, it is not possible that a first interval can be assumed;
since every interval is divisible in infinitum. Besides, if we any
how distinguish a first interval, and through it draw a right line, a
triangle is produced, and not one angle. But Carpus Antiochenus says,
that an angle is quantity, and is the distance of its comprehending
lines, or superficies; and that this is distant by one interval, and
yet an angle is not on that account a line: since it is not true that
every thing which is distant by only one interval, is a line. But this
surely is the most absurd of all, that there should be any magnitude
except a line, which is distant only by one interval. And thus much
concerning the nature of an angle. But with respect to the division
of angles, some consist in superficies, but others in solids. And of
those which are in superficies, some are in simple ones, but others in
such as are mixt. For an angle may be produced in a cylindric, conic,
spherical, and plane superficies. But of those which consist in simple
superficies, some are constituted in the spherical; but others in the
plane. For the zodiac itself forms angles, dividing the equinoctial
in two parts, at the vertex of the cutting superficies. And angles
of this kind subsist in a spherical superficies. But of those which
are in planes, some are comprehended by simple lines, others by mixt
ones; and others, again, by both. For in the shield-like figure[155],
an angle is comprehended by the axis, and the line of the shield: but
one of these lines is mixt, and the other simple. But if a circle
cuts the shield, the angle will be comprehended by the circumference,
and the ellipsis. And when cissoids, or lines similar to an ivy leaf,
closing in one point like the leaves of ivy (from whence they derive
their appellation) make an angle, such an angle is comprehended by
mixt lines. Also, when the hippopede, or line familiar to the foot
of a mare, which is one of the spirals, inclining to another line,
forms an angle, it is comprehended by mixt lines. Lastly, the angles
contained by a circumference and a right line, are comprehended by
simple lines. But of these again, some are contained by such as are
similar in species, but others by such as are dissimilar. For two
circumferences, mutually cutting, or touching each other, produce
angles: and these triple, for they are either on both sides convex,
when the convexities of the circumferences are external: or on both
sides concave, when both the concavities are external; which they call
sistroides; or mixt from convex and concave lines, as the lines called
lunulas. But besides this, angles are contained in a twofold manner,
by a right line and a circumference: for they are either contained by
a right line, and a concave circumference, as the semi-circular angle;
or by a right line and a convex circumference, as the cornicular angle.
But all those which are comprehended by two right lines, are called
rectilinear angles, which have likewise a triple difference[156]. The
geometrician, therefore, in the present hypothesis, defines all those
angles which are constituted in plane superficies, and gives them the
common name of a plane angle. And the genus of these he denominates
inclination: but the place, the plane itself, for angles have position:
but their origin such, that it is requisite there should be two lines
at least, and not three as in a solid. And that these should touch
each other, and by touching, must not lie in a right line, as an angle
is the inclination and comprehension of lines: but is not distance
only, according to one interval. But if we examine this definition,
in the first place it appears that it does not admit, an angle can be
perfected by one line; though a cissoid, which is but one, perfects
an angle. And, in like manner, the hippopede. For we call the whole
a cissoid, and not its portions (lest any one should say, that the
conjunction of these forms an angle) and the whole a spiral, but not
its parts. Each, therefore, since it is one, forms an angle to itself,
and not to another. But after this, he is faulty, in defining an angle
to be inclination. For how, on this hypothesis, will there be two
angles, from one inclination? How can we call angles equal and unequal?
And whatever else is usually objected against this opinion. Thirdly,
and lastly, that part of the definition, which says, _and not placed
in a right line_, is superfluous in certain angles, as in those
which are formed from orbicular lines. For without the assistance of
this part, the definition is perfect; since the inclination of one of
the lines to the other, forms the angle. And it is not possible that
orbicular angles should be placed in a right line. And thus much we
have thought proper to say concerning the definition of Euclid; partly,
indeed, interpreting, and partly doubting its truth.




                            DEFINITION IX.

        But when the Lines containing the Angle, are right, the
                     Angle is called RECTILINEAR.


An angle is the symbol and image of the connection and compression,
which subsists in the divine genera, and of that order which collects
divisibles into one, particles into an impartible nature, and the many
into conciliating community. For it is the bond of a multitude of lines
and superficies, the collector of magnitude into the impartibility of
points, and the comprehender of every figure which is composed by its
confining nature. On which account, the oracles[157] call the angular
junctions of figures, knots, so far as they bring with them an image of
connecting union, and divine conjunctions, by which discrete natures
mutually cohere with each other. The angles, therefore, subsisting in
superficies, express the more immaterial, simple, and perfect unions
which superficies contain: but those which are in solids, represent
the unions, which proceed even to inferiors, and supply a community
to things disjunct, and a construction of the same nature, to things
which on every side receive a perfect partition. But of the angles in
superficies, some shadow forth primary and unmixt unions; but others,
such as comprehend in themselves, an infinity of progressions. And
some, indeed, are the sources of union to intellectual forms; but
others, to sensible reasons; and others, again, are copulative of those
forms which obtain between these, a middle situation. Hence, the angles
which are made from circumferences, imitate those causes which envelop
intellectual variety in coercive union; for circumferences, hastening
to coalesce with each other, are images of intellect, and intellectual
forms. On the contrary, rectilineal angles, are the symbols of those
unions which preside over sensibles, and afford a conjunction of the
reasons subsisting in these: but mixt angles represent the preservers
of the communion, as well of sensible, as of intellectual forms,
according to one immoveable union. It is requisite, therefore, by
regarding these paradigms, or exemplars, to render the causes of each.
For among the Pythagoreans we shall find various angles dedicated to
various gods. Thus, Philolaus, consecrates to some a triangular, but to
others a quadrangular angle; and to others, again, different angles.
Likewise, he permits the same to many gods, and many to the same god,
according to the different powers which they contain. And with a view
to this, and to the demiurgic triangle, which is the primary cause of
all the ornament of the elements, it appears to me, that Theodorus
Asinæus the philosopher, constitutes some of the gods, according to
sides; but others, according to angles. The first, indeed, supplying
progression and power; but the second, the conjunction of the universe,
and the collection of progressive natures again into one. But these,
indeed, direct us to the knowledge of the things which are. And we must
not wonder that lines are here said to contain an angle. For the one
and impartible nature which is found in these, is adventitious: but
in the gods themselves, and in true beings, the whole, and impartible
good, precedes things many, and divided.




                             DEFINITION X.

  When a Right Line standing on a Right Line, makes the successive
  Angles on each side equal to one another, each of the equal
  Angles, is a Right Angle; and the insisting Right Line, is called
  a PERPENDICULAR to that upon which it stands.

                            DEFINITION XI.

         An OBTUSE ANGLE is that which is greater than a RIGHT
                                ANGLE.

                            DEFINITION XII.

           But an ACUTE ANGLE, is that which is less than a
                             RIGHT ANGLE.


These are the triple species of angles, which Socrates speaks of in
the Republic, and which are received by geometricians from hypothesis;
a right-line constituting these angles, according to a division into
species; I mean, the right, the obtuse, and the acute. The first of
these being defined by equality, identity and similitude; but the
others being composed through the nature of the greater and the lesser;
and lastly, through inequality and diversity, and through the more and
the less, indeterminately assumed. But many geometricians, are unable
to render a reason of this division, and use the assertion, that there
are three angles, as an hypothesis[158]. So that, when we interrogate
them concerning its cause, they answer, this is not to be required
of them as geometricians. However, the Pythagoreans, referring the
solution of this triple distribution to principles, are not wanting in
rendering the causes of this difference of right-lined angles. For,
since one of the principles subsists according to bound, and is the
cause of limitation, identity, and equality, and lastly, of the whole
of a better co-ordination: but the other is of an infinite nature,
and confers on its progeny, a progression to infinity, increase,
and decrease, inequality, and diversity of every kind, and entirely
presides over the worse series; hence, with great propriety, since
the principles of a right-lined angle are constituted by these, the
reason proceeding from bound, produces a right angle, one, with respect
to the equality of every right angle, endued with similitude, always
finite and determinate, ever abiding the same, and neither receiving
increment nor decrease. But the reason proceeding from infinity,
since it is the second in order, and of a dyadic nature, produces
twofold angles about the right angle, distinguished by inequality,
according to the nature of the greater and the lesser, and possessing
an infinite motion, according to the more and the less, since the one
becomes more or less obtuse; but the other more or less acute. Hence,
in consequence of this reason, they ascribe right angles to the pure
and immaculate gods of the divine ornaments, and divine powers which
proceed into the universe, as the authors of the invariable providence
of inferiors; for rectitude, and an inflexibility and immutability to
subordinate natures accords with these gods: but they affirm, that the
obtuse and acute angles should be ascribed to the gods, who afford
progression, and motion, and a variety of powers. Since obtuseness is
the image of an expanded progression of forms; but acuteness possesses
a similitude to the cause dividing and moving the universe. But
likewise, among the things which are, rectitude is, indeed, similar to
essence, preserving the same bound of its being; but the obtuse and
acute, shadow forth the nature of accidents. For these receive the more
and the less, and are indefinitely changed without ceasing. Hence,
with great propriety, they exhort the soul to make her descent into
generation, according to this invariable species of the right angle,
by not verging to this part more than to that; and by not affecting
some things more, and others less. For the distribution of a certain
convenience and sympathy of nature, draws it down into material error,
and indefinite variety[159]. A perpendicular line is, therefore, the
symbol of inflexibility, purity, immaculate, and invariable power,
and every thing of this kind. But it is likewise the symbol of divine
and intellectual measure: since we measure the altitudes of figures
by a perpendicular, and define other rectilineal angles by their
relation to a right angle, as by themselves they are indefinite and
indeterminate. For they are beheld subsisting in excess and defect,
each of which is, by itself, indefinite. Hence they say, that virtue
also stands according to rectitude; but that vice subsists according
to the infinity of the obtuse and acute, that it produces excesses and
defects, and that the more and the less exhibit its immoderation, and
inordinate nature. Of rectilineal angles, therefore, we must establish
the right angle, as the image of perfection, and invariable energy,
of limitation, intellectual bound, and the like; but the obtuse and
acute, as shadowing forth infinite motion, unceasing progression,
division, partition and infinity. And thus much for the theological
speculation of angles. But here we must take notice, that the genus
is to be added to the definitions of an obtuse and acute angle; for
each is right-lined, and the one is greater, but the other less than a
right-angle. But it is not absolutely true, that every angle which is
less than a right one, is acute. For the cornicular is less than every
right-angle, because less than an acute one, yet is not on this account
an acute angle. Also, a semi-circular is less than any right-angle, yet
is not acute. And the cause of this property is because they are mixt,
and not rectilineal angles. Besides, many curve-lined angles appear
greater than right-lined angles, yet are not on this account obtuse;
because it is requisite that an obtuse should be a right-lined angle.
Secondly, as it was the intention of Euclid, to define a right-angle,
he considers a right-line standing upon another right-line, and making
the angles on each side equal. But he defines an obtuse and acute
angle, not from the inclination of a right line to either part, but
from their relation to a right-angle. For this is the measure of
angles deviating from the right, in the same manner as equality of
things unequal. But lines inclined to either part, are innumerable,
and not one alone, like a perpendicular. But after this, when he says,
(_the angles equal to one another_) he exhibits to us a specimen
of the greatest geometrical diligence; since it is possible that angles
may be equal to others, without being right. But when they are equal
to one another, it is necessary they should be right. Besides, the
word _successive_ appears to me not to be added superfluously, as
some have improperly considered it; since it exhibits the reason of
rectitude. For it is on this account that each of the angles is right;
because, when they are _successive_, they are equal. And, indeed,
the insisting right-line, on account of its inflexibility to either
part, is the cause of equality to both, and of rectitude to each.
The cause, therefore, of the rectitude of angles, is not absolutely
mutual equality, but position in a consequent order, together with
equality. But, besides all this, I think it here necessary to call to
mind, the purpose of our author; I mean, that he discourses in this
place, concerning the angles consisting in one plane. And hence, this
definition is not of every perpendicular; but of that which is in one
and the same plane. For it is not his present design to define a solid
angle. As, therefore, he defines, in this place, a plane angle, so
likewise a perpendicular of this kind. Because a solid perpendicular
ought not to make right angles to one right-line only; but to all which
touch it, and are contained in its subject plane: for this is its
necessary peculiarity.




                           DEFINITION XIII.

       A BOUND is that which is the Extremity of any thing[160].


A Bound, in this place, is not to be referred to all magnitudes, for
there is a bound and extremity of a line; but to the spaces which are
contained in superficies, and to solid bodies. For he now calls a
bound, the ambit which terminates and distinguishes every space. And
a bound of this kind, he defines to be an extremity: but not after
the manner in which a point is called the extremity of a line, but
according to its property of including and excluding from circumjacent
figures. But this name is proper to geometry in its infant state, by
which they measured fields, and preserved their boundaries distinct and
without confusion, and from which they arrived at the knowledge of the
present science. Since, therefore, Euclid calls the external ambit,
a bound, it is not without propriety that he, by this means, defines
the extremity of spaces. For by this, every thing comprehended is
circumscribed. I say, for example, in a circle, its bound and extremity
is the circumference; but itself, a certain plane space: and so of the
rest.




                            DEFINITION XIV.

         A FIGURE is that which is comprehended by one or more
                              Boundaries.


