The Project Gutenberg eBook of Marks’ First Lessons in Geometry., by Bernhard Marks

How it ever came to pass that Arithmetic should be taught to the extent attained in the grammar schools of the civilized world, while Geometry is almost wholly excluded from them, is a problem for which the author of this little book has often sought a solution, but with only this result; viz., that Arithmetic, being considered an elementary branch, is included in all systems of elementary instruction; but Geometry, being regarded as a higher branch, is reserved for systems of advanced education, and is, on that account, reached by but very few of the many who need it.

The error here is fundamental. Instead of teaching the elements of all branches, we teach elementary branches much too exhaustively.

The elements of Geometry are much easier to learn, and are of more value when learned, than advanced Arithmetic; and, if a boy is to leave school with merely a grammar-school education, he would be better prepared for the active duties of life with a little Arithmetic and some Geometry, than with more Arithmetic and no Geometry.

Thousands of boys are allowed to leave school at the age of fourteen or sixteen years, and are sent into the carpenter-shop, the machine-shop, the mill-wright’s, or the surveyor’s office, stuffed to repletion with Interest and Discount, but so 4utterly ignorant of the merest elements of Geometry, that they could not find the centre of a circle already described, if their lives depended upon it.

Unthinking persons frequently assert that young children are incapable of reasoning, and that the truths of Geometry are too abstract in their nature to be apprehended by them.

To these objections, it may be answered, that any ordinary child, five years of age, can deduce the conclusion of a syllogism if it understands the terms contained in the propositions; and that nothing can be more palpable to the mind of a child than forms, magnitudes, and directions.

There are many teachers who imagine that the perceptive faculties of children should be cultivated exclusively in early youth, and that the reason should be addressed only at a later period.

It is certainly true that perception should receive a larger share of attention than the other faculties during the first school years; but it is equally certain that no faculty can be safely disregarded, even for a time. The root does not attain maturity before the stem appears; neither does the stem attain its growth before its branches come forth to give birth in turn to leaves; but root, stem, and leaves are found simultaneously in the youngest plant.

That the reason may be profitably addressed through the medium of Geometry at as early an age as seven years is asserted by no less an authority than President Hill of Harvard College, who says, in the preface to his admirable little Geometry, that a child seven years old may be taught Geometry more easily than one of fifteen.

The author holds that this science should be taught in all 5primary and grammar schools, for the same reasons that apply to all other branches. One of these reasons will be stated here, because it is not sufficiently recognized even by teachers. It is this:—

The prime object of school instruction is to place in the hands of the pupil the means of continuing his studies without aid after he leaves school. The man who is not a student of some part of God’s works cannot be said to live a rational life. It is the proper business of the school to do for each branch of science exactly what is done for reading.

Children are taught to read, not for the sake of what is contained in their readers, but that they may be able to read all through life, and thereby fulfil one of the requirements of civilized society. So, enough of each branch of science should be taught to enable the pupil to pursue it after leaving school.

If this view is correct, it is wrong to allow a pupil to reach the age of fourteen years without knowing even the alphabet of Geometry. He should be taught at least how to read it.

It certainly does seem probable, that if the youth who now leave school with so much Arithmetic, and no Geometry, were taught the first rudiments of the science, thousands of them would be led to the study of the higher mathematics in their mature years, by reason of those attractions of Geometry which Arithmetic does not possess.

This little book is constructed for the purpose of instructing large classes, and with reference to being used also by teachers who have themselves no knowledge of Geometry.

The first statement will account for the many, and perhaps seemingly needless, repetitions; and the second, for the suggestive style of some of the questions in the lessons which develop the matter contained in the review-lessons.

Here the attention is directed first to this particular angle; then this is taken as an example of its kind, and the idea generalized by describing the class. See also page 29.

Many of the questions are intended to test the vividness of the pupil’s conception. See page 29.

Also page 78. If the circumference were divided into 360 equal parts, would each arc be large or small?

The thing is not to be recognized by the definition; but the definition is to be a description of the thing, a description of the conception brought to the mind of the pupil by means of the name.

LESSON FIRST.

LINES.

PART I.

Lines	9

Points	9

Crooked Lines	10

Curved Lines	11

Straight Lines	11

Other Lines	11

Positions of Lines	14

Angles	17

Relations of Angles	20

Adjacent Angles	20

Vertical Angles	21

Kinds of Angles	23

Right Angles	23

Acute Angles	24

Obtuse Angles	24

Relations of Lines	27

Perpendicular Lines	27

Parallel Lines	28

Oblique Lines	28

Interior Angles	30

Exterior Angles	31

Opposite Angles	32

Alternate Angles	33

Problems relating to Angles	38

Polygons	40

Triangles	44

Isosceles Triangles	48

8Problems relating to Triangles	53

Quadrilaterals	55

Parallelograms	59

Comparison and Contrast of Figures	62

Measurement of Surfaces	66

Problems relating to Surfaces	71

The Circle and its Lines	73

Arcs and Degrees	78

Parts of the Circle	82

PART II.

AXIOMS AND THEOREMS.

Axioms. Illustrated	85

Theorems. Illustrated	88

Note to the Teacher.—In all the development-lessons, the pupils are to be occupied with the diagrams, and not with the printed matter.

POINTS

CROOKED LINES.

STRAIGHT LINES.

OTHER LINES.

LESSON SECOND.

REVIEW.

LESSON THIRD.

POSITIONS OF LINES.

