the partial arcs will also be equal to each other (Pr. 4), 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 angles are incommensurable; that is, their ratio can not be expressed exactly in numbers.

Suppose the angle DEF to be divided into any number n of equal parts; then ACB will contain a certain number m of these parts, plus a remainder which is less than one of the parts. The ACB

m

numerical expression of the ratio will be correct within

1

DEF

part (B. II., Art. 10). Draw radii to the several points of division of the arcs. The arc DF will be divided into n equal parts, and the arc AB will contain m such parts, plus a remainder which is Therefore the numerical expression

less than one of the parts.

DEF

AB however small the parts into which DEF is divided. ThereDF' fore these ratios must be absolutely equal; and hence, whatever may be the ratio of the two angles, we have the proportion

angle ACB: angle DEF ;: arc AB : arc DF.

Therefore, in the same circle, etc.

Scholium. Since the angle at the centre of a circle and the are intercepted by its sides are so related that when one is increased or diminished, the other is increased or diminished in the same ratio, an angle at the centre is said to be measured by its inter cepted arc.

It should, however, be observed that, since angles and arcs are unlike quantities, they are necessarily measured by different units. The most simple unit of measure for angles is the right angle, and the corresponding unit of measure for arcs is a quadrant. An acute angle would accordingly be expressed by some number between 0 and 1; an obtuse angle by some number between 1 and 2.

The unit, however, most commonly employed for angles is an angle equal toth part of a right angle, called a degree. The corresponding unit of measure for arcs is th part of a quadrant, and is also called a degree. An angle or an arc is thus numerically expressed by the unit degree and its subdivisions. A right angle and a quadrant are both expressed by 90 degrees. If an angle is ths of a right angle, it is expressed by 72 degrees.

Cor. Since in equal circles sectors are equal when their angles are equal, it follows that in equal circles sectors are to each other as their arcs.

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 centre of the circle, be within the angle BAD. Draw the diameter AE, also the radii CB, CD.

Because CA is equal to CB, the angle CAB is equal to the angle CBA (B. I., Pr. 10); therefore the angles CAB, CBA are together

B

double the angle CAB. But the angle BCE is equal (B. I., Pr. 27) to the angles CAB, CBA; therefore, also, the angle BCE is double of the angle BAC. Now the angle B CE, being an angle at the centre, 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 centre of the circle, be without the angle BAD. Draw the diameter AE.

B

DE

B

DE

C

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 are DE; hence their difference, BAD, is measured by half of BD. Therefore, an inscribed angle,

etc.

Cor. 1. All the angles BAC, BDC, etc., inscribed in the same segment, are equal, for they are all measured by half the same arc B

EC. (See next fig.)

Cor. 2. An angle BCD at the centre of a circle is double of the angle BAD at the circumference, subtended by the same arc. Cor. 3. Every angle inscribed in a semicircle is a right angle, because it is measured by half a semi-circumference; that is, the fourth part of a circumference.

ference.

Cor. 4. 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 in an obtuse angle, for it is measured by half an arc greater than a semi-circum

Cor. 5. The opposite angles of an inscribed quadrilateral, ABE C, are supplements of each other; 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 two right angles.

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.

From the point A draw the diameter A G D. The angle BAD is a right angle (Pr. 9), and is measured by half the semi-circumference AFD; also, the angle DAC is

measured by half the arc DC (Pr. 15); therefore the sum of the angles BAD, DAC is measured by half the entire arc AFDC.

In the same manner, it may be shown that the angle CAE is measured by half the arc AGC, included between its sides.

Cor. The angle BAC is equal to an angle inscribed in the seg ment AGC, and the angle EAC is equal to an angle inscribed in the segment AFC.

PROPOSITION XVII. THEOREM.

The angle formed by two chords which cut each other is measured by one half the sum of the arcs intercepted between its sides and between the sides of its vertical angle.

Let AB, CD be two chords which cut each other at E; then will the angle AED be measured by one half the sum of the arcs AD and BC, intercepted between the sides of AED and the sides of its vertical angle BEC.

Join AC; the angle AED is equal to the sum of the angles ACD and BAC (B. I., Pr. 27). But A

B

CD is measured by half the arc AD (B. III., Pr. 15), and the angle BAC is measured by half the arc BC. Therefore AED is measured by half the sum of the arcs AD and BC. Therefore the angle, etc.

PROPOSITION XVIII. THEOREM.

The angle formed by two secants intersecting without the circumference, is measured by one half the difference of the intercepted arcs.

Let AB, AC be two secants which intersect at A; then will the angle BAC be measured by one half the difference of the arcs BC and DE.

B

D

Join CD; the angle BDC is equal to the sum of the angles DAC and ACD (B.I., Pr. 27); therefore the angle A is equal to the difference of the angles BDC and ACD. But the angle BDC is measured by one half the arc BC (B. III., Pr. 15), and the angle A CD is measured by one half the arc DE. Therefore the angle A is measured by one half the difference of the arcs BC and DE. Therefore the angle, etc.

BOOK IV.

COMPARISON AND MEASUREMENT OF POLYGONS.

Definitions.

1. The area of a figure is its superficial content. The area is expressed numerically by the number of times that the figure contains some other surface which is assumed for its measuring unit; that is, it is the ratio of its surface to that of the unit of surface. A unit of surface is called a superficial unit. The most convenient superficial unit is the square, whose side is the linear unit, as a square foot or a square yard.

2. 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.

3. 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, etc.

4. Similar polygons 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 polygons, or which are adjacent to equal angles, are called homologous. The equal angles may also be called homologous angles.

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

5. Two sides of one polygon 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.

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

A

D

H

centre.

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