Suppose, for example, that the angles ACB, DEF are to each other as 7 to 4; or, which is the same thing, suppose that the angle M, which may serve as a common measure, is contained seven times in the angle ACB, and four times in the angle DEF. The seven partial angles into which ACB is divided, being each equal to any of the four partial angles into which DEF is divided, the partial arcs will also be equal to each other (Prop. IV.), and the entire arc AB will be to the entire arc DF as 7 to 4. Now the same reasoning would apply, if in place of 7 and 4 any whole numbers whatever were employed; therefore, if the ratio of the angles ACB, DEF can be expressed in whole numbers, the arcs AB, DF will be to each other as the angles ACB, DEF.

Case second. When the ratio of the angles can not be ex pressed by whole numbers.

Let ACB, ACD be two angles having any ratio whatever. Suppose ACD to be the smaller angle, and let it be placed on the greater; then will the angle ACB angle A ACD arc AB arc AD.

C

A A

DRI

For, if this proportion is not true, the first three terms remaining the same, the fourth must be greater or less than AD. Suppose it to be greater, and that we have

Angle ACB angle ACD :: arc AB : arc AI. Conceive the arc AB to be divided into equal parts, each less than DI; there will be at least one point of division betweet. D and I. Let H be that point, and join CH. The arcs AB, AH will be to each other in the ratio of two whole numbers, and, by the preceding case, we shall have

Angle ACB angle ACH: : arc AB : arc AH. Comparing these two proportions with each other, and observing that the antecedents are the same, we conclude that the consequents are proportional (Prop. IV., Cor., B. II.); therefore,

Angle ACD: angle ACH :: arc AI : arc AH.

But the arc AI is greater than the arc AH; therefore the angle ACD is greater than the angle ACH (Def. 2, B. II.), that is, a part is greater than the whole, which is absurd. Hence the angle ACB can not be to the angle ACD as the arc AB to an arc greater than AD.

In the same manner, it may be proved that the fourth term of the proportion can not be less than AD; therefore, it must be AD, and we have the proportion

Angle ACB angle ACD:: arc AB: arc AD.

Cor. 1. Since the angle at the center of a circle, and the

arc intercepted by its sides, are so related, that when one is increased or diminished, the other is increased or diminished in the same ratio, we may take either of these quantities as the measure of the other. Henceforth we shall take the arc AB to measure the angle ACB. It is important to observe, that in the comparison of angles, the arcs which measure them must be described with equal radii.

Cor. 2. In equal circles, sectors are to each other as their. arcs; for sectors are equal when their angles are equal.

PROPOSITION XV. THEOREM.

An inscribed angle is measured by half the arc included between its sides.

Let BAD be an angle inscribed in the circle BAD. The angle BAD is measured by half the arc BD.

First. Let C, the center of the circle, be within the angle BAD. Draw the diameter AE, also the radii CB, CD.

E

Because CA is equal to CB, the angle CAB is equal to the angle CBA (Prop. X., B. I.); therefore the angles CAB, CBA are together double the angle CAB. But B the angle BCE is equal (Prop. XXVII., B. I.) to the angles CAB, CBA; therefore, also, the angle BCE is double of the angle BAC. Now the angle BCE, being an angle at the center, is measured by the arc BE; hence the angle BAE is measured by the half of BE. For the same reason, the angle DAE is measured by half the arc DE. Therefore, the whole angle BAD is measured by half the arc BD.

Second. Let C, the center of the circle, be without the angle BAD. Draw the diameter AE. It may be demonstrated, as in the first case, that the angle BAE is measured by half the arc BE, and the angle DAE by half the arc DE; hence their difference, BAD, is measured by half of B BD. Therefore, an inscribed angle, &c.

Cor. 1. All the angles BAC, BDC, &c.,

DE

inscribed in the same segment are equal, for they are all measured by half the same arc BEC. (See next fig.)

Cor. 2. Every angle inscribed in a semicircle is a right angle, because it is measured by half a semicircumference that is, the fourth part of a circumference.