Let the pupils hold their books so that they will be straight up and down like the wall.

VERTICAL LINES.

A vertical line is a straight line that points to the center of the earth.

HORIZONTAL LINES.

OBLIQUE LINES.

An oblique line is a straight line that points neither to the horizon nor to the center of the earth.

Note.—After going through with the lessons on angles, the pupils may be told that oblique lines are so called because they form oblique angles with the horizon.

LESSON FOURTH.

REVIEW.

LESSON FIFTH.

ANGLES.

Do the lines a b and c d (Diagram 5.) point in the same direction? (See note, page 15.)

Which of the two lines c d, e f, has the greater difference of direction from the line a b?

Then does the size of an angle depend upon the length of the lines that form it?

If the lines a b, e f, were shortened, would the angle formed by them be any smaller?

If two lines form an angle with each other, and meet, the point of meeting is called the vertex.

What is the vertex of the angle formed by the lines k j, i j?—i j, i l?

An angle is named by three letters, that which denotes the vertex being in the middle. Thus, the angle formed by k j, i j, is read k j i, or i j k.

LESSON SIXTH.

REVIEW.

Which angle is the greater, l o m, or m o j?—i o k, or i o l?—l o j, or m o j?

If the lines x y and v w were lengthened or produced, would the angles v z x, y z w be any greater?

LESSON SEVENTH.

RELATIONS OF ANGLES.

ADJACENT ANGLES.

Are the angles a e c, c e b (Diagram 7.), on the same side of any line? What line?

Then, because they are both on the same side of the same straight line a b, and are both formed by the second straight line c d, they are called “adjacent angles.”

VERTICAL ANGLES.

Vertical angles are angles formed by the same straight lines, but not adjacent to each other.

LESSON EIGHTH.

REVIEW.

LESSON NINTH.

KINDS OF ANGLES.

RIGHT ANGLES.

ACUTE ANGLES.

OBTUSE ANGLES.

LESSON TENTH.

REVIEW.

LESSON ELEVENTH.

RELATIONS OF LINES.

PERPENDICULAR LINES.

What kind of angles do the lines a b and c d make with each other? (Diagram 11.)

PARALLEL LINES.

Do the lines k l, m n, differ in direction? Then do they form any angle with each other?

OBLIQUE LINES.

LESSON TWELFTH.

REVIEW.

LESSON THIRTEENTH.

RELATIONS OF ANGLES.

INTERIOR ANGLES.

EXTERIOR ANGLES.

LESSON FOURTEENTH.

REVIEW.

LESSON FIFTEENTH.

RELATIONS OF ANGLES.

OPPOSITE ANGLES.

Are the angles e m b, b m n, on the same side of the intersecting line e f?

Opposite angles lie on the same side of the intersecting line, but are not adjacent.

Read two pairs of opposite exterior and interior angles on the right of e f. On the left.

ALTERNATE ANGLES.

Do the angles b m n, m n c, lie on the same side of the intersecting line e f?

34Alternate angles lie on different sides of the intersecting line, and are neither adjacent nor vertical.

LESSON SIXTEENTH.

REVIEW.

Why are b m n, m n c, interior angles? Why alternate? What are alternate angles?

Read the acute interior alternate angles of the parallels t u, v w. The obtuse.

Read the pair of opposite exterior angles on the right of the line e f. On the left.

LESSON SEVENTEENTH.

REVIEW.

LESSON EIGHTEENTH.

PROBLEMS.

LESSON NINETEENTH.

POLYGONS.

If the length of each side of triangle A is one inch, how long are the three sides together?

LESSON TWENTIETH.

REVIEW.

LESSON TWENTY-FIRST.

TRIANGLES.

ACUTE-ANGLED TRIANGLES.

In the triangle 1, what kind of an angle is b a c? a c b? c b a? (Diagram 17.)

OBTUSE-ANGLED TRIANGLES.

RIGHT-ANGLED TRIANGLES.

LESSON TWENTY-SECOND.

REVIEW.

LESSON TWENTY-THIRD.

TRIANGLES. (Continued.)

ISOSCELES TRIANGLES.

An equilateral triangle is one that has its three sides equal to each other.

What kind on account of its two equal sides s t, s u, or t s, t u, or u s, u t?

If the side e f is longer than either of the other two sides, is it an equilateral triangle?

What is Fig. 8 called on account of its three equal sides? On account of its three equal angles?

LESSON TWENTY-FOURTH.

REVIEW.

From the same point draw two straight lines of any length, making an acute angle with each other.

Draw an isosceles triangle whose equal sides shall each be less than the third side.

At one end draw a straight line perpendicular to it, and three times as long as the same measure.

Draw a right-angled triangle whose base is of any length, and its perpendicular twice as long.

Draw a right-angled triangle whose base is three times as long as any short measure, and its perpendicular five times as long as the same measure or unit.

The quadrilateral a b d c, or c d b a, may be read a d, or b c, or c b, or d a.

Has the quadrilateral e h any two sides parallel? Which two? Are the other two sides parallel?

LESSON TWENTY-FIFTH.

REVIEW.

LESSON TWENTY-SIXTH.

KINDS OF PARALLELOGRAMS.

RHOMBOIDS.

RHOMBS.

RECTANGLES.

SQUARES.

LESSON TWENTY-SEVENTH.

REVIEW.

LESSON TWENTY-EIGHTH.

COMPARISON AND CONTRAST.

TRAPEZIUM AND TRAPEZOID.

RHOMBOID AND RHOMBUS.

