AB to be divided into equal parts, each of which is less than DO; there will be at least one point of division between D and O; let I be that point; and draw CI. The arcs AB, AI, will be to each other as two whole numbers, and by the preceding theorem, we shall have, the angle ACB: angle ACI:: arc AB arc AI. Comparing these two proportions with each other, we see that the antecedents are the same: hence, the consequents are proportional (Book II. Prop. IV.); and thus we find the angle ACD: angle ACI :: arc AO: arc AI. But the arc AO is greater than the arc AI; hence, if this proportion is true, the angle ACD must be greater than the angle ACI: on the contrary, however, it is less; hence the angle ACB cannot be to the angle ACD as the arc AB is to an arc greater than AD.

By a process of reasoning entirely similar, it may be shown that the fourth term of the proportion cannot be less than AD; hence it is AD itself; therefore we have

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

Cor. Since the angle at the centre of a circle, and the arc intercepted by its sides, have such a connexion, that if the one be augmented or diminished in any ratio, the other will be augmented or diminished in the same ratio, we are authorized to establish the one of those magnitudes as the measure of the other; and we shall henceforth assume the arc AB as the measure of the angle ACB. It is only necessary that, in the comparison of angles with each other, the arcs which serve to measure them, be described with equal radii, as is implied in all the foregoing propositions.

Scholium 1. It appears most natural to measure a quantity by a quantity of the same species; and upon this principle it would be convenient to refer all angles to the right angle; which, being made the unit of measure, an acute angle would be expressed by some number between 0 and 1; an obtuse angle by some number between 1 and 2. This mode of expressing angles would not, however, be the most convenient in practice. It has been found more simple to measure them by arcs of a circle, on account of the facility with which arcs can be made equal to given arcs, and for various other reasons. At all events, if the measurement of angles by arcs of a circle is in any degree indirect, it is still equally easy to obtain the direct and absolute measure by this method; since, on comparing the arc which serves as a measure to any angle, with the fourth part of the circumference, we find the ratio of the given angle to a right angle, which is the absolute

measure.

E*

Scholium 2. All that has been demonstrated in the last three propositions, concerning the comparison of angles with arcs, holds true equally, if applied to the comparison of sectors with arcs; for sectors are not only equal when their angles are so, but are in all respects proportional to their angles; hence, two sectors ACB, ACD, taken in the same circle, or in equal circles, are to each other as the arcs AB, AD, the bases of those sectors. It is hence evident that the arcs of the circle, which serve as a measure of the different angles, are proportional to the different sectors, in the same circle, or in equal circles.

PROPOSITION XVIII. THEOREM.

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

Let BAD be an inscribed angle, and let us first suppose that the centre of the circle lies within the angle BAD. Draw the diameter AE, and the radii CB, CD.

The angle BCE, being exterior to the triangle ABC, is equal to the sum of the two interior angles CAB, ABC (Book I. Prop. XXV. Cor. 6.): but the triangle BAC being isosceles, the angle CAB is equal to ABC; hence the angle BCE is double of BAC. Since BCE lies at the centre, it is measured by the arc BE; hence BAC will be measured by the half of BE. For a like reason, the angle CAD will be measured by the half of ED; hence BAC+ CAD, or BAD will be measured by half of BE+ED, or of BED.

Suppose, in the second place, that the centre C lies without the angle BAD. Then drawing the diameter AE, the angle BAE will be measured by the half of BE; the angle DAE by the half of DE: hence their difference BAD will be measured by the half of BE minus the half of ED, or by the B half of BD.

Hence every inscribed angle is measured by half of the arc included between its sides.

DE

C

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

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

And every angle BOC, inscribed in a segment less than a semicircle, is an obtuse angle; for it is measured by half of the arc BAC, greater than a semicircumference.

B

B

A

A

A

Cor. 4. The opposite angles A and C, of an inscribed quadrilateral ABCD, are together equal to two right angles: for the angle BAD is measured by half the arc BCD, the angle BCD is measured by half the arc BAD; hence the two angles BAD, BCD, taken together, are measured by the half of the circumference; hence their sum is equal to two right angles.

PROPOSITION XIX. THEOREM.

The angle formed by two chords, which intersect each other, is measured by half the sum of the arcs included between its sides

Let AB, CD, be two chords intersecting each other at E: then will the angle AEC, or DEB, be measured by half of AC+DB.

Draw AF parallel to DC: then will the arc DF be equal to AC (Prop. X.); and the angle FAB equal to the angle DEB (Book I. Prop. XX. Cor. 3.). But F the angle FAB is measured by half the arc FDB (Prop. XVIII.); therefore, DEB

is measured by half of FDB; that is, by half of DB+DF, or half of DB+ÁC. In the same manner it might be proved that the angle AED is measured by half of AFD+BC.

PROPOSITION XX. THEOREM.

The angle formed by two secants, is measured by half the difference of the arcs included between its sides.

Let AB, AC, be two secants: then will the angle BAC be measured by half the difference of the arcs BEČ and DF.

Draw DE parallel to AC: then will the arc EC be equal to DF, and the angle BDE equal to the angle BAC. But BDE is measured by half the arc B BE; hence, BAC is also measured by half the arc BE; that is, by half the difference of BEC and EC, or half the difference of BEC and DF.

PROPOSITION XXI. THEOREM.

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

Let BE be the tangent, and AC the chord. From A, the point of contact, draw the diameter AD. The angle BAD is a right angle (Prop. IX.), and is measured by half the semicircumference AMD; the angle DAC is measured by the half of DC: hence, BAD+DAC, or BAC, is measured by the half of AMD plus the half of DC, or by half the whole arc AMDC.

M

It might be shown, by taking the difference between the angles DAE, DAC, that the angle CAE is measured by half the arc AC, included between its sides.

PROBLEMS RELATING TO THE FIRST AND THIRD BOOKS

PROBLEM I.

To divide a given straight line into two equal parts.

Let AB be the given straight line.

From the points A and B as centres, with a radius greater than the half of AB, describe two arcs cutting each other in D; the point D will be equally distant from A and B. Find, in like manner, above or beneath the line AB, a second point E, equally distant from the points A and B; through the two points D and E, draw the line DE: it will bisect the line AB in C.

For, the two points D and E, being each equally distant from the extremities A and B, must both lie in the perpendicular raised from the middle of AB (Book I. Prop. XVI. Cor.). But only one straight line can pass through two given points; hence the line DE must itself be that perpendicular, which divides AB into two equal parts at the point C.