Marks' first lessons in geometry In two parts. Objectively presented, and designed for the use of primary classes in grammar schools, academies, etc. — Reading Companion

FIRST LESSONS IN GEOMETRY.

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.

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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.

POSITIONS OF LINES 14

RELATIONS OF ANGLES 20

RELATIONS OF LINES 27

PERPENDICULAR LINES 27

PROBLEMS RELATING TO ANGLES 38

ISOSCELES TRIANGLES 48

PROBLEMS RELATING TO TRIANGLES 53

COMPARISON AND CONTRAST OF FIGURES 62

MEASUREMENT OF SURFACES 66

PROBLEMS RELATING TO SURFACES 71

THE CIRCLE AND ITS LINES 73

PARTS OF THE CIRCLE 82

AXIOMS. ILLUSTRATED 85

THEOREMS. ILLUSTRATED 88

FIRST LESSONS IN GEOMETRY.

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._

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.

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.

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_.

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_.

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_.

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.

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_?

What kind of a line is 9 10?

Why? What is a spiral line?

What kind of a line is _w x_?

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?

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

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._

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._

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.

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.

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_.

Read all the lines that form angles with the line _a b_. (DIAGRAM 6.)

Which of them forms the greatest angle with it?

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?

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

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_.

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?

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_.

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_.

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_.

Why is _r p s_ an acute angle?

What is an acute angle?

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_?

Read three obtuse angles whose vertices are at _x_.

Acute and obtuse angles are also called oblique angles.

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?

Which of the two acute angles _v f w_, _y x z_ is the greater?

Read four obtuse angles whose vertices are at _n_.

Why is _f n m_ an obtuse angle?

What is an obtuse angle?

What does obtuse mean? Acute?

By what other name are both called?

Which is greater, the large acute angle 1 4 2, or the small obtuse angle 1 4 3?

How much greater than the right angle is the obtuse angle _f n l_?

How much less than a right angle is _f n i_?

What kind of angles do the lines _a b_ and _c d_ make with each other? (DIAGRAM 11.)

Then they are perpendicular to each other.

What line is perpendicular to _x y_?

Why is it perpendicular to it?

What line is perpendicular to _z_ 1?

When is a line said to be perpendicular to another?

Can a line standing alone be properly called a perpendicular line?

What two lines are perpendicular to the lines _r s_?

Is the line _g h_ perpendicular to the line _i j_? Why?

What other line is perpendicular to the line _i j_?

Read three lines that are perpendicular to the line _a b_.

Do the lines _k l_, _m n_, differ in direction? Then do they form any angle with each other?

They are said to be _parallel_ to each other.

Read four other lines that are parallel with _k l_.

What line is parallel with 2 10?

_Lines are parallel with each other when they do not differ in direction._

What kind of angles do the lines _u t_ and 8 9 form with each other?

Then they are said to be oblique to each other.

_Lines are oblique to each other when they form oblique angles._

See Note C, Appendix.

Read five lines that are perpendicular to the line _a b_. (DIAGRAM 12.)

Five that are perpendicular to _c d_.

Two that are perpendicular to _u v_, and meet it. Three that do not meet it.

Why are _o p_ and _m n_ perpendicular to each other?

When are lines said to be perpendicular to each other?

Read four lines that are parallel with _e f_.

Why are the lines _e f_ and _g h_ said to be parallel to each other?

When are lines said to be parallel to each other?

Read four lines that are parallel to 5 6.

Four that are parallel to _o p_.

Is any line parallel to _u v_?

Can a single line be properly called perpendicular? Parallel?

If two lines are perpendicular to each other, what angle do they form?

If parallel, what angle? If oblique?

Is the angle _a m n_ between the parallels, or outside of them? (DIAGRAM 13.)

It is called an _interior angle_.

Read three other interior angles between the same parallels.

Why is _b m n_ an interior angle?

_An interior angle is one that lies between parallel lines._

Read the interior angles between the parallel lines _g h_ and _k l_.

Why is _o p l_ an interior angle?

What is an interior angle?

Is the angle _a m e_ between the parallels, or outside of them?

It is called an _exterior angle_.