Third. If the two parallels DE, IL, are tangents, the one at H, the other at K, draw the parallel secant AB; and, from what has just been shewn, we shall have MH=HP, MK=KP; and hence the whole arc HMK=HPK. It is farther evident that each of these arcs is a semicircumference.

113. If two circles cut each other in two points, the line which passes through their centres, will be perpendicular to the chord which joins the points of intersection, and will divide it into two equal parts.

For the line AB, which joins the points of intersection, is a chord common to the two circles. And if a perpendicular

D

be erected from the middle of this chord, it will pass (106.) through each of the two centres C and D. But no more than one straight line can be drawn through two points; hence the straight line, which passes through the centres, will bisect the chord at right angles.

7

114. If the distance between the centres of two circles is less than the sum of the radii, the greater radius being at the same time less than the sum of the smaller and the distance between the centres, the two circles will cut each other.

For, to make an intersection possible, the triangle CAD (see the preceding figure) must be possible. Hence, not only must we have CDZAC+AD, but also the greater radius ADZAC+CD. And, whenever the triangle CAD can be constructed, it is plain that the circles described from the centres C and D, will cut each other in A and B.

115. If the distance between the centres of two circles is equað to the sum of their radii, those two circles will touch each other externally.

Let C and D be the centres at a distance from each other=CA+AD.

The circles will evidently have the point A common, and they will have no other; because, if they had two points common, the distance between their centres must be less than the sum of their radii.

C

THEOREM.

116.

If the distance between the centres of two circles, is equal to the difference of their radii, those two circles will touch each other internally.

Let C and D be the centres at a dis- E tance from each other=AD-CA.

It is evident, as before, that they will have the point A common: they can have no other; because, if they had, the greater radius AD (114.) must be less than the sum of the radius AC and the distance between the centres, which is contrary to the supposition.

117. Cor. Hence, if two circles touch each other, either externally or internally, their centres and the point of contact will be in the same right line.

118. Scholium. All circles which have their centres on the right line CD, and which pass through the point A, are tangent to each other; they have only the point A common. And if through the point A, AE be drawn perpendicular to CD, the straight line AE will be a common tangent to all the circles.

119.

In the same circle, or in equal circles, equal angles having their vertices at the centre, intercept equal arcs on the circumference: and conversely, if the arcs intercepted are equal, the angles contained by the radii will also be equal.

Let C and C be the centres of equal circles, and the angle ACB=DCE.

First. Since the angles ACB, DCE, are equal, they may be placed upon each other; and since their sides are equal, the point A will evidently fall on D,

and the point B on E. But, in A

B

D

I

that case, the arc AB must also fall on the arc DE; for if they did not exactly coincide, there would, in the one or the other, be points unequally distant from the centre; which is impossible: hence the arc AB is equal to DE.

Secondly. If we suppose AB-DE, the angle ACB will be equal to DCE. For if those angles are not equal, suppose ACB to be the greater, and let ACI be taken equal to DCE. From what has just been shewn, we shall have AI=DE: but, by hypothesis, AB is equal to DE; hence AI must be equal to AB, or a part, to the whole, which is absurd: hence the angle ACB is equal to DCE.

120. In the same circle, or in equal circles, if two angles at the centre are to each other in the proportion of two whole numbers, the intercepted arcs will be to each other in the proportion of the same numbers, and we shall have the angle: the angle :: the corresponding arc: the corresponding arc.

M

E

Suppose, for example, that the angles ACB, DCE, are to each other as 7 is to 4; or, which is the same thing, suppose that the angle M, which may serve as a common measure, is contained 7 times in the angle ACB and 4 times in DCE. The seven partial angles ACm, mCn, nCp, &c., into which ACB is divided, being each equal to any of the four partial angles into which DCE is divided; each of the partial ares Am, mn, np, &c., will (119.) be equal to each of the partial arcs Ax, xy, &c. Therefore the whole arc AB will be to the whole arc DE, as 7 is to 4. But the same reasoning would evidently apply, if in place of 7 and 4 any numbers whatever were employed; hence, if the ratio of the angles ACB, DCE, can be expressed in whole numbers, the arcs AB, DE, will be to each other as the angles ACB, DCE.

121. Scholium. Conversely, if the arcs AB, DE, are to each other as two whole numbers, the angles ACB, DCE, will be to each other as the same whole numbers, and we shall have ACB: DCE:: AB: DE. For the partial arcs, Am, mn, &c. and Dx, xy, &c. being equal, the partial angles ACm, mCn, &c. and DCx, xCy, &c. will also be equal.

122.

THEOREM.

Whatever be the ratio of two angles, those two angles will always be to each other as the arcs intercepted between their sides, and described from their vertices as centres, with equal radii.

Let ACB be the greater and ACD the less angle.

Let the less angle be placed on the greater. If the proposition is not true, the angle ACB will be to the angle ACD as the arc AB is to an arc great-A er or less than AD. Suppose

DIO

B

D

this arc to be greater, and let it be represented by AO; we shall thus have the angle ACB : angle ACD :: arc AB : arc AO. Next conceive the arc 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 join 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, and observing that the antecedents are the same, we infer that the consequents are proportional, 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 shewn 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.

123. 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 required that, in the comparison of angles with each other, the arcs which serve to measure them, be described with equal radii, as all the foregoing propositions imply.

124. 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