Bernhard Marks.

Marks' first lessons in geometry, objectively presented online

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MARKS' FIRST LESSONS IN GEOMETRY ***




Produced by Richard Tonsing and the Online Distributed
Proofreading Team at http://www.pgdp.net (This file was
produced from images generously made available by The
Internet Archive)









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.

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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.


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

PROBLEMS 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




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.

[Illustration: 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.

Footnote 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?

[Illustration: Diagram 2.]

[Illustration: 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.

[Illustration: Diagram 4.]

[Illustration: 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?

[Illustration: 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?

[Illustration: 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?

[Illustration: 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_.

[Illustration: 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?

[Illustration: Diagram 10.]

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_?

[Illustration: Diagram 11.]




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.)

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_.


PARALLEL LINES.

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?

Why?

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


OBLIQUE LINES.

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.

[Illustration: Diagram 12.]




LESSON TWELFTH.


REVIEW.

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.


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Online LibraryBernhard MarksMarks' first lessons in geometry, objectively presented → online text (page 1 of 6)