It is called a “_radius_.”—(Plural, _radii_.)
_A radius is a straight line that joins the centre to a point of the
circumference._
Read five radii.
Which is farther from the centre, the point _a_ or the point _d_?
Can the radius _c d_ be greater than the radius _c a_? Or greater than
_c v_, or _c o_?
_Then all radii of the same circle are equal to each other._
What do we call the lines _o d_, _c d_, _c o_?
What part of the diameter _o d_ is the radius _o c_?
Name a chord that is produced without the circle.
It is called a “_secant_.”
_A secant is a chord produced._
Name two secants.
If the chord _d i_ were made a secant, would it become longer or
shorter?
In how many points does the straight line _l m_ touch the
circumference?
It is called a “_tangent_.”
_A tangent is a straight line that touches a circumference in only
one point._
Name three tangents.
LESSON THIRTY-THIRD.
REVIEW.
Read six chords. (DIAGRAM 24.)
Why is _i d_ a chord?
What is a chord?
Name two diameters.
Why is _a j_ a diameter?
What is a diameter?
Is every chord a diameter?
Is every diameter a chord?
Name five radii.
Why is _c a_ a radius?
What is a radius?
A diameter is equal to how many radii?
Are all radii equal to each other?
Are all chords equal to each other?
Are all diameters equal to each other?
Name two secants.
Why is either one a secant?
What is a secant?
Name three tangents.
Why is _a b_ a tangent?
What is a tangent?
Is a tangent inside of a circle or outside of it?
Is a chord inside or outside of a circle?
Is a secant within or without a circle?
If the radius is three inches, how long is the diameter?
[Illustration: Diagram 25.]
LESSON THIRTY-FOURTH.
ARCS AND DEGREES.
What small part of the circumference of circle 1 (DIAGRAM 25.) is
marked?
It is called an “_arc_.”
_An arc is any part of a circumference._
Read five arcs that are marked.
Which is longer, the arc _e d_, or the arc _e f_? _b d_, or _b d e_?
_a b d_, or _a b d e_?
Name an arc which is half of the circumference.
It is called a “_semi-circumference_.”
“Semi” means “half.”
_A semi-circumference is half of a circumference._
Read three arcs, each of which is one-fourth of the circumference.
If the whole circumference were divided into three hundred and sixty
equal arcs, would each arc be large or small?
Each of these arcs would be called a “_degree_.” [Degrees are marked
(°).]
_A degree of a circumference is a three hundred and sixtieth part of
it._
How many degrees in a semi-circumference?
How many degrees in one-fourth of a circumference?
If a fourth of a circumference were divided into three equal parts,
how many degrees would there be in each part?
Into how many parts would each third of a quarter have to be again
divided to make single degrees?
Is an arc of ninety-one degrees greater or less than one-fourth of a
circumference?
Is an arc of a hundred and seventy-nine degrees greater or less than a
semi-circumference?
Can there be more than three hundred and sixty degrees in a
circumference?
If the circumference of circle 1 were divided into degrees, each
degree would be so small an arc that it would look like a dot.
If a degree were divided into sixty equal parts, each part would be
called a minute.
If a minute were divided into sixty equal parts, each part would be
called a second.
How many degrees in the large circle of Fig. 2?
How many in the smaller one?
Has a large circle any more degrees than a small circle?
In the large circle how many degrees from _a_ to _b_?
In the small circle how many from _a_ to _b_?
Which is greater, an arc of ninety degrees of the large circle, or one
of ninety degrees of the small one?
Which is greater, an arc of a degree of the large circle, or one of a
degree of the small one?
The angle _a o b_ has its vertex at what part of the larger circle?
At what part of the smaller circle?
On how many degrees of the larger circle does the angle stand?
On how many degrees of the smaller circle does it stand?
Then it is said to be an angle of 90°.
If the angle _a o f_ is an angle of 30°, how many degrees must there
be in the arc _a f_?
If the arc _f e_ is an arc of 60°, what is the size of the angle _f o
e_?
An angle of 10° stands upon an arc of how many degrees? Of 8°? Of 1°?
