PROPOSITION XIII. THEOREM. If the distance between the centres of two circles is equal to the sum of their radii, the circumferences will touch each other externally. Let C and D be the centres of two circles at a distance from each other equal to CA+AD. The circles will evidently have the point A common, and they will have no other; because if they have two points common, the distance between their centres must be less than the sum of their radii, which is contrary to the supposition. E -D Cor. If the distance between the centres of two circles is greater than the sum of their radii, the two circumfer ences will be exterior the one to the other. PROPOSITION XIV. THEOREM. If the distance between the centres of two circles is equal to the difference of their radii, the two circumferences will touch each other internally. Let C and D be the centres of two circles at a distance from each other equal to AD- CA. It is evident, as before, that the two circumferences will have the point A common: they can have no other; because if they had, the distance between the centres would be greater than AD-CA (P. 12); which is contrary to the supposition. E A C D Cor. 1. Hence, if two circles touch each other, either externally or internally, their centres and the point of contact will be in the same straight line. Cor. 2. If the distance between the centres of two circles is less than the difference of their radii, one circle will be entirely within the other. Scholium 1. All circles which have their centres on the right line AD, and which pass through the point A, are tangent to each other at the point A. For, they have only the point A common, and if through A, AE be drawn perpendicular to AD, it will be a common tangent to all the circles. Scholium. 2. Two circumferences must occupy with respect to each other, one of the five positions above indicated. 1st. They may intersect each other in two points: 3d. They may be external, the one to the other: PROPOSITION XV. THEOREM. In the same circle, or in equal circles, radii making equal angles at the centre, intercept equal arcs on the circumference. And conversely: If the arcs intercepted are equal, the angles contained by the radii are also equal. Let C and C be the centres of equal circles, and the angle ACB=DCE. First. Since the angles B D ACB, DCE, are equal, one of them may be placed upon the other. Let the angle ACB be placed on DCE. Then since their sides are equal, the point A will evidently fall on D, and the point B on E. The arc AB will also fall on the arc DE; for, if the arcs 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 (A. 14). Second. If the arc AB=DE, the angle ACB is equal to DCE For, if these angles are not equal, suppose one of them, as ACB, to be the greater, and let ACI be taken equal to DCE. From what has just been shown, we shall then have AI= DE; but, by hypothesis, AB is equal to DE; hence, AI must be equal to AB, or a part equal to the whole, which is absurd (A. 8); hence, the angle ACB is equal to DCE. PROPOSITION XVI. THEOREM. In the same circle, or in equal circles, if two angles at the centre have to each other the ratio of two whole numbers, the intercepted arcs will have to each other the same ratio: or, we shall have the angle to the angle, as the corresponding arc to the corresponding arc. Suppose, for example, that the angles ACB, DCE, are to each other as 7 is to 4; or, what 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 M C times in DCE. The seven partial angles ACm, mCn, nCp, &c., into which ACB is divided, are each equal to any of the four partial angles into which DCE is divided; and each of the partial arcs, Am, mn, np, &c., is equal to each of the partial arcs Dx, xy, &c. (P. 15). 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 angles ACB, DCE, are to each other as two whole numbers, they will also be to each other as the arcs AB, DE. Cor. 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 AB : DE :: ACB DCE. For, the partial arcs, Am, mn, &c., and Dx, xy, &c., being equal, the partial angles ACm, mCn, &c., and DCx, x Cy, &c., will also be equal, and the entire arcs will be to each other as the entire angles. PROPOSITION XVII. THEOREM. In the same circle, or in equal circles, any two angles at the centre are to each other as the intercepted arcs. Let ACB and ACD be two angles at the centres of equal circles: then will ACB : ACD :: AB AD. For, if the angles are equal, the arcs will be equal (P.15). If they are unequal, let the less be placed on the greater. Then, if the proposition is not true, the A C : DIOB A angle ACB will be to the angle ACD as the arc AB is to an arc greater or less than AD. Suppose such 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 0; let I be that point; and draw CI. Then the arcs AB, AI, will be to each other as two whole numbers, and by the preceding theorem, we shall have, angle ACB angle ACT :: arc AB : arc AI Comparing the two proportions with each other, we see that the antecedents in each are the same: hence, the consequents are proportional (B. II., P. 4); 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 ess than AD; hence, it is AD itself; therefore, we have angle ACB angle ACD :: arc AB : arc AD. Scholium 1. Since the angle at the centre of a circle, and the arc intercepted by its sides, have such a connec tion, that if the one be augmented or diminished, the other will be augmented or diminished in the same ratio, we are authorized to assume the one of these 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 neces sary, in the comparison of angles with each other, that the arcs which serve to measure them, be described with equal radii. Scholium 2. An angle less than a right angle will be measured by an arc less than a quarter of the circumfer ence: a right angle, by a quarter of the circumference: and an obtuse angle by an arc greater than a quarter, and less than half the circumference. Scholium 3. 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. This 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 the arcs of a circle, on account of the facility with which arcs can be made to correspond to angles, and for various other reasons. At all events, if the measurement of angles by the arcs of a circle is in any degree indirect, it is still very easy to obtain the direct and absolute measure by this method; since, by comparing the fourth part of the circumference with the arc which serves as a measure of any angle, we find the ratio of a right angle to the given angle, which is the absolute measure. |