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The cover image was created by the transcriber and is placed in the public domain.

MARKS’
FIRST LESSONS IN GEOMETRY.
IN TWO PARTS.
OBJECTIVELY PRESENTED,
AND DESIGNED FOR
THE USE OF PRIMARY CLASSES IN GRAMMAR SCHOOLS, ACADEMIES, ETC.

BY

BERNHARD MARKS,

PRINCIPAL OF LINCOLN SCHOOL, SAN FRANCISCO.

NEW YORK:

PUBLISHED BY IVISON, PHINNEY, BLAKEMAN, & CO.

PHILADELPHIA: J. B. LIPPINCOTT & CO.

CHICAGO: S.C. GRIGGS & CO.

1869.

Entered, according to Act of Congress, in the year 1868, by

BERNHARD MARKS,

In the Clerk’s Office of the District Court of the United States for the District of California.

Geo. C. Rand & Avery, Electrotypers and Printers,

3 Cornhill, Boston.

PREFACE.

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 utterly 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 primary 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.

TO THE PROFESSIONAL READER.

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.

Attention is respectfully directed to the following points:—

First the particular, then the general. See page [25].

Why is m n g an acute angle?

What is an acute angle?

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].

Why are the lines e f and g h said to be parallel?

When are lines said to be parallel?

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?

Many of the questions are intended to test the attention of the pupil.

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.

CONTENTS.

FIRST LESSONS IN GEOMETRY.

PART FIRST.

LESSON FIRST.

LINES.

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

See Note [A], Appendix.

Refer to Diagram [1], and show that

What are here drawn are intended to represent length only.

They have a little width, that they may be seen.

They are called lines.

A line is that which has length only.

POINTS

Show that

Position is denoted by a point.

It occupies no space.

It has some size, that it may be seen.

The ends of a line are points.

A line may be regarded as a succession of points.

The intersection of two lines is a point.

A point is named by placing a letter near it.

Diagram 1.

A point may be represented by a dot. The point is in the center of the dot.

A point is that which denotes position only.

A line is named by naming the points at its ends.

Read all the lines in Diagram [1].

CROOKED LINES.

See Note [B], Appendix.

Does the line m n change direction at the point 1?

At what other points does it change direction?

It is called a crooked line.

A crooked line is one that changes direction at some of its points.

CURVED LINES.

The line o p changes direction at every point.

It is called a curved line.

A curved line is one that changes direction at every point.

STRAIGHT LINES.

Does the line i j change direction at any point?

It is called a straight line.

A straight line is one that does not change direction at any point.

OTHER LINES.

The line q r winds about a line.

It is called a spiral line.

The line w x winds about a point.

It also is called a spiral line.

A spiral line is one that winds about a line or point.

The line 7 8[^[1]] looks like waves.

[1]. To be read seven, eight, not seventy-eight.

It is called a wave line.

What kind of a line is a b?

Why? What is a straight line?

What kind of a line is 11 16?

Why? What is a crooked line?

What kind of a line is o p?

Why? What is a curved line?

What kind of a line is s t?

Why?

What kind of a line is 9 10?

Why? What is a spiral line?

What kind of a line is w x?

Why?

LESSON SECOND.

REVIEW.

Read all the straight lines. (Diagram [2].)

Why is m n a straight line?

Define a straight line.

Read all the crooked lines.

Why is 7 8 a crooked line?

Define a crooked line.

Read all the curved lines.

Why is 5 6 a curved line?

What is a curved line?

Read all the wave lines.

Read all the spiral lines.

Why is 3 4 a spiral line?

Why is u v a spiral line?

What is a spiral line?

Diagram 2.

Diagram 3.

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.

The straight line a b points to the center of the earth. (Diagram [3].)

It is called a vertical line.

Name all the vertical lines.

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

HORIZONTAL LINES.

The straight line o p points to the horizon.

It is called a horizontal line.

Read all the horizontal lines.

A horizontal line is a straight line that points to the horizon.

OBLIQUE LINES.

The line s t points neither to the center of the earth nor to the horizon.

It is called an oblique line.

Read all the 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.

Read all the vertical lines. (Diagram [4].)

Why is q r a vertical line?

What is a vertical line?

Read all the horizontal lines.

Why is 5 6 a horizontal line?

Define a horizontal line.

Read all the oblique lines.

Why is s t an oblique line.

What is an oblique line?

Note.—Lines that point in the same direction do not approach the same point.

