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THEOREM.-If two equal spheres cut one another, their surfaces

coincide only in one self-rejoining line. Let two equal spheres whose centres are A and B, cut one another. Their surfaces coincide only in one self-rejoining line.

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Because the spheres cut one another, the coincidence of their surfaces is not in a point only; for if it was in a point only, the spheres would meet but not cut one another. And because the coincidence of the surfaces is not in a point, it is in a figure or figures of some kind, either surface or line. Also the figure or

figures will be such, that if the two spheres are united as one INTERC. 2. body and turned* about the two points A and B which are the

centres, the figure, or figures severally, shall be without change

of place. For because each sphere is turned about its own +INTIAC. 4. centre, and is consequentlyt without change of place, the surfaces

will at all times during the turning coincide in the whole of such figure or figures and not elsewhere. For if they did not, they would at one instant coincide in some point in fixed space and both of them pass through it, and at another instant they would not; which cannot be without one or both of the spheres having

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suffered change of place. Such figure therefore (or each such, if there can be more than one) will be either a self-rejoining line or a belt of surface, which on the united body being turned about A and B, revolves on its own ground in fixed space

without change of place. Take then any point C in which the surfaces of the two spheres at any time coincide ; and on the united spheres being turned as before till the united body returns to the situation from which it set out, the point C, whether it be in a line or in a belt of surface, will describe a self-rejoining line as CD. [The other half of CD is supposed to be on the other side the spheres, and consequently is not represented.] The surfaces of the spheres shall coincide no where but in CD,

For if this be disputed,
First Case ; let it be assumed that they coincide in a belt of

surface having breadth, as that contained between the lines
CD and EF; and not elsewhere. In EF take any point as
G; and on the other side of CD take any point as H, in the
surface of the hollow sphere whose centre is A; and by the
turning of the substantial sphere which is its reciprocal, about
the points A and G while the hollow sphere remains at rest,
let the point H trace on the surface of the substantial sphere
the line HOPLNM cutting EF in N. In like manner let
the substantial sphere whose centre is B, be turned about the
points B and G; and let the point N in its reciprocal which
remains at rest, trace on its surface the line IOPKNM.
There are therefore two surfaces HOPLNM and IOPKNM,
which coincide with one another in the portion MOPGN,
but in their remaining portions MOH, MOI, and NPL,
NPK, they do not coincide. Let now the sphere whose cen-
tre is B, be turned about the point G, till its centre B is

applied to the centre A of the other sphere (which can be * INTERC.10. done inasmuch as the radii of these equal spheres are* Cor. 4. +INTERC. 3.

equal); whereupon the surfaces of the two spheres willt Cor. 5.

coincide throughout. And because the lines HOPLNM and IOPKNM are now traced on a common surface by a common point N, these lines will coincide with one another, and the surfaces inclosed by them will coincide throughout. Also if on the one sphere the points M, N, O, P, do not coincide with the same points on the other respectively, they may be made to do so by turning one of the spheres about the point G

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* I. Nom. 3. +1.Nom. 26.

and the common centre A, and the whole surfaces HOPLNM
and IOPKNM will continue to coincide throughout. Which
having been done, let the spheres be returned to their original
situation. Wherefore the two surfaces HOPLNM and
IOPKNM which, when the centres of the spheres were
applied together, coincided entirely with one another, do now
coincide in the portion MOPGN as before, but in their
remaining portions MOH, MOI, and NPL, NPK, they do
not. Which is impossible. For if the two surfaces coincide
entirely in the one situation, the portions MOH, MOI, and
NPL, NPK, 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 hard* bodies. The assumptiont,
therefore, which involves this impossible consequence, cannot
be true; or the two spheres cannot coincide in the belt
CDFE. And in like manner may be shown that they
cannot coincide in any other belt; and this equally whether

CD be one of its boundaries or not.
Second Case ; let it be assumed that besides coinciding in CD

they also coincide in some other self-rejoining line as EF, but
not in the surface between. Whereupon may be shown by
the same process as in the preceding case, that the two lines
HOPLNM and IOPKNM which when the centres of the
spheres are applied together coincide entirely with one another,
do afterwards coincide in the points M, N, O, P as before,
but in the rest of their extent they do not coincide. Which
is impossible. For if the two lines coincide entirely in the one
situation, the portions MHO, MIO, and NLP, NKP, can
in no way be made to cease coinciding and be separate
while in the points M, N, O, P the lines continue to coin-
cide, 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 hardf bodies. The
assumption", therefore, which involves this impossible conse-
quence, cannot be true; or the two spheres cannot coincide
in the self-rejoining line EF in addition to CD. And in
like manner may be shown that they cannot coincide in any
other self-rejoining line in addition to CD.

