PROPOSITION XI. THEOREM. 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, let the line AB join the points of intersection. It will be a common chord to the two circles. Now if a perpendicular be erected from the middle of this chord, it will pass through each of the two centres C and D (Prop. VI. Sch.). 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. PROPOSITION XII. THEOREM. 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 circumferences will cut each other. For, to make an intersection possible, the triangle CAD must be possible. Hence, not only must we have CD<AC+AD, but also the greater radius AD< AC+CD (Book I. Prop. VII.). And, whenever the triangle CAD can be constructed, it is plain C. D B that the circles described from the centres C and D, will cut each other in A and B. D PROPOSITION XIII. THEOREM. If the distance between the centres of two circles is equa sum of their radii, the two circles will touch each other ex nally. Let C and D be the centres 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 had two points common, the distance between C E their centres must be less than the sum of their radii. PROPOSITION XIV. THEOREM. If the distance between the centres of two circles is equal to the difference of their radii, the two circles will touch each other internally. Let C and D be the centres at a distance from each other equal to 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 must be less than the sum of the radius AC and the distanceCD between the centres (Prop. XII.); which is contrary to the supposition. A 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. Scholium. All circles which have their centres on the right line AD. and which pass through the point A, are tangent to each other. For, they have only the point A common, and if through the point A, AE be drawn perpendicular to AD, the straight line AE will be a common tangent to all the circles. PROPOSITION XV THEOREM. 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 that case, the arc AB must 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. Secondly. If we suppose AB=DE, the angle ACB will be equal to DCE. For, if these angles are not equal, suppose ACB to be the greater, and let ACI be taken equal to DCE. From what has just been shown, 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 (Ax. 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 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 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, 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 M 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 arcs Am, mn, np, &c., will be equal to each of the partial arcs Dr, xy, &c. (Prop. XV.). 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. 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. PROPOSITION XVII. THEOREM. Whatever be the ratio of two angles, they will always be to each other as the arcs intercepted between their sides; the arcs being described from the vertices of the angles 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 truc, the angle ACB will be to the angle ACD as the arc AB is to an arc DIOB A greater or less than AD. Suppose 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 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. |