Because figure is predicated in various ways, and is divided into
different species, it is requisite, in the first place, to behold
its differences; and afterwards to discourse concerning that figure
which is proposed in this Definition. There is, then a certain figure
which is constituted by mutation, and is produced from passion,
while the recipients of the figure are disturbed, divided, or taken
away; while they receive additions, or are altered, or suffer other
various affections. There is also a figure, which is produced by the
potter’s, or statuary’s art, according to the pre-existent reason,
which art itself contains: art, indeed, producing the form, but matter
receiving from thence, form, and beauty, and elegance. But there are
still more noble and more illustrious figures than these, the skilful
operations of nature. Some, indeed, existing in the elements under
the moon[161], and having a power of comprehending the reasons those
elements contain: but others are situated in the celestial regions,
distinguishing their powers, and endless revolutions. For the heavenly
bodies, both when considered by themselves, and with relation to each
other, exhibit an abundant and admirable variety of figures; and at
different times they present to our view different forms, bringing with
them a splendid image of intellectual species; and, by their elegant
and harmonious revolutions, describing the incorporeal and immaterial
powers of figures. But there are, again, besides all these, most pure
and perfect beauties, the figures of souls, which, because they are
full of life, and self-motive, have an existence prior to things moved
by another; and which, because they subsist immaterially, and without
any dimension, excel the forms which are endued with dimension and
matter. In the nature of which we are instructed by Timæus, who has
explained to us the demiurgic, and essential figure of souls[162].
But again, the figures of intellects are by far more divine than the
figures of souls; for these, on every side, excel partible essences;
are every where resplendent with impartible and intellectual light;
are prolific, effective, and perfective of the universe; are equally
present, and firmly abide in all things; and procure union to the
figures of souls; but recall the mutation of sensible figures to the
limitation of their proper bound. Lastly, there are, separate from
all these, those perfect, uniform, unknown, and ineffable figures of
the gods, which are resident, indeed, in the figures of intellects;
but jointly terminate all figures, and comprehend all things in their
unifying boundaries. The properties of which the theurgic art, also
expressing, surrounds various resemblances of the gods, with various
figures. And some, indeed, it fashions by characters, in an ineffable
manner; for characters of this kind, manifest the unknown powers of the
gods: but others it imitates by forms and images; fashioning some of
them erect, and others fitting; and some similar to a heart, but others
spherical, and others expressed by different figures. And again, some
it fabricates of a simple form; but others it composes from a multitude
of forms; and some are sacred and venerable; but others are domestic,
exhibiting the peculiar gentleness of the gods. And some it constructs
of a severe aspect; and lastly, attributes to others, different
symbols, according to the similitude and sympathy pertaining to the
gods[163]. Since, therefore, figure derives its origin from the gods
themselves, it arrives, by a gradual progression, even to inferiors,
in these also appearing from primary causes. Since it is requisite to
suppose the perfect before the imperfect, and things situated in the
stability of their own essence, prior to those which subsist in others,
and previous to things full of their own privation, such as preserve
their proper nature sincere. Such figures, therefore, as are material,
participate of material inelegance, and do not possess a purity
convenient to their nature. But the celestial figures are divisible,
and subsist in others. And the figures of souls are endued with
division, and variety, and involution of every kind; but the figures
of intellects, together with immaterial union, possess a progression
into multitude. And lastly, the figures of the gods are free, uniform,
simple, and generative; they subsist before all things, containing all
perfection in themselves, and extending from themselves to all things,
the completion of forms. We must not, therefore, listen to, and endure
the opinions of many, who affirm, that certain additions, ablations,
and alterations, produce sensible figures, (for motions, since they are
imperfect, cannot possess the principle and primary cause of effects;
nor could the same figures often be produced from contrary motions; for
the same form is sometimes generated from addition and detraction,)
but we must consider operations of this kind as subservient to other
purposes in generation, and derive the perfection of figure from other
primogenial causes. Nor must we subscribe to their opinion, who assert
that figures destitute of matter can have no subsistence; but those
only which appear in matter. Nor to theirs, who acknowledge, indeed,
that they are external to matter, but consider them as subsisting
alone, according to thought and abstraction. For where shall we
preserve in safety, the certainty, beauty, and order of figures, among
things which subsist by abstraction? For, since they are of the same
kind with sensibles, they are far distant from indubitable and pure
certainty. But from whence do they derive the certainty, order, and
perfection which they receive? For they either derive it from sensibles
(but they have no subsistence in these), or from intelligibles (but in
these they are more perfect), since, to say from that which is not, is
the most absurd of all. For nature does not produce imperfect figures,
and leave the perfect without any subsistence. Nor is it lawful, that
our soul should fabricate more certain, perfect, and orderly figures,
than intellect and the gods themselves. There are, therefore, prior
to sensible figures, self-moving, intellectual, and divine reasons of
figures. And we are excited, indeed, from the obscurity of sensible
forms, but we produce internal reasons, which are the lucid images
of others. And we possess a knowledge of sensible figures, by their
exemplars resident in soul (παραδειγματικῶς), but we comprehend by
images (εἰκονικῶς) such as are intellectual and divine. For the reasons
we contain, emerging from the dark night of oblivion, and propagating
themselves in sciential variety, exhibit the forms of the gods, and
the uniform bounds of the universe, by which they ineffably convert
all things into themselves. In the gods, therefore, there is both an
egregious knowledge of universal figures, and a power of generating and
constituting all inferiors. But in natures, figures are endued with
a power generative of apparent forms; but are destitute of cognition
and intellectual perception. And, in particular souls, there is,
indeed, an immaterial intellection, and a self-energizing knowledge;
but there is wanting a prolific, and efficacious cause. As, therefore,
nature, by her forming power presides over sensible figures, in the
same manner, soul, by her gnostic energy, drops in the phantasy as in
a mirror, the reasons of figures. But the phantasy receiving these in
her shadowy forms, and possessing images of the inherent reasons of
the soul, affords by these the means of inward conversion to the soul,
and of an energy directed to herself, from the spectres of imagination.
Just as if any one beholding his image in a mirror, and admiring the
power of nature, and his own beauty, should desire to see himself in
perfection, and should receive a power of becoming, at the same time,
the perceiver, and the thing perceived. For the soul, after this
manner, looking abroad into the bright mirror of the phantasy, and
surveying the shadowy figures it contains, and admiring their beauty
and order, pursues, in consequence of her admiration, the reasons from
which these images proceed; and being wonderfully delighted, dismisses
their beauty, as conversant about spectres alone; but afterwards
seeks her own purer beauty, and desires to pass into her own profound
retreats, and there to perceive the circle and the triangle, and all
things subsisting together, in an impartible manner, and to insert
herself in the objects, to contract her multitude into one; and lastly,
to behold the occult and ineffable figures of the gods, seated in the
most sacred and divine recesses of her nature. She is likewise desirous
of bringing into light, from its awful concealment, the solitary
beauty, of the gods, and of perceiving the circle, subsisting in its
true perfection, more impartible than any centre, and the triangle
without interval; and lastly, by ascending into an union with herself,
of surveying every object which is subject to the power of cognition.
The figure, therefore, which is self-motive, precedes that which is
moved by another; and the impartible that which is self-motive: but
that which is the same with _one_, precedes the impartible itself.
For all things are bounded, when they return to the unities of their
nature; since all things pass through these as a divine entrance into
being. And thus much for this long digression, which we have delivered
according to the sentiments of the Pythagoreans. But the geometrician,
contemplating that figure which is seated in the phantasy, and defining
this, in the first place, (since this definition agrees with sensibles,
in the second place) says, that figure is that which is comprehended by
one or more boundaries. For, since he receives it together with matter,
and conceives of it as distant with intervals, he does not improperly
call it finite and terminated[164]. [Since every thing which contains
either intelligible or feasible matter, is allotted an adventitious
bound; and is not itself bound, but that which is bounded.] Nor is it
the bound of itself; but one of its powers is terminating, and the
other terminated. Nor does it subsist in bound itself, but is contained
by bound. For figure is joined to quantity, and subsists together
with it; and, at the same time, quantity is subjected to figure; but
the reason and aspect of that quantity is nothing else than figure
and form. Since, indeed, reason terminates quantity, and adds to it
a particular character and bound, either simple or composite. For,
since this also exhibits the twofold progression of _bound_ and
_infinite_ in its proper forms, (in the same manner as the reason
of an angle,) it invests the objects of its comprehension with one
boundary and simple form, according to _bound_, but with many,
according to infinity[165]. Hence, every thing figured, vindicates to
itself either one boundary, or a many. Euclid, therefore, denominating
that which is figured and material, and annexed to quantity figure,
does not improperly say, that it is contained by one or more terms.
But Possidonius defines figure to be concluding bound, separating
the reason of figure from quantity; and considering it as the cause
of terminating, defining, and comprehending quantity. For that which
encloses, is different from that which is enclosed; and bound from that
which is bounded. And Possidonius, indeed, seems to regard the external
surrounding bound; but Euclid, the whole subject. Hence, the one calls
a circle a figure, with relation to its whole plane, and exterior
ambit; but the other with relation to its circumference only. And the
one defines that which is figured, and which is beheld together with
its subject: but the other desires to define the reason of the circle;
I mean that which terminates and concludes its quantity. But if any
logician, and captious person, should blame the definition of Euclid,
because he defines genus from species (for things contained by one or
more terms, are the species of figure,) we shall assert, in opposition
to such an objection, that genera also pre-occupy in themselves the
powers of species. And when men of ancient authority, were willing to
manifest genera themselves, from those powers which genera contain,
they appeared, indeed, to enter on their design from species, but, in
reality, they explained genera from themselves, and from the powers
which they contain. The reason of figure, therefore, since it is one,
comprehends the differences of many figures, according to the bound
and infinity residing in its nature. And he who defined this reason,
was not void of understanding, whilst he comprehended in a definition,
the differences of the powers it contained. But you will ask, From
whence does the reason of figure originate, and by what causes is
it perfected? I answer, that it first arises from _bound_ and
_infinite_, and that which is mixed from these. Hence it produces
some species from _bound_, others from _infinite_, and others
from the _mixt_. And this it accomplishes by bringing the form of
bound to circles; but that of infinite, to right lines: and that of
the mixt to figures composed from right and circular lines. But, in
the second place, this reason is perfected from that totality, which
is separated into dissimilar parts. From whence, indeed, it occasions
a whole to every form, and each figure is cut into different species.
For a circle, and every right-lined figure may be divided, by reason or
proportion, into dissimilar figures; which is the business of Euclid
in his book of divisions, where he divides one figure into figures
similar to each as are given; but another into such as are dissimilar.
In the third place, it is invigorated from accumulated multitude, and,
on account of this, extends forms of every kind, and produces the
multiform reasons of figures. Hence, in propagating itself, it does
not cease till it arrives at something last, and has unfolded all the
variety of forms. And, as in the intelligible world, _one_ is
shewn to abide in that which _is_; and, at the same time, that
which _is_ in _one_, so likewise, reason exhibits circular in
right-lined figures; and on the contrary, rectilinear comprehended in
circular figures. And it peculiarly manifests its whole nature in each,
and all these in all. Since the whole subsists in all collectively, and
in each separate and apart. From that order, therefore, it is endued
with this power. In the fourth place, it receives from the first of
numbers[166], the measures of the progression of forms. From whence it
constitutes all figures according to numbers; some, indeed, according
to the more simple, but others according to the more composite. For
triangles, quadrangles, quinquangles, and all multangles, proceed in
infinitum, together with the mutations of numbers. But the cause of
this is, indeed, unknown to the vulgar, though, to those who understand
where number and figure subsist, the reason is manifest. Fifthly,
it is replete with that division of forms, which divides forms into
other similar forms, from another second totality, which is also
distributed into similar parts. And by this, a triangular reason is
divided into triangles, and a quadrangular reason into quadrangles.
And hence, exercising our inward powers, we effect what I have said
in images, since it pre-existed by far the first in its principles.
But by regarding these distributions, we may render many causes of
figures, reducing them to their first principles. And the more common,
or geometrical figure, is allotted an order of this kind, and from so
many causes, receives the perfection of its nature. But, from hence
it advances to the genera of the gods, and is variously attributed
according to its various forms, and energizes differently in different
gods. To some, indeed, affording more simple figures; but to others,
such as are more composite. And to some, again, assigning primary
figures, and those which are produced in superficies; but to others
(entering the tumor of solid bodies) such figures, as in solids are
convenient to themselves. For all figures, indeed, subsist in all,
since the forms of the gods are accumulated, and full of universal
powers: but, by their peculiarity, they produce one thing according
to another. For one possesses all things circularly, another in a
triangular manner, but another according to a quadrangular reason. And
in a similar manner in solids.

                            DEFINITION XV.

   A CIRCLE is a Plane Figure, comprehended by one Line, which is
   called the Circumference, to which all Right Lines falling from
   a certain Point within the Figure, are equal to each other.

                            DEFINITION XVI.

          And that Point is called the CENTRE of the CIRCLE.


A Circle is the first, most simple, and most perfect of figures. For
it excels all solids, because it exists in a more simple place; but
it is superior to the figures subsisting in planes, on account of
its similitude and identity. And it has a corresponding proportion
to bound, and unity, and a better co-ordination of being. Hence, in
a distribution of mundane and super-mundane figures, you will always
find that the circle is of a diviner nature. For if you make a division
into the heavens, and the universal regions of generation, you must
assign to the heavens a circular form; but to generation, that of a
right line, For whatever among generable natures is circular, descends
from the heavens; since generation revolves into itself, through
their circumvolutions, and reduces its unstable mutation to a regular
and orderly continuance. But if you distribute incorporeal natures
into soul and intellect, you will say, that the circle belongs to
intellect, and the right line to the soul. And on this account, the
soul, by its conversion to intellect, is said to be circularly moved;
and it possesses the same proportion to intellect, as generation to
the heavens. For it is circularly moved, (says Socrates[167],) because
it imitates intellect. But the generation and progression of soul is
made according to a right-line. For it is the property, of the soul
to apply herself at different times to different forms. But if you
wish to divide into body and soul, you must constitute every thing
corporeal, according to the right line; but you must assign to every
animal a participation of the identity and similitude of the circle.
For body is a composite, and is endued with various powers, similar to
right-lined figures: but soul is simple and intelligent; self-motive,
and self-operative; converted into, and energizing in herself. From
whence, indeed, Timæus also, when he had composed the elements of
the universe from right-lined figures, assigned to them a circular
motion and formation, from that divine soul which is seated in the
bosom of the world. And thus, that the circle every where holds the
first rank, in respect of other figures, is sufficiently evident from
the preceding observations. But it is requisite to survey its whole
series, beginning supernally, ending in inferiors, and perfecting all
things, according to the aptitude of the natures which receive its
alliance. To the gods, therefore, it affords a conversion to their
causes, and ineffable union: it occasions their abiding in themselves,
prevents their departing from their own beatitude, strengthens their
highest unions, as centres desirable to inferior natures; and stably
places about these the multitude of the powers which the gods possess,
containing them in the simplicity of their essences. But the circle
affords to intellectual natures, a perpetual energy in themselves, is
the cause of their being filled with knowledge from themselves, and
of possessing in their essences, intelligibles contractedly; and of
perfecting intellections in themselves. For every intellect, proposes
to itself that which is intelligible; and this is as a centre to
intellect, about which it continually revolves: for intellect folds
itself, and operates about this, and is united within itself on all
sides, by universal intellectual energies. But it extends to souls by
illumination, a self-vital, and self-motive power, and an ability of
turning, and leaping round intellect, and of returning according to
proper convolutions, unfolding the impartibility of intellect. Again,
the intellectual orders excel souls after the manner of centres, but
souls energize circularly about their nature. For every soul, according
to its intellectual part, and the supreme _one_, which is the
very flower of its essence, receives a centre: but, according to its
multitude, it has a circular revolution, desiring, by this means, to
embrace the intellect which it participates. But, to the celestial
bodies, the circle affords an assimilation to intellect, equality, a
comprehension of the universe, in proper limits, revolutions which
take place in determinate measures, a perpetual subsistence, a nature
without beginning and end, and every thing of this kind. And to the
elements under the concave of the moon’s orb, it is the cause of a
period, conversant with mutations; an assimilation to the heavens; that
which is without generation, in generated natures; that which abides in
things which are moved; and whatever is bounded in partible essences.
For all things are perpetual, through the circle of generation; and
equability is every where preserved on account of the reciprocation
of corruption. Since, if generation did not return, in a circular
revolution, in a short space of time, the order, and all the ornament
of the elements would vanish. But again, the circle procures to animals
and plants, that similitude which is found in generations; for these
are produced from seeds, and seeds from these. Hence, generation here,
and a circumvolution, alternately takes place, from the imperfect to
the perfect, and the contrary; so that corruption subsists together
with generation. But, besides this, to unnatural productions it
imposes order, and reduces their indeterminate variety to the
limitation of bound; and, through this, nature herself is gracefully
ornamented in the last vestiges of her powers. Hence, things contrary
to nature have a revolution according to determinate numbers, and not
only fertility, but also sterility, subsists according to the alternate
convolutions of circles (as the discourse of the Muses evinces), and
all evils though they are dismissed from the presence of the gods,
into the place of mortals, yet these roll round, says Socrates, and to
these there is present a circular revolution, and a circular order;
so that nothing immoderate and evil is deserted by the gods; but that
providence, which is perfective of the universe, reduces also the
infinite variety of evils, to bound, and an order convenient to their
nature. The circle, therefore, is the cause of ornament to all things,
even to the last participations, and leaves nothing destitute of
itself, since it supplies beauty, similitude, formation, and perfection
to the universe. Hence too, in numbers it contains the middle centres
of the whole progression of numbers, which revolves from unity to the
decad (or ten). For five and six exhibit a circular power, because, in
the progressions from themselves, they return again into themselves,
as is evident in the multiplication of these numbers. Multiplication,
therefore, is an image of progression, since it is extended into
multitude; but an ending in the same species, is an image of regression
into themselves. But a circular power affords each of these, exciting,
indeed, as from an abiding centre, those causes which are productive
of multitude; but converting multitude after the productions to their
causes. Two numbers, therefore, having the properties of a circle,
possess the middle place between all numbers: of which one, indeed,
precedes every convertible genus of males and an odd nature; but the
other, recalls every thing feminine and even, and all prolific series,
to their proper principles, according to a circular power. And thus
much concerning the perfection of the circle. Let us now contemplate
the mathematical definition of the circle, which is every way perfect.
In the first place, therefore, he defines it a figure, because, indeed,
it is finite, and every where comprehended by one limit, and is not of
an infinite nature, but associated to bound. Likewise plane, because,
since figures are either beheld in superficies, or in solid bodies, a
circle is the first of plane figures, excelling solids in simplicity,
but possessing the proportion of unity to planes. But comprehended by
one line, because it is similar to one, by which it is defined, and
because it does not extrinsically receive a variety of surrounding
terms. And again, that this line makes all the lines drawn to it from
a certain point within equal, because of the figures which are bounded
by one line, some have all the lines proceeding from the middle equal;
but others not at all. For the ellipsis is comprehended by one line,
yet all the lines issuing from the centre, and bounded by is curvature,
are not equal, but only two. Also the plane, which is included by the
line called a cissoid, has one containing line, yet it does not contain
a centre, from which all the lines are equal. But, because the centre
in a circle is entirely one point (for there are not many centres of
one circle), on this account, the geometrician adds, that lines falling
from one point to the bound of the circle, are equal. For there are
infinite points within it, but of all these, one only has the power of
a centre. And because this one point, from which all the lines drawn to
the circumference of the circle are equal, is either within the circle,
or without (for every circle has a pole, from which all the lines drawn
to its circumference are equal), on this account he adds, _of the
points within the figure_, because, here he receives the centre
alone, and not the pole. For he wishes to behold all its properties in
one plane, but the pole is more elevated than the subject plane. Hence,
he necessarily adds, in the end of the definition, that this point,
which is placed within the circle, and to which all right lines drawn
from it to the circumference, are equal, is the centre of the circle.
For there are only two points of this kind, the pole and the centre.
But the former is without, and the other within the plane. Thus, for
instance, if you conceive a perpendicular standing on the centre of a
circle, its superior extremity is the pole: for all lines drawn from
it to the circumference of the circle, are demonstrated to be equal.
And, in like manner, in a cone, the vertex of the whole cone, is the
pole of the circle at the base. And thus far we have determined what a
circle is, and its centre, and what the nature is of its circumference,
and the whole circular figure. Again, therefore, from these, let us
return to the speculation of their exemplars, contemplating in them
the centre, according to one impartible and stable excellence. But the
distances from the centre, according to the progressions which are made
from one, to multitude infinite in capacity. And the circumference of
the circle, according to the regression of the progressions to the
centre, by means of which the multitude of powers are rolled round
their union, and all of them hasten to its comprehension, and desire
to energize about its indivisible embrace. And, as in the circle
itself, all things subsist together, the centre, intervals, and
external circumference; so in these which are its image, one thing
has not an essence pre-existent, and another consequent in time;
but all things are, indeed, together, permanency, progression, and
regression. But these differ from those, because the former subsist
indivisibly, and without any dimension; but the latter with dimension;
and in a divisible manner; the centre existing in one place, the lines
emanating from the centre, in another; and the external circumference
terminating the circle, having a still different situation. But there
all things abide in one: for if you regard that which performs the
office of a centre, you will find it the receptacle of all things. If
the progression distant from the centre, in this, likewise, you will
find all things contained. And, in a similar manner, if you regard
its regression. When, therefore, you are able to perceive all things
subsisting together, and have taken away the defect proceeding from
dimension, and have removed from your inward vision, the position about
which partition subsists, you will find the true circle, advancing to
itself, bounding, and energizing in itself, existing both one and many,
and abiding, proceeding, and returning; likewise firmly establishing
that part of its essence which is most impartible, and especially
singular; but advancing from this according to rectitude, and the
infinity which it contains; and rolling itself from itself to one, and
exciting itself by similitude and identity to the impartible centre
of its nature, and to the occult power of the one which it contains.
But this one, which the circle contains, and environs in its bosom, it
emulates according to the multitude of its own nature. For that which
is convolved, imitates that which abides, and the periphery is as a
centre which is distant with interval, and nods to itself, hastening
to receive, and to become one with the centre, and to terminate its
regress where it received the principle of its progression. For the
centre is every where in the place of that which is lovely, and the
object of desire, presiding over all things which subsist about its
nature, and existing as the beginning and author of all progressions.
And this the mathematical centre also expresses, by terminating all the
lines falling from itself to the circumference, and by affording to
them equality, as an image of proper union. But the oracles likewise
define the centre, after this manner: _The centre is that from which
and[168] to which all the lines to the circumference are equal_.
Indicating the beginning of the distance of the lines, by the particle
_from which_; but the middle of the circumference by the particle
_to which_: for this, in every part, is joined with the centre.
But if it be necessary to declare the first cause, through which a
circular figure appears and receives its perfection, I affirm, that
it is the supreme order of intelligibles. For the centre, indeed, is
assimilated to the cause of bound; but the lines emanating from this,
and which are infinite, with respect to themselves, both in multitude
and magnitude, represent infinity; and the line which terminates their
extension, and conjoins the circular figure with the centre, is similar
to that occult ornament, consisting from the intelligible orders;
which Orpheus also says, is circularly borne, in the following words,
_But it is carried with an unwearied energy, according to an infinite
circle_. For, since it is moved intelligibly, about that which is
intelligible, having it for the centre of its motion, it is, with great
propriety, said to energize in a circular manner. Hence, from these
also, the triadic god[169] proceeds, who contains in himself the cause
of the progression of right-lined figures. For on this account, wise
men, and the most mystic of theologists, have fabricated his name.
[[170]Hence too, it is manifest, that a circle is the first of all
figures:] but a triangle is the first of such as are right-lined.
Figures, therefore, appear first in the regular ornaments of the gods;
but they have a latent subsistence, according to pre-existent causes,
in intelligible essences.