RECTANGLE AND SQUARE.

RHOMBUS AND SQUARE.

Because both are parallelograms, and both have their sides equal, what name may be given to each?

LESSON TWENTY-NINTH.

REVIEW.

LESSON THIRTIETH.

MEASUREMENT OF SURFACES.

How many squares in that part which is two units long, m n, and one unit wide, m l?

On account of the second unit in width, l k, how many times two squares are there?

How many squares in that part which is three units long, o p, and one unit wide, o t?

The second unit in width, t q, gives how many more squares? How many times three squares?

If a rectangle were five units long and one unit wide, how many square units would it contain?

If your ruler is ten inches long and only one inch wide, how many square inches are there in it?

If your arithmetic-cover is seven inches long and five inches wide, how many square inches are there in it?

If a wall of this room is twenty feet long, how many square feet are there in that part which is one foot high? Two high? Three high? Four high?

What difference is there between two times three squares and three times two squares?

LESSON THIRTY-FIRST.

REVIEW.

The cover of a geography is one foot long and one foot wide, how many square feet in it?

An equilateral triangle has each of its sides one inch long, what is its perimeter?

An isosceles triangle has its two equal sides each three inches long, and its third side five inches long, what is its perimeter?

A right-angled isosceles triangle has its base five inches, and its hypothenuse seven inches long, what is its perimeter?

A room is twenty-four feet long and twenty-one feet wide, how many feet all round it?

72Charles and Henry start from the same place, and walk in opposite directions; Charles goes twenty yards, and Henry fifteen, how many yards apart are they?

If they start from opposite ends of a straight walk twenty-five feet long, and walk towards each other, how many feet will Charles have to walk to meet Henry who has walked fifteen feet?

LESSON THIRTY-SECOND.

THE CIRCLE AND ITS LINES.

If the straight line c a were a string made fast at c, with a sharp pencil-point at the other end a, and the pencil-point were moved towards d, what line would be drawn?

If the pencil-point continued to move in the same direction until it returned to the starting-point a, what curved line would be drawn, naming it by all the points in it which are marked?

A circle is a plane figure bounded by a curved line, all points of which are equally distant from the centre.

A radius is a straight line that joins the centre to a point of the circumference.

Can the radius c d be greater than the radius c a? Or greater than c v, or c o?

A tangent is a straight line that touches a circumference in only one point.

LESSON THIRTY-THIRD.

REVIEW.

LESSON THIRTY-FOURTH.

ARCS AND DEGREES.

Which is longer, the arc e d, or the arc e f? b d, or b d e? a b d, or a b d e?

If the whole circumference were divided into three hundred and sixty equal arcs, would each arc be large or small?

If a fourth of a circumference were divided into three equal parts, how many degrees would there be in each part?

79Into how many parts would each third of a quarter have to be again divided to make single degrees?

Is an arc of ninety-one degrees greater or less than one-fourth of a circumference?

Is an arc of a hundred and seventy-nine degrees greater or less than a semi-circumference?

If the circumference of circle 1 were divided into degrees, each degree would be so small an arc that it would look like a dot.

If a degree were divided into sixty equal parts, each part would be called a minute.

If a minute were divided into sixty equal parts, each part would be called a second.

Which is greater, an arc of ninety degrees of the large circle, or one of ninety degrees of the small one?

Which is greater, an arc of a degree of the large circle, or one of a degree of the small one?

If the angle a o f is an angle of 30°, how many degrees must there be in the arc a f?

LESSON THIRTY-FIFTH.

REVIEW.

If the arc f h were divided into three equal parts, how many degrees would there be in each?

If the angle f o g is an angle of 130°, what is the size of the arc f i h g?

LESSON THIRTY-SIXTH.

PARTS OF THE CIRCLE.

The part of the circle bounded by the chord a b and the arc a b is called a segment.

Read three segments, each less than half a circle, thus,—the segment bounded by the chord a d and the arc a b d.

83A segment bounded by a diameter and a semi-circumference is a “semicircle.”

The part of the circle between the two radii o f, o i, and the arc f i, is called a “sector.”

If the chord e f be regarded a diameter, what do you call the semicircle below it?

LESSON THIRTY-SEVENTH.

REVIEW.

AXIOMS ILLUSTRATED.

Axiom 1.

If two things are separately equal to the same thing, they are equal to each other.

Axiom 2.

To the rectangle C add the square A, and we have an L pointing in what direction?

To the same rectangle C add the square B, and we have an L pointing in what direction?

Axiom 3.

From the inverted T take away the square A, and we have an L pointing in what direction?

From the same Fig. T take away the square B, and we have an L pointing in what direction?

Axiom 5.

Axiom 6.

Axiom 7.

Axiom 9.

Axiom 10.

Axiom 11.

THEOREMS ILLUSTRATED.

DEVELOPMENT LESSON.

Then are not the angles Blue, Yellow, Green, Red, equal to all the space on the line a b?

Note.—The word space, as here used, means angular space; and it is indispensable that the teacher impress this fact upon the learner.

By means of former lessons, the pupil has learned positively, that an angle is the difference between the directions of two lines; and, impliedly, that the included space has nothing to do with the size of the angle. There cannot, therefore, be much danger that the pupil will imbibe any erroneous notion from this style of expression, which is very much more simple than to say that the difference of direction of two given lines is equal to the difference of direction of two other given lines, which style will be used somewhat later in these lessons.

PROPOSITION I. THEOREM.

