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.
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.
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?
[Illustration: Diagram 13.]
LESSON THIRTEENTH.
RELATIONS OF ANGLES.
INTERIOR ANGLES.
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?
EXTERIOR ANGLES.
Is the angle _a m e_ between the parallels, or outside of them?
It is called an _exterior angle_.
Read three other exterior angles formed by the lines _a b_, _c d_, and
_e f_.
Why is the angle _c n f_ an exterior angle?
_An exterior angle is one that lies outside of the parallels._
LESSON FOURTEENTH.
REVIEW.
Read all the interior angles formed by the lines _a b_, _c d_, and _e
f_.
Why is _m n d_ an interior angle?
What is an interior angle?
Read all the exterior angles formed by the same lines.
Why is _d n f_ an exterior angle?
What is an exterior angle?
Read all interior angles formed by the lines _g h_, _k l_, and _i j_.
All the remaining interior angles in the diagram. All the exterior
angles.
[Illustration: Diagram 14.]
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_?
Are they adjacent?
Are _e m b_, _m n d_, on the same side of the intersecting line _e f_?
Are they adjacent?
Then they are called opposite angles.
_Opposite angles lie on the same side of the intersecting line, but
are not adjacent._
Are the angles _e m b_, _f n d_, on the same side of the intersecting
line?
Are they adjacent?
Then are they opposite?
Are they interior or exterior angles?
Then they are “_opposite exterior angles_.”
Why are they exterior?
Why are they opposite?
Are the angles _b m n_, _m n d_, opposite angles?
Are they interior or exterior angles?
Then they are “_opposite interior angles_.”
Why are they opposite? Why interior?
Read the opposite exterior angles on the left of the line _e f_.
Read the opposite interior angles on the same side.
Are the opposite angles _e m a_, _m n c_, both exterior or interior?
Then they are _opposite exterior and interior angles_.
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_?
Are they adjacent to each other?
Are they vertical angles?
Then they are alternate angles.
_Alternate angles lie on different sides of the intersecting line,
and are neither adjacent nor vertical._
Are the alternate angles _b m n_, _m n c_, exterior or interior?
Then they are called “_interior alternate angles_.”
Read another pair of interior alternate angles between _a b_ and _c
d_.
Are the angles _e m b_, _c n f_, alternate angles? Why?
Are they exterior or interior?
Then what may they be called?
Read another pair of exterior alternate angles.
Why are _e m a_, _d n f_, alternate angles? Why exterior alternate?
LESSON SIXTEENTH.
REVIEW.
Read the exterior opposite angles on the right of the line _e f_.
(DIAGRAM 14.)
On the left. On the right of _r s_. On the left.
Why are _e m a_, _c n f_, exterior angles?
Why are they opposite angles?
What are opposite angles?
Read the interior opposite angles on the right of the intersecting
line _e f_.
On the left of it. On the right of _r s_. On the left.
Read the interior alternate angles formed by the lines _a b_, _c d_,
and _e f_.
Which pair are acute angles?
Which pair are obtuse angles?
Why are _b m n_, _m n c_, interior angles? Why alternate? What are
alternate angles?
Read the exterior alternate angles of the same lines.
Read the acute interior alternate angles of the parallels _t u_, _v
w_. The obtuse.
The acute exterior alternate angles. Obtuse.
Read the pair of opposite exterior angles on the right of the line _e
f_. On the left.
On the right of _r s_. On the left.
[Illustration: Diagram 15.]
LESSON SEVENTEENTH.
REVIEW.
Read thirteen or more angles whose vertices are at _c_. (DIAGRAM 15.)
Read four obtuse angles.
Read two right angles.
What three acute angles equal one right angle?
Which is greater, the right angle 4, or the right angle 5?
The obtuse angle 6, or the acute angle 7?
Read twelve pairs of adjacent angles formed by the lines _w x_, &c.
Read six pairs of vertical angles formed by the same lines.
Read all the interior angles formed by the lines _i j_, _k l_, and _m
n_.
Read all the exterior angles formed by the same lines.
Two pairs of opposite exterior angles.
Two pairs of opposite interior angles.
Four pairs of opposite exterior and interior angles.
Two pairs of alternate interior angles.
Two pairs of alternate exterior angles.
Why are the angles _i o m_, _m o j_, called adjacent?
What are adjacent angles?
