does not coincide with some point in the other surface, then let the straight line CFL be brought into the situation in which it passes through the point N, and after transposition of the united spheres let them be turned round BA till the point F in CFL occupies the same place it occupied when CFL passed through N; and because the point N is not in the second surface, the straight line CFL after the transposition cannot pass through N, for if it did, the transposed surface would pass through N, and it does not ; and because in one situation of the straight line CFL the points C and F coincide with the points C and F in the other situation of it, but one of these straight lines passes through the point N and the other does not, the two straight lines CFL coincide in two points C and F, but do not coincide to the extent of the : Interc.12, length common to both, which is" impossible. And in the same Cor. 1. manner may be shown that there is no other point in the one surface, either within or without the line FJHK, which does not after transposition of the united spheres coincide with some point in the other surface. 32ert them, in the straight line of unlimited length CL (See Fig. 2 in the next page) take any point other than F, as M ; and if about the centres A and B, with the radii AM and BM, spheres be described, THE Point M, AT ALL TIMEs DURING THE REvolution of CL, SHALL BE Found IN THE INTERSECTION of THESE spherEs. For, join AM, B.M.; and afterwards let the united spheres be transposed, and turned round BA till M be in the same place as before. Because the extremities A and M occupy the places previously occupied by the extremities B and M, AM and BM are equal; wherefore, if about the centres A and B, with the equal radii AM and BM, be described two spheres, these too. willt be equal to one another; and because they are equal but have * * different centres A and B, and have the point M in their surfaces in goal. common, they willf cut one another in one self-rejoining line passl's ing through M, or else touch externally in the point M only; but they cannot touch externally in M, for then the straight line AB *INTERc.10. which joins their centres must” pass through M, and it does not ; ** wherefore they will cut one another in one self-rejoining line passing through M. Let this line be MQOP; and if afterwards the united spheres together with the straight line CL be turned round AB, M shall always be found in some part of the line MQOP and no where else; for because AM and BM are the radii of the two equal spheres that were described, the point M is always in the surface of both; whence, if it be not always in some part of MQOP, the surfaces of these spheres must coincide elsewhere than in the *INTEnc.11, one self-rejoining line MQOP, which cannot” be. And in the same way of every other point in CL, or in the surface described by it. tinterc.9. Wilbereto mert, in MQCP take any second point as Q, and joint Cor. MQ; THE STRAIGHT LINE BETwFEN M AND Q, with its prolongation either way, SHALL LIE wholly IN THE SURFACE DEscFIBED BY CL. For if MQ pass through the point C, it will without further question lie wholly in the surface described by CL, inasmuch as each half of it will coincide with part of CL as it was found at some instant of the revolution by which the surface was described. But if MQ do not pass through C, join *INTERc.10, AM, AQ, BM, BQ, and these straight lines will bef all equal one to another, because they are radii of the same or equal spheres; where Cor. 4 *INTERc. 12 Cor. 1. +INTERC.10, Cor. 5. 1 INTERc. 3. Cor. 4. *INTERC.12. Cor. 8. See Note 1. +INTER c. 1. Cor. 3. fore if about the centres M and Q, and with the radii MA and QA, were described two spheres, it might be shown as has been done in like circumstances before, that the surfaces of these would cut one another in one self-rejoining line in which is the point A, and that if the figure MBQ was turned round the straight line MQ, the point B would always be found in the self-rejoining line which passes through A, and consequently the turning might be continued till the point B coincided with A, every point in the straight line MQ remaining without changeof place; whence, if in MQ had been taken any point as R, and RA and RB had been joined, when the point B was made to coincide with the point A the straight line RB would at the same time have* coincided with the straight line RA, and therefore RA and RB are equal. About the centres A and B, with the radii AR and BR, describe two spheres [omitted in the figure]; which, it may be shown as has been done before, will cut one another in one self-rejoining line. Because AR and BR are equal, they are necessarily each greater than AC or BC (unless the point R coincides with C, in which event it is without further question in the surface described by CL); for if they were equal to AC and BC, they could meet in no point but C (inasmuch as the spheres AC and BC meet no where but in that point), and if they were less they could not meet at all. Wherefore the sphere described about the centre A with the radius AR, will bet greater than the sphere described about the same centre with the radius AC; and its surface will not coincide at all with the surface of the other, but be exterior ; for if it coincided at all, the surfaces would: coincide throughout and the spheres be equal; and if it did not coincide but was interior, the sphere would be the less, which cannot be, for it is the greater. And because C is a point within the sphere described with the radius AR, the straightlineofunlimited length CL drawn from it will” cut the surface of such sphere and be cut by it. Let it be met in S; and join AS, BS. Because S is a point in CL, it may be shown as has been done before, that AS and BS are equal; wherefore the point S shall be in the self-rejoining line which is the intersection of the spheres described with the radii AR and BR, for if not, then AS would be equal to AR, but BS would not be equal to BR which is equal to AR; which isł impossible, for AS and BS are equal. And because, as has been shown before, the point S will at all times during the revolution of CL be found in some part of the intersection of the spheres which pass through S, which intersection also passess through R; the point S in CL, will by the revolution of CL be made to pass through R; wherefore R is in the surface described by the revolution of CL. And in like manner may be shown of every other point in the straight line MQ or in its prolongation either way, that some point in the straight line of unlimited length CL passes through it on the united spheres being turned round AB; wherefore every point in MQ or in its prolongation either way, is in the surface described by CL. And if from M be drawn a straight line to any other point in MQCP, and prolonged to an unlimited length, in like manner may be shown that every point in this straight line or in its prolongation either way, is in the surface described by CL. Wherefore IF A straight LINE OF UNLIMITED LENGTH As MT, BE MADE To TURN continuoUSLY ABOUT M AND PASS ALways THROUGH soME SEcond PoinT IN MQOP, this straight line (both the portion which may lie within MQCP, and its prolongation either way), will at all times lie wholly in the surface described by CL. And because MT may be turned till it passes through any point in the surface which is described by CL in or nithin the self-rejoining line MQOP; the straight line from M to any such point, as also its prolongation either way may be shown to lie wholly in the surface described by CL. 35ut every point in the surface described by CL shall be in or nithin the self-rejoining line MQOP, which is not farther from the centres A and B than the point M. For if N be any point in the surface described by CL, it may be shown as before, that N is in the intersection of the spheres that should be described with the equal radii AN and BN; wherefore if AN and BN were respectively equal to AM and BM, N would be in MQOP, for the spheres respectively would” coincide, and consequently their intersections would coincide; and if AN and BN are respectively less than AM and BM, the sphere with the radius AN will be less than the sphere with the radius AM, and its surface willf beinterior to the surface of the other throughout, and in the same manner the surface of the sphere with the radius BN will be interior to the surface of the sphere with the radius BM; and because the surfaces of the two spheres are respectively interior to the surfaces of the two others, their intersection, in which is N, will not coincide with the intersection of the others but be nithin it. Wherefore the straight line from M To ANY PoinT IN THE SUR *INTERC.12. FACE DESCRIBED BY CL which Is NoT FARTHER FROM THE plane. described, may be" prolonged to any length. B |