                           DEFINITION XVII.

   A DIAMETER of a Circle is a certain straight Line, drawn through
   the Centre, which is terminated both ways by the Circumference
   of the Circle, and, divides the Circle into two equal Parts.


Euclid here perspicuously shews, that he does not define every
diameter, but that which belongs to a circle only. Because there
is a diameter of quadrangles and all parallelograms, and likewise
of a sphere among solid figures. But in the first of these, it is
denominated a diagonal: but in a sphere, the axis; and in circles
the diameter only. Indeed, we are accustomed to speak of the axis of
an ellipsis, cylinder, and cone; but of a circle, with propriety,
the diameter. This, therefore, in its genus, is a right-line; but as
there are many right-lines in a circle, as likewise infinite points,
one of which is a centre, so this only is called a diameter, which
passes through the centre, and neither falls within the circumference,
nor transcends its boundary; but is both ways terminated by its
comprehensive bound. And these observations exhibit its origin. But
that which is added in the end, that it also divides the circle into
two equal parts, indicates its proper energy in the circle, exclusive
of all other lines drawn through the centre, which are not terminated
both ways by the circumference. But they report, that Thales first
demonstrated, that the circle was bisected by the diameter. And the
cause of this bisection, is the indeclineable transit of the right
line, through the centre. For, since it is drawn through the middle,
and always preserves the same inflexible motion, according to all its
parts, it cuts off equal portions on both sides to the circumference
of the circle. But if you desire to exhibit the same mathematically,
conceive the diameter drawn, and one part of the circle placed on
the other[171]. Then, if it is not equal, it either falls within, or
without; but the consequence either of these ways must be, that a less
right-line will be equal to a greater. Since all lines from the centre
to the circumference are equal. The line, therefore, which tends to
the exterior circumference, will be equal to that which tends to the
interior. But this is impossible. These parts of the circle, then,
agree, and are on this account equal. But here a doubt arises, if two
semi-circles are produced by one diameter, and infinite diameters may
be drawn through the centre, a double of infinities will take place,
according to number. For this is objected[172] by some against the
section of magnitudes to infinity. But this we may solve by affirming,
that magnitude may, indeed, be divided infinitely, but not into
infinites. For this latter mode produces infinites in energy, but the
former in capacity only. And the one affords essence to infinite,
but the other is the source of its origin alone. Two semi-circles,
therefore, subsist together with one diameter, yet there will never be
infinite diameters, although they may be infinitely assumed. Hence,
there can never be doubles of infinites; but the doubles which are
continually produced, are the doubles of finites; for the diameters
which are always assumed, are finite in number. And what reason can be
assigned why every magnitude should not have finite divisions, since
number is prior to magnitudes, defines all their sections, pre-occupies
infinity, and always determines the parts which rise into energy, from
dormant capacity?




                           DEFINITION XVIII.

   A SEMI-CIRCLE is the Figure contained by the Diameter, and that
   Part of the Circumference which is cut off by the Diameter.

                         DEFINITION XIX.[173]

      But the CENTRE of the Semi-circle, is the same with that of
                              the Circle.


From the definition of a circle Euclid finds out the nature of the
centre, differing from all the other points which the circle contains.
But from the centre he defines the diameter, and separates it from the
other right lines, which are described within the circle. And from
the diameter, he teaches the nature of the semi-circle; and informs
us, that it is contained by two terms, always differing from each
other, viz. a right-line and a circumference: and that this right-line
is not any one indifferently, but the diameter of the circle. For
both a less and a greater segment of a circle, are contained by a
right-line and circumference; yet these are not semi-circles, because
the division of the circle is not made through the centre. All these
figures, therefore, are biformed, as a circle was monadic, and are
composed from dissimilars. For every figure which is comprehended by
two terms, is either contained by two circumferences, as the lunular:
or by a right-line and circumference, as the above mentioned figures;
or by two mixt lines, as if two ellipses intersect each other (since
they enclose a figure, which is intercepted between them), or by a
mixt line and circumference, as when a circle cuts an ellipsis; or by
a mixt and right-line, as the half of an ellipsis. But a semi-circle
is composed from dissimilar lines, yet such as are, at the same time,
simple, and touching each other by apposition. Hence, before he defines
triadic figures, he, with great propriety, passes from the circle to
a biformed figure. For two right-lines can, indeed, never comprehend
space. But this may be effected by a right-line and circumference.
Likewise by two circumferences, either making angles, as in the
lunular figure; or forming a figure without angles, as that which is
comprehended by concentric circles. For the middle space intercepted
between both, is comprehended by two circumferences; one interior,
but the other exterior, and no angle is produced. For they do not
mutually intersect, as in the lunular figure, and that which is on
both sides convex. But that the centre of the semi-circle is the same
with that of the circle, is manifest. For the diameter, containing in
itself the centre, completes the semi-circle, and from this all lines
drawn to the semi-circumference are equal. For this is a part of the
circumference of the circle. But equal right lines proceed from the
centre to all parts of the circumference. The centre, therefore, of the
circle and semi-circle is one and the same. And it must be observed,
that among all figures, this alone contains the centre in its own
perimeter, I say, among all plane figures. Hence you may collect, that
the centre has three places. For it is either within a figure, as in
the circle; or in its perimeter, as in the semi-circle; or without
the figure, as in certain conic lines[174]. What then is indicated by
the semi-circles, having the same centre with the circle, or of what
things does it bear an image, unless that all figures which do not
entirely depart from such as are first, but participate them after
a manner, may be concentric with them, and participate of the same
causes? For the semi-circle communicates with the circle doubly, as
well according to the diameter, as according to the circumference.
On this account, they possess a centre also in common. And perhaps,
after the most simple principles, the semi-circle is assimilated to
the second co-ordinations, which participate those principles; and by
their relation to them, although imperfectly, and by halves, they are,
nevertheless, reduced to that which is, and to their first original
cause.

                            DEFINITION XX.

        RECTILINEAR FIGURES are those which are comprehended by
                            Straight Lines.

                            DEFINITION XXI.

          TRILATERAL FIGURES, or TRIANGLES, by three Straight
                                Lines.

                           DEFINITION XXII.

                QUADRILATERAL, by four Straight Lines.

                           DEFINITION XXIII.

         MULTILATERAL FIGURES, or POLYGONS, by more than four
                            Straight Lines.


After the monadic figure having the relation of a principle to all
figures, and the biformed semi-circle, the progression of right-lined
figures in infinitum, according to numbers, is delivered. For on this
account also, mention was made of the semi-circle, as communicating
according to terms or boundaries; partly, indeed, with the circle,
but partly with right-lines: just as the duad is the medium between
unity and number. For unity, by composition, produces more than by
multiplication; but number, on the contrary, is more increased by
multiplication than composition: and the duad, whether multiplied into,
or compounded with itself, produces an equal quantity. As, therefore,
the duad is the middle of unity and number, so likewise, a semi-circle
communicates, according to its base, with right-lines; but according
to its circumference, with the circle. But right-lined figures proceed
orderly to infinity, attended by number and its bounding power,
which begins from the triad. On this account, Euclid also begins
from hence[175]. For he says, trilateral and quadrilateral, and the
following figures, called by the common name of multilateral: since
trilateral figures are also multilaterals; but they have likewise
a proper, besides a common denomination. But, as we are but little
able to pursue the rest, on account of the infinite progression of
numbers, we must be content with a common denomination. But he only
makes mention of trilaterals and quadrilaterals, because the triad and
tetrad are the first in the order of numbers; the former being a pure
odd among the odd; but the latter, an entire even among even numbers.
Euclid, therefore, assumes both in the origin of right-lined figures,
for the purpose of exhibiting their subsistence, according to all even
and odd numbers. Besides, since he is about to teach concerning these
in the first book, as especially elementary (I mean triangles and
parallelograms) he does not undeservedly, as far as to these, establish
a proper enumeration: but he embraces all other right-lined figures by
a common name, calling them multilaterals: but of these enough. Again,
assuming a more elevated exordium, we must say, that of plane figures,
some are contained by simple lines, others by such as are mixt, but
others again by both. And of those which are comprehended by simple
lines, some are contained by similars in species, as right-lines; but
others by dissimilars in species, as semi-circles, and segments, and
apsides, which are less than semi-circles. Likewise of those which
are contained by similars in species, some are comprehended by a
circular line; but others by a right-line. And of those comprehended
by a circular line, some are contained by one, others by two, but
others by more than two. By one, indeed, the circle itself. But by
two, some without angles, as the crowns[176] terminated by concentric
circles; but others angular (γεγωνιωμένα) as the lunula. And of those
comprehended by more than two, there is an infinite procession. For
there are certain figures contained by three and four and succeeding
circumferences. Thus, if three circles touch each other, they will
intercept a certain trilateral space; but if four, one terminated by
four circumferences, and in like manner, by a successive progression.
But of those contained by right lines, some are comprehended by three,
others by four, and others by a multitude of lines. For neither is
space comprehended by two right-lines, nor much more by one right-line.
Hence, every space comprehended by one boundary, or by two, is either
mixt or circular. And it is mixt in a twofold manner, either because
the mixt lines comprehend it, as the space intercepted by the cissoidal
line; or because it is contained by lines dissimilar in species,
as the apsis: since mingling is twofold, either by apposition or
confusion. Every right-lined figure, therefore, is either trilateral,
or quadrilateral, or gradually multilateral; but every trilateral, or
quadrilateral, or multilateral figure, is not right-lined; since so
great a number of sides is also produced from circumferences. And thus
much concerning the division of plane figures. But we have already
asserted[177], that rectitude of progression is both a symbol of motion
and infinity, and that it is peculiar to the generative co-ordinations
of the gods, and to the producers of difference, and to the authors
of mutation and motion. Right-lined figures, therefore, are peculiar
to these gods, who are the principles of the prolific energy of the
whole progression of forms. On which account, generation also, was
principally adorned by these figures, and is allotted its essence from
these, so far as it subsists in continual motion and mutation without
end.




                           DEFINITION XXIV.

          Of three-sided FIGURES: an EQUILATERAL TRIANGLE is
                   that which has three equal Sides.

                            DEFINITION XXV.

        An ISOSCELES TRIANGLE, is that which has only two Sides
                                equal.

                           DEFINITION XXVI.

          A SCALENE TRIANGLE, is that which has three unequal
                                Sides.

                           DEFINITION XXVII.

           A RIGHT-ANGLED TRIANGLE is that which has a Right
                                Angle.

                          DEFINITION XXVIII.

            An OBTUSE-ANGLED TRIANGLE is that which has an
                             Obtuse Angle.

                           DEFINITION XXIX.

           An ACUTE-ANGLED TRIANGLE is that which has three
                             Acute Angles.


The division of triangles sometimes commences from angles, but
sometimes from sides. And that, indeed, which originates from
sides, precedes as known; but that from angles follows as a proper
distribution. For these three angles alone belong to right-lined
figures, viz. the right, the obtuse, and the acute: but the equality
and inequality of sides subsist also in non-rectilinear figures.
Euclid says, therefore, that of triangles, some are equilateral,
others isosceles, and others scalene: for they have either all their
sides equal, or all unequal, or only two equal. And again, that of
triangles some are right-angled, others obtuse-angled, and others
acute-angled. And he defines a right-angled triangle, that which has
one right angle, as likewise an obtuse-angled triangle, that which has
one obtuse angle: for it is impossible that a triangle can have more
than one right, or obtuse angle[178]. But he defines an acute-angled
triangle, that which has all its angles acute. For here it is not
sufficient that it should have only one acute; since, in this case,
all triangles would be acute-angled, as every triangle has necessarily
two acute angles[179]. But, to possess three acute angles, is the
property of an acute-angled triangle alone. But Euclid appears to me to
have made a separate division into angles and sides, from considering
this alone, that every triangle is not also trilateral. For there are
quadrilateral triangles, which are called by mathematicians themselves
(ἀκιδοειδῆ) that is, similar to the point of a spear[180]: but by
Zenodorus (κοιλογώνια) that is, having an hollow angle. For on one of
the sides of a trilateral figure, constitute two right-lines inwardly;
by this means a certain space will be enclosed, which is comprehended
by external and internal right-lines, and which has three angles;
one, indeed, contained by the external lines; but two comprehended by
these and the internal lines, at the extremities in which these lines
are conjoined. A figure of this kind, therefore, is a quadrilateral
triangle. And hence, it does not immediately follow, that because a
figure has three angles (whether they are all acute, or one right,
or one obtuse), we shall find it trilateral; for it may be, perhaps,
quadrilateral. In like manner, you may also find quadrangles having
more than four sides. And therefore, we must not rashly determine the
number of sides from the multitude of angles. But of this enough. But
the Pythagoreans affirm that the triangle is simply the principle
of generation, and of the formation of generable natures. On which
account, Timæus says, that natural reasons, as well as those of the
construction of the elements, are triangular. For they are distant by a
triple interval, are on all sides collective of partible, and variously
mutable natures, are replete with material infinity, and bear before
themselves the conjunctions of material bodies, loosened and free: as,
indeed, triangles also are comprehended by three right-lines, but they
possess angles which collect the multitude of lines, and afford to them
an adventitious angle and conjunction. With great propriety, therefore,
Philolaus has consecrated the angle of a triangle to four gods,
Saturn, Pluto, Mars, and Bacchus, comprehending in these the whole
quadripartite ornament of the elements descending from the heavens,
or from the four segments of the zodiac. For Saturn constitutes an
essence wholly humid and frigid; but Mars a nature totally fiery; and
Pluto contains the whole terrestrial life; but Bacchus governs a humid
and hot generation; of which wine also is a symbol, for this is humid
and hot. Hence, all these gods differ according to their operations
in inferior concerns: but they are mutually united according to their
proper natures. And on this account, Philolaus collects their union
according to one angle. But if the differences of triangles contribute
to generation, we shall very properly confess that a triangle is
the principle and author of the constitution of sublunary natures.
For a right angle, indeed, affords them essence, and determines the
measure of being; and the reason of a right-angled triangle produces
the essence of the elements of generable natures; but an obtuse angle
assigns to them universal distance; and the reason of an obtuse-angled
triangle increases material forms in magnitude, and in mutation of
every kind. But an acute angle effects their divisible nature; and the
reason of an acute-angled triangle prepares them to receive infinite
division. But simply, a triangular reason constitutes the essence of
material bodies distant with interval, and on all sides divisible.
And thus much should we speculate concerning the nature of triangles.
But from these divisions you may understand, that all the species of
triangles are neither more nor less than seven. For the equilateral
triangle is one, since it is acute-angled only; but each of the rest
is triple. For the isosceles is either right-angled, or obtuse-angled,
or acute-angled; and, in like manner, the scalene triangle possesses
this triple difference. If then, these have a triple distinction, but
the equilateral has but one mode of existence, all the species of
triangles will be seven. But again, you will understand the proportion
of triangles to the things which _are_, according to the division
of sides; for the equilateral, entirely excelling in equality and
simplicity, is allied to divine souls; since it is the measure and
equality of things unequal, in the same manner as divinity of all
inferior concerns. But the isosceles triangle is allied to the better
genera, which govern a material nature, the greater part of which
genera is held by the limitation of measure; but their extremes extend
to inequality and material immoderation; for the two sides of an
isosceles triangle are equal, but the base is unequal. But a scalene
triangle symbolizes with partible lives, which are on all sides lame
and defective, which prepare themselves for generation, and are replete
with matter and material imperfection.




                            DEFINITION XXX.

   Of Quadrilateral Figures, a QUADRANGLE or SQUARE is that which
   has all its Sides equal, and all its Angles Right Angles.

                           DEFINITION XXXI.

     An OBLONG is that which has all its Angles right Angles, but
                     has not all its Sides equal.

                           DEFINITION XXXII.

       A RHOMBUS, is that which has all its Sides equal, but its
                     Angles are not right Angles.

                          DEFINITION XXXIII.

   A RHOMBOID is that which has its opposite Sides equal to one
   another, but all its Sides are not equal, nor its Angles Right
   Angles.

                           DEFINITION XXXIV.