The angle _a o b_ is what kind of an angle?
Upon how many degrees does it stand?
Then a right angle is an angle of how many degrees?
If an angle stand upon less than 90°, what kind of an angle is it?
If an angle stand upon more than 90°, what kind of an angle is it?
Can an angle have as many degrees as a hundred and eighty?
LESSON THIRTY-FIFTH.
REVIEW.
Read nine arcs whose ends are marked. (DIAGRAM 26.)
Read three arcs each of which is one-fourth of a circumference.
Read two arcs each of which is one-half of a circumference.
Why is _e g_ an arc?
What is an arc?
How many degrees in the arc _f h_? In _e h_?
If the arc _f h_ were divided into three equal parts, how many degrees
would there be in each?
How many degrees in a circumference?
In a semi-circumference?
How many more degrees in a large circumference than in a small one?
If the arc _i f_ is 40°, what is the size of the angle _f o i_?
If the angle _f o g_ is an angle of 130°, what is the size of the arc
_f i h g_?
How many degrees in each of the adjacent angles _f o h_, _h o e_?
When two adjacent angles are equal to each other, what is each called?
How many degrees in a right angle?
[Illustration: Diagram 26.]
LESSON THIRTY-SIXTH.
PARTS OF THE CIRCLE.
The part of the circle bounded by the chord _a b_ and the arc _a b_ is
called a segment.
Read three segments, each less than half a circle, thus,—the segment
bounded by the chord _a d_ and the arc _a b d_.
_A segment is a part of a circle bounded by an arc and a chord._
Read two segments that are each half a circle.
What is the chord called?
What is the arc called?
A segment bounded by a diameter and a semi-circumference is a
“_semicircle_.”
_A semicircle is half a circle._
Read four segments each larger than a semicircle.
The part of the circle between the two radii _o f_, _o i_, and the arc
_f i_, is called a “_sector_.”
Read four sectors each less than one-fourth of a circle.[2]
Footnote 2:
Thus, a sector bounded by the two radii _o g_, _o h_, and the arc _g
h_.
_A sector is a part of a circle bounded by two radii and an arc._
What part of the whole circle is the sector _f o h_?
It is called a “_quadrant_.”
_A quadrant is a sector which is one-fourth of a circle._
Read a sector which is greater than a quadrant.
If the chord _e f_ be regarded a diameter, what do you call the
semicircle below it?
If it be regarded as two radii, what is the semicircle called?
Then a semicircle is both a segment and a sector.
LESSON THIRTY-SEVENTH.
REVIEW.
Name ten segments. (DIAGRAM 26.)
What is a segment?
Of the segments named, which are less than a semicircle?
Which are greater?
Which two are semicircles?
Which two are on the chord _a f_?
Name nine sectors.
Why is _g o i_ a sector?
What is a sector?
Which four of the sectors named are each less than a quadrant?
Which three are quadrants?
Which two are greater than a quadrant?
What part of the circle is both a segment and a sector?
How many quadrants in a circle?
How many semicircles?
PART SECOND.
AXIOMS AND THEOREMS.
AXIOMS ILLUSTRATED.
AXIOM 1.
The triangle A is equal to the triangle C.
The triangle B is also equal to the triangle C.
What do you think of the two triangles A and B? Why?
[Illustration]
_If two things are separately equal to the same thing, they are equal
to each other._
AXIOM 2.
The square A is equal to the square B.
To the rectangle C add the square A, and we have an L pointing in what
direction?
To the same rectangle C add the square B, and we have an L pointing in
what direction?
[Illustration]
Which is larger, the L pointing to the left, or that pointing to the
right?
To what same thing did you add two equals?
What two equals did you add to it?
What was the first sum?
The second?
What do you think of the two sums?
_If equals be added to the same thing, the sums will be equal._
AXIOM 3.
[Illustration]
The square A is equal to the square B.
From the inverted T take away the square A, and we have an L pointing
in what direction?
From the same Fig. T take away the square B, and we have an L pointing
in what direction?
