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• I. Nom.3. + I. Nom.26.

the second, and vice versa. Which is impossible. For if the two lines IESF and KETF are made to coincide entirely as in the situation CEGDHF, the two portions EKF and ETF of KETF can in no way be turned about the points E and F so as to be both of them above or both below the line IESF or its portions, other than by their particles, or some of them, being moved among themselves; which cannot be, for the bodies on which the lines are exhibited, are hard* bodies. The assumptiont, therefore, which involves the impossible consequence, cannot be true; or the two spheres cannot coincide in the self-rejoining line CEGDHF. And in like manner may be shown that they cannot coincide in any other

self-rejoining line. Third Case; let it be assumed that they coincide in the line CD

in Fig. 3. below, which is not a self-rejoining line; and not elsewhere. But if so, let the two spheres be united as one body [as may be supposed to be done by their being imbedded in one inclosing body of hard, and for convenience, transparent matter), and let this body be turned always in the same direction about the points A and B in it which remain at rest, till it returns into the situation from which it was first moved. And because one face of the line CD, as for instance

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* 1. Nom.26.

the face towards A, will be turned to all sides in succession ; the line CD, which is not a self-rejoining line, must change its place; for it cannot have its face turned to all sides in succession and be without change of place. Let it therefore, at some period during the turning, be in the situation EF. But if the spheres do at first coincide in CD, it may be shown as has been done before, that during the turning they must always continue to coincide in CD and not elsewhere. But they also, at one period during the turning, coincide in EF. Which is impossible. The assumption*, therefore, which involves this impossible consequence, cannot be true; or the two spheres cannot coincide in the line CD. And in like manner may be shown that they cannot coincide in any other

line which is not a self-rejoining line. Fourth Case; let it be assumed that they coincide in insulated

points more than one, as C and D in Fig. 4. below; and not elsewhere. But if so, let the two spheres be united as one body, and turned about the points A and B, as before. But if during such turning, one of the points C and D remains at rest, then the other must revolve round it and change its place; and if neither of them remains at rest, then both must change their places. Let one of them, therefore, at some period during

Fig. l.

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the turning be in the new situation E. But if the spheres do at first coincide in C and D, it may be shown as has been done before, that during the turning they must always continue to coincide in C and D, and not elsewhere. But they also, at

one period during the turning, coincide in E. Which is • I. Nom.26.

impossible. The assumption*, therefore, which involves the impossible consequence, cannot be true; or the two spheres cannot coincide in the insulated points C and D. And in like manner may be shown that they cannot coincide in

any

other insulated points more than one at a time. But if the two spheres cannot coincide in any surface, nor in any line, nor in any insulated points more than one at a time; they can touch only in a point.

And by parity of reasoning, the like may be proved of any other spheres. Wherefore, universally, if two spheres touch one another externally, they touch only in a point. Which was to be demonstrated.

CoR. Two spheres may be applied to one another externally, so that they shall touch in a point assigned on the surface of one, or of each severally.

For in whatever manner the spheres are made to touch externally, (by Prop. V. above) they will touch in a point. Where

upon either or each of them may be turned about its centre + INTERC. 4. withoutt change of place, till any assigned point on its surface be

made to coincide with the point of contact.

PROPOSITION VI.

THEOREM.--A sphere cannot have more than one centre.

B в

Let there be a sphere de-
scribed about the centre A. No
other point, as B, can also be

E
a centre (that is, can also be
equidistant from all the points
in the surface of the sphere.]

In the surface of the sphere
described about A, take any
point as C; and about C as

•D
a centre, with a central dis-
HINTERC.3. tance equal to CA, describe
Cor. 2.

another sphere. Because the
central distance of this sphere is equal to CA, A will be a point in

its surface; for if not, its central distance would not be equal to CA.

If now the point B is not in the surface of the sphere whose centre * INTERC. 2. is C, let the sphere whose centre is A be turned* about A till the + INTERC.3. Cor. 2.

point B is in that surface. About any centre as D, describet another

sphere with a central distance equal to AC; and apply this sphere INTERC. 5. to the sphere whose centre is C, in such manner$ that they shall Cor.

touch in the point A. Whereupon the point D in this last-described
sphere will be found in the surface of the sphere whose centre is
A; for if not, their central distances would not be equal. Let it
be found then in E. But because B is in the surface of the sphere

whose centre is C, and the two spheres whose centres are C and *Interc.5. E touch only* in the point A, B is not in the surface of the

sphere whose centre is E; wherefore the distance BE is not + Constr. equal to the distance AE, or AC. But BC ist equal to AC ;

wherefore BC is not equal to BE; for if they were equal, BE INTERC. 1. would also be f equal to AC, and it is not. And because BC is Cor. 1.

not equal to BE, the point B is not equidistant from all the points
in the surface of the sphere. And in like manner may be shown
that any other point which is not-A, is not equidistant from all
the points in the surface of the sphere.

And by parity of reasoning, the like may be proved of every
other sphere. Wherefore, universally, a sphere cannot have &c.
Which was to be demonstrated.

PROPOSITION VII.
PROBLEM.-A point being assigned outside a given sphere, to de-

scribe about it a sphere which shall touch the given sphere

externally.
[N.B. For brevity, a sphere may be named by the letter which is at its

centre alone, when no obscurity arises therefrom.]
Let the sphere whose centre is
A, be the given sphere; and B
the assigned point outside. It is
required to describe about B,
sphere which shall touch the sphere
whose centre is A externally.

•B
The given sphere and the point
B being supposed imbedded in one

continuous hard body, let the *INTERC. 2. whole be turned* about B, in such

sort that the sphere A shall be moved to a new situation having no

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part in common with the old, as for instance to the situation of the sphere whose centre is C; and let the line then described by the

point A from A to C, be afterwards turned about B in various * Interc.3. directions as called for, till a sphere is described* with a central

distance equal to BA. In the surface of the sphere A, in that portion of it which is within the sphere so described about B, let any point be taken, as D. But when the point A (in the process of describing a sphere about B as above) was made to describe a line, the point D at the same time described a line of some kind from D to some other point E, inasmuch as every point in the sphere whose centre is A was moved from its place; and by the after turning of this line about B, (if the turning was continued in various directions as called for,) was described a sphere with a central distance equal to BD; and in the same manner with any other point in the sphere A. By the motion of every point therefore in the sphere A, was described a sphere about the centre

And because the concentric spheres so described either + INTERC.3. coincidet throughout their whole surfaces or not at all, there is Cor. 4.

some one (whether traced by the motion of a single point or of more) which has none other of the concentric spheres within it either totally or in part. [As would be mechanically exhibited, if the space about B were filled with some yielding substance like plaster, out of which a sphere was to be formed through the scraping away of the greatest possible quantity of matter by the turning of the sphere A in different directions about B.] Let then this sphere, which has none other of the concentric spheres within it, be the sphere BF. It touches the sphere whose centre is A, as required.

For the sphere A, during its turning about B, is or may be applied to every point in the surface of the sphere BF (as by Prop. III); wherefore they meet. Also it does at no time cut the sphere

for if any point in the sphere A ever fell within the sphere BF, there would be traced by it a concentric sphere interior to the

sphere BF, and there is not. Wherefore because it always meets INTERC. 3. the sphere BF but does not cut it, it always touchesť it. And Nom.

because the spheres A and BF touch one another in every situation of the sphere A, they touch one another when the sphere A is returned to the situation from which it was first moved. Wherefore there has been described a sphere BF, touching the given sphere externally. Which was to be done.

BF;

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