What kind of an angle is _i o m_? Why?
What is an acute angle?
What kind of an angle is _m o j_? Why?
What is an obtuse angle?
Why are _a c f_, _f c b_, right angles?
What is a right angle?
Why are _m o i_, _j o p_, vertical angles?
What are vertical angles?
Why is _m o i_ an exterior angle?
What is an exterior angle?
Why is _j o p_ an interior angle?
What is an interior angle?
Why are _m o i_, _o p k_, opposite angles?
What are opposite angles?
Why are _j o p_, _o p k_, alternate angles?
What are alternate angles?
LESSON EIGHTEENTH.
PROBLEMS.
Draw an obtuse angle which shall be only a little larger than a right
angle.
Draw one which shall be much greater than a right angle.
Draw an acute angle which shall be only a little less than a right
angle.
Draw one which shall be much less than a right angle.
Draw an obtuse angle with lines about one inch long.
Draw an acute angle with sides three inches long.
Which is greater, the obtuse angle, or the acute angle?
Draw a right angle with lines an inch long.
Draw one with lines five inches long.
Which is the greater, first or the second?
[Illustration: Diagram 16.]
LESSON NINETEENTH.
POLYGONS.
Name any thing besides your desk that has a flat surface.
A flat surface is called a plane.
How many sides has the plane Fig. A? (DIAGRAM 16.)
It is called a triangle. “Tri” means “three.”
What other triangles do you see.
Triangles are sometimes called trigons.
_A triangle is a plane figure having three sides._
How many sides has the plane figure marked B? How many angles?
It is called a quadrangle, or quadrilateral. “Quad” denotes “four.”
What other quadrangles do you see?
Why is Fig. B a quadrangle?
_A quadrangle is a plane figure having four sides._
How many sides has the Fig. C?
It is called a _pentagon_.
What other pentagon do you see?
Why is Fig. C a pentagon?
_A pentagon is a plane figure having five sides._
In like manner,—
_A hexagon is a plane figure having six sides._
_A heptagon is a plane figure having seven sides._
An octagon has eight sides.
A nonagon has nine sides.
A decagon has ten sides.
All these figures are called _polygons_.
“Poly” means “many.”
What do you call a polygon of three sides? Of four sides? Of six
sides? &c.
If the length of each side of triangle A is one inch, how long are the
three sides together?
The sum of the sides of a polygon is its perimeter.
Which of the triangles has unequal sides? Which has equal sides?
The latter is called a _regular polygon_.
Which pentagon has one side longer than any one of its other sides?
Which has its sides all equal to each other? Are its angles also
equal?
It is therefore a _regular polygon_, or _regular pentagon_.
Name a hexagon that is not regular.
Name a regular hexagon.
A regular octagon. A regular heptagon.
_A polygon is a plane figure bounded by straight lines._
LESSON TWENTIETH.
REVIEW.
Name all the triangles. (DIAGRAM 16.)
Why is Fig. A a triangle?
What is a triangle?
What other name is sometimes given to triangles?
Name all the quadrilaterals.
Why is Fig. B a quadrilateral?
What is a quadrilateral, or quadrangle?
Name all the pentagons, hexagons, heptagons, octagons, and nonagons.
Why is C a pentagon? What is a pentagon? A hexagon? A heptagon? &c.
How many polygons in the diagram?
What is a polygon?
If each side of Fig. B is one inch, how many inches are there in its
perimeter?
When is a polygon regular?
Name all the regular polygons in diagram 16.
Name all the irregular polygons.
[Illustration: Diagram 17.]
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.)
Then it is called an _acute-angled triangle_.
_An acute-angled triangle is one whose angles are all acute._
Read three other acute-angled triangles.
OBTUSE-ANGLED TRIANGLES.
In the triangle 4, what kind of an angle is _l k m_?
Then it is called an _obtuse-angled triangle_.
_An obtuse-angled triangle is one that has one obtuse angle._
Name two others.
RIGHT-ANGLED TRIANGLES.
In the triangle 3, what kind of an angle is _g i j_?
Then it is called a _right-angled triangle_.
_A right-angled triangle is one that has one right angle._
Name three other right-angled triangles.
Upon which side does the triangle 3 seem to stand?
Then _i j_ is called the _base_ of the triangle.
What letter marks the vertex of the angle opposite the base?
Then the point _g_ is said to be the vertex of the triangle.
