let the turning be continued in various directions as called for, until the line AC shall have been applied to the entire surface of a solid (as would be done if it were employed to scoop out a hollow figure from the interior of some yielding substance like plaster]. There shall be described the solid required. * By Con For, the point B remaining at rest, the point A can* be struction. applied to every point in the line AC, in any of its situations. + Constr. Also every point in the surface of the described solid, hast had some point in the line AC applied to it. Wherefore the point A can be applied to every point in the described surface, the point B remaining at rest ; consequently each point in the surface 11. Nom.11. and B, are equally distant with A and B, or all the points in *1.Nom.12. the surface are equidistant* from B. Which was to be done. And by parity of reasoning, the like may be done in every other instance. CoR. 1. Instead of being described about B, the solid, if required, may be described about any other assigned point as D. For the hard body in which are the points A and B, may be moved till the point B is applied to D; and the solid described as before. NOMENCLATURE-A solid figure of which all the points in the surface are equidistant from a certain point within, is called a sphere. And such point within, is called the centre of the sphere; and the distance from it to every point in the surface, is called the central distance. Spheres are said to touch one another, which meet but do not cut one another. Spheres described about the same centre, are called concentric. Cor. 2. A sphere may be described about any centre, and with a central distance equal to the distance of any two points that shall have been assigned. For it may be done as in Cor. 1 above. CoR. 3. When a sphere is described as above, the interior sphere, and the hollow one formed by the substance out of which +1.Nom.14. it may be supposed to be taken, are reciprocalst. For their boundaries are in contact in every part at once. COR. 4. The surfaces of concentric spheres either coincide throughout or not at all. I Constr. For if they coincide in one point, they willi coincide throughout. Cor. 5. If the centres of two spheres coincide, and their central distances are equal ; their surfaces will coincide throughout. And if their central distances are not equal, their surfaces will not coincide at all, but one be interior to the other. For (by Cor. 4 above) their surfaces either coincide throughout or not at all. But if their central distances are equal, then two points on the surfaces may be made to coincide, and the surfaces (by Cor. 4 above) will coincide throughout. And if their central distances are not equal, no two points in the surfaces can coincide ; for if they did, the central distances would be equal. Cor. 6. Spheres having equal central distances, are equal. And equal spheres, have equal central distances. For if spheres baving equal central distances, have their centres applied together, their surfaces (by Cor. 5 above) will coincide *1.Nom.14. throughout; wherefore the spheres are equal*. And if equal spheres have not equal central distances, they must have unequal; wherefore, if their centres be applied to gether, their surfaces (by Cor. 5) will not coincide at all, but one +I. Nom.15. be interior to the other; that is to say, one sphere will be greatert than the other; which is impossible, for they are equal. The central distances, therefore, cannot be unequal ; that is, they are equal. PROPOSITION IV. the centre which remains at rest, the sphere will be without Let the substantial sphere whose ner whatsoever about the centre of place. posed to remain unmoved, any par+ Constr. ticular point in the surface of the substantial sphere as A, will+ at all times be coincident with some point or other in the reciprocal; because they are equally distant from the centre. And in like manner every other point in the surface of either sphere, will at all times be coincident with some point in the surface of the other; wherefore the two surfaces will be always everywhere coincident. And because the surface of the substantial sphere which is turned, is always everywhere coincident with that of the reciprocal which is without change of place; the substantial sphere is without change of place. So also of the hollow sphere, if it be turned while the substantial one which is its reciprocal remains unmoved. And by parity of reasoning, the like may be proved of every other sphere. Wherefore, universally, if a sphere be turned &c. Which was to be demonstrated. PROPOSITION V. THEOREM.--If two spheres touch one another externally, they touch only in a point. Let there be two spheres whose centres are A and B, touching one another externally. The one cannot touch the other in more than a single point at once. For if this be disputed, CEGDHF in Fig. 1 below, and not elsewhere. But if so, • INTERC, 4. INTERC, 2. either or both of the spheres may be turned in any manner whatsoever about its centre, and they must still always coincide in the surface in fixed space CEGDHF and not elsewhere. For each of them will be* without change of place ; wherefore they must at all times coincide in that surface and not elsewhere ; for if they do not, one or both must have suffered change of place. Let then the sphere whose centre is A, be turnedt about A, till the portion of its surface which was originally in the situation CEGDHF, is brought into the situation IESF; and because the spheres will still coincide in CEGDHF and not elsewhere, the portion CESF of IESF will coincide with the sphere whose centre is B, and the portion IECF will not. Let now the surface IESF be returned into its original situation (by turning the sphere back again about its centre A); and let the portion of CEGDHF with which CESF thereupon coincides, be OGDH. And after this, let the sphere be turned about its centre again, till the portion of its surface OGDH coincides with CESF as before; and let the sphere whose centre is B be turned about its centre till the portion of its surface OGDH coincides also with CESF, and let the remainder of its surface which formerly coincided with the surface of the other sphere in CÉGDHF, be KECF. Wherefore the two surfaces which in the situation CEGDHF coincided entirely with one another, do now coincide in the portion CESF as before, but in their remaining portions IECF and KECF they do not. Which is impossible. For if the two surfaces are made to coincide entirely as in the situation CEGDHF, the portions IECF and KECF can in no way be made to cease coinciding and be separate while the remainders continue to coincide, other than by their particles, or some of them, being moved among themselves; which cannot be, for the bodies on which the surfaces are exhibited are hardt bodies. The assumption", therefore, which involves this impossible consequence, cannot be true ; or the two spheres cannot coincide in the surface CEGDHF. And in like manner may be shown that they cannot coincide in any other surface. II. Nom. 3. * I.Nom.26. See Note. Second Case ; let it be assumed that they coincide in the self rejoining line CEGDHF in Fig. 2. below; and not elsewhere. But if so, it may be shown as before, that they must continue to coincide in the same line in fixed space CEGDHF and not elsewhere, in whatsoever manner they may be turned about their respective centres. Let then the sphere whose centre is A, be turned about A, till the line on its surface which originally coincided with CEGDHF, is brought into the situation IESF; and on the sphere being returned to its former situation, let the points E and F in IESF fall on G and H. And after this let the sphere be turned round its centre again, till the points on its surface G and H coincide with E and F as before; and let the sphere whose centre is B be likewise turned about its centre till the points on its surface G and H coincide also with E and F, the line on its surface which originally coincided with CEGDHF being thereby brought into the situation KETF. Wherefore the two lines on the surfaces of the different spheres, which in the situation CEGDHF coincided entirely with each other, do now coincide in the points E and F only, and their portions which are on different sides of those points are so posited, that if the portion EKF of the one line lie above the portion EIF of the other line, the portion ETF of the first also lies above the portion ESF of |