Which is larger, the L pointing to the right, or that pointing to the
left?
What two equal things did we take away from the same thing?
From what same thing did we take them away?
What did we find true of the two remainders?
_If equals be taken from the same thing, the remainders will be
equal._
AXIOM 4.
[Illustration]
The rectangle 1 2 is equal to the rectangle 1 3.
From the rectangle 1 2 take away the square A, and what rectangle
remains?
From the rectangle 1 3 take away the same square A, and what rectangle
remains?
Which is greater, the rectangle B, or the rectangle C?
What same thing did we take away from equals?
From what did we first take it?
What remained?
From what did we next take it?
What remained?
What did we find true of the two remainders?
_If the same thing be taken from equals, the remainders will be
equal._
AXIOM 5.
_If equals be added to equals, the sums will be equal._
AXIOM 6.
_If equals be subtracted from equals, the remainders will be equal._
AXIOM 7.
_If the halves of two things are equal, the wholes will be equal._
AXIOM 8.
_Every Whole is equal to the sum of all its parts._
AXIOM 9.
_From one point to another only one straight line can be drawn._
AXIOM 10.
_A straight line is the shortest distance between two points._
AXIOM 11.
_If two things coincide throughout their whole extent, they are
equal._
THEOREMS ILLUSTRATED.
[Illustration: Diagram 29.]
DEVELOPMENT LESSON.
Do the angles Blue, Red, take up all the space on the line _a b_?
Do the angles Blue, Yellow, Red, take up all the space on the line?
Do the angles Blue, Yellow, Green, Red, take up all the space on the
line?
Is there room between any two of the angles to put in another angle?
Then are not the angles Blue, Yellow, Green, Red, equal to all the
space on the line _a b_?
NOTE.—The word _space_, as here used, means _angular space_; and it
is indispensable that the teacher impress this fact upon the
learner.
By means of former lessons, the pupil has learned positively, that
an angle is the difference between the directions of two lines; and,
impliedly, that the included space has nothing to do with the size
of the angle. There cannot, therefore, be much danger that the pupil
will imbibe any erroneous notion from this style of expression,
which is very much more simple than to say that the difference of
direction of two given lines is equal to the difference of direction
of two other given lines, which style will be used somewhat later in
these lessons.
[Illustration: Diagram 30.]
PROPOSITION I. THEOREM.
DEVELOPMENT LESSON.
Are the adjacent angles Green, Red, equal to all the angular space on
the line _a b_?
Place a paper square corner or right angle on the line _a b_ at the
_left_ of _c d_ with its vertex at _c_.
It will cover all the angle Green and part of the angle Red up to the
line _c d_.
Now place another square corner on the line _a b_ to the _right_ of
the line _c d_, and with its vertex at the point _c_.
It will cover the remaining part of the angle Red, and two edges of
the square corners will meet along the line _c d_.
Are the two right angles equal to all the angular space on the line _a
b_?
Then if the two adjacent angles Green, Red, are equal to all the
angular space on the line _a b_, and the two right angles are also
equal to the same space, what do you infer concerning the _adjacent
angles_ and the _two right angles_?
What axiom do you apply when you say that the _adjacent_ angles are
equal to the _two right angles_?
To what _same thing_ did you find two things separately equal?
What did you first see equal to it?
What did you next see equal to it?
Then what did you _find_ true?
If the angle Red were smaller, and the angle Green larger, would the
adjacent angles still be equal to two right angles?
Then,—
_Any two adjacent angles are equal to two right angles._
If we draw the straight line _c d_ where the edges of the square
corners come together, what kind of angles will _a c d_, _d c b_,
be?
See now if you can understand the following demonstration:—
DEMONSTRATION.
We wish to prove that
_Any two adjacent angles are equal to two right angles._
Let the two straight lines _a b_, _m n_, intersect each other in the
point _c_. (DIAGRAM 30.)
Then will any two adjacent angles, as Green, Red, be equal to two
right angles?
For, from the point _c_, draw the straight line _c d_ so as to make
the angles _a c d_, _d c b_, right angles.