       All other Quadrilateral Figures besides these, are called
                              TRAPEZIUMS.


It is requisite that the first division of quadrilateral figures should
take place in two numbers; and that some of them should be called
parallelograms, but others non-parallelograms. But of parallelograms
some are rectangular and equilateral, as quadrangles; but others
neither of these, as rhomboids: others again, are rectangular, but not
equilateral, as oblongs: but others, on the contrary, are equilateral,
but not rectangular, as the rhombuses. For it is requisite either
to possess both, viz. equality of sides and rectitude of angles, or
neither; or one of these, and this in a twofold respect. Hence a
parallelogram has a quadruple subsistence. But of non-parallelograms,
some have only two parallel sides, and not the rest; but others have
none of their sides parallel. And those are called Trapeziums, but
these Trapezoids. But of Trapeziums, some, indeed, have the sides
equal, by which the parallel sides of this kind are conjoined;
but others unequal; and the former of these are called isosceles
trapeziums; but the latter scalene trapeziums. A quadrilateral figure,
therefore, is constituted by us according to a seven-fold distribution.
For one is a quadrangle; but the other an oblong; the third a rhombus;
the fourth a rhomboides; the fifth an isosceles trapezium; the sixth
a scalene trapezium; the seventh a trapezoid. But Possidonius makes
a perfect division of right-lined quadrilateral figures into so many
members; for he establishes seven species of these; as likewise
of triangles. But Euclid could not divide into parallelograms and
non parallelograms, because he neither mentions parallels, nor
teaches us concerning the parallelogram itself. But trapeziums, and
all trapezoids, he calls by a common name, describing trapeziums
themselves, according to the difference of those four figures[181], in
which the property of parallelograms is verified. And this is to have
the opposite sides and angles equal. For a quadrangle and an oblong,
and a rhombus, have their opposite sides and angles equal. But in a
rhomboides he only adds this, _that its opposite sides are equal_,
lest he should define it by negations alone, since he neither calls it
equilateral, nor rectangular. For where we want proper appellations,
it is necessary to use such as are common. But we should hear Euclid
shewing that this is common to all parallelograms. But a rhombus
appears to be a quadrangle having its sides moved, and a rhomboides
a moved oblong. Hence, according to sides, these do not differ from
those; but they vary only according to the obtuseness and acuteness
of angles; since the quadrangle and the oblong are rectangular. For
if you conceive a quadrangle or an oblong, having its sides drawn in
such a manner, that while two of its opposite angles are dilated, the
other two are contracted; then the dilated angles will appear obtuse,
and the contracted, acute. And the appellation of rhombus[182] seems
to have been imposed from motion. For if you conceive a quadrangle
moving after the manner of a rhombus, it will appear to you changed in
order, according to its angles: just as if a circle is moved after the
manner of a sling, it will immediately exhibit the appearance of an
ellipsis. But here you may perhaps enquire concerning the quadrangle,
why it has this denomination? and why the appellation of quadrangle
may not be applied to other quadrilateral figures, as the name of
triangle is common to all those which are neither equiangular nor
equilateral, and in like manner of quinquangles or pentagons; for
the geometrician, in these, adds only the particle _an equilateral
triangle_, or a _quinquangle_, _which is equilateral and
equiangular_, as if these could not be otherwise than such as they
are? But when he mentions a quadrangle, he immediately indicates that
it must be equilateral and rectangular. But the reason of this is as
follows: a quadrangle alone has the best space, both according to
its sides and angles. For each of the latter is right, intercepting
a measure of angles, which neither receives intention nor remission.
As it excels, therefore, in both respects, it deservedly obtains a
common appellation. But a triangle, though it may have equal sides,
yet will in this case have all its angles acute, and a quinquangle all
its angles obtuse. Since, therefore, of all quadrilateral figures,
a quadrangle alone is replete with equality of sides, and rectitude
of angles, it was not undeservedly allotted this appellation: for,
to excellent forms, we often dedicate the name of the whole. But it
appeared also to the Pythagoreans, that this property of quadrilateral
figures, principally conveyed an image of a divine essence. For they
particularly signified by this, a pure and immaculate order. Since
rectitude imitates inflexibility, but equality a firm and permanent
power: for motion emanates from inequality, but quiet from equality
itself. The gods, therefore, who are the authors to all things
of stable disposition, of pure and uncontaminated order, and of
indeclinable power, are deservedly manifested as from an image, by a
quadrangular figure. But, besides these, Philolaus also, according to
another apprehension, calls a quadrangular angle, the angle of Rhea,
Ceres and Vesta. For, since a quadrangle constitutes the earth, and is
its proximate element, as we learn from Timæus, but the earth herself
receives from all these divinities, genital seeds, and prolific powers,
he does not unjustly consecrate the angle of a quadrangle to these
goddesses, the bestowers of life. For some call both the earth and
Ceres, Vesta[183], and they say that Rhea totally participates her
nature, and that all generative causes are contained in her essence.
Philolaus, therefore, says that a quadrangular angle comprehends, by
a certain terrestrial power, one union of the divine genera. But some
assimilate a quadrangle to universal virtue, so far as every quadrangle
from its perfection has four right angles. Just as we say that each of
the virtues is perfect, content with itself, the measure and bound of
life, and the middle of every thing which, in morals, corresponds to
the obtuse and acute. But it is by no means proper to conceal, that
Philolaus attributes a triangular angle to four, but a quadrangular
angle to three gods, exhibiting their alternate transition, and the
community of all things in all, of odd natures in the even, and of even
in the odd. Hence, the tetradic ternary, and the triadic quaternary,
participating of prolific and efficacious goods, contain the whole
ornament of generable natures, and preserve them in their proper
state. From which the duodenary, or the number twelve, is excited
to a singular unity, viz. the government of Jupiter. For Philolaus
says, that the angle of a dodecagon (or twelve-sided figure) belongs
to Jove, so far as Jupiter contains and preserves, by his singular
union, the whole number of the duodenary. For also, according to Plato,
Jupiter presides over the duodenary[184], and governs and moderates
the universe with absolute sway. And thus much we have thought proper
to discourse concerning quadrilateral figures, as well declaring the
sense of our author, as likewise affording an occasion of more profound
inspections to such as desire the knowledge of intelligible and occult
essences.




                           DEFINITION XXXV.

   PARALLEL RIGHT LINES are such as being in the same Plane, and
   produced both ways infinitely, will in no part mutually coincide.


What the elements of parallels are, and by what accidents in these
they may be known, we shall afterwards learn: but what parallel right
lines are, he defines in these words: “It is requisite, therefore (says
he), that they should be in one plane, and while they are produced
both ways have no coincidence, but be extended in infinitum.” For
non-parallel lines also, if they are produced to a certain distance,
will not coincide. But to be produced infinitely, without coincidence,
expresses the property of parallels. Nor yet this absolutely, but to be
extended both ways infinitely, and not coincide. For it is possible
that non-parallel lines may also be produced one way infinitely, but
not the other; since, verging in this part, they are far distant from
mutual coincidence in the other. But the reason of this is, because two
right-lines cannot comprehend space; for if they verge to each other
both ways, this cannot happen. Besides this, he very properly considers
the right-lines as subsisting in the same plane. For if the one should
be in a subject plane, but the other in one elevated, they will not
mutually coincide according to every position, yet they are not on this
account parallel. The plane, therefore, should be one, and they should
be produced both ways infinitely, and not coincide in either part. For
with these conditions, the right-lines will be parallel. And agreeable
to this, Euclid defines parallel right-lines. But Posidonius says,
parallel lines are such as neither incline nor diverge in one plane;
but have all the perpendiculars equal which are drawn from the points
of the one to the other. But such lines as make their perpendiculars
always greater and less, will some time or other coincide, because
they mutually verge to each other. For a perpendicular is capable of
bounding the altitudes of spaces, and the distances of lines. On which
account, when the perpendiculars are equal, the distances of the right
lines are also equal; but when they are greater and less, the distance
also becomes greater and less, and they mutually verge in those parts,
in which the lesser perpendiculars are found. But it is requisite to
know, that non-coincidence does not entirely form parallel lines. For
the circumferences of concentric circles do not coincide: but it is
likewise requisite that they should be infinitely produced. But this
property is not only inherent in right, but also in other lines: for
it is possible to conceive spirals described in order about right
lines, which if produced infinitely together with the right lines,
will never coincide[185]. Geminus, therefore, makes a very proper
division in this place, affirming from the beginning, that of lines
some are bounded, and contain figure, as the circle and ellipsis,
likewise the cissoid, and many others; but others are indeterminate,
which may be produced infinitely, as the right-line, and the section
of a right-angled, and obtuse angled cone; likewise the conchoid
itself. But again, of those which may be produced in infinitum, some
comprehend no figure, as the right-line and the conic sections; but
others, returning into themselves, and forming figure, may afterwards
be infinitely produced. And of these some will not hereafter coincide,
which resist coincidence, how far soever they may be produced; but
others are coincident, which will some time or other coincide. But
of non-coincident lines, some are mutually in one plane; and others
not. And of non-coincidents subsisting in one plane, some are always
mutually distant by an equal interval; but others always diminish the
interval, as an hyperbola in its inclination to a right-line, and
likewise the conchoid[186]. For these, though they always diminish
the interval, never coincide. And they mutually converge, indeed,
but never perfectly nod to each other; which is indeed a theorem in
geometry especially admirable, exhibiting certain lines endued with a
non-assenting nod. But the right-lines, which are always distant by an
equal interval, and which never diminish the space placed between them
in one plane, are parallel lines. And thus much we have extracted from
the studies of the elegant Geminus, for the purpose of explaining the
present definition.


                       END OF THE FIRST VOLUME.


FOOTNOTES:

[1] The Grecian literature of this writer will now prove of real
utility; and the graces and the sublimities of PLATO will
soon be familiarised to the English reader, by a hand that I am
persuaded will not appear inferior to his great original. Let me also
be permitted to recommend his version of PLOTINUS on THE
BEAUTIFUL.

[2] i.e. Capable of parts.

[3] i.e. Not capable of parts.

[4] Dr. Young, in his Night Thoughts.

[5] See book the second, of Aristotle’s Metaphysics.

[6] Ennead vi. lib. vii.

[7] In his commentary on the 2d, 12th, and 13th books of Aristotle’s
Metaphysics, page 60. A Latin translation only of this invaluable work
is extant; but I have fortunately a copy in my possession, with the
version every where corrected by the learned Thomas Gale, and with
large extracts from the Greek.

[8] See Proclus on Plato’s Theology, p. 226.

[9] Ennead vi. lib. 6.

[10] In giving monadic number a subsistence in opinion, I have followed
the distribution of Proclus, in the conclusion of his comment on a
point; and, I think, not without sufficient reason. For since monadic
numbers are more immaterial than geometrical lines and figures, they
must have a more immaterial subsistence. But as they are correspondent
to matter, they cannot reside in the essential reasons of the soul;
nor can they subsist in the phantasy, because they are superior to
geometrical figures. It remains, therefore, that we must place them
between διάνοια or cogitation, and the phantasy; and this middle
situation is that of opinion. For cogitation, which Plato defines,
in his Sophista, to be an inward discourse, without voice, is an
energy of the rational soul, extending itself from propositions to
conclusions. And, according to Plato, in the same place, opinion is
the silent affirmation, or negation of διάνοια, or thought. Hence,
says he, “opinion is the conclusion of cogitation; but imagination,
the mutual mixture of sense and opinion.” So that opinion may, with
great propriety, be said to contain monadic number, to which it bears
the proportion of matter. And hence the reason is obvious, why the
Pythagoreans called the duad opinion.

[11]

     Ἄτροπον, ἀκαμάτον Δεκάδα κλείουσιν μιν ἁγιὴν,
    Ἀθάνατοί τε θεοὶ καὶ γηγενέεις ἃνθρωποι.

    Syrian. in Meta. Aristot. p. 113. Gr.

i.e. (According to the Pythagoreans) “the immortal gods and earth-born
men, call the venerable decad, immutable and unwearied.”

[12] Αυτὸς μὲν Πυθαγόρας ἐν τῷ ἱερῷ λόγῳ διαῤῥηδην μορφῶν καὶ ἰδεῶν
κράντορα τὸν ἀριθμόν ἔλεγεν εἶναι.

    Vid. Syrian. in Arist. Meta. p. 85. Gr.


[13] Φιλόλαος δέ, τῆς τῶν κοσμικὼν αἰωνίας διαμονῆς τὴν κρατιστεύουσαν
καὶ αὐτογειῆ συνοχὴν εἶναι ἀπεφήνατο τὸν ἀριθμόν.

    Syrian. in eodem loco.


[14] Οἱ δὲ περὶ Ἴππασον ἀκουσματικοὶ, ἀριθμόν εἶπον παράδειγμα πρῶτον
κοσμοποιίας. Καὶ πάλιν κριτικὸν κοσμουργοῦ θεοῦ ὄργανον.

    Jamb. in Nicomach. Arith. p. 11.


[15] In his Mathematical Lectures, page 48.

[16] In Arithmet. p. 23.

[17] In Aristot. Meta. p. 113. Gr. vel 59. b. Lat.

[18] For the tetrad contains all numbers within its nature, in the
manner of an exemplar; and hence it is, that in monadic numbers, 1, 2,
3, 4, are equal to ten.

[19] Notes to Letters on Mind, page 83.

[20] This bright light is no other than that of ideas themselves;
which, when it is once enkindled, or rather re-kindled in the soul,
becomes the general standard, and criterion of truth. He who possesses
this, is no longer the slave of opinion; puzzled with doubts, and lost
in the uncertainties of conjecture. Here the fountain of evidence is
alone to be found.--This is the true light, whose splendors can alone
dispel the darkness of ignorance, and procure for the soul undecaying
good, and substantial felicity. Of this I am certain, from my own
experience; and happy is he who acquires this invaluable treasure. But
let the reader beware of mixing the extravagancies of modern enthusiasm
with this exalted illumination. For this light is alone brought into
the mind by science, patient reflection, and unwearied meditation:
it is not produced by any violent agitation of spirits, or extasy of
imagination; for it is far superior to the energies of these: but it is
tranquil and steady, intellectual and divine. Avicenna, the Arabian,
was well acquainted with this light, as is evident from the beautiful
description he gives of it, in the elegant introduction of Ebn Tophail,
to the Life of Hai Ebn Yokdhan. “When a man’s desires (says he) are
considerably elevated, and he is competently well exercised in these
speculations, there will appear to him some small glimmerings of the
truth, as it were flashes of lightning, very delightful, which just
shine upon him, and then become extinct. Then the more he exercises
himself, the oftener will he perceive them, till at last he will
become so well acquainted with them, that they will occur to him
spontaneously, without any exercise at all; and then as soon as he
perceives any thing, he applies himself to the divine essence, so as to
retain some impression of it; then something occurs to him on a sudden,
whereby he begins to discern the _truth_ in every thing; till
through frequent exercise he at last attains to a perfect tranquillity;
and that which used to appear to him only by fits and starts, becomes
habitual, and that which was only a glimmering before, a constant
light; and he obtains a constant and steady knowledge.” He who desires
to know more concerning this, and a still brighter light, that arising
from an union with the supreme, must consult the eighth book of
Plotinus’ fifth Ennead, and the 7th and 9th of the sixth, and his book
on the Beautiful, of which I have published a translation.

[21] Lest the superficial reader should think this is nothing more than
declamation, let him attend to the following argument. If the soul
possesses another eye different from that of sense (and that she does
so, the sciences sufficiently evince), there must be, in the nature
of things, species accommodated to her perception, different from
feasible forms. For if our intellect speculates things which have no
real subsistence, such as Mr. Locke’s ideas, its condition must be much
more unhappy than that of the sensitive eye, since this is co-ordinated
to beings; but intellect would speculate nothing but illusions. Now, if
this be absurd, and if we possess an intellectual eye, which is endued
with a visive power, there must be forms correspondent and conjoined
with its vision; forms immoveable, indeed, by a corporeal motion, but
moved by an intellectual energy.

[22] The present section contains an illustration of almost all the
first book of Aristotle’s last Analytics. I have for the most part
followed the accurate and elegant paraphrase of Themistius, in the
execution of this design, as the learned reader will perceive: but I
have likewise everywhere added elucidations of my own, and endeavoured
to render this valuable work intelligible to the thinking mathematical
reader.

[23] See the twenty-eighth proposition of the first book of Euclid’s
Elements.

[24] We are informed by Simplicius, in his Commentary on Aristotle’s
third Category of Relation, “that though the quadrature of the
circle seems to have been unknown to Aristotle, yet, according to
Jamblichus, it was known to the Pythagoreans, as appears from the
sayings and demonstrations of Sextus Pythagoricus, who received (says
he) by succession, the art of demonstration; and after him Archimedes
succeeded, who invented the quadrature by a line, which is called
the line of Nicomedes. Likewise, Nicomedes attempted to square the
circle by a line, which is properly called τεταρτημόριον, or _the
quadrature_. And Apollonius, by a certain line, which he calls the
sister of the curve line, similar to a cockle, or tortoise, and which
is the same with the _quadratix_ of Nicomedes. Also Carpus wished
to square the circle, by a certain line, which he calls simply formed
from a twofold motion. And many others, according to Jamblichus, have
accomplished this undertaking in various ways.” Thus far Simplicius.
In like manner, Boethius, in his Commentary on the same part of
Aristotle’s Categories (p. 166.) observes, that the quadrature of
the circle was not discovered in Aristotle’s time, but was found out
afterwards; the demonstration of which (says he) because it is long,
must be omitted in this place. From hence it seems very probable,
that the ancient mathematicians applied themselves solely to squaring
the circle geometrically, without attempting to accomplish this by an
arithmetical calculation. Indeed, nothing can be more ungeometrical
than to expect, that if ever the circle be squared, the square to which
it is equal must be commensurable with other known rectilineal spaces;
for those who are skilled in geometry know that many lines and spaces
may be exhibited with the greatest accuracy, geometrically, though they
are incapable of being expressed arithmetically, without an infinite
series. Agreeable to this, Tacquet well observes (in lib. ii. Geom.
Pract. p. 87.) “Denique admonendi hic sunt, qui geometriæ, non satis
periti, sibi persuadent ad quadraturam necessarium esse, ut ratio lineæ
circularis ad rectam, aut circuli ad quadratum in numeris exhibeatur.
Is sane error valde crassus est, et indignus geometrâ, quamvis enim
irrationalis esset ea proportio, modo in rectis lineis exhibeatur,
reperta erat quadratura.” And that this quadrature is possible
geometrically, was not only the opinion of the above mentioned learned
and acute geometrician, but likewise of Wallis and Barrow; as may be
seen in the Mechanics of the former, p. 517 and in the Mathematical
Lectures of the latter, p. 194. But the following discovery will, I
hope, convince the liberal geometrical reader, that the quadrature of
the circle may be obtained by means of a circle and right-line only,
which we have no method of accomplishing by any invention of the
ancients or moderns. At least this method, if known to the ancients, is
now lost, and though it has been attempted by many of the moderns, it
has not been attended with success.