If, in the triangle 7, we consider _t v_ the base, what point is the
vertex?
If _v_ be considered the vertex, which side will be the base?
In the triangle 3, what side is opposite the right angle?
Then _g j_ is called the _hypothenuse_ of the triangle.
_The hypothenuse of a triangle is the side opposite the right
angle._
Read the hypothenuse of each of the triangles 5, 6, and 11.
Either side about the right angle may be considered the base.
Then the other side will be the perpendicular.
In the triangle 3, if _i j_ is the base, which side is the
perpendicular?
If _g i_ be considered the base, which side is the perpendicular?
In triangle 5, if _n o_ is the base, which side is the perpendicular?
LESSON TWENTY-SECOND.
REVIEW.
Name four acute-angled triangles. (DIAGRAM 17.)
Why is the triangle 8 acute-angled?
What is an acute-angled triangle?
Name three obtuse-angled triangles.
Why is the triangle 9 an obtuse-angled triangle?
What is an obtuse-angled triangle?
Name four right-angled triangles.
Why is the triangle 6 a right-angled triangle?
What is a right-angled triangle?
In the triangle 6, which side is the hypothenuse?
Why?
What is the hypothenuse?
What two sides of the triangle 6 may be regarded as the base?
If _q r_ be considered the base, what do you call the side _q s_?
Read the hypothenuse of each of the triangles 3, 5, 6, and 11.
[Illustration: Diagram 18.]
LESSON TWENTY-THIRD.
TRIANGLES. (_Continued._)
ISOSCELES TRIANGLES.
Of the triangle 1, which two sides are equal to each other?
Then it is called an _isosceles triangle_.
_An isosceles triangle is one that has two equal sides._
Name eight isosceles triangles.
Why is the triangle 2 an isosceles triangle?
What kind of a triangle is it on account of its angles?
Then it is an _acute-angled isosceles triangle_.
Name four acute-angled isosceles triangles.
What kind of a triangle is Fig. 4 on account of the angle _k j l_?
What kind on account of its equal sides?
Then it is called an _obtuse-angled isosceles triangle_.
Name one other obtuse-angled isosceles triangle.
What kind of a triangle is Fig. 6 on account of the angle _q p r_?
What kind on account of its equal sides?
Then it is called a _right-angled isosceles triangle_.
Name one other right-angled isosceles triangle.
Why is Fig. 12 a right-angled triangle? Why isosceles?
EQUILATERAL TRIANGLES.
Which of the isosceles triangles has all its three sides equal to each
other?
It is called an _equilateral triangle_.
“Equi” means “equal.” “Latus” means a “side.”
_An equilateral triangle is one that has its three sides equal to
each other._
What kind of a triangle is Fig. 7 on account of its three equal sides?
What kind on account of its two equal sides _s t_, _s u_, or _t s_, _t
u_, or _u s_, _u t_?
Then must not every equilateral triangle be also isosceles?
What kind of a triangle is Fig. 2 on account of its equal sides _d e_,
_d f_?
If the side _e f_ is longer than either of the other two sides, is it
an equilateral triangle?
Then is every isosceles triangle also equilateral?
Name another isosceles triangle that is _not_ equilateral.
Name one that _is_ equilateral.
In any equilateral triangle the three angles are equal to each other.
On account of its equal angles, it is also called an _equiangular
triangle_.
What is Fig. 8 called on account of its three equal sides? On account
of its three equal angles?
Every equilateral triangle is also equiangular.
Every equiangular triangle is also equilateral.
Name a triangle that has no two sides equal to each other.
It is called a _scalene triangle_.
What kind of a triangle is Fig. 5 on account of its right angle?
What kind on account of its three unequal sides?
Then it is a _right-angled scalene triangle_.
What name can you give Fig. 11 on account of the angle _g e f_?
On account of its three unequal sides?
Then what may it be called?
[Illustration: Diagram 19.]
LESSON TWENTY-FOURTH.
REVIEW.
Name eight isosceles triangles. (DIAGRAM 19.)
Why is Fig. 2 an isosceles triangle?
What is an isosceles triangle?
Name two right-angled isosceles triangles.
Name five acute-angled isosceles triangles.
Name one obtuse-angled isosceles triangle.
Name two isosceles triangles that are also equilateral.
Are all isosceles triangles equilateral?