The adjacent angles Green, Red, are equal to all the angular space on
the line _a b_.
The right angles _a c d_, _d c b_, are also equal to all the angular
space on the line _a b_.
Therefore the adjacent angles Green, Red, are equal to two right
angles.
TEST QUESTIONS.
To what same thing did you find two things equal?
What did you first see equal to it?
What did you next see equal to it?
Then what new thing did you find true?
What axiom did you make use of?
[Illustration: Diagram 31.]
TEST LESSON.
By means of Fig. A,—
1. Prove that the adjacent angles Green, Red, are equal to two right
angles.
2. Prove that the adjacent angles Blue, Yellow, are equal to two right
angles.
By means of Fig. B,—
3. Prove that the adjacent angles Green, Red, are equal to two right
angles.
4. Prove that the adjacent angles Yellow, Blue, are equal to two right
angles.
By means of Fig. C,—
5. Prove that the adjacent angles Red, Blue, are equal to two right
angles.
6. Prove that the adjacent angles Green, Yellow, are equal to two
right angles.
7. Give the preceding demonstrations again, but name the angles by
their letters instead of by their colors.
[Illustration: Diagram 32.]
TEST LESSON.
By means of Fig. A prove,—
1. That the adjacent angles _a c m_, _m c b_, are equal to two right
angles.
2. That the adjacent angles _a c n_, _n c b_, are equal to two right
angles.
By means of Fig. B prove,—
3. That the adjacent angles _a c n_, _n c b_, are equal to two right
angles.
4. That the adjacent angles _a c m_, _m c b_, are equal to two right
angles.
By means of Fig. C prove,—
5. That the adjacent angles _a c m_, _m c b_, are equal to two right
angles.
6. That the adjacent angles _a c n_, _n c b_, are equal to two right
angles.
By means of Fig. D prove,—
7. That the adjacent angles _a c n_, _n c b_, are equal to two right
angles.
8. That the adjacent angles _b c m_, _m c a_, are equal to two right
angles.
[Illustration: Diagram 33.]
PROPOSITION II. THEOREM.
DEVELOPMENT LESSON.
What kind of angles are P and S?
How do the adjacent angles Yellow, Blue, compare with the right angles
P, S?
How do the adjacent angles Blue, Red, compare with the two right
angles?
Then if the adjacent angles Yellow, Blue, are equal to two right
angles, and the adjacent angles Blue, Red, are also equal to two
right angles, what do you think of the two pairs of adjacent angles,
Yellow, Blue, and Blue, Red?
If, from the adjacent angles Yellow, Blue, we take away the angle
Blue, what remains?
If, from the adjacent angles Blue, Red, we take away the same angle
Blue, what remains?
Then, since the same angle Blue has been taken from equal pairs of
adjacent angles, what do you think of the two remainders, Yellow,
Red?
Suppose the lines _a b_ and _m n_ were so drawn that the angles
Yellow, Red, were larger or smaller, would they still be equal to
each other?
Then,—
_All vertical angles are equal to each other._
[Illustration: Diagram 34.]
DEMONSTRATION.
We wish to prove that
_All vertical angles are equal to each other._
Let the straight lines _a b_, _m n_, intersect each other at the point
_c_, then will any two vertical angles, as Yellow, Red, be equal to
each other.
For the adjacent angles Yellow, Blue, are equal to two right
angles.[3]
Footnote 3:
When this comparison is made, let the pupil look at the right angles
P and S.
The adjacent angles Blue, Red, are also equal to two right angles.
Therefore the adjacent angles Yellow, Blue, are equal to the adjacent
angles Blue, Red.
If, from the adjacent angles Yellow, Blue, we take away the angle
Blue, we shall have left the angle Yellow.
If, from the adjacent angles Blue, Red, we take away the same angle
Blue, we shall have left the angle Red.
Therefore the vertical angles Yellow, Red, are equal to each other.
TEST QUESTIONS.
When you say that the adjacent angles Yellow, Blue, are equal to two
right angles, do you know it because you _see_ it, or because you
have _proved_ it?