DEVELOPMENT LESSON.

Are the adjacent angles Green, Red, equal to all the angular space on the line a b?

Place a paper square corner or right angle on the line a b at the left of c d with its vertex at c.

It will cover all the angle Green and part of the angle Red up to the line c d.

Now place another square corner on the line a b to the right of the line c d, and with its vertex at the point c.

It will cover the remaining part of the angle Red, and two edges of the square corners will meet along the line c d.

Then if the two adjacent angles Green, Red, are equal to all the angular space on the line a b, and the two right angles are also equal to the 91same space, what do you infer concerning the adjacent angles and the two right angles?

What axiom do you apply when you say that the adjacent angles are equal to the two right angles?

If the angle Red were smaller, and the angle Green larger, would the adjacent angles still be equal to two right angles?

If we draw the straight line c d where the edges of the square corners come together, what kind of angles will a c d, d c b, be?

DEMONSTRATION.

Let the two straight lines a b, m n, intersect each other in the point c. (Diagram 30.)

92For, from the point c, draw the straight line c d so as to make the angles a c d, d c b, right angles.

The adjacent angles Green, Red, are equal to all the angular space on the line a b.

The right angles a c d, d c b, are also equal to all the angular space on the line a b.

TEST QUESTIONS.

TEST LESSON.

7. Give the preceding demonstrations again, but name the angles by their letters instead of by their colors.

PROPOSITION II. THEOREM.

DEVELOPMENT LESSON.

Then if the adjacent angles Yellow, Blue, are equal to two right angles, and the adjacent angles Blue, Red, are also equal to two right angles, what do you think of the two pairs of adjacent angles, Yellow, Blue, and Blue, Red?

If, from the adjacent angles Yellow, Blue, we take away the angle Blue, what remains?

If, from the adjacent angles Blue, Red, we take away the same angle Blue, what remains?

98Then, since the same angle Blue has been taken from equal pairs of adjacent angles, what do you think of the two remainders, Yellow, Red?

Suppose the lines a b and m n were so drawn that the angles Yellow, Red, were larger or smaller, would they still be equal to each other?

DEMONSTRATION.

Let the straight lines a b, m n, intersect each other at the point c, then will any two vertical angles, as Yellow, Red, be equal to each other.

Therefore the adjacent angles Yellow, Blue, are equal to the adjacent angles Blue, Red.

If, from the adjacent angles Yellow, Blue, we take away the angle Blue, we shall have left the angle Yellow.

100If, from the adjacent angles Blue, Red, we take away the same angle Blue, we shall have left the angle Red.

TEST QUESTIONS.

When you say that the adjacent angles Yellow, Blue, are equal to two right angles, do you know it because you see it, or because you have proved it?

How do you know that the adjacent angles Blue, Red, are equal to two right angles?

When you say the adjacent angles Yellow, Blue, are equal to the adjacent angles Blue, Red, what axiom do you use?

When you take the angle Blue from the adjacent angles Yellow, Blue, what is the remainder?

When you take the same angle Blue from the adjacent angles Blue, Red, what is the remainder?

OTHER METHODS OF DEMONSTRATION.

Then what do you know of the two pairs of adjacent angles Yellow, Green, and Green, Red?

From the adjacent angles Yellow, Green, take away the angle Green. What remains?

In the last lesson, when you proved the vertical angles Yellow, Red, equal to each other, you made use of the angle Blue; now prove the same two angles equal by means of the angle Green.

Then what do you know of the two pairs of adjacent angles Blue, Red, and Red, Green?

From the adjacent angles Red, Green, take away the same angle Red. What remains?

Prove the vertical angles Blue, Green, equal to each other by means of the angle Yellow.

TEST LESSON.

1. Prove that the vertical angles Yellow, Red, are equal to each other, using the angle Green.

3. Prove that the vertical angles Blue, Green, are equal to each other, using the angle Yellow.

5. Prove the vertical angles Yellow, Red, equal to each other, using the angle Green.

Go through the preceding eight demonstrations again, calling the angles by their letters instead of by their colors.

PROPOSITION III. THEOREM.

DEVELOPMENT LESSON.

In the above diagram, the lines a b, c d, are parallel, and are intersected by the line e f at the points m and n.

The angle Red measures the difference of direction between the line m b and what other line?

The angle Yellow measures the difference of direction between the line n d and what other line?

Then, as the lines m b and n d are parallel, must there not be the same difference of direction between them and the line e f?

Then can there be any difference between the angles which measure those equal directions?

Then what do you think of the opposite exterior and interior angles Red, Yellow?

Let the straight line e f intersect the two parallel straight lines a b, c d, at the points m and n.

Then will any two opposite exterior and interior angles, as Red, Yellow, be equal to each other.

For the angle Red measures the difference of direction of the lines m b and e f.

And the angle Yellow measures the difference of direction of the lines n d and e f.

TEST LESSON.

1. Prove that the opposite exterior and interior angles Green, Blue, are equal to each other.

2. Prove that the opposite exterior and interior angles Red, Yellow, are equal to each other.

PROPOSITION IV. THEOREM.

DEVELOPMENT LESSON.

Then if the interior alternate angles Green, Yellow, are separately equal to the angle Red, what new fact do you know?

Let the straight line e f intersect the two parallel straight lines a b, c d, in the points m and n.

Then will any two interior alternate angles, as Green, Yellow, be equal to each other.

Then because the interior alternate angles Green, Yellow, are separately equal to the angle Red, they are equal to each other.

