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 urcs; 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.

A

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.

A

DE

C

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

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

E

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 BEC 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 tw 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.

F

B

D

A

-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. therefore, the sum of the angles BAD, DAC is measured by half the entire arc AFDC.

V.);

In the same manner, it may be shown that the angle AE is measured by half the arc AC, included between its siens. 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

B

CE

F

G

H

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

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 triangle ABC have the same base, AB, and the same altitude; the triangle is half of the parallelogram.

C

FE

D

Complete the parallelogram ABFC; then the parallelogram ABFC is equivalent to the parallelogram ABDE, because they have the same base and the same altitude (Prop. I.). But the triangle ABC is half of the parallelogram ABFC (Prop. XXIX., Cor., B. I.); wherefore the triangle ABC is also half of the parallelogram ABDE. Therefore, every triangle, &c.

A

B

Cor. 1. Every triangle is half of the rectangle which has the same base and altitude.

Cor. 2. Triangles which have equal bases and equal alti tudes are equivalent.

PROPOSITION III. THEOREM.

Two rectangles of the same altitude, are to each other as their

bases.

Let ABCD, AEFD be two rectangles which have the common altitude AD; they are to each other s their bases AB, AE.

D

A

E

B

Case first. When the bases are in the ratio of two whole numbers, for example, as 7 to 4. If AB be divided into seven equal parts, AE will contain four of those parts. At each point of division, erect a perpendicular to the base; seven partial rectangles will thus be formed, all equal to each other, since they have equal bases and altitudes (Prop. I.). The rectangle ABCD will contain seven partial rectangles, while AEFD will contain four; therefore the rectangle ABCD is to the rectangle AEFD as 7 to 4, or as AB to AE. The same reasoning is applicable to any other ratio than that of 7 to 4; therefore, whenever the ratio of the bases can be expressed in whole numbers, we shall have

ABCD: AEFD::AB: AE.

Case second. When the ratio of the bases can not be expressed in whole numbers, it is still true that

ABCD: AEFD::AB: AE.

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

ABCD: AEFD::AB: AG.

Conceive the line AB to be divided into A