How do you know that the adjacent angles Blue, Red, are equal to two
right angles?
When you say the adjacent angles Yellow, Blue, are equal to the
adjacent angles Blue, Red, what axiom do you use?
What same thing do you take away from equals?
From what equals do you take it away?
When you take the angle Blue from the adjacent angles Yellow, Blue,
what is the remainder?
When you take the same angle Blue from the adjacent angles Blue, Red,
what is the remainder?
What do you find true of the two remainders?
What axiom do you use?
[Illustration: Diagram 35.]
OTHER METHODS OF DEMONSTRATION.
The adjacent angles Yellow, Green, are equal to what?
The adjacent angles Green, Red, are equal to what?
Then what do you know of the two pairs of adjacent angles Yellow,
Green, and Green, Red?
From the adjacent angles Yellow, Green, take away the angle Green.
What remains?
From the adjacent angles Green, Red, take the same angle Green. What
remains?
What do you know of the two remainders?
Why?
What axiom do you use?
In the last lesson, when you proved the vertical angles Yellow, Red,
equal to each other, you made use of the angle Blue; now prove the
same two angles equal by means of the angle Green.
The adjacent angles Blue, Red, are equal to what?
The adjacent angles Red, Green, are equal to what?
Then what do you know of the two pairs of adjacent angles Blue, Red,
and Red, Green?
From the adjacent angles Blue, Red, take away the angle Red. What
remains?
From the adjacent angles Red, Green, take away the same angle Red.
What remains?
Then what do you know of the two remainders, Blue, Green?
Now apply the preceding demonstration to the vertical angles Blue,
Green.
Prove the vertical angles Blue, Green, equal to each other by means of
the angle Yellow.
[Illustration: Diagram 36.]
TEST LESSON.
By means of Fig. A,—
1. Prove that the vertical angles Yellow, Red, are equal to each
other, using the angle Green.
2. Prove the same thing, using the angle Blue.
3. Prove that the vertical angles Blue, Green, are equal to each
other, using the angle Yellow.
4. Prove the same thing, using the angle Red.
By means of Fig. B,—
5. Prove the vertical angles Yellow, Red, equal to each other, using
the angle Green.
6. Prove the same thing, using the angle Blue.
7. Prove the vertical angles Green, Blue, equal by means of the angle
Red.
8. Prove the same thing by means of the angle Yellow.
Go through the preceding eight demonstrations again, calling the
angles by their letters instead of by their colors.
By means of Fig. C, prove that
9. _a c n_ equals _m c b_, by means of _a c m_.
10. _a c n_ equals _m c b_, by means of _b c n_.
11. _a c m_ equals _n c b_, by means of _a c n_.
12. _a c m_ equals _n c b_, by means of _m c b_.
By means of Fig. D, prove that
13. _m c a_ equals _b c n_, by means of _a c n_.
14. _m c a_ equals _b c n_, by means of _m c b_.
15. _m c b_ equals _a c n_, by means of _m c a_.
16. _m c b_ equals _a c n_, by means of _b c n_.
[Illustration: Diagram 37.]
PROPOSITION III. THEOREM.
DEVELOPMENT LESSON.
In the above diagram, the lines _a b_, _c d_, are parallel, and are
intersected by the line _e f_ at the points _m_ and _n_.
The angle Red measures the difference of direction between the line _m
b_ and what other line?
The angle Yellow measures the difference of direction between the line
_n d_ and what other line?
Then, as the lines _m b_ and _n d_ are parallel, must there not be the
same difference of direction between them and the line _e f_?
Then can there be any difference between the angles which measure
those equal directions?
Then what do you think of the opposite exterior and interior angles
Red, Yellow?
DEMONSTRATION.
We wish to prove that
_Opposite exterior and interior angles are equal to each other._
Let the straight line _e f_ intersect the two parallel straight lines
_a b_, _c d_, at the points _m_ and _n_.
Then will any two opposite exterior and interior angles, as Red,
Yellow, be equal to each other.