TEST LESSON.

Then if the interior alternate angles Green, Yellow, 112are separately equal to the angle Red, what do you infer?

1. Prove that the interior alternate angles Green, Yellow, are equal, using the angle Red.

3. Go through the same demonstrations again, calling the angles by their letters instead of by their colors.

4. Prove the interior alternate angles Red, Blue, equal, using the angle Yellow.

6. Go through the same two demonstrations again, naming the angles by their letters instead of by their colors.

7. Prove the interior alternate angles c n m, n m b, equal, using the angle f n d.

9. Prove the interior alternate angles a m n, m n d, equal, using the angle e m b.

PROPOSITION V. THEOREM.

DEVELOPMENT LESSON.

Then if the exterior alternate angles Red, Green, are separately equal to the angle Yellow, what new thing do you know to be true?

Let the straight line e f intersect the two parallel straight lines a b, c d, at the points m and n.

For the opposite exterior and interior angles Red, Yellow, are equal to each other.

Then because the exterior alternate angles Red, Green, are separately equal to the angle Yellow, they are equal to each other.

TEST LESSON.

116Then if the exterior alternate angles Yellow, Blue, are separately equal to the angle Red, what do you know of them?

1. Prove that the exterior alternate angles Yellow, Blue, are equal, using the angle Red.

4. Prove the exterior alternate angles e m b, c n f, equal, using the angle a m n.

6. Prove that the exterior alternate angles c m e, f n b, are equal, using the angle n m d.

8. Prove the exterior alternate angles e m d, a n f, equal, using the angle c m n.

PROPOSITION VI. THEOREM.

DEVELOPMENT LESSON.

Then, having added equals to the same thing, what do you think of the two sums,—the adjacent angles Green, Yellow, and the interior opposite angles Green, Red?

What do you know of the adjacent angles Green, Yellow, and the right angles P, S?

118Then if the interior opposite angles Green, Red, and the two right angles P, S, are separately equal to the adjacent angles Green, Yellow, what new thing do you know?

DEMONSTRATION.

Let the straight line e f intersect the two parallel straight lines a b, c d, in the points m and n.

If to the angle Green we add the angle Yellow, we shall have the adjacent angles Green, Yellow.

If to the same angle Green we add the equal angle Red, we shall have the interior opposite angles Green, Red.

Then the adjacent angles Green, Yellow, are equal to the interior opposite angles Green, Red.

Then because the interior opposite angles Green, Red, and two right angles, are separately equal to the two adjacent angles Green, Yellow, they are equal to each other.

TEST LESSON.

1. Prove the interior opposite angles Green, Yellow, equal to two right angles, using the angle Red.

1205. Go through the same demonstrations again, naming the angles by their letters instead of by their colors.

6. Prove the interior opposite angles Red, Blue, equal to two right angles, using the angle Yellow.

10. Go through the same demonstrations again, calling the angles by their letters instead of by their colors.

11. Prove the interior opposite angles a g h, g h c, equal to two right angles, using the angle g h d.

14. Prove the interior opposite angles b g h, g h d, equal to two right angles, using the angle a g h.

Compare the angles Yellow, Green, each with its exterior opposite angle, and see if you can prove that the exterior opposite angles e g b, f h d, are also equal to two right angles.

PROPOSITION VII. THEOREM.

DEVELOPMENT LESSON.

But, by measuring, we find that the interior angles Blue, Yellow, on the same side of the secant ^[4] line e f, are equal to two right angles:

From the adjacent angles Blue, Red, take away the same angle Blue: what remains?

Then if the lines g b, h d, have the same direction from the line e f, what do you call them?

DEMONSTRATION.

If a straight line intersects two other straight lines so that two interior angles on the same side of the intersecting line are equal to two right angles, the two lines are parallel.

Let the straight line e f intersect the two straight lines a b, c d, in the points g and h, so that the angles Red, Blue, are equal to two right angles.

Then the interior angles Red, Blue, are equal to the adjacent angles Red, Green.

If from the interior angles Red, Blue, we take away the angle Red, we have left the angle Blue.

If from the adjacent angles Red, Green, we take the same angle Red, we shall have left the angle Green.

But the angle Blue measures the direction of the line h d from the line e f.

And the angle Green measures the direction of the line g b from the line e f.

TEST LESSON.

3. Prove the same, supposing the angles a g h, g h c, equal to two right angles, and using the angle a g e.

The following demonstration is very easy. Read it once, and see if you can go through it without a second reading:—

DEMONSTRATION.

Let the figure a b c be a triangle, then will the sum of any two sides, as a c, c b, be greater than the third side a b.

For the straight line a b is the shortest distance between the two points a and b, and is therefore less than the broken line a c b.

PROPOSITION IX. PROBLEM.

SOLUTION.

With the point a as a centre, and a b as a radius, draw the circumference of the circle, or a part of one.

With the point b as a centre, and the same radius a b, draw another circumference, or a part of one.

From the point c, in which the circumferences or arcs intersect, draw the straight lines a c and b c.

Now, because the lines a b and a c are radii of the same circle, they are equal.

And, because the lines a b and b c are radii of the same circle, they are also equal.

Then, because the two lines a c, b c, are separately equal to the line a b, they are equal to each other, and the triangle is equilateral.

PROPOSITION X. THEOREM.

DEVELOPMENT LESSON.

We have now another angle, b c d, and we wish to find out if it is equal to any of the angles of the triangle.

