marked.

So I want to find the centre. I draw any two chords, as AB and CD (the nearer they are at right angles to each other the better for accuracy). I then bisect each chord with a perpendicular, as AB with the perpendicular MN, and CD with RS (39). The intersection of these two perpendiculars, as O, is the centre of the circle. [The pupil must do everything with his pencil, ruler, and dividers, just as he says. He must not be of those who “ say and do not." He must do the things told," over and over," till he can do them neatly and easily.]

N

B

FIG. 32.

58. Prob.-To pass a circumference through three given points. SOLUTION.-I wish to pass a circumference through the three given points A, B, and C. [The pupil should first designate three points by dots on his paper, slate, or board, and then proceed according to the solution.] In order to do this, I join A and B with a line, and also B and C. I now bisect these lines with the perpendiculars MN and RS, as in the last problem. The intersection of these perpendiculars, O, is the centre of the required circle. Now setting the sharp point of the dividers upon O and opening them till the pencil point just reaches A (B or C will answer as well), I draw the circumference with O as its centre and the radius OA, and find that it passes through the three given points A, B, and C.

A

N

FIG. 3.

Ex. 1. To pass a circumference through the three vertices of a triangle, i. e., to circumscribe a circumference about a triangle, as this operation is technically called.

SUG. This is just like the last, A, B, and C being the vertices of the triangle. The four figures in the margin represent the successive steps in the solution. First draw the given triangle. Then take the first step in the solution, then the second, etc.

Ex. 2. Given the centre of a circle and a point in the circumference, to draw the circle.

SUG.-Make a dot on the board to indicate the centre, and another dot to indicate the point in the circumference to be found. This is what is given. You are

GIVEN

IT STEP

2 NO STEP

X

FIG. 24.

then to draw the circumference, which shall pass through the latter point, and have the former for its centre.

Ex. 3. Draw an arc of a circle, and rub out the mark, if you make any, at the centre, so that you cannot see where the centre is. Then find the centre, and complete the circumference according to these problems.

SUG.-Mark three points in the given arc, and then the example is just like the last. [Do not fail to do it, over and over," till you can do it quickly and neatly. These exercises require much care in order to get good figures.]

59. Theorem.-The circumference of a circle is about 3.1416 times its diameter. The Greek letter π (called p) is used to represent this number; and hence the circumference is said to be п times the diameter.

ILL. The pupil can illustrate this fact by taking any wheel which is a true circle, and measuring the diameter with a narrow band of paper (something that will not stretch), and then wrapping this measure about the circumference. He will find that it takes a little more than three diameters to go around. Of course he cannot tell exactly how much more. In fact, nobody knows exactly. But the number given above is near enough for most purposes. For many purposes 34 is sufficiently accurate.

29

$31.4

11

By drawing a circle very carefully, say 1 inch in diameter, as in the margin, and dividing the diameter into 10ths inches, a nice pair of dividers can be opened one 10th inch and made to step around the 1 2 3 4 5 6 7 8 903 circumference. If it is all done with nicety, it will be found to be a little over 31 steps around, when it is 10 across.

24 23 22 21

20

14

75

16

19

Ex. 1. The distance across a wagon-wheel (the diameter) is 4 feet, how long a bar of iron will it take to make the tire?

FIG. 35.

Ex. 2. Suppose the crown of your hat is a circular cylinder 7 inches in diameter, how much ribbon will it take for a band, allowing of a yard for the knot?

Ex. 3. How many times will the driving-wheel of an engine, which is 6 feet in diameter, revolve in going from Detroit to Chicago, a distance of 288 miles, allowing nothing for slipping?

Ex. 4. A boy's hoop revolved 200 times in going around a citysquare, a distance of 140 rods. What was the diameter of his hoop?

Ex. 5. What is the radius of a circle whose semi-circumference is πρ In a circle whose radius is 1, what part of the circumference

π

represent? What part? What part does 27 represent?

SECTION III.

ABOUT ANGLES.

60. Prob.-To show how angles are generated and measured.

D

ILL.-An angle is generated by a line revolving about one of its extremities. Thus, suppose OB to have started from coincidence with OA, and, O remaining fixed, the line to have revolved to the position OB, the angle BOA would have been generated. When the revolving line has passed one-quarter the way around, as to DO, it has generated a right angle; when one-half way around, as to FO, two right angles; when entirely around, four right angles.

E

e

kz

B

ke A

Now, if any circle be described from O as a centre, the arc included by the sides of any angle having its vertex at O, is the same part of a quarter of this circumference as the angle is of a right angle. Hence the angle is said to be measured by the arc included by its sides. Thus, the angle COA is measured by the arc ac; i. e., it is the same part of a right angle that are ac is of arc ad. (See Trigonometry, 3-10.)

FIG. 36.

61. Theorem.-The relative lengths of arcs described with the same radius can be found in a manner altogether similar to that given in (36) for comparing straight lines.

b

ILL.-If I wish to compare the two arcs ab and cd described with the same radii, I take the dividers, and placing the sharp point on d (one end of the shorter arc), open them till the other point is at c. I then measure this distance off on ab as many times as I can,-in this case 2 times, with a remainder fb. This remainder, fb, I measure off in the same way upon dc, and find it goes once with a remainder gc. This remainder, gc, I apply to the arc fb, and find it goes once with a remainder hb. This last remainder I find is contained in the last preceding, gc, 2 times. Then, counting up the parts, I find that de is made up of 5 parts each equal to hb, and ab of 13 such parts. Therefore, ab is 23 times as long as dc. [The angle O is therefore 23 times the angle C.]

2

5

9

Fra. 37.

Ex. 1. Draw an acute angle and also an obtuse angle, and then compare them as above.

Ex. 2. Draw a small acute angle and a large acute one, and then compare them as above.

Ex. 3. Draw a small acute angle, and then draw another angle 3 times as large.

Ex. 4. Draw an acute angle, and also a right angle, and compare them as above.

SUG.-Article (39) shows how to draw a right angle.

Ex. 5. Draw any angle, and then draw another equal to it.

Ex. 6. Show that the angles a, b, and c are respectively,, and .6 of a right angle.*

Ex. 7. Show that angles a and b, Fig. 39, are respectively 14 and 11⁄2 times a right angle.

Ex. 8. Draw a regular inscribed hexagon, as in Fig. 31, and then comparing any one of its angles with a right angle, find that it is 13 times a right angle.

* Of course, absolute accuracy is not to be expected in such solutions.

62. An Inscribed Angle is an angle whose vertex is in the circumference of a circle, and whose sides are chords, as A, Fig. 41.

63. Theorem.-An inscribed angle is measured by one-half the arc included between its sides.

ILL.-The meaning of this is that an inscribed angle like A, which includes any particular arc, as cd, is only half as large as an angle would be at the centre, as cod, whose sides included the same arc, cd, or an equal arc. Thus, in this case, drawing the arc ab from A as a centre, with the same radius, Od, as cd is drawn with, I find that ab which measures A is of cd which measures cod.

Ex. 1. Which of the angles a, b, c, d, e is the largest? measured by? What b? What c? What d? What e?

What is a
Fig. 42.

By what

Ex. 2. Which is the greatest angle, a, b, or c, Fig. 43? is a measured? By what b? By what c? What is the measure of a right angle? [See Example 10 in the preceding set.]

Ex. 3. Suppose I take a square card like CEDF, with a hole in one corner as at C, and sticking two pins firmly in my paper, as at A and B, place the corner of the card between them, as in Fig 44, and then, keeping the sides of the card snug against the pins, put a