#1. Nom.3. * I. Nom.26.

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Third Case ; let it be assumed that besides coinciding in the

line CD, they coincide in some insulated point, or in some surface or line which is not a belt or a self-rejoining line. Then, if the two spheres be united as one body and turned about the points A and B as before, their surfaces must coincide in all the belt or self-rejoining line which will be described by such surface, line, or point during the turning; as was shown of the point C during the construction of the figure. But it has been shown that they cannot coincide in any belt or self-rejoining line in addition to CD. Wherefore

they cannot coincide in any point, surface, or line as assumed. But if the surfaces of the two spheres cannot coincide in any surface, line, or point, in addition to the line CD; they coincide only in CD.

And by parity of reasoning, the like may be proved of any other equal spheres. Wherefore, universally, if two equal spheres cut one another, &c. Which was to be demonstrated.

Cor. If two equal spheres with different centres, have a point in their surfaces in common, they will cut one another in one self-rejoining line passing through that point, or else touch one another externally in that point only.

For their surfaces cannot coincide throughout; because then * INTERC. 6. they could not have* different centres. And if they cut, their

surfaces (by Prop. XI above) will coincide only in one selfrejoining line; wherefore any point which they have in common,

must be in that line. And if they do not cut, but still have a +Ínterc.5. point in common; they touch in that point, andt that only.

PROPOSITION XII.

See Note. THEOREM.-Two straight lines, which are not in one and the

same line, cannot have a common segment.

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For if this be disputed, let it be assumed that ABD is also a

straight line. About B as a centre, with the radii BC and BD *Interc.10. respectively, describe* concentric spheres. In BC take any point Cor. 6.

as E, between B and the surfaces of both these concentric spheres on the side of B which is towards C; and about B as a centre, with the radius BE, describe a sphere, whose surface (because it

lies within the surfaces which pass through the extremities C and +INTERC.10. D) will cut BC and BD. Let it cut them in thet points E and F; Cor. 9. INTERC.10.

wherefore BE, BF aref equal. About E as a centre, with the Cor. 4. radius EB, describe a sphere; and about A as a centre, with the

radius AB, describe another sphere. • Constr.

Because the surfaces of these two spheres pass* through the

point B, which is a point in the straight line joining their centres; + INTERC. 10. the spheres toucht one another externally in the point B. And if Cor. 8.

the straight line AC, together with the two substantial spheres

whose centres are A and E, be turned about the points A and C; * INTERC. 9.

because AC is a* straight line, every point in it, and among Nom.

others the point E, will be without change of place. Wherefore the substantial spheres whose centres are A and E and which touch one another externally in the point B, are turned about

their centres which remain at rest, and consequently the spheres PINTERC. 4. arewithout change of place ; and if these spheres are supposed

imbedded in one inclosed body of hard matter which is turned *Interc. 9. along with them round* the straight line AC, this inclosing Nom. + Interc. 8. body willt also be without change of place, as respects the parts Cor. bounded by the hollow spherical surfaces. And because the

inclosing body being turned round the straight line AC is without change of place, its solid parts which lie in any one direction from

the point B, as for instance towards M and H, are capable of coin11. Nom.9. ciding# with, and consequently are equal* to, its solid parts which • I. Nom. 14.

lie in any other direction from B, as for instance towards N and I.; for in the course of the turning, one set of these is made to occupy the place of the other. If then ABD is also a straight line;

about F as a centre, with a radius equal to FB or EB, de1 INTERC.10. scribet a sphere. Whereupon may be shown as before, that if Cor. 6.

ABD is a straight line, the sphere so described will touch the sphere whose centre is A externally in the point B, and if the body of hard matter inclosing the two spheres whose centres are A and F be turned round the line ABD,this body also will be without change of place. Which is impossible. For since the spheres described

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