In the circle _g o e f_, let _g o_ be the quadrantal arch, and the
right-line _g x_ its tangent. Then conceive that the central point
_a_ flows uniformly along the radius _a e_, infinitely produced; and
that it is endued with an uniform impulsive power. Let it likewise be
supposed, that during its flux, radii emanate from it on all sides,
which enlarge themselves in proportion to the distance of the point
_a_ from its first situation. This being admitted, conceive that the
point _a_ by its impulsive power, through the radii _a n_, _a m_, &c.
acting every where equally on the arch _g o_, impells it into its equal
tangent arch _g r_. And when, by its uniform motion along the infinite
line _a_ φ, it has at the same time arrived at _b_, the centre of the
arch _g r_, let it impel in a similar manner the arch _g r_, into its
equal tangent arch _g s_, by acting every where equally through radii
equal to _b r_. Now, if this be conceived to take place infinitely
(since a circular line is capable of infinite remission) the arch _g
o_ will at length be unbent into its equal, the tangent line _g x_;
and the extreme point _o_, will describe by such a motion of unbending
a circular line _o x_. For since the same cause, acting every where
similarly and equally, produces every where similar and equal effects;
and the arch _g o_, is every where equally remitted or unbent, it
will describe a line similar in every part. Now, on account of the
simplicity of the impulsive motion, such a line must either be straight
or circular; for there are only three lines every where similar, i. e.
the right and circular line, and the cylindric helix; but this last,
as Proclus well observes in his following Commentary on the fourth
definition, is not a simple line, because it is generated by two simple
motions, the rectilineal and circular. But the line which bounds more
than two equal tangent arches cannot be a right line, as is well known
to all geometricians; it is therefore a circular line. It is likewise
evident, that this arch _o x_ is concave towards the point _g_: for if
not, it would pass beyond the chord _o x_, which is absurd. And again,
no arch greater than the quadrant can be unbent by this motion: for any
one of the radii, as _a p_ beyond _g o_, has a tendency from, and not
to the tangent _g x_, which last is necessary to our hypothesis. Now if
we conceive another quadrantal arch of the circle _g o e f_, that is
_g y_, touching the former in _g_ to be unbent in the same manner, the
arch _x y_ shall be a continuation of the arch _x o_; for if _γ x κ_ be
drawn perpendicular to _x g_, as in the figure, it shall be a tangent
in _x_ to the equal arches _y x_, _x o_; because it cannot fall within
either, without making the sine of some one of the equal arches, equal
to the right-line _x g_, which would be absurd. And hence we may easily
infer, that the centre of the arch _y x o_, is in the tangent line _x
g_. Hence too, we have an easy method of finding a tangent right-line
equal to a quadrantal arch: for having the points _y_, _o_ given, it is
easy to find a third point, as _s_; and then the circle passing through
the three points _o_, _s_, _y_, shall cut off the tangent _x g_,
equal to the quadrantal arch _g o_. And the point _s_ may be speedily
obtained, by describing the arch _g s_ with a radius, having to the
radius _a g_ the proportion of 6 to 4; for then _g s_ is the sixth part
of its whole circle, and is equal to the arch _g o_. And thus, from
this hypothesis, which, I presume, may be as readily admitted as the
increments and decrements of lines in fluxions, the quadrature of the
circle may be geometrically obtained; for this is easily found, when a
right-line is discovered equal to the periphery of a circle. I am well
aware the algebraists will consider it as useless, because it cannot be
accommodated to the farrago of an arithmetical calculation; but I hope
the lovers of the ancient geometry will deem it deserving an accurate
investigation; and if they can find no paralogism in the reasoning,
will consider it as a legitimate demonstration.

  [Illustration]

  [Illustration]

[25] Axioms have a subsistence prior to that of magnitudes and
mathematical numbers, but subordinate to that of ideas; or, in other
words, they have a middle situation between essential and mathematical
magnitude. For of the reasons subsisting in soul, some are more simple
and universal, and have a greater ambit than others, and on this
account approach nearer to intellect, and are more manifest and known
than such as are more particular. But others are destitute of all
these, and receive their completion from more ancient reasons. Hence it
is necessary (since conceptions are then true, when they are consonant
with things themselves) that there should be some reason, in which the
axiom asserting, _if from equals you take away equals_, _&c._
is primarily inherent; and which is neither the reason of magnitude,
nor number, nor time, but contains all these, and every thing in which
this axiom is naturally inherent. Vide Syrian. in Arith. Meta. p. 48.

[26] Geometry, indeed, wishes to speculate the impartible reasons
of the soul, but since she cannot use intellections destitute of
imagination, she extends her discourses to imaginative forms, and
to figures endued with dimension, and by this means speculates
immaterial reasons in these; and when imagination is not sufficient
for this purpose, she proceeds even to external matter, in which she
describes the fair variety of her propositions. But, indeed, even then
the principal design of geometry is not to apprehend sensible and
external form, but that interior vital one, resident in the mirror of
imagination, which the exterior inanimate form imitates, as far as its
imperfect nature will admit. Nor yet is it her principal design to
be conversant with the imaginative form; but when, on account of the
imbecility of her intellection, she cannot receive a form destitute of
imagination, she speculates the immaterial reason in the purer form of
the phantasy; so that her principal employment is about universal and
immaterial forms. Syrian. in Arist. Meta. p. 49.

[27] Syrianus, in his excellent Commentary on Aristotle’s Metaphysics,
(which does not so much explain Aristotle, as defend the doctrine of
ideas, according to Plato, from the apparent if not real opposition of
Aristotle to their existence), informs us that it is the business of
wisdom, properly so called, to consider immaterial forms or essences,
and their essential accidents. By the method of resolution receiving
the principles of being; by a divisive and and definitive method,
considering the essences of all things; but by a demonstrative process,
concluding concerning the essential properties which substances
contain. Hence (says he) because intelligible essences are of the
most simple nature, they are neither capable of definition nor
demonstration, but are perceived by a simple vision and energy of
intellect alone. But middle essences, which are demonstrable, exist
according to their inherent properties: since, in the most simple
beings, nothing is inherent besides their being. On which account we
cannot say that _this_ is their essence, and _that_ something
else; and hence they are better than definition and demonstration.
But in universal reasons, considered by themselves, and adorning a
sensible nature, essential accidents supervene; and hence demonstration
is conversant with these. But in material species, individuals, and
sensibles, such things as are properly accidents are perceived by the
imagination, and are present and absent without the corruption of their
subjects. And these again being worse than demonstrable accidents, are
apprehended by signs, not indeed by a wise man, considered as wise, but
perhaps by physicians, natural philosophers, and all of this kind.

[28] See Note to Chap. i. Book i. of the ensuing Commentaries.

[29] Page 227.

[30] Page 250.

[31] Methodus hæc cum algebrâ speciosâ facilitate contendit, evidentiâ
vero et demonstrationum elegantiâ eam longe superare videtur: ut abunde
constabit, si quis conferat hanc Apollonii doctrinam _de Sectione
Rationis_ cum ejusdem Problematis Analysi Algebraicâ, quam exhibuit
clarissimus Wallisius, tom. ii. Operum Math. cap. liv. p. 220.

[32] Verum perpendendum est, aliud esse problema aliqualiter resolutum
dare, quod modis variis, plerumque fieri potest, aliud methodo
elegantissimâ ipsum efficere; Analysi brevissimâ et simul perspicuâ,
Synthesi concinnâ et minime operosâ.

[33] In his Mathematical Lectures, p. 44.

[34] Lib. iv.

[35] Lib. i. p. 30.

[36] In Theæteto.

[37] In his most excellent work on Abstinence, lib. i. p. 22, &c.

[38] See the Excerpta of Ficinus from Proclus, on the first Alcibiades
of Plato; his Latin version only of which is extant. Ficini Opera, tom.
ii.

[39] Marinus, the author of the ensuing life, was the disciple of
Proclus; and his successor in the Athenian school. His philosophical
writings were not very numerous, and have not been preserved. A
commentary ascribed to him, on Euclid’s data, is still extant; but
his most celebrated work, appears to have been, the present life of
his master. It is indeed in the original elegant and concise; and may
be considered as a very happy specimen of philosophical biography.
Every liberal mind must be charmed and elevated with the grandeur and
sublimity of character, with which Proclus is presented to our view. If
compared with modern philosophical heroes, he appears to be a being of
a superior order; and we look back with regret on the glorious period,
so well calculated for the growth of the philosophical genius, and the
encouragement of exalted merit. We find in his life, no traces of the
common frailties of depraved humanity; no instances of meanness, or
instability of conduct: but he is uniformly magnificent, and constantly
good. I am well aware that this account of him will be considered by
many as highly exaggerated; as the result of weak enthusiasm, blind
superstition, or gross deception: but this will never be the persuasion
of those, who know by experience what elevation of mind and purity of
life the Platonic philosophy is capable of procuring; and who truly
understand the divine truths contained in his works. And the testimony
of the multitude, who measure the merit of other men’s characters by
the baseness of their own, is surely not to be regarded. I only add,
that our Philosopher flourished 412 years after Christ, according to
the accurate chronology of Fabricius; and I would recommend those who
desire a variety of critical information concerning Proclus, to the
Prolegomena prefixed by that most learned man to his excellent Greek
and Latin edition of this work, printed at London in 1703.

[40] Plato in Phædro. Meminit et Plutarch. VIII. Sympos. Suidas in
μήτοι. Fabricius.

[41] For a full account of the distribution of the virtues according to
the Platonists, consult the sentences of Porphyry, and the Prolegomena
of Fabricius to this work.

[42] See the sixth book of his Republic, and the Epinomis.

[43] We are informed by Fabricius, that the Platonic Olympiodorus in
his MS. Commentary on the Alcibiades of Plato, divides the orders of
the Gods, into ὑπερκόσμιοι, or super-mundane, which are separate from
all connection with body; and into ἐγκόσμιοι, or mundane. And that
of these, some are οὐράνιοι, or celestial, others αἰθέριοι, or, or
etherial, or πύριοι, fiery, others ἀέριοι, or aerial, others ἔνυδροι,
or watry, others χθόνιοι, or earthly; and others ὑποταρτάριοι, or
subterranean. But among the terrestrial, some are κλιματάρχαι, or
governors of climates, others πολιοῦχοι, or rulers over cities, and
others lastly κατοικίδιοι, or governors of houses.

[44] This epithet is likewise ascribed by Onomacritus to the Moon, as
may be seen in his hymn to that deity; and the reason of which we have
given in our notes to that hymn.

[45] Divine visions, and extraordinary circumstances, may be fairly
allowed to happen to such exalted geniuses as Proclus; but deserve
ridicule when ascribed to the vulgar.

[46] What glorious times! when it was considered as an extraordinary
circumstance for a teacher of rhetoric to treat a noble and wealthy
pupil as his domestic. When we compare them with the present, we can
only exclaim, _O tempora! O mores!_ Philosophy sunk in the ruins
of ancient Greece and Rome.

[47] Fabricius rightly observes, that this Olympiodorus is not the
same with the Philosopher of that name, whose learned commentaries, on
certain books of Plato, are extant in manuscript, in various libraries.
As in these, not only Proclus himself, but Damascius, who flourished
long after Proclus, is celebrated.

[48] Concerning the various mathematicians of this name, see Fabricius
in Bibliotheca Græca.

[49] The word in the original is λογικὰ, which Fabricius rightly
conjectures has in this place a more extensive signification than
either Logic, or Rhetoric: but I must beg leave to differ from that
great critic, in not translating it simply _philosophical_, as I
should rather imagine, Marinus intended to confine it to that part of
Aristotle’s works, which comprehends only logic, rhetoric, and poetry.
For the verb ἐξεμάνθανω, or _to learn_, which Marinus uses on
this occasion, cannot with propriety be applied to the more abstruse
writings of Aristotle.

[50] Hence Proclus was called, by way of eminence, διάδοχος Πλατωνικός,
or the Platonic Successor.

[51] Concerning Polletes, see Suidas; and for Melampodes, consult
Fabricius in Bibliotheca Græca.

[52] This Syrianus was indeed a most excellent philosopher, as we may
be convinced from his commentary on the metaphysics of Aristotle,
a Latin translation only of which, by one Hieronimus Bagolinus,
was published at Venice in 1558. The Greek is extant, according to
Fabricius, in many of the Italian libraries, and in the Johannean
library at Hamburg. According to Suidas, he writ a commentary on the
whole of Homer in six books; on Plato’s politics, in four books; and
on the consent of Orpheus, Pythagoras, and Plato, with the Chaldean
Oracles, in ten books. All these are unfortunately lost; and the
liberal few, are by this means deprived of treasures of wisdom, which
another philosophical age, in some distant revolution, is alone likely
to produce.

[53] Socrates, in the 6th book of Plato’s Republic, says, that from
great geniuses nothing of a middle kind must be expected; but either
great good, or great evil.

[54] The reader will please to take notice, that this great man is not
the same with Plutarch the biographer, whose works are so well known;
but an Athenian philosopher of a much later period.

[55] Aristotle’s philosophy, when compared with the discipline of
Plato is, I think, deservedly considered in this place as bearing the
relation of the proteleia to the epopteia in sacred mysteries. Now the
proteleia, or things previous to perfection, belong to the initiated,
and the mystics; the former of whom were introduced into some lighter
ceremonies only: but the mystics, were permitted to be present with
certain preliminary and lesser sacred concerns. On the other hand, the
epoptæ were admitted into the sanctuary of the greater sacred rites;
and became spectators of the symbols, and more interior ceremonies.
Aristotle indeed appears to be every where an enemy to the doctrine of
ideas, as understood by Plato; though they are doubtless the leading
stars of all true philosophy. However, the great excellence of his
works, considered as an introduction to the divine theology of Plato,
deserves the most unbounded commendation. Agreeable to this, Damascius
informs us that Isidorus the philosopher, “when he applied himself to
the more holy philosophy of Aristotle, and saw that he trusted more to
necessary reasons than to his own proper sense, yet did not entirely
employ a divine intellection, was but little solicitous about his
doctrine: but that when he had tasted of Plato’s conceptions, he no
longer deigned to behold him in the language of Pindar. But hoping he
should obtain his desired end, if he could penetrate into the sanctuary
of Plato’s mind, he directed to this purpose the whole course of his
application.” Photii Bibliotheca. p. 1034.

[56] _according to the oracle._

[57] Nothing is more celebrated by the ancients than that strict
friendship which subsisted among the Pythagoreans; to the exercise of
which they were accustomed to admonish each other, _not to divide
the god which they contained_, as Jamblichus relates, lib. i. c.
33. De Vita Pythagoræ. Indeed, true friendship can alone subsist in
souls, properly enlightened with genuine wisdom and virtue; for it then
becomes an union of intellects, and must consequently be immortal and
divine.

[58] Pythagoras, according to Damascius, said, that friendship was the
mother of all the political virtues.

[59] A genuine modern will doubtless consider the whole of Proclus’
religious conduct as ridiculously superstitious. And so, indeed,
at first sight, it appears; but he who has penetrated the depths
of ancient wisdom, will find in it more than meets the vulgar ear.
The religion of the Heathens, has indeed, for many centuries, been
the object of ridicule and contempt: yet the author of the present
work is not ashamed to own, that he is a perfect convert to it in
every particular, so far as it was understood and illustrated by
the Pythagoric and Platonic philosophers. Indeed the theology of
the ancient, as well as of the modern vulgar, was no doubt full of
absurdity; but that of the ancient philosophers, appears to be worthy
of the highest commendations, and the most assiduous cultivation.
However, the present prevailing opinions, forbid the defence of such
a system; for this must be the business of a more enlightened and
philosophic age. Besides, the author is not forgetful of Porphyry’s
destiny, whose polemical writings were suppressed by the decrees of
emperors; and whose arguments in defence of his religion were so
very futile and easy of solution, that, as St. Hierom informs us,
in his preface on Daniel, Eusebius answered him in twenty-five, and
Apollinaris in thirty volumes!

[60] See Proclus on Plato’s Politics, p. 399. Instit. Theolog. num.
196; and the extracts of Ficinus from Proclus’s commentary on the first
Alcibiades, p. 246. &c.

[61] Alluding to the beautiful description given of Ulysses, in the 3d
book of the Iliad, v. 222.

    Καί ἔπεα νιφάδεσιν ἐοικότα χειμερίησιν.

Which is thus elegantly paraphrased by Mr. Pope.

    But when he speaks, what elocution flows!
    Soft as the fleeces of descending snows
    The copious accents fall, with easy art;
    Melting they fall, and sink into the heart! &c.