Because the straight line a d intersects the two parallels a b, c e, the angle a is equal to what other angle?

Because the straight line b c intersects the two parallels a b, c e, the angle b is equal to what other angle?

Then, if the angles a and b, on the one hand, and the angle b c d, on the other, are separately equal to the angles b c e, e c d,

DEMONSTRATION.

If any side of a triangle be produced, the new angle formed will be equal to the sum of the angles that are not adjacent to it.

Produce the side a c to d; then will the new angle b c d be equal to the sum of the angles a and b.

Then, because the straight line a d intersects the two parallels a b, c e, in the points a and c,

The opposite exterior and interior angles a and e c d are equal to each other.

And because the straight line b c intersects the same parallels in the points b and c,

Then the angles a and b of the triangle are equal to the angles b c e and e c d.

Then because the new angle b c d, and the angles a and b are separately equal to the angles b c e, e c d, they are equal to each other.

PROPOSITION XI. THEOREM.

DEVELOPMENT LESSON.

What angle must we add to these angles to make up the three angles of the triangle?

Then the three angles of the triangle, a, b, and c, are equal to what two angles?

Then, because the three angles of the triangle, a, b, and c, and two right angles, are separately equal to the two adjacent angles c and b c d.

Let the figure a b c be a triangle; then will the sum of the angles a, b, and c, be equal to two right angles.

If to the angles a and b we add the angle c, we shall have the three angles of the triangle.

If to the angle b c d we add the same angle c, we shall have the adjacent angles c and b c d.

Then the three angles of the triangle a, b, c, are equal to the adjacent angles c and b c d.

Then, because the three angles of the triangle are equal to the adjacent angles c and b c d, they are equal to two right angles.

PROPOSITION XII. THEOREM.

DEVELOPMENT LESSON.

Because the straight line B D intersects the parallels A B and C F, the angle B is equal to what other angle?

Because the straight line C F intersects the parallels A C and B D, the angle C is equal to what other angle?

DEMONSTRATION.

Then, because the angles B and C are separately equal to the angle E, they are equal to each other.

3. Prove the angles A and D equal to each other by producing the line C D towards the left.

5. See if you can prove the same by drawing a diagonal through the points A and D.

PROPOSITION XIII. THEOREM.

DEVELOPMENT LESSON.

In these two triangles we have tried to make the side a b of the one equal to the side d e of the other; the side a c of the one equal to the side d f of the other; and the included angle b a c of the one equal to the included angle e d f of the other.

We now wish to find out if the third side b c of the one is equal to the third side e f of the other, and if the two remaining angles b and c of the one are equal to the two remaining angles e and f of the other.

Suppose we were to cut the triangle d e f out of the page, and place it upon the triangle a b c, so that the line d e should fall upon the line a b, and the point d upon the point a.

As the line d e is equal to the line a b, upon what point will the point e fall?

If the angle e d f were less than the angle b a c, 133would the line d f fall within or without the triangle?

If the angle e d f were greater than the angle b a c, where would the line d f fall?

As the line d f is equal to the line a c, upon what point will the point f fall?

Then, if the point e falls upon the point b, and the point f upon the point c, where will the line e f fall?

Now, because the three sides of the triangle d e f exactly fall upon the three sides of the triangle a b c, we say the two magnitudes coincide throughout their whole extent, and are therefore equal.

What three parts of the triangle a b c did we suppose to be equal to three corresponding parts of the triangle d e f before we placed one upon the other.

DEMONSTRATION.

If two triangles have two sides, and the included angle of the one equal to two sides and the included angle of the other, each to each, the two triangles are equal in all respects.

Let the triangles a b c and d e f have the side a b of the one equal to the side d e of the other; the side a c of the one equal to the side d f of the other; and the included angle b a c of the one equal to the included angle e d f of the other, each to each; then will the two triangles be equal in all their parts.

For, place the triangle d e f upon the triangle a b c, so that the line d e shall fall upon the line a b, with the point d upon the point a.

Because the line d e is equal to the line a b, the point e will fall upon the point b.

Because the angle e d f is equal to the angle b a c, the line d f will fall upon the line a c.

Because the line d f is equal to the line a c, the point f will fall upon the point c.

135Then, because the point e is on the point b, and the point f on the point c, the line e f will coincide with the line b c, and the two triangles will be found equal in all their parts;

That is, the angle e is found to be equal to the angle b, the angle f to the angle c, the line e f to the line b c, and the area of the triangle a b c to the area of the triangle d e f.

PROPOSITION XIV. THEOREM.

DEVELOPMENT LESSON.

In these two triangles we have tried to make the angle b of the one equal to the angle e of the other; the angle c of the one equal to the angle f of the other; and the included side b c of the one equal to the included side e f of the other.

We now wish to find out if the remaining angle a of the one is equal to the remaining angle d of the other, and if the two remaining sides a b and a c of the one are equal to the two remaining sides d e and d f of the other.

Suppose we were to cut the triangle d e f out of the page and place it upon the triangle a b c, so that the line e f shall fall upon the line b c, with the point e upon the point b.

Because the line e f is equal to the line b c, upon what point will the point f fall?

Then, if the line d e falls upon the line a b and the line d f upon the line a c, where will the point d fall?

Now because the three sides of the triangle d e f exactly fall upon the three sides of the triangle a b c, we say the two magnitudes coincide throughout their whole extent, and are therefore equal.

Suppose the angle e were greater than the angle b, would the line e d fall within or without the triangle?

DEMONSTRATION.