Cor. 3. Every angle inscribed in a segment greater than a semicircle is an acute angle, for it is measured by half an arc less than semicircumference.

Every angle inscribed in a segment less than a semicircle is an obtuse an- B gle, for it is measured by half an arc greater than a semicircumference.

E

D

Cor. 4. The opposite angles of an inscribed quadrilateral, ABEC, are together equal to two right angles; for the angle BAC is measured by half the arc BEC, and the angle BEČ is measured by half the arc BAC; therefore the two angles BAC, BEC, taken together, are measured by half the circumference; hence their sum is equal to two right angles.

PROPOSITION XVI. THEOREM.

The angle formed by a tangent and a chord, is measured by half the arc included between its sides.

Let the straight line BE touch the circumference ACDF in the point A, and from A let the chord AC be drawn; the angle BAC is measured by half the arc AFC.

B

D

G

E

From the point A draw the diameter AD. The angle BAD is a right angle (Prop. IX.), and is measured by half the semicircumference AFD; also, the angle DAC is measured by half the arc DC (Prop. V.); therefore, the sum of the angles BAD, DAC is measured by half the entire arc AFDC.

A

A

In the same manner, it may be shown that the angle AE is measured by half the arc AC, included between its siers. Cor. The angle BAC is equal to an angle inscribed ir he segment AGC; and the angle EAC is equal to an angl scribed in the segment AFC.

BOOK IV.

THE PROPORTIONS OF FIGURES.

Definitions.

1. Equal figures are such as may be applied the one to the other, so as to coincide throughout. Thus, two circles having equal radii are equal; and two triangles, having the three sides of the one equal to the three sides of the other, each to each, are also equal.

2. Equivalent figures are such as contain equal areas. Two figures may be equivalent, however dissimilar. Thus, a circle may be equivalent to a square, a triangle to a rectangle, &c.

3. Similar figures are such as have the angles of the one equal to the angles of the other, each to each, and the sides about the equal angles proportional. Sides which have the same position in the two figures, or which are adjacent to equal angles, are called homologous. The equal angles may also be called homologous angles.

Equal figures are always similar, but similar figures may be very unequal.

4. Two sides of one figure are said to be reciprocally proportional to two sides of another, when one side of the first is to one side of the second, as the remaining side of the second is to the remaining side of the first.

5. In different circles, similar arcs, sectors, or segments, are those which correspond to equal angles at the center.

Thus, if the angles A and D are equal, the arc BC will be similar to the arc EF, the sector ABC to the sector DEF, and the segment BGC to the segment EHF.

A

D

ДД

B

6. The altitude of a triangle is the perpendicular let ll from the vertex of an angle on the opposite side, taken as a base, or on the base produced.

G

CE

F

H

7. The altitude of a parallelogram is the perpendicular drawn to the base from the opposite side.

8. The altitude of a trapezoid is the distance between its parallel sides.

base; and, since the two parallelograms are supposed to have the same altitude, their upper bases, DC, FE, will be in the same straight line parallel to AB.

Now, because ABCD is a parallelogram, DC is equal to AB (Prop. XXIX., B. I.). For the same reason, FE is equal to AB, wherefore DC is equal to FE; hence, if DC and FE be taken away from the same line DE, the remainders CE and DF will be equal. But AD is also equal to BC, and AF to BE; therefore the triangles DAF, CBE are mutually equi lateral, and consequently equal.

Now if from the quadrilateral ABED we take the triangle ADF, there will remain the parallelogram ABEF; and if from the same quadrilateral we take the triangle BCE, there will remain the parallelogram ABCD. Therefore, the two parallelograms ABCD, ABEF, which have the same base and the same altitude, are equivalent.

Cor. Every parallelogram is equivalent to the rectangle which has the same base and the same altitude.

PROPOSITION II. THEOREM.

Every triangle is half of the parallelogram which has the same base and the same altitude.

Let the parallelogram ABDE and the triang ABC have the same base, AB, and the same altitude; the triangle is half of the parallelogram,