Name six isosceles triangles that are _not_ equilateral.
What does “equi” mean? “Latus”?
What are equilateral triangles called on account of their equal
angles?
Are all equilateral triangles equiangular?
Are all equiangular triangles equilateral?
What are equilateral triangles?
Name four scalene triangles.
Name two right-angled scalene triangles.
Why is Fig. 3 a right-angled triangle? Why scalene?
What is a scalene triangle?
Name one obtuse-angled scalene triangle.
Name one acute-angled scalene triangle.
PROBLEMS.
From the same point draw two straight lines of any length, making an
acute angle with each other.
Make them equal to each other by measuring.
Join their ends.
What kind of a triangle is it on account of its angles?
On account of its two equal sides?
Write its two names inside of it.
Draw an isosceles triangle whose equal sides shall each be less than
the third side.
Write its two names within it.
Draw an oblique straight line twice as long as any short measure or
unit.
At one end draw a straight line perpendicular to it, and three times
as long as the same measure.
Connect the ends of the two lines by a straight line.
What kind of an angle is that opposite the last line drawn?
Are any two of its sides equal?
Write its two names under it.
Draw a horizontal straight line of any length.
At one end draw a vertical line of equal length.
Complete the triangle, and write two names inside.
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.
[Illustration: Diagram 20.]
QUADRILATERALS.
How many sides has the figure _a b d c_?
What is it called on account of the number of its sides?
Name three other quadrilaterals whose vertices are marked.
Name seven by numbers.
Quadrilaterals are sometimes named by means of two opposite vertices.
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_.
Name the quadrilateral, _g h f e_, four ways.
How many angles has each figure?
On account of the number of their angles they are called
_quadrangles_.
Has the quadrilateral _a d_ any two sides parallel to each other?
Then it is called a _trapezium_.
_A trapezium is a quadrilateral that has no two sides parallel._
Name two other trapeziums.
Why is Fig. 7 a trapezium?
Has the quadrilateral _e h_ any two sides parallel? Which two? Are the
other two sides parallel?
It is called a “_trapezoid_.”
“Oid” means like. What does “trapezoid” mean?
_A trapezoid is a quadrilateral that has only one pair of sides
parallel._
Name another trapezoid.
Why is Fig. 6 a trapezoid?
How many pairs of parallel sides has the quadrilateral _i l_?
Name the horizontal parallels.
Name the oblique parallels.
It is called a “_parallelogram_.”
_A parallelogram is a quadrilateral whose opposite sides are
parallel._
Name five other parallelograms.
Why is Fig. 4 a parallelogram?
Why is not Fig. 6 a parallelogram?
Why is not _e h_ a parallelogram?
What two names may you give to Fig. 5?
Why is it a quadrilateral? Why a trapezium?
What two names may we give to Fig. 6?
Why is it a quadrilateral? Why a trapezoid?
What two names may we give to Fig. 3?
Why is it a parallelogram? Why a quadrilateral?
LESSON TWENTY-FIFTH.
REVIEW.
How many quadrilaterals in the diagram. (DIAGRAM 20.)
Why is Fig. _a d_ a quadrilateral?
What is a quadrilateral?
On account of the number of its angles, what may it be called?
Name all the quadrilaterals.
Name three trapeziums.
Why is Fig. 5 a trapezium?
What is a trapezium?
Name two trapezoids.
Why is Fig. 6 a trapezoid?
Name its parallel sides.
What is a trapezoid?
Name six parallelograms.
Why is Fig. 4 a parallelogram?
Name its two pairs of parallel sides.
What is a parallelogram?
What two names can you give to Fig. 4?
Why the first? Why the second?
What two names may be given to Fig. 7?
Why the first? Why the second?
What two to Fig. 6?
Why the first? Why the second?
[Illustration: Diagram 21.]
LESSON TWENTY-SIXTH.
KINDS OF PARALLELOGRAMS.
RHOMBOIDS.
How many quadrilaterals in the diagram? (DIAGRAM 21.)
How many parallelograms?
Has the parallelogram _a d_ any right angle?
It is called a “_rhomboid_.”
_A rhomboid is a parallelogram which has no right angle._
Name five other rhomboids.
What three names may be given to Fig. 2?
Why is it a quadrilateral?
Why a parallelogram? Why a rhomboid?
RHOMBS.
Are the four sides of the rhomboid _a d_ equal to each other?