[62] Concerning Domninus, see Photius and Suidas from Damascius in his
Life of Isidorus.

[63] Nicephorus, in his commentary on Synesius de Insomniis, p. 562.
informs us, that the hecatic orb, is a golden sphere, which has a
sapphire stone included in its middle part, and through its whole
extremity, characters and various figures. He adds, that turning this
sphere round, they perform invocations, which they call Jyngæ. Thus
too, according to Suidas, the magician Julian of Chaldea, and Arnuphis
the Egyptian, brought down showers of rain, by a magical power. And by
an artifice of this kind, Empedocles was accustomed to restrain the
fury of the winds; on which account he was called ἀλεξάνεμος, or a
chaser of winds.

[64] No opinion is more celebrated, than that of the metempsychosis
of Pythagoras: but perhaps, no doctrine is more generally mistaken.
By most of the present day it is exploded as ridiculous; and the few
who retain some veneration for its founder, endeavour to destroy the
literal, and to confine it to an allegorical meaning. By some of the
ancients this mutation was limited to similar bodies: so that they
conceived the human soul might transmigrate into various human bodies,
but not into those of brutes; and this was the opinion of Hierocles, as
may be seen in his comment on the Golden Verses. But why may not the
human soul become connected with subordinate as well as with superior
lives, by a tendency of inclination? Do not similars love to be united;
and is there not in all kinds of life, something similar and common?
Hence, when the affections of the soul verge to a baser nature, while
connected with a human body, these affections, on the dissolution of
such a body, become enveloped as it were, in a brutal nature, and the
rational eye, in this case, clouded with perturbations, is oppressed by
the irrational energies of the brute, and surveys nothing but the dark
phantasms of a degraded imagination. But this doctrine is vindicated
by Proclus with his usual subtilty, in his admirable commentary on the
Timæus, lib. v. p. 329, as follows, “It is usual, says he, to enquire
how souls can descend into brute animals. And some, indeed, think
that there are certain similitudes of men to brutes, which they call
savage lives: for they by no means think it possible that the rational
essence can become the soul of a savage animal. On the contrary,
others allow it may be sent into brutes, because all souls are of
one and the same kind; so that they may become wolves and panthers,
and ichneumons. But true reason, indeed, asserts that the human soul
way be lodged in brutes, yet in such a manner, as that it may obtain
its own proper life, and that the degraded soul may, as it were, be
carried above it, and be bound to the baser nature, by a propensity and
similitude of affection. And that this is the only mode of insinuation,
we have proved by a multitude of reasons, in our commentaries on the
Phædrus. But if it is requisite to take notice, that this is the
opinion of Plato, we add, that in his politics, he says, that the soul
of Thersites assumed an ape, but not the body of an ape: and in the
Phædrus, that the soul descends into a savage life, but not into a
savage body; for life is conjoined with its proper soul. And in this
place he says it is changed into a brutal nature: for a brutal nature
is not a brutal body, but a brutal life.”

[65] Pericles Lydus, a Stoic philosopher.

[66] Vide Pausan. lib. i. Atticorum, cap. 21. et 20.

[67] He means the Christians.

[68] Proclus was born in the year of Christ 412, on the 6th of the Ides
of February. But, for the sake of the astrologers, I have subjoined
the following figure from the Prolegomena of Fabricius to this life:
and though I am not skilled in the art myself, I am persuaded, from
the arguments of Plotinus, that it contains many general truths; but
when made subservient to particulars, is liable to great inaccuracy
and error. In short, its evidence is wholly of a physiognomic nature;
for such is the admirable order and connection of things, that
throughout the universe, one thing is signified by another, and wholes
are after a manner contained in their parts. So that the language of
the obscure and profound Heraclitus is perfectly just, when he says,
“You must connect the perfect and the imperfect, the agreeing and
the disagreeing, the consonant and the dissonant, and out of one all
things, and out of all things one.”

  Illustration: A Scheme of the situation of the Stars, such as it
  was at Byzantium, when the philosopher Proclus was born.

[69] It was formerly the custom of almost all nations, to have their
burial places in the suburbs, and not in the city itself.

[70] This eclipse happened, according to Fabricius, in A. C. 484. 19
Cal. Feb. at sun-rise.

[71] All the ancient theologists, among whom Plato holds a
distinguished rank, affirmed that the soul was of a certain middle
nature and condition between intelligibles and sensibles: agreeable
to which doctrine, Plotinus divinely asserts that she is placed in
the horizon, or in the boundary and isthmus, as it were, of eternal
and mortal natures; and hence, according to the Magi, she is similar
to the moon, one of whose parts is lucid, but the other dark. Now the
soul, in consequence of this middle condition, must necessarily be the
receptacle of all middle energies, both vital and gnostic; so that her
knowledge is inferior to the indivisible simplicity of intellectual
comprehension, but superior to the impulsive perceptions of sense.
Hence the mathematical genera and species reside in her essence, as
in their proper and natural region; for they are entirely of a middle
nature, as Proclus proves in this and the sixth following chapter.
But this doctrine of Plato’s, originally derived from Brontinus and
Archytas, is thus elegantly explained by that philosopher, in the
concluding part of the sixth book of his Republic. “Socrates, know
then, they are, as we say, two (the Good itself, and the Sun,) and
that the one reigns over the intelligible world, but the other over
the visible, not to say the heavens, lest I should deceive you by the
name. You comprehend then, these two orders of things, I mean the
visible and the intelligible?--Glauco. I do.--Socrates. Continue this
division then, as if it were a line divided into two unequal segments;
and each part again, i. e. the sensible and intelligible, divided
after a similar manner, and you will have evidence and obscurity
placed by each other. In the visible segment, indeed, one part will
contain images. But I call images, in the first place, shadows;
afterwards, the resemblances of things appearing in water, and in
dense, smooth, and lucid bodies, and every thing of this kind, if you
apprehend me?--Glauco. I apprehend you.--Socrates. Now conceive that
the other section comprehends the things, of which these images are
nothing more than similitudes, such as the animals around us, together
with plants, and whatever is the work of nature and art.--Glauco. I
conceive it.--Socrates. Do you consider this section then, as divided
into true and false? And that the hypothesis of opinion is to the
knowledge of science, as a resemblance to its original?--Glauco. I
do, very readily.--Socrates. Now then, consider how the section of
the intelligible is to be divided.--Glauco. How?--Socrates. Thus:
one segment is that which the soul enquires after, using the former
divisions as images, and compelled to proceed from hypotheses, not
to the principle, but to the conclusion. The other is that which
employs the cogitative power of the soul, as she proceeds from
an hypothesis to a principle no longer supposed, and, neglecting
images, advances through their _obscurity_ into _the light of_ ideas
themselves.--Glauco. I do not, in this, sufficiently understand
you.--Socrates. But again, for you will more easily understand me from
what has been already premised. I think you are not ignorant, that
those who are conversant in geometry, arithmetic, and the like, suppose
even and odd, together with various figures, and the three species of
angles, and other things similar to these, according to each method of
proceeding. Now, having established these, as hypotheses sufficiently
known, they conceive that no reason is to be required for their
position: but beginning from these, they descend through the rest, and
arrive at last, at the object of their investigation.--Glauco. This
I know perfectly well.--Socrates. This also you know, that they use
visible forms, and make them the subject of their discourse, at the
same time not directing their intellect to the perception of these,
but to the _originals_ they resemble; I mean the square itself, and
the diameter itself; and not to the figures they delineate. And thus,
other forms, which are represented by shadows and images in water,
are employed by them, merely as resemblances, while they strive to
behold that which can be seen by cogitation alone.--Glauco. You speak
the truth.--Socrates. This is what I called above a species of the
intelligible, in the investigation of which, the soul was compelled to
use hypotheses; not ascending to the principle, as incapable of rising
above hypotheses, but using the images formed from inferior objects,
to a similitude of such as are superior, and which are so conceived
and distinguished by opinion, as if they perspicuously contributed to
the knowledge of things themselves.--Glauco. I understand indeed, that
you are speaking of the circumstances which take place in geometry,
and her kindred arts.--Socrates. Understand now, that by the other
section of the intelligible, I mean that which reason herself reaches,
by her power of demonstrating, when no longer esteeming hypotheses for
principles, but receiving them in reality for hypotheses, she uses
them as so many steps and handles in her ascent, until she arrives at
that which is no longer hypothetical, the principle of the universe;
and afterwards descending, holding by ideas which adhere to the
principle, she arrives at the conclusion, employing nothing sensible
in her progress, but proceeding through ideas, and in these at last
terminating her descent.--Glauco. I understand you, but not so well
as I desire: for you seem to me to propose a great undertaking. You
endeavour, indeed, to determine that the portion of true being and
intelligible, which we speculate by the science of demonstration, is
more evident than the discoveries made by the sciences called arts;
because in the first hypotheses are principles, and their masters are
compelled to employ the eye of cogitation, and not the perceptions
of the senses. Yet, because they do not ascend to the principle,
but investigate from hypotheses, they seem to you not to have
intelligence concerning these, though they are intelligible, through
the light of the principle. But you seem to me to call the habit of
reasoning on geometrical and the like concerns, cogitation, rather
than intelligence, as if cogitation held the middle situation between
opinion and intellect.--Socrates. You understand me sufficiently well.
And again: with these four proportions take these four corresponding
affections of the soul: with the highest intelligence; with the second
cogitation; against the third set opinion; and against the fourth
assimilation, or imagination. Besides this, establish them in the
order of alternate proportion, so that they may partake of evidence,
in the same manner as their corresponding objects participate of
reality.” I have taken the liberty of translating this fine passage
differently from both Petvin and Spens; because they have neglected
to give the proper meaning of the word διάνοια, or cogitation, the
former translating it _mind_, and the _eye of the mind_, and by
this means confounding it with intellect; and the latter calling it
_understanding_. But it is certain that Plato, in this place, ranks
intellect as the first, on account of the superior evidence of its
perceptions; in the next place, cogitation; in the third, opinion;
and in the fourth, imagination. However, the reader will please to
remember, that by διάνοια, or cogitation, in the present work, is
understood that power of the soul which reasons from premises to
conclusions, and whose syllogistic energy, on active subjects, is
called prudence; and on such as are speculative, science. But for
farther information concerning its nature, see the dissertation
prefixed to this work, and the following fifth chapter.

[72] These two principles, _bound_ and _infinite_, will
doubtless be considered by the unthinking part of mankind, as
nothing more than general terms, and not as the most real of beings.
However, an accurate contemplation of the universe, will convince
every _truly_ philosophic mind of their reality. For the heavens
themselves, by the coherence and order of their parts, evince their
participation of bound. But by their prolific powers, and the
unceasing revolutions of the orbs they contain, they demonstrate their
participation of _infinity_. And the finite and perpetually
abiding forms with which the world is replete, bear a similitude _to
bound_: while, on the contrary, the variety of particulars, their
never-ceasing mutation, and the connection of more and less in the
communion of forms, represents an image of _infinity_. Add too,
that every natural species, by its form is similar to _bound_;
but by its matter, to _infinity_. For these two, form and matter,
depend on _bound_ and _infinity_, and are their ultimate
progressions. And each of these, indeed, participates of unity; but
form is the measure and bound of matter, and is more _one_. But
matter is in capacity all things, because it subsists by an emanation
from the first capacity, or the _infinite itself_.

[73] Of human disciplines, those alone deserve to be called sciences
which use no hypotheses, which resolve things into their principles,
which are conversant with true being, and elevate us to ideas
themselves. Dialectic is wholly of this kind (I mean the dialectic of
Plato); for this alone uses no suppositions, but, neglecting shadows
and images, raises us, by a sublime investigation, to the principle
of the universe; and on this account, deserves to be called the
very apex of disciplines. But we must not imagine, that by the word
dialectic here, is meant logic, or any part of logic, or that method
of disputation, by which we fabricate probable reasons; but we must
conceive it as signifying a discipline, endued with the greatest
acuteness; neglecting all hypotheses, truly soaring to primary causes,
and ultimately reposing in their contemplation. Plotinus has given us
most happy specimens of this method, in his books _on the genera of
being_.

[74] See note to the first chapter.

[75] I would particularly recommend this chapter to modern
mathematicians, most of whom, I am afraid, have never considered
whether or not the subjects of their speculation have any real
subsistence: though it is surely an enquiry worthy the earnest
attention of every liberal mind. For if the objects of mathematical
investigation are merely imaginary, I mean the point without parts,
the line without breadth, &c. the science, founded on these false
principles, must of course be entirely delusive. Indeed, an absolutely
true conclusion, can never flow from an erroneous principle, as from
its cause: as the stream must always participate of its source. I
mean such a conclusion as is demonstrated by the proper cause, πλὴν
οὐ διότι, ἀλλ’ ὅτι, says Aristotle, in his first Analytics; that is,
a syllogism from false principles will not prove the _why_, but
only simply _that it is_: indeed it can only simply prove _that
it is_, to him who admits the false propositions; because he who
allows the premises, cannot deny the conclusion, when the syllogism is
properly constructed. Thus we way syllogize in the first figure,

               Every thing white, is an animal:
               Every bird is white:
    Therefore, Every bird is an animal.

And the conclusion will be true, though the major and minor terms are
false; but then these terms are not the causes of the conclusion, and
we have an inference without a proof. In like manner, if mathematical
species are delusive and fictitious, the conclusions deduced from them
as principles, are merely hypothetical, and not demonstrative.

[76] Aristotle, in his last Analytics. The reader will please to
observe, that the whole force of this nervous, accurate, and elegant
reasoning, is directed against Aristotle; who seems unfortunately to
have considered, with the moderns, that mathematical species subsist
in the soul, by an abstraction from sensibles. See the preceding
Dissertation.

[77] Viz. 1, 2, 4, 8, 3, 9, 27. Concerning which, see lib. iii. of
Proclus’s excellent Commentary on the Timæus.

[78] Plato frequently, both in the Meno and elsewhere, shews that
science is Reminiscence; and I think not without the strongest
reason. For since the soul is immaterial, as we have demonstrated
in the dissertation to this work, she must be truly immortal, i. e.
both _a parte ante_, & _a parte post_. That she must be
eternal, indeed, with respect to futurity, if immaterial, is admitted
by all; and we may prove, with Aristotle, in his first book de Cœlo,
that she is immortal, likewise _a parte ante_, as follows.
Every thing without generation, is incorruptible, and every thing
incorruptible, is without generation: for that which is without
generation, has a necessity of existing infinitely _a parte ante_
(from the hypothesis); and therefore, if it possesses a capacity of
being destroyed, since there is no greater reason why it should be
corrupted now, rather than in some former period, it is endued with
a capacity of being destroyed and ceasing to be, in every instant of
infinite time, in which it necessarily is. In like manner, that which
is incorruptible, has a necessity of existing infinitely _a parte
post_; therefore, if it possesses a capacity of being generated,
since there is no greater reason why it should be generated now rather
than afterwards, it possesses a capacity of being generated, in every
instant of time, in which it necessarily is. If then the soul is
essentially immortal, with respect to the past and future circulations
of time; and if she is replete with forms or ideas of every kind, as
we have proved in the dissertation, she must, from her circulating
nature, have been for ever conversant in alternately possessing and
losing the knowledge of these. Now, the recovery of this knowledge by
science, is called by Plato, reminiscence; and is nothing more than
a renewed contemplation of those divine forms, so familiar to the
soul, before she became involved in the dark vestment of an earthly
body. So that we may say, with the elegant Maximus Tyrus, (Disser.
28.) “Reminiscence is similar to that which happens to the corporeal
eye, which, though always endued with a power of vision, yet darkness
sometimes obstructs its passage, and averts it from the perception of
things. Art therefore, approaches, which though it does not give to the
eye the power of vision, yet removes its impediments, and affords a
free egress to its rays. Conceive now, that our rational soul is such a
power of perceiving, which sees and knows the nature of beings. To this
the common calamity of bodies happens, that darkness spreading round
it, hurries away its aspect, blunts its sharpness, and extinguishes its
proper light. Afterwards, the art of reason approaches, which, like
a physician, does not bring or afford it a new science, but rouses
that which it possesses, though very slender, confused, and unsteady.”
Hence, since the soul, by her immersion in body, is in a dormant state,
until she is roused by science to an exertion of her latent energies;
and yet even previous to this awakening, since she contains the
vivid sparks, as it were, of all knowledge, which only require to be
ventilated by the wings of learning, in order to rekindle the light of
ideas, she may be said in this case to know all things as in a dream,
and to be ignorant of them with respect to vigilant perceptions. Hence
too, we may infer that time does not antecede our essential knowledge
of forms, because we possess it from eternity: but it precedes our
knowledge with respect to a production of these reasons into perfect
energy. I only add, that I would recommend the liberal English reader,
to Mr. Sydenham’s excellent translation of Plato’s Meno, where he
will find a familiar and elegant demonstration of the doctrine of
Reminiscence.

[79] Concerning this valuable work, entitled ΙΕΡΟ‘Σ ΛΟΓΟ’Σ, see the
Bibliotheca Græca of Fabricius, vol. i. p. 118 and 462, and in the
commentary of Syrianus on Aristotle’s metaphysics, p. 7, 71, 83, and
108, the reader will find some curious extracts from this celebrated
discourse; particularly in p. 83. Syrianus informs us, “that he who
consults this work will find all the orders both of Monads and Numbers,
without neglecting one, fully celebrated (ὐμνουμένας.)” There is no
doubt, but that Pythagoras and his disciples concealed the sublimest
truths, under the symbols of numbers; of which he who reads and
understands the writings of the Platonists will be fully convinced.
Hence Proclus, in the third book of his excellent commentary on the
Timæus, observes, “that Plato employed mathematical terms for the sake
of mystery and concealment, as certain veils, by which the penetralia
of truth might be secluded from vulgar inspection, just as the
theologists made fables, but the Pythagoreans symbols, subservient to
the same purpose: for in images we may speculate their exemplars, and
the former afford us the means of access to the latter.”