Diagram 4.

Diagram 5.

LESSON FIFTH.

ANGLES.

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

Then they form an angle with each other.

What other line forms an angle with a b?

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

Then which one forms the greater angle with a b?

What line forms a still greater angle with the line a b?

An angle is the difference of direction of two straight lines.

If the lines a b, e f, were made longer, would their direction be changed?

Then would there be any greater or less difference of direction?

Then would the angles formed by them be any greater or less?

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.

Read the four angles formed by the lines m n and o p.

The eight formed by r s, t u, and v w.

LESSON SIXTH.

REVIEW.

Read all the lines that form angles with the line a b. (Diagram [6].)

Which of them forms the greatest angle with it?

Diagram 6.

Which the least?

Of the two lines c d, g h, which forms the greater angle with e f?

Read all the angles whose vertices are at o on i j.

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?

Read all the angles formed by the lines v w and x y.

Read all the angles above the line n p.

Below the line n p. Above the line q r.

At the right of the line 5 u.

At the left. At the right of the line s t.

At the left of the line s t.

Which angle is the greater, n 1 3, or n 2 4?

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

If they were shortened, would the angles be any less?

What is an angle?

Does the size of an angle depend upon the length of the lines which form it?

Diagram 7.

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?

By what other straight line are they both formed?

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.”

The angles c e b, b e d are both on the same side of what straight line?

They are both formed by what second straight line?

Then what kind of angles are they?

Why are they called adjacent angles?

Read the adjacent angles below the line a b. Below the line c d.

How many pairs of adjacent angles can be formed by two straight lines?

Read all the adjacent angles formed by the lines l m and n p.

VERTICAL ANGLES.

Are the angles a e c, b e d formed by the same straight lines?

Are they adjacent angles?

They are called “vertical angles.”

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

Read the other pair of vertical angles formed by the lines a b, c d.

Read all the vertical angles formed by the lines f g, i h. By l m, n p.

Why are the angles l o n, n o m adjacent angles?

Why are the angles l o n, p o m vertical angles?

Diagram 8.

LESSON EIGHTH.

REVIEW.

Read the pairs of adjacent angles above the line a b. (Diagram [8].)

Why are they adjacent?

What are adjacent angles?

Read the adjacent angles below the line a b.

On the right of the line c d. On the left.

How many pairs of adjacent angles are formed by the intersection of two lines.

Read the pairs of adjacent angles formed by the lines f g and i h.

Read all the adjacent angles formed by the lines l m, n p.

Read all the pairs of vertical angles formed by the lines a b, c d.

Why are c e b and a e d called vertical angles?

What are vertical angles?

Read all the pairs of vertical angles formed by the lines h i, f g.

How many pairs of vertical angles are formed by the intersection of two lines?

Read all the pairs of vertical angles formed by the lines l m, n p.

LESSON NINTH.

KINDS OF ANGLES.

RIGHT ANGLES.

What do we call the angles a o c, c o b? (Diagram [9].)

Are they equal to each other?

Then they are called right angles.

A right angle is one of two adjacent angles that are equal to each other.

Are the adjacent angles c o b, b o d equal to each other?

Then what are they called?

Read the right angles below the line a b. On the left of c d.

Read three right angles whose vertices are at p.

Diagram 9.

ACUTE ANGLES.

Is the angle m p q greater or less than the right angle m p r?

Then it is called an acute angle.

An acute angle is one which is less than a right angle.

Read four acute angles whose vertices are at p.

Acute means sharp.

Why is r p s an acute angle?

What is an acute angle?

OBTUSE ANGLES.

Is the angle m p s greater or less than the right angle m p r?

Then it is called an obtuse angle.

An obtuse angle is one which is greater than a right angle.

What other obtuse angle has its vertex at p?

Obtuse means blunt.

Read three obtuse angles whose vertices are at x.

Acute and obtuse angles are also called oblique angles.

LESSON TENTH.

REVIEW.

Read all the right angles formed by the lines a b and c d. (Diagram [10].)

Why are the adjacent angles c e b, b e d, right angles?

What is a right angle?

Read four right angles whose vertices are at n.

Which is the greater, the right angle p q r, or the right angle t s u?

Can one right angle be greater than another?

Read six acute angles whose vertices are at n.

Why is m n g an acute angle?

What is an acute angle?

Which is greater, the acute angle m n g, or the acute angle l n m?

May one acute angle be greater than another?

What three acute angles are equal to one right angle?