Are the four sides of the rhomboid _e h_ equal to each other?
If a triangle has its three sides equal to each other, what do you
call it?
Then when a rhomboid has its sides equal to each other, what may it be
called?
An equilateral rhomboid is called a rhombus.
_A rhombus is an equilateral rhomboid._
See Note D, Appendix.
Name two other rhombuses, or rhombs.
What four names can you give to Fig. _e h_?
Why a quadrilateral? Why a parallelogram? Why a rhomboid? Why a
rhombus?
RECTANGLES.
Has the parallelogram _i l_ any right angles?
How many?
It is called a “_rectangle_.”
_A rectangle is a right-angled parallelogram._
Name four other rectangles.
What three names may be given to Fig. _i l_?
Why a quadrilateral? Why a parallelogram? Why a rectangle?
SQUARES.
Has the rectangle _i l_ its four sides equal?
Has the rectangle _m p_ its four sides equal?
It is called a “square.”
_A square is an equilateral rectangle._
Name another “_square_.”
What four names may be given to Fig. _m p_?
Why a quadrilateral? Why a parallelogram? Why a rectangle? Why a
square?
LESSON TWENTY-SEVENTH.
REVIEW.
Name six rhomboids. (DIAGRAM 21.)
What three names may be given to Fig. 3?
Why a quadrilateral? Why a parallelogram? Why a rhomboid?
What is a quadrilateral? Parallelogram? Rhomboid?
Name three rhombs.
What four names may you give Fig. 5?
Why a quadrilateral? Why a parallelogram? Why a rhomboid? Why a rhomb?
What is a rhomboid? A rhomb?
Name five rectangles.
What three names may be given to Fig. 1?
Why a quadrilateral? Why a parallelogram? Why a rectangle?
What is a rectangle?
Name two squares?
By what four names may Fig. 7 be called?
Why by the first? By the second? By the third? By the fourth?
What is a square?
What is a rectangle?
What is a parallelogram?
What is a quadrilateral?
[Illustration: Diagram 22.]
LESSON TWENTY-EIGHTH.
COMPARISON AND CONTRAST.
TRAPEZIUM AND TRAPEZOID.
In what respect are Figs. A and B alike?
On this account, what name may be given to each?
How does Fig. B differ from Fig. A?
What particular name may you give to Fig. B?
What one to Fig. A?
RHOMBOID AND RECTANGLE.
In what two respects are Figs. C and D alike?
On account of the number of their sides, what may each be called?
Because their opposite sides are parallel, what may each be called?
In what respect do they differ?
What particular name may be given to Fig. C?
What one to Fig. D?
What three names may you give to the figure with right angles?
What three to the one _without_ right angles?
RHOMBOID AND RHOMBUS.
In what three things are Figs. E and F alike?
What three names may be given to each?
How do they differ from each other?
What particular name may you give to Fig. F?
What four names has Fig. F?
RECTANGLE AND SQUARE.
In what three things are Figs. G and H alike?
On account of the number of their sides, what may each be called?
Because their opposite sides are parallel, what may each be called?
Because they have right angles, what may they be called?
In what respect is Fig. H different from Fig. G?
On this account, what particular name may be applied to Fig. H?
What three names may be applied to Fig. G?
What _four_ to Fig. H?
RHOMBUS AND SQUARE.
In what three things are Figs. F and H alike?
On account of the number of their sides, what name may be given to
each?
Because their opposite sides are parallel, what name may be given to
each?
Because both are parallelograms, and both have their sides equal, what
name may be given to each?
What particular name has Fig. F?
What particular name has Fig. H?
What four names may be given to Fig. F?
What four to Fig. H?
LESSON TWENTY-NINTH.
REVIEW.
What two names may be given to Fig. A. (DIAGRAM 22.)
To Fig. B?
In what are they alike?
In what do they differ?
By what three names may Fig. C be called?
By what three names may Fig. D be called?
In what two things are they alike?
In what one thing do they differ?
What particular name has C? What one has D?
What three names may be applied to Fig. E?
What four to Fig. F?
What property has F that E has not?
What particular name has it on that account?
What three names has Fig. G?
What four has Fig. H?
What property has Fig. H that G has not?
What particular name has it in consequence?
What four names may you give to Fig. F?
What four to Fig. H?
What three names may be applied to either?
In what three things are they alike?