[80] Concerning this Geometric Number, in the 8th book of Plato’s
Republic, than which Cicero affirms there is nothing more obscure, see
the notes of Bullialdus to Theo. p. 292.

[81] I am sorry to say, that this part of the enemies to pure geometry
and arithmetic, are at the present time very numerous; conceptions
of utility in these sciences, extending no farther than the sordid
purposes of a mere animal life. But surely, if intellect is a part of
our composition, and the noblest part too, there must be an object of
its contemplation; and this, which is no other than truth in the most
exalted sense, must be the most noble and useful subject of speculation
to every rational being.

[82] In the 13th book of his Metaphysics, cap. iii.

[83] In. I. De Partib. Animalium, et in primo Ethic. cap. iii.

[84] See more concerning this in the Dissertation.

[85] Since number is prior to magnitude, the demonstrations of
arithmetic must be more intellectual, but those of geometry more
accommodated to the rational power. And when either arithmetic or
geometry is applied to sensible concerns, the demonstrations, from the
nature of the subjects, must participate of the obscurity of opinion.
If this is the case, a true mathematician will value those parts of his
science most, which participate most of evidence; and will consider
them as degraded, when applied to the common purposes of life.

[86] This division of the mathematical science, according to the
Pythagoreans, which is nearly coincident with that of Plato, is blamed
by Dr. Barrow in his Mathematical Lectures, p. 15. as being confined
within too narrow limits: and the reason he assigns for so partial a
division, is, “because, in Plato’s time, others were either not yet
invented, or not sufficiently cultivated, or at least were not yet
received into the number of the mathematical sciences.” But I must
beg leave to differ from this most illustrious mathematician in this
affair; and to assert that the reason of so confined a distribution
(as it is conceived by the moderns) arose from the exalted conceptions
these wise men entertained of the mathematical sciences, which they
considered as so many preludes to the knowledge of divinity, when
properly pursued; but they reckoned them degraded and perverted, when
they became mixed with sensible objects, and were applied to the common
purposes of life.

[87] That is, a right and circular line.

[88] I am afraid there are few in the present day, who do not consider
tactics as one of the most principal parts of mathematics; and who
would not fail to cite, in defence of their opinions, that great
reformer of philosophy, _as he is called_, Lord Bacon, commending
pursuits which come home to men’s businesses and bosoms. Indeed, if
what is lowest in the true order of things, and best administers to the
vilest part of human nature, is to have the preference, their opinion
is right, and Lord Bacon is a _philosopher_!

[89] By this is to be understood the art new called Perspective: from
whence it is evident that this art was not unknown to the ancients,
though it is questioned by the moderns.

[90] From hence it appears, that it is doubtful whether Plato is the
author of the dialogue called Epinomis; and I think it may with great
propriety be questioned. For though it bears evident marks of high
antiquity, and is replete with genuine wisdom, it does not seem to be
perfectly after Plato’s manner; nor to contain that great depth of
thought with which the writings of this philosopher abound. Fabricius
(in his Bibliotheca Græca, lib. i. p. 27.) wonders that Suidas should
ascribe this work to a philosopher who distributed Plato’s laws into
twelve books, because it was an usual opinion; from whence it seems,
that accurate critic had not attended to the present passage.

[91] This proximate conjunction of the mathematical sciences, which
Proclus considers as subordinate to dialectic, seems to differ from
that vertex of science in this, that the former merely embraces the
principles of all science, but the latter comprehends the universal
genera of being, and speculates the principle of all.

[92] In the Meno.

[93] This is certainly the true or philosophical employment of the
mathematical science; for by this means we shall be enabled to
ascend from sense to intellect, and rekindle in the soul that divine
light of truth, which, previous to such an energy, was buried in the
obscurity of a corporeal nature. But by a contrary process, I mean,
by applying mathematical speculations, to experimental purposes, we
shall blind the liberal eye of the soul, and leave nothing in its stead
but the darkness of corporeal vision, and the phantoms of a degraded
imagination.

[94] The design of the present chapter is to prove that the figures
which are the subjects of geometric speculation, do not subsist in
external and sensible matter, but in the receptacle of imagination,
or the matter of the phantasy. And this our philosopher proves with
his usual elegance, subtilty, and depth. Indeed, it must be evident
to every attentive observer, that sensible figures fall far short
of that accuracy and perfection which are required in geometrical
definitions: for there is no sensible circle perfectly round, since
the point from which it is described is not without parts; and, as
Vossius well observes, (de Mathem. p. 4.) there is not any sphere in
the nature of things, that only touches in a point, for with some
part of its superficies it always touches the subjected plane in a
line, as Aristotle shews Protagoras to have objected against the
geometricians. Nor must we say, with that great mathematician Dr.
Barrow, in his Mathematical Lectures, page 76, “that all imaginable
geometrical figures, are really inherent in every particle of matter,
in the utmost perfection, though not apparent to sense; just as the
effigies of Cæsar lies hid in the unhewn marble, and is no new thing
made by the statuary, but only is discovered and brought to sight by
his workmanship, i. e. by removing the parts of matter by which it is
overshadowed and involved. Which made Michael Angelus, the most famous
carver, say, _that sculpture was nothing but a purgation from things
superfluous. For take all that is superfluous_, (says he) _from
the wood or stone, and the rest will be the figure you intend_. So,
if the hand of an angel (at least the power of God) should think fit to
polish any particle of matter, without vacuity, a spherical superficies
would appear to the eyes, of a figure exactly round; not as created
anew, but as unveiled and laid open from the disguises and covers of
its circumjacent matter.” For this would be giving a perfection to
sensible matter, which it is naturally incapable of receiving: since
external body is essentially full of pores and irregularities, which
must eternally prevent its receiving the accuracy of geometrical body,
though polished by the hand of an angel. Besides, what polishing
would ever produce a point without parts, and a line without breadth?
For though body may be reduced to the greatest exility, it will not
by this means ever pass into an incorporeal nature, and desert its
triple dimension. Since external matter, therefore, is by no means the
receptacle of geometrical figures, they must necessarily reside in the
catoptric matter of the phantasy, where they subsist with an accuracy
sufficient for the energies of this science. It is true, indeed, that
even in the purer matter of imagination, the point does not appear
perfectly impartible, nor the line without latitude: but then the
magnitude of the point, and the breadth of the line is indefinite, and
they are, at the same time, unattended with the qualities of body, and
exhibit to the eye of thought, magnitude alone. Hence, the figures in
the phantasy, are the proper recipients of that universal, which is
the object of geometrical speculation, and represent, as in a mirror,
the participated subsistence of those vital and immaterial forms which
essentially reside in the soul.

[95] This division is elegantly explained by Ammonius, (in Porphyr. p.
12.) as follows, “Conceive a seal-ring, which has the image of some
particular person, for instance, of Achilles, engraved in its seal, and
let there be many portions of wax, which are impressed by the ring.
Afterwards conceive that some one approaches, and perceives all the
portions of wax, stamped with the impression of this one ring, and
keeps the impression of the ring in his mind: the seal engraved in the
ring, represents the universal, prior to the many: the impression in
the portions of wax, the universal in the many: but that which remains
in the intelligence of the beholder, may be called the universal,
after and posterior to the many. The same must we conceive in genera
and species. For that best and most excellent artificer of the world,
possesses within himself the forms and exemplars of all things: so that
in the fabrication of man, he looks back upon the form of man resident
in his essence, and fashions all the rest according to its exemplar.
But if any one should oppose this doctrine, and assert that the forms
of things do not reside with their artificer, let him attend to the
following arguments. The artificer either knows, or is ignorant of that
which he produces: but he who is ignorant will never produce any thing.
For who will attempt to do that, which he is ignorant how to perform?
since he cannot act from an irrational power like nature, whose
operations are not attended with animadversion. But if he produces
any thing by a certain reason, he must possess a knowledge of every
thing which he produces. If, therefore, it is not impious to assert,
that the operations of the Deity, like those of men, are attended with
knowledge, it is evident that the forms of things must reside in his
essence: but forms are in the demiurgus, like the seal in the ring;
and these forms are said to be prior to the many, and separated from
matter. But the species man, is contained in each particular man, like
the impression of the seal in the wax, and is said to subsist in the
many, without a separation from matter. And when we behold particular
men, and perceive the same form and effigy in each, that form seared
in our soul, is said to be after the many, and to have a posterior
generation: just as we observed in him, who beheld many seals impressed
in the wax from one and the same ring. And this one, posterior to the
many, may be separated from body, when it is conceived as not inherent
in body, but in the soul: but is incapable of a real separation from
its subject.” We must here, however, observe, that when Ammonius speaks
of the knowledge of the Deity, it must be conceived as far superior
to ours. For he possesses a nature more true than all essence, and a
perception clearer than all knowledge. And as he produced all things
by his unity, so by an ineffable unity of apprehension, he knows the
universality of things.

[96] In lib. vii. Metaphys. 35 & 39.

[97] In lib. iii. de Anima, tex. 20.

[98] That is, geometry first speculates the circle delineated on
paper, or in the dust: but by the medium of the circular figure in
the phantasy, contemplates the circle resident in cogitation; and by
that universal, or circular reason, participated in the circle of the
phantasy, frames its demonstrations.

[99] In his first Analytics, t. 42. See the Dissertation to this work.

[100] Such as the proportion of the diagonal of a square to its side;
and that of the diameter of a circle, to the periphery.

[101] The gnomons, from which square numbers are produced, are odd
numbers in a natural series from unity, i. e. 1, 3, 5, 7, 9, 11, &c.
for these, added to each other continually, produce square numbers
_ad infinitum_. But these gnomons continually decrease from the
highest, and are at length terminated by indivisible unity.

[102] This doctrine of ineffable quantities, or such whose proportion
cannot be expressed, is largely and accurately discussed by Euclid, in
the tenth book of his Elements: but its study is neglected by modern
mathematicians, _because it is of no use_, that is, because it
contributes to nothing mechanical.

[103] This proposition is the 11th of the second book: at least,
the method of dividing a line into extreme and mean proportion, is
immediately deduced from it; which is done by Euclid, in the 30th, of
the sixth book. Thus, Euclid shews (11. 2.) how to divide the line (A
G B)[diagram] A B, so that the rectangle under the whole A B, and
the segment G B, may be equal to the square made from A G: for when
this is done, it follows, that as A B is to A G, so is A G to G B; as
is well known. But this proposition, as Dr. Barrow observes, cannot be
explained by numbers; because there is not any number which can be so
divided, that the product from the whole into one part, may be equal to
the square from the other part.

[104] All polygonous figures, may, it is well known, be resolved
into triangles; and this is no less true of polygonous numbers,
as the following observations evince. All number originates from
indivisible unity, which corresponds to a point: and it is either
linear, corresponding to a line; or superficial, which corresponds to
a superficies; or solid, which imitates a geometrical solid. After
unity, therefore, the first of linear numbers is the duad; just as
every finite line is allotted two extremities. The triad is the first
of superficial numbers; as the triangle of geometrical figures. And
the tetrad, is the first of solids; because a triangular pyramid, is
the first among solid numbers, as well as among solid figures. As,
therefore, the monad is assimilated to the point, so the duad to the
line, the triad to the superficies, and the tetrad to the solid. Now,
of superficial numbers, some are triangles, others squares, others
pentagons, hexagons, heptagons, &c. Triangular numbers are generated
from the continual addition of numbers in a natural series, beginning
from unity. Thus, if the numbers 1, 2, 3, 4, 5, &c. be added to each
other continually, they will produce the triangular numbers 1, 3, 6,
10, 15, &c. and if every triangular number be added to its preceding
number, it will produce a square number. Thus 3 added to 1 makes 4; 6
added to 3 is equals 9; 10 added to 6 is equal to 16; and so of the
rest. Pentagons, are produced from the junction of triangular and
square numbers, as follows. Let there be a series of triangular numbers
1, 3, 6, 10, 15, &c.

And of squares 1, 4, 9, 16, 25, &c.

Then the second square number, added to the first triangle, will
produce the first pentagon from unity, i.e. 5. The third square added
to the second triangle, will produce the second pentagon, i.e. 12;
and so of the rest, by a similar addition. In like manner, the second
pentagon, added to the first triangle, will form the first hexagon from
unity; the third pentagon and the second triangle, will form the second
hexagon, &c. And, by a similar proceeding, all the other polygons may
be obtained.

[105] Intellections are universally correspondent to their objects,
and participate of evidence or the contrary, in proportion as their
subjects are lucid or obscure. Hence, Porphyry, in his sentences,
justly observes, that “we do not understand in a similar manner with
all the powers of the soul, but according to the particular essence
of each. For with the intellect we understand intellectually; and
with the soul, rationally: our knowledge of plants is according to a
seminal conception; our understanding of bodies is imaginative; and our
intellection of the divinely solitary principle of the universe, who is
above all things, is in a manner superior to intellectual perception,
and by a super-essential energy.” Ἀφορμαὶ πρὸς τὰ Νοητὰ, (10.) So that,
in consequence of this reasoning, the speculations of geometry are then
most true, when most abstracted from sensible and material natures.

[106] See Plutarch, in the life of Marcellus.

[107] In lib. i. de Cælo, tex. 22. et lib. i. Meteo. cap. 3. Aristotle
was called demoniacal by the Platonic philosophers, in consequence of
the encomium bestowed on him by his master, Plato, “That he was the
dæmon of nature.” Indeed, his great knowledge in things subject to
the dominion of nature, well deserved this encomium; and the epithet
_divine_, has been universally ascribed to Plato, from his
profound knowledge of the intelligible world.

[108] Εἰς νοῦν, is wanting in the original, but is supplied by the
excellent translation of Barocius.

[109] Ἀλόγων, in the printed Greek, which Fabricius, in his Bibliotheca
Græca, vol. i. page 385, is of opinion, should be read ἀναλόγων; but
I have rendered the word according to the translation of Barocius,
who is likely to have obtained the true reading, from the variety of
manuscripts which he consulted.

[110] The quadrature of the Lunula is as follows.

  [Illustration]

Let A B C be a right-angled triangle, and B A C a semi-circle on the
diameter B C: B N A a semi-circle described on the diameter A B; A M
C a semi-circle described on the diameter A C. Then the semi-circle B
A C is equal to the semi-circle B N A, and A M C together: (because
circles are to each other as the squares of their diameters, 31, 6.)
If, therefore, you take away the two spaces B A, A C common on both
sides, there will remain the two lunulas B N A, A M C, bounded on both
sides with circular lines, equal to the right-angled triangle B A C.
And if the line B A, be equal to the line A C, and you let fall a
perpendicular to the hypotenuse B C, the triangle B A O will be equal
to the lunular space B N A, and the triangle C O A will be equal to
the lunula C M A. Those who are curious, may see a long account of an
attempt of Hippocrates to square the circle, by the invention of the
lunulas, in Simplicius on Aristotle’s Physics, lib. i.

[111] So Barocius reads, but Fabricius Μεδμᾶιος.

[112] i. e. The five regular bodies, the pyramid, cube, octaedron,
dodecaedron and icosaedron; concerning which, and their application to
the theory of the universe, see Kepler’s admirable work, De Harmonia
Mundi.

[113] It may be doubted whether the optics and catoptrics, ascribed to
Euclid in the editions of his works are genuine: for Savil, and Dr.
Gregory, think them scarcely worthy so great a man.

[114] There are two excellent editions of this work, one by Meibomius,
in his collection of ancient authors on harmony; and the other by Dr.
Gregory, in his collection of Euclid’s works.

[115] This work is most probably lost. See Dr. Gregory’s Euclid.

[116] All this is shewn by Proclus in the following Commentaries; and
is surely most admirable and worthy the investigation of every liberal
mind; but I am afraid modern mathematicians very little regard such
knowledge, because it cannot be applied to practical and mechanical
purposes.

[117] This work is unfortunately lost.

[118] Because this is true only in isosceles and equilateral triangles.

[119] This follows from the 32d proposition of the first book of
Euclid; and is demonstrated by Dr. Barrow, in his scholium to that
proposition.

[120] The method of constructing these is shewn by our philosopher,
in his comment on the first proposition, as will appear in the second
volume of this work.

[121] The reader will please to observe, that the definitions are,
indeed, hypotheses, according to the doctrine of Plato, as may be seen
in the note to chap, i. book I. of this work.

[122] In his last Analytics. See the preceding Dissertation.

[123] That part of this work enclosed within the brackets, is wanting
in the original; which I have restored from the excellent version of
Barocius. The philosophical reader, therefore, of the original, who may
not have Barocius in his possession, will, I hope, be pleased, to see
so great a vacancy supplied; especially, as it contains the beginning
of the commentary on the definition of a point.

[124] I do not find this ænigma among the Pythagoric symbols which are
extant; so that it is probably no where mentioned but in the present
work. And I am sorry to add, that a _figure and three oboli_, in
too much the general cry of the present times.

[125] The present Comment, and indeed most of the following, eminently
evinces the truth of Kepler’s observation, in his excellent work,
_De Harmonia Mundi_, p. 118. For, speaking of our author’s
composition in the present work, which he every where admires and
defends, he remarks as follows, “oratio fluit ipsi torrentis instar,
ripas inundans, et cæca dubitationum vada gurgitesque occultans, dum
mens plena majestatis tantarum rerum, luctatur in angustiis linguæ, et
conclusio nunquam sibi ipsi verborum copiâ satisfaciens, propositionum
simplicitatem excedit.” But Kepler was skilled in the Platonic
philosophy, and appears to have been no less acquainted with the great
depth of our author’s mind than with the magnificence and sublimity of
his language. Perhaps Kepler is the only instance among the moderns,
of the philosophical and mathematical genius being united in the same
person.