If two triangles have two angles, and the included side of the one equal to two angles and the included side of the other, each to each, the two triangles are equal to each other in all respects.

Let the triangles a b c and d e f have the angle b of the one equal to the angle e of the other; the angle c of the one equal to the angle f of the other; and the included side b c of the one equal to the included side e f of the other, each to each; then will the two triangles be equal in all their parts.

For place the triangle d e f upon the triangle a b c, so that the line e f shall fall upon the line b c, with the point e upon the point b.

Because the line e f is equal to the line b c the point f will fall upon the point c.

Because the angle e is equal to the angle b, the line e d will fall upon the line b a, and the point d will be somewhere in the line b a.

139Because the angle f is equal to the angle c, the line f d will fall upon the line c a, and the point d will be somewhere in the line c a.

Then, because the point d is in the two lines, b a and c a, it must be in their intersection, or upon the point a.

And, as the two triangles coincide throughout their whole extent, they are equal in all their parts.

That is, the angle a is found to be equal to the angle d; the side b a to the side e d; the side c a to the side f d; and the area of the triangle a b c to the area of the triangle d e f.

PROPOSITION XV. THEOREM.

DEMONSTRATION.

Let the figure a b c d be a parallelogram; then will the sides a b and c d be equal to each other; likewise the sides a d and b c.

Because the figure is a parallelogram, the sides a b and d c are parallel, and the interior alternate angles n and o are equal.

Because the figure is a parallelogram, the interior alternate angles r and m are equal.

Then the two triangles a d b, b d c, have two angles and the included side of the one equal to two angles and the included side of the other, each to each, and are therefore equal;

And the side a b opposite the angle m is equal to the side c d opposite the equal angle r;

And the side a d opposite the angle n is equal to the side b c opposite the equal angle o.

TEST.

PROPOSITION XVI. THEOREM.

DEVELOPMENT LESSON.

Let us see if this perpendicular is not shorter than any other line we can draw from the same point to the same line.

Now, as C E is any line whatever other than a perpendicular, if we find that the perpendicular C F is shorter than it we must conclude that it is the shortest line that can be drawn from C to A B.

If the two triangles E F C, E F D, have the side C F of the one equal to the side F D of the other, the side E F common to both, and the included angle E F C of the one equal to the included angle E F D of the other, each to each, what do you infer?

Then what third side of the one have you found equal to a third side of the other?

If C E is any line whatever other than a perpendicular, what may we now say of the perpendicular from the point C to the straight line A B?

DEMONSTRATION.

Let A B be a straight line, and C A point out of it; then will the perpendicular C E be the shortest line that can be drawn from the point to the line.

The two triangles F E C, F E D, have the side C E of the one equal to the side E D of the other, the side F E common, and the included angle F E C of the one equal to the included angle F E D of the other, they are therefore equal, and the side C F equals the side F D.

144But the straight line C D is the shortest distance between the two points C D; therefore it is shorter than the broken line C F D.

And, as C F is any line other than a perpendicular, the perpendicular C E is the shortest line that can be drawn from C to A B.

PROPOSITION XVII. THEOREM.

DEMONSTRATION.

A tangent to a circumference is perpendicular to a radius at the point of contact.

Let the straight line A B be tangent at the point D to the circumference of the circle whose centre is C.

145Join the centre C with the point of contact D, the tangent will be perpendicular to the radius C D.

As the point D is the only one in which the tangent touches the circumference, any other point, as F, must be without the circumference.

Then the line C F, reaching beyond the circumference, must be longer than the radius C D, which would reach only to it; therefore C D is shorter than any other line which can be drawn from the point C to the straight line A B; therefore it is perpendicular to it.

DEMONSTRATION.

Let the triangle A B C be isosceles, having the side A B equal to the side A C; then will the angle B, opposite the side A C, be equal to the angle C, opposite the equal side A B.

For draw the line A D so as to divide the angle A into two equal parts, and let it be long enough to divide the side B C at some point as D.

Now the two triangles A D B, A D C, have the side A B of the one equal to the side A C of the other, the side A D common to both, and the included angle B A D of the one equal to the included angle C A D of the other; therefore the two triangles are equal in all respects, and the angle B, opposite the side A C, is equal to the angle C, opposite the side A B.

PROPOSITION XIX. THEOREM.

DEMONSTRATION.

If two triangles have the three sides of the one equal to the three sides of the other, each to each, they are equal in all their parts.

Let the two triangles A B C, A D C, have the side A B of the one equal to the side A D of the other; the side B C of the one equal to the side D C of the other, and the third side likewise equal; then will the two triangles be equal in all their parts.

For place the two triangles together by their longest side, and join the opposite vertices B and D by a straight line.

Because the side A B is equal to the side A D, the triangle B A D is isosceles, and the angles A B D, A D B, opposite the equal sides are equal.

148Because the side B C is equal to the side D C, the triangle B C D is isosceles, and the angles C B D, C D B, opposite the equal sides are equal.

And if to the equal of A B D, that is, A B D, we add the equal of D B C, that is, B D C, we shall have the angle A D C.

Then the two triangles A B C, A D C, have two sides, and the included angle of the one equal to two sides and the included angle of the other, each to each, and are equal in all their parts; that is, the three angles of the one are equal to the three angles of the other, and their areas are equal.

PROPOSITION XX. THEOREM.

DEMONSTRATION.

An angle at the circumference is measured by half the arc on which it stands.