In what respect do they differ?
What particular name has Fig. F?
What particular name has Fig. H?
[Illustration: Diagram 23.]
LESSON THIRTIETH.
MEASUREMENT OF SURFACES.
In Fig. 1 (DIAGRAM 23.) call the line _a b a_ unit.
Rectangle 1 is how many units long?
How many high?
Because its sides are equal, what is it called?
Rectangle 2 is how many units long?
How many high or wide?
How many squares does it contain?
Rectangle 3 is how many units long?
How many wide?
How many squares does it contain?
If it were four units long and one wide, how many squares would it
contain?
If it were five long and one wide? Six long? &c.
Rectangle 4 is how many units long?
How many wide?
How many squares does it contain?
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?
If the width were one unit more, how many times two squares would
there be?
Rectangle 5 is how many units long?
How many units wide or high?
How many squares does it contain?
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 another unit were added to the width, how many more squares would
be made?
How many times three squares?
If it were four units wide, how many times three squares would there
be?
Rectangle 6 is how many units long?
How many units high or wide?
How many squares in that part which is four units long and one high?
How many times four squares in that part which is four long and two
high?
How many times four squares when it is four long and three high?
If another unit were added to the height, how many more squares would
be added?
How many times four squares would there be?
If a rectangle were five units long and one unit wide, how many square
units would it contain?
If it were two units wide, how many times five square units would it
contain?
If it were three units wide? Four? &c.
If your ruler is ten inches long and only one inch wide, how many
square inches are there in it?
If it were two inches wide, how many times ten square inches would it
contain?
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?
If the same wall is sixteen feet high, how many square feet in it?
Fig. 5 has how many times three squares?
Fig. 7 has how many times two squares?
Which has the greater number of squares?
What difference is there between two times three squares and three
times two squares?
LESSON THIRTY-FIRST.
REVIEW.
Draw a rectangle of any width whose length is three times the width.
How many squares has it if the width be taken as the unit?
Make it twice as wide as before.
How many squares has it now?
What two numbers multiplied together will give the number of squares?
Make it three times as wide.
How many squares has it now?
What two numbers multiplied together will give the number of squares?
The cover of a geography is one foot long and one foot wide, how many
square feet in it?
How many inches long is the same cover? How many wide?
How many square inches does it contain?
How many square inches are equal to one square foot?
A table is one yard long and one yard wide, how many square yards in
it?
How many feet long is the same table?
How many feet wide?
How many square feet does it contain?
One square yard equals how many square feet?
Draw a square whose side is a unit of any length.
Draw another whose side is two units of the same length.
The second square is how many times as large as the first one?
How many squares in half the second square?
Which is greater, two square inches, or two inches square?
Two inches square is how many times two square inches?
Draw a square whose side is three inches.
How many square inches does it contain?
How many times as many squares as the square of one inch?
How many square inches in the bottom row?
How many in all?
Which is greater, three inches square, or three square inches?
Three inches square is how many times three square inches?
PROBLEMS.
An equilateral triangle has each of its sides one inch long, what is
its perimeter?
If each side were two inches long, what would be 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 square geography-cover is nine inches long on one side, how long all
round?
How many square inches in it?
A slate is sixteen inches long and twelve wide, how many inches all
round it?
A rectangle is five inches long and three wide, how long all round?
How many square inches in it?
A slate is one foot long and eight inches wide, what is its perimeter?
A room is twenty-four feet long and twenty-one feet wide, how many
feet all round it?
How many square feet in the floor?
How many pieces of paper each a foot square would exactly cover it?
A yard of carpet is two feet wide, how many square feet in it?
Charles 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?
A lot is forty rods long and thirty wide, how long must the fence be?
What length of fence will divide it into four equal parts?
[Illustration: Diagram 24.]
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?
What kind of a line would it be?
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?
The plane figure bounded by this curve is called a “_circle_.”
What point is at the centre of this figure?
_A circle is a plane figure bounded by a curved line, all points of
which are equally distant from the centre._
The curved line is called a “_circumference_.”
_The circumference of a circle is the curve which bounds it._
Name a straight line that joins two points in the circumference.
It is called a “_chord_.”
_A chord is a straight line that joins two points of a
circumference._
Read six chords in the diagram.
Which two of these chords pass through the centre?
They are called “_diameters_.”
_A diameter is a chord that passes through the centre._