[126] That is, the reason of a triangular figure (for instance) in the
phantasy, or triangle itself, is superior to the triangular nature
participated in that figure.

[127] In the tenth book of his Republic.

[128] See the Hymn to the Mother of the Gods, in my translation of the
Orphic Initiations.

[129] The philosopher here seems to contradict what he asserts in the
end of his comment on the 13th Definition: for there he asserts, that
the circle is a certain plane space. Perhaps he may be reconciled, by
considering, that as the circle subsists most according to bound, when
we speculate its essence in this respect we may define it according
to the circumference, which is the cause of its bound. But when we
consider it as participating of infinity also, though not in so eminent
a degree, and view it from its emanations from the centre as well as in
its regressions, we may define it a plane space.

[130] That is, the essential one of the soul is the mother of number;
but that which subsists in opinion is nothing more than the receptacle
of the former; just as matter is the seat of all forms. For a farther
account of the subsistence of numbers, see the first section of the
preceding Dissertation.

[131] That is, number composed from units.

[132] This sentence within the brackets, is wholly omitted in the
printed Greek.

[133] In i. De Cælo.

[134] This sentence within the brackets, which is very imperfect in the
Greek, I have supplied from the excellent translation of Barocius. In
the Greek there is nothing more than λὲγω δὲ ἑνὸν τῂν γραμμὴν δυαδός
πρὸς τὸ στερεόν.

[135] In the Greek, γὰρ ἡ μονὰς ἐκεῖ πρῶτον, ὅπου πατρικὴ μονάς ἐστι
φησὶ τὸ λόγιον. The latter part only of this oracle, is to be found
in all the printed editions of the Zoroastrian oracles; though it is
wonderful how this omission could escape the notice of so may able
critics, and learned men. It seems probable, from hence, that it is
only to be found perfect in the present work.

[136] The word τανάη, is omitted in the Greek.

[137] This and the following problems, are the 1st, 22d, and 12th
propositions of the first book. But in the two last, instead of the
word ἄπειρος or infinite, which is the term employed by Euclid, Mr.
Simson, in his edition of the Elements, uses the word unlimited. But
it is no unusual thing with this great geometrician, to alter the
words of Euclid, when they convey a philosophical meaning; as we shall
plainly evince in the course of these Commentaries. He certainly
deserves the greatest praise for his zealous attachment to the ancient
geometry: but he would (in my opinion) have deserved still more, had
he been acquainted with the Greek philosophy; and fathomed the depth
of Proclus; for then he would never have attempted to restore Euclid’s
Elements, by depriving them of some very considerable beauties.

[138] This is doubtless the reason why the proportion between a right
and circular line, cannot be exactly obtained in numbers; for on this
hypothesis, they must be incommensurable quantities; because the one
contains property essentially different from the other.

[139]

  [Illustration]

The cornicular angle is that which is made from the periphery of a
circle and its tangent; that is, the angle comprehended by the arch
L A, and the right line F A, which Euclid in (16. 3.) proves to be
less than any right-lined angle. And from this admirable proposition
it follows, by a legitimate consequence, that any quantity may
be continually and infinitely increased, but another infinitely
diminished; and yet the augment of the first, how great soever it
may be, shall always be less than the decrement of the second: which
Cardan demonstrates as follows. Let there be proposed an angle of
contact B A E, and an acute angle H G I. Now if there be other lesser
circles described A C, A D, the angle of contact will be evidently
increased. And if between the right lines G H, G I, there fall other
right lines G K, G L, the acute angle shall be continually diminished:
yet the angle of contact, however increased, is always less than the
acute angle, however diminished. Sir Isaac Newton likewise observes,
in his Treatise on Fluxions, that there are angles of contact made
by other curve lines, and their tangents infinitely less than those
made by a circle and right line; all which is demonstrably certain:
yet, such is the force of prejudice, that Mr. Simson is of opinion,
with Vieta, that this part of the 16th proposition is adulterated; and
that the space made by a circular line and its tangent, is no angle.
At least his words, in the note upon this proposition, will bear such
a construction. Peletarius was likewise of the same opinion; but is
elaborately confuted by the excellent Clavius, as may be seen in his
comment on this proposition. But all the difficulties and paradoxes in
this affair, may be easily solved and admitted, if we consider, with
our philosopher, that the essence of an angle does not subsist in ether
quantity, quality, or inclination, taken singly, but in the aggregate
of them all. For if we regard the inclination of a circular line to
its tangent, we shall find it possess the property, by which Euclid
defines an angle: if we respect its participation of quantity, we shall
find it capable of being augmented and diminished; and if we regard
it as possessing a peculiar quality, we shall account for its being
incommensurable with every right-lined angle. See the Comment on the
8th Definition.

[140] In i. De Cælo.

[141] It is from this cylindric spiral that the screw is formed.

[142]

  [Illustration]

The present very obscure passage, may be explained by the following
figure. Let A B C, be a right angle, and D E the line to be moved,
which is bisected in G. Now, conceive it to be moved along the lines A
B, B C, in such a manner, that the point D may always remain in A B,
and the point E in B C. Then, when the line D E, is in the situations
_d e_, _δ ε_, the point G, shall be in _g, γ_, and these
points G, _g, γ_, shall be in a circle. And any other point F in
the line D E, will, at the same time, describe an ellipsis; the greater
axis being in the line A B, when the point F is between D and G; and in
the line B C, when the point F is between G and E.

[143] That is, the soul of the world.

[144] In Timæo.

[145] The ellipsis.

[146] The cissoid. For the properties of this curve, see Dr. Wallis’s
treatise on the cycloid, p. 81.

[147] The conchoid.

[148] Thus, a right line, when considered as the side of a
parallelogram, moving circularly, generates a cylindrical superficies:
when moving circularly, as the side of a triangle, a conical surface;
and so in other lines, the produced superficies varying according to
the different positions of their generative lines.

[149] Inv ii. De Rep.

[150] In multis locis.

[151] This definition is the same with that which Mr. Simson has
adopted instead of Euclid’s, expressed in different words: for he says,
“a plane superficies is that in which any two points being taken, the
straight line between them lies wholly in that superficies.” But he
does not mention to whom he was indebted for the definition; and this,
doubtless, because he considered _it was not worth while to relate
the trifles of Proclus at full length_: for these are his own words,
in his note to proposition 7, book i. Nor has he informed us in what
respect Euclid’s definition is _indistinct_.

[152] In the Greek ἐννοιὰς, but it should doubtless be read εἰκόνας,
_images_, as in the translation of Barocius.

[153] Mr. Simson, in his note on this definition, supposes it to
be the addition of some less skilful editor; on which account, and
because it is quite useless (in his opinion) he distinguishes it from
the rest by inverted double commas. But it is surely strange that the
definition of angle in general should be accounted useless, and the
work of an unskilful geometrician. Such an assertion may, indeed, be
very suitable to a professor of experimental philosophy, who considers
the useful as inseparable from practice; but is by no means becoming
a restorer of the liberal geometry of the ancients. Besides, Mr.
Simson seems continually to forget that Euclid was of the Platonic
sect; and consequently was a philosopher as well as a mathematician.
I only add, that the commentary on the present definition is, in my
opinion, remarkably subtle and accurate, and well deserves the profound
attention of the greatest geometricians.

[154] For a philosophical discussion of the nature of quality and
quantity, consult the Commentaries of Ammonius, and Simplicius on
Aristotle’s Categories, Plotinus on the genera of beings, and Mr.
Harris’s Philosophical Arrangements.

[155] That is, the ellipsis.

[156] That is, they are either right, acute, or obtuse.

[157] This oracle is not mentioned by any of the collectors of the
Zoroastrian oracles.

[158] This, indeed, must always be the case with those geometricians,
who are not at the same time, philosophers; a conjunction no less
valuable than rare. Hence, from their ignorance of principles and
intellectual concerns, when any contemplative enquiry is proposed,
they immediately ask, in what its utility consists; considering every
thing as superfluous, which does not contribute to the solution of some
practical problem.

[159] Concerning the soul’s descent into body, see lib. ix., Ennead
iv. of Plotinus; and for the method by which she may again return to
her pristine felicity, study the first book of Porphyry’s Treatise on
Abstinence.

[160] This Definition too, is marked by Mr. Simson with inverted
commas, as a symbol of its being interpolated. But for what reason
I know not, unless because it is useless, that is, because it is
_philosophical_!

[161] That is, the various species of forms, with which the four
elements are replete.

[162] That is, the circle.

[163] An admirer of the moderns, and their pursuits, will doubtless
consider all this as the relics of heathen superstition and ignorance;
and will think, perhaps, he makes a great concession in admitting
the existence of one supreme god, without acknowledging a multitude
of deities subordinate to the first. For what the ancients can urge
in defence of this obsolete opinion, I must beg leave to refer the
reader to the dissertation prefixed to my translation of Orpheus; in
addition to which let him attend to the following considerations. Is it
possible that the machinery of the gods in Homer could be so beautiful,
if such beings had no existence? Or can any thing be beautiful which
is destitute of all reality? Do not things universally please in
proportion as they resemble reality? Perhaps it will be answered,
that the reverse of this is true, and that fiction more generally
pleases than truth, as is evident from the great avidity with which
romances are perused. To this I reply, that fiction itself ceases to be
pleasing, when it supposes absolute impossibilities: for the existence
of genii and fairies cannot be proved impossible; and these compose all
the marvellous of romance. This observation is verified in Spencer’s
Fairy Queen: for his allegories, in which the passions are personified,
are tedious and unpleasant, because they are not disguised under the
appearance of reality: while the magic of Circe, the bower of Calypso,
the rocks of Scylla and Charybdis, and the melody of the Syrens, in the
Odyssey of Homer, though nothing but allegories, universally enchant
and delight, because they are covered with the semblance of truth.
It is on this account that Mikon’s battles in heaven are barbarous
and ridiculous in the extreme; for every one sees the impossibility
of supposing gun-powder and cannons in the celestial regions: the
machinery is forced and unnatural, contains no elegance of fancy, and
is not replete with any mystical information. On the contrary, Homer’s
machinery is natural and possible, is full of dignity and elegance, and
is pregnant with the sublimest truths; it delights and enobles the mind
of the reader, astonishes him with its magnificence and propriety, and
animates him with the fury of poetic inspiration. And this, because it
is possible and true.

[164] The sentence within the brackets is omitted in the Greek.

[165] That is, the circular form proceeds from _bound_, but
right-lined figures from _infinity_.

[166] That is, the number three.

[167] In Timæo.

[168] πρὸς ὃ, or, _to which_, is wanting in the original, and in
all the published collections of the Zoroastrian oracles.

[169] That is Jupiter, who is called triadic, because he proceeds from
Saturn and Rhea; and because his government is participated by Neptune
and Pluto, for each of these is called Jupiter by Orpheus.

[170] This sentence, within the brackets, is omitted in the printed
Greek.

[171]

  [Illustration: Fig. I.]

  [Illustration: Fig. II.]

Thus let a part A E B cut off by the diameter A B (fig. I.) of the
circle A E B D be placed on the other part A D B, as in fig. II.. Then,
if it is not equal to the other part, either A E B will fall within A D
B, or A D B within A E B: but in either case, C E will be equal to C D,
which is absurd.

[172] This objection is urged by Philoponus, in his book against
Proclus on the eternity of the world; but not, in my opinion, with
any success. See also Simplicius, in his third digression against
Philoponus, in his commentary on the 8th book of Aristotle’s Physics.

[173] This definition is no where extant but in the commentaries of
Proclus. Instead of it, in almost all the printed editions of Euclid,
the following is substituted. _A segment of a circle is the figure
contained by a diameter, and the part of the circumference cut off by
the diameter._ This Mr. Simson has marked with commas, as a symbol
of its being interpolated: but he has taken no notice of the different
reading in the commentaries of Proclus. And what is still more
remarkable, this variation is not noticed by any editor of Euclid’s
Elements, either ancient or modern.

[174] As in every hyperbola.

[175] The Platonic reader must doubtless be pleased to find that Euclid
was deeply skilled in the philosophy of Plato, as Proclus every where
evinces. Indeed, the great accuracy, and elegant distribution of these
Elements, sufficiently prove the truth of this assertion. And it is no
inconsiderable testimony in favour of the Platonic philosophy, that its
assistance enabled Euclid to produce such an admirable work.

[176] Concerning these crowns, or annular spaces, consult the great
work of that very subtle and elegant mathematician Tacquet, entitled
_Cylindrica et Annularia_.

[177] In the preceding tenth commentary.

[178] This in consequence of every triangle possessing angles alone
equal to two right.

[179] This too, follows from the same cause as above.

[180] Thus the following figure A B D C has four sides, and but three
angles.

  [Illustration]

[181] The Greek in this place is very erroneous, which I have restored
from the version of Barocius.

[182] For the Greek word ῥόμβος is derived from the verb ῥέμβω, which
signifies to have a circumvolute motion.

[183] See the Orphic Hymns of Onomacritus to these deities; my
translation of which I must recommend to the English reader, because
there is no other.

[184] These twelve divinities, of which Jupiter is the head, are,
_Jupiter_, _Neptune_, _Vulcan_, _Vesta_, _Minerva_, _Mars_, _Ceres_,
_Juno_, _Diana_, _Mercury_, _Venus_, and _Apollo_. The first triad of
these is demiurgic, the second comprehends guardian deities, the third
is vivific, or zoogonic, and the fourth contains elevating gods. But,
for a particular theological account of these divinities, study Proclus
on Plato’s Theology, and you will find their nature unfolded, in page
403, of that admirable work.

[185] For it is easy to conceive a cylindric spiral described about a
right-line, so as to preserve an equal distance from it in every part;
and in this case the spiral and right-line will never coincide though
infinitely produced.

[186]

  [Illustration]

As the conchoid is a curve but little known, I have subjoined the
following account of its generation and principal property. In
any given right line A P, call P the pole, A the vertex, and any
intermediate point C the centre of the conchoid: likewise, conceive
an infinite right line C H, which is called a rule, perpendicular to
A P. Then, if the right line A _p_ continued at _p_ as much
as is necessary, is conceived to be so turned about the abiding pole
_p_, that the point C may perpetually remain in the right line C
H, the point A will describe the curve A _o_, which the ancients
called a conchoid.

In this curve it is manifest (on account of the right line P O, cutting
the rule in H that the point _o_ will never arrive at rule C H;
but because _h_ O is perpetually equal to C A, and the angle of
section is continually more acute, the distance of the point O from C
H will at length be less than any given distance, and consequently the
right line C H will be an asymptote to the curve A O.

When the pole is at P, so that P C is equal to C A, the conchoid A O
described by the revolution of P A, is called a primary conchoid, and
those described from the poles _p_, and _π_, or the curves
A _o_, A _ω_, secondary conchoids; and these are either
contracted or protracted, as the eccentricity P C, is greater or less
than the generative radius C A, which is called the altitude of the
curve.

Now, from the nature of the conchoid, it may be easily inferred, that
not only the exterior conchoid A _ω_ will never coincide with
the right line C H, but this is likewise true of the conchoids A O, A
_o_; and by infinitely extending the right-line A _π_, an
infinite number of conchoids may be described between the exterior
conchoid A _ω_, and the line C H, no one of which shall ever
coincide with the asymptote C H. And this paradoxical property of
the conchoid which has not been observed by any mathematician, is a
legitimate consequence of the infinite divisibility of quantity. Not,
indeed, that quantity admits of an actual division in infinitum, for
this is absurd and impossible; but it is endued with an unwearied
capacity of division, and a power of being diffused into multitude,
which can never be exhausted. And this infinite capacity which it
possesses arises from its participation of the indefinite duad; the
source of boundless diffusion, and innumerable multitude.

  [Illustration]

But this singular property is not confined to the conchoid, but is
found in the following curve. Conceive that the right line A C which is
perpendicular to the indefinite line X Y, is equal to the quadrantal
arch H D, described from the centre C, with the radius C D: then from
the same centre C, with the several distances C E, C F, C G, describe
the arches E _l_, F _n_, G _p_, each of which must be conceived equal
to the first arch H D, and so on infinitely. Now, if the points H, _k_,
_l_, _n_, _p_, be joined, they will form a curve line, approaching
continually nearer to the right-line A B (parallel to C Y) but never
effecting a perfect coincidence. This will be evident from considering
that each of the sines of the arches H D, _l_ E, _n_ F, &c. being less
than its respective arch, must also be less than the right-line A C,
and consequently can never coincide with the right-line A B.

But if other arches D _i_, E _m_, F _o_, &c. each of them equal to
the right-line A C, and described from one centre, tangents to the
former arches H D, _l_ E, _n_ F, &c. be supposed; it is evident that
the points H, _i_, _m_, _o_, &c. being joined, will form a curve line,
which shall pass beyond the former curve, and converge still nearer
to the line A B, without a possibility of ever becoming coincident:
for since the arches D _i_, E _m_, F _o_, &c. have less curvature than
the former arches, but are equal to them in length, it is evident that
they will be subtended by longer lines, and yet can never touch the
right-line A B. In like manner, if other tangent arches be drawn to the
former, and so on infinitely, with the same conditions, an infinite
number of curve-lines will be formed, each of them passing between
H _p_ and A B, and continually diverging from the latter, without a
possibility of ever coinciding with the former. This curve, which I
invented some years since, I suspect to be a parabola; but I have not
yet had opportunity to determine it with certainty.


Transcriber’s Notes:

1. Obvious printers’, punctuation and spelling errors have been
corrected silently.

2. Where hyphenation is in doubt, it has been retained as in the
original.

3. Some hyphenated and non-hyphenated versions of the same words have
been retained as in the original.

4. The errata have been soilently corrected.

5. Italics are shown as _xxx_.





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