Let B A D be an angle whose vertex is in the circumference of the circle whose centre is C; then will it be measured by half the arc B D.

Because the sides C A, C B, are radii of the circle, they are equal, the triangle is isosceles, the angles B and B A C opposite the equal sides are equal, and the angle B A C is half of both.

Then, because the angle B A C is half of B and B A C, it must be half of their equal E C B.

150But E C B, being at the centre, is measured by B E; then half of it, or B A C, must be measured by half B E.

Then, because B A C is measured by half B E, and C A D by half E D, the whole angle B A D must be measured by half the whole arc B D.

SECOND CASE.

Draw another radius from C to F, and prove that F A E is measured by half the arc F E.

Then, because the angle F A E is measured by half the arc F E, and the angle B A E is measured by half the arc B E,

The difference of the angles, or F A B, must be measured by half the difference of the arcs, or half of F B.

PROPOSITION XXI. THEOREM.

DEMONSTRATION.

Let the chords A B, C D, be parallel; then will the intercepted arcs A C and B D be equal.

Because the lines A B and C D are parallel, the interior alternate angles A B C, B C D, are equal.

Then, because the angles are equal, the half arcs which measure them must be equal, and the whole arcs themselves must be equal.

PROPOSITION XXII. THEOREM.

DEMONSTRATION.

The angle formed by a tangent and a chord meeting at the point of contact is measured by half the intercepted arc.

Let the tangent C A B and the chord A D meet at the point of contact A; then will the angle B A D be measured by half the intercepted arc A D.

Because A B is a tangent, and A E a radius at the point of contact, the angle B A F is a right angle, and is measured by the semicircle A D F.

Because the angle F A D is at the circumference, it is measured by half the arc D F.

153Then the difference between the angles B A F and D A F, or B A D, must be measured by half the difference of the arcs A D F and D F, or A D;

PROPOSITION XXIII. THEOREM.

DEMONSTRATION.

A tangent and chord parallel to it intercept equal arcs of the circumference.

Let A B be tangent to the circumference at the point D, and let C F be a chord parallel to the tangent; then will the intercepted arcs C D and D F be equal.

Because the tangent and chord are parallel, the interior alternate angles A D C and D C F are equal.

But the angle A D C, being formed by the tangent D A and the chord D C, is measured by half the intercepted arc D C;

And the angle D C F, being at the circumference, is measured by half the arc on which it stands, D F:

Then, because the angles are equal, the half arcs which measure them are equal, and the arcs themselves are equal.

PROPOSITION XXIV. THEOREM.

DEMONSTRATION.

The angle formed by the intersection of two chords in a circle is measured by half the sum of the intercepted arcs.

Let the chords A B and C D intersect each other in the point E; then will the angle B E D or A E C be measured by half the sum of the arcs A C, B D.

Add each of these equals to B D, and we have B D plus A C equal to B D plus B F; that is, the sum of the arcs B D, A C, is equal to the arc F D.

Because the chords A B, C F, are parallel, the opposite exterior and interior angles D E B, D C F, are equal.

156But D C F is an angle at the circumference, and is therefore measured by half the arc F D.

Then the equal angle D E B must be measured by half of the arc F D, or its equal B D, plus A C.

PROPOSITION XXV. THEOREM.

DEMONSTRATION.

The angle formed by two secants meeting without a circle is measured by half the difference of the intercepted arcs.

Let the secants A B, A C, intersect the circumference in the points D and E; then will the angle B A C 157be measured by half the difference between the arcs B C and D E.

Because A C and D F are parallel, the opposite exterior and interior angles B D F and B A C are equal.

If from the arc B C we take the arc D E, or its equal F C, we shall have left the arc B F;

But the angle B D F, being at the circumference, is measured by half the arc B F:

Then the equal of B D F, or B A C, must be measured by half the arc B F, or half the difference between the intercepted arcs B C and D E.

Note A.—To those teachers who think that the line should be derived from a surface, and the surface from a solid, the author would say, that, according to his experience, children apprehend the ideas conveyed by the terms line and surface as readily as they do any ideas whatever; and that, therefore, there seems to be no necessity for extraordinary care in this case to avoid giving wrong impressions.

Still, if it be considered desirable in this manner to derive lines and surfaces, it will be apparent that all that can be done in the matter is to give such instruction only by way of a preliminary lesson.

Note B.—Crooked and curved lines are here treated of before straight lines, because the first two are defined by means of an affirmative property,—they do change direction; while the last is defined by means of the absence of one,—they do not change direction. It is easier for a child to comprehend what is than what is not.

Note C.—If the pupils are old enough, they may be shown that vertical lines cannot be parallel, but only seem so on account of their shortness and nearness to each other.

Note D.—This definition may be considered objectionable because rhomboid means like a rhomb. That the more general figure, the rhomboid, has been named from the more restricted one, the rhomb, is unfortunate, because it interferes with the symmetry of the nomenclature. The rhomb possesses all the properties of the rhomboid, and should, therefore, when these are considered, be called by the same name; its additional property entitles it to a name which should comprehend the other names. If the rectangle had been called a squaroid, the difficulty would have been repeated.

Note E.—If teachers consider it desirable, they may require the class to prove, by way of corollary, such propositions as assert the parallelism of the lines when the interior alternate angles are equal, when the opposite exterior and interior angles are equal, &c., in continuation of what has already been done.

*** END OF THE PROJECT GUTENBERG EBOOK MARKS' FIRST LESSONS IN GEOMETRY ***

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