For the angle Red measures the difference of direction of the lines _m
b_ and _e f_.
And the angle Yellow measures the difference of direction of the lines
_n d_ and _e f_.
But because the lines _m b_, _n d_, are parallel, these differences
are equal.
Therefore the angles which measure them are equal; that is,
The opposite exterior and interior angles Red, Yellow, are equal to
each other.
[Illustration: Diagram 38.]
TEST LESSON.
By means of Fig. A,—
1. Prove that the opposite exterior and interior angles Green, Blue,
are equal to each other.
2. Prove that the opposite exterior and interior angles Red, Yellow,
are equal to each other.
3. Prove the opposite exterior and interior angles _c n e_, _a m n_,
equal.
4. Prove the opposite exterior and interior angles _e n d_, _n m b_,
equal.
By means of Fig. B,—
5. Prove the opposite exterior and interior angles _e m a_, _m n d_,
equal.
6. Prove the opposite exterior and interior angles _a m n_, _d n f_,
equal.
7. Prove the opposite exterior and interior angles _e m b_, _m n c_,
equal.
8. Prove the opposite exterior and interior angles _b m n_, _c n f_,
equal.
[Illustration: Diagram 39.]
PROPOSITION IV. THEOREM.
DEVELOPMENT LESSON.
What do you know of the opposite exterior and interior angles Red,
Yellow?
What do you know of the vertical angles Red, Green?
Then if the interior alternate angles Green, Yellow, are separately
equal to the angle Red, what new fact do you know?
What axiom do you employ?
To what same thing did you find two things equal?
What two things did you find equal to it?
DEMONSTRATION.
We wish to prove that
_Any two interior alternate angles are equal to each other._
Let the straight line _e f_ intersect the two parallel straight lines
_a b_, _c d_, in the points _m_ and _n_.
Then will any two interior alternate angles, as Green, Yellow, be
equal to each other.
For the opposite exterior and interior angles Red, Yellow, are equal.
The vertical angles Red, Green, are also equal.
Then because the interior alternate angles Green, Yellow, are
separately equal to the angle Red, they are equal to each other.
[Illustration: Diagram 40.]
TEST LESSON.
What do you know of the vertical angles Green, Red, in Fig. A?
What do you know of the opposite exterior and interior angles Red,
Yellow?
Then if the interior alternate angles Green, Yellow, are separately
equal to the angle Red, what do you infer?
By means of Fig. A,—
1. Prove that the interior alternate angles Green, Yellow, are equal,
using the angle Red.
2. Prove the same angles equal, using the angle Blue.
3. Go through the same demonstrations again, calling the angles by
their letters instead of by their colors.
By means of Fig. B,—
4. Prove the interior alternate angles Red, Blue, equal, using the
angle Yellow.
5. Prove the same angles equal, using the angle Green.
6. Go through the same two demonstrations again, naming the angles by
their letters instead of by their colors.
By means of Fig. C,—
7. Prove the interior alternate angles _c n m_, _n m b_, equal, using
the angle _f n d_.
8. Prove the same, using the angle _a m e_.
9. Prove the interior alternate angles _a m n_, _m n d_, equal, using
the angle _e m b_.
10. Prove the same, using the angle _c n f_.
[Illustration: Diagram 41.]
PROPOSITION V. THEOREM.
DEVELOPMENT LESSON.
What do you know of the opposite exterior and interior angles Red,
Yellow?
What do you know of the vertical angles Yellow, Green?
Then if the exterior alternate angles Red, Green, are separately equal
to the angle Yellow, what new thing do you know to be true?
What axiom do you employ?
To what same thing did you know two things to be equal?
What two things did you know to be equal to it?
Then what new thing did you _find_ to be true?
DEMONSTRATION.
We wish to prove that
_Any two exterior alternate angles are equal to each other._
Let the straight line _e f_ intersect the two parallel straight lines
_a b_, _c d_, at the points _m_ and _n_.
Then will any two exterior alternate angles, as Red, Green, be equal.
For the opposite exterior and interior angles Red, Yellow, are equal
to each other.