Book XII. of BZ; therefore the square of KV is greater than

the square of BZ: And because BA is equal to AK, and
that the squares of BZ, ZA, are equal together to the square
of BA, and the squares of KV, VA, to the square of AK;
therefore the squares of BZ, ZA are equal to the squares
of KV, VA; and of these the square of KV is greater than
the square of BZ; therefore the square of VA is less than
the square of ZA, and the straight line AZ greater than VA;
Much more then is AZ greater than AG; because, in the
preceding proposition, it was shown that KV falls without
the circle FGH: And AZ is perpendicular to the plane
KBOS, and is therefore the shortest of all the straight lines
that can be drawn from A, the centre of the sphere, to
that plane. Therefore the plane KBOS does not meet the
lesser sphere.

And that the other planes between the quadrants BX,
KX, fall without the lesser sphere, is thus demonstrated :
From the point A draw AI perpendicular to the plane of
the quadrilateral SOPT, and join 10; and, as was demon-
strated of the plane KBOS and the point Z, in the same
way it may be shown that the point I is the centre of a cir-
cle described about SOPT: and that OS is greater than
PT; and PT was shown to be parallel to OS: Therefore
because the two trapeziums KBOS, SOPT inscribed in
circles have their sides BK, OS, parallel, as also OS, PT;
and their other sides BO, KS, OP, ST, all equal to one

another, and that BK is greater than OS, and OS greater * 2 Lem. 12. than PT, therefore the straight line ZB is greatera than 10.

Join A which will be equal to AB; and because AIO,
AZB are right angles, the squares of AI, IO are equal to
the square of A0 or of AB; that is, to the squares of AZ,
ZB; and the square of ZB is greater than the square of
10, therefore the square of AZ is less than the square of AI;
and the straight line AZ less than the straight line AI: And
it was proved, that AZ is greater than AG; much more
then is Al greater than AĞ: Therefore the plane SOPT
falls wholly without the lesser sphere. In the same manner
it may be demonstrated, that the plane TPRY falls without
the same sphere, as also the triangle YRX, viz. by the Cor.
of 2d Lemma. And after the same way it may be demon-
strated, that all the planes which contain the solid polyhe-
dron, fall without the lesser sphere. Therefore in the greater
of two spheres, which have the same centre, a solid polyhe-
dron is described, the superficies of which does not meet
the lesser sphere. Which was to be done..

* But the straight line AZ may be demonstrated to be Book XII. greater than AG otherwise, and in a shorter manner, without the help of Prop. 16, as follows. From the point G draw GU at right angles to AG, and join AU. If then the circumference BE be bisected, and its half again bisected, and so on, there will at length be left a circumference less than the circumference which is subtended by a straight line equal to GU, inscribed in the circle BCDE: Let this be the circumference KB: Therefore the straight line KB is less than GU: And because the angle BZK is obtuse, as was proved in the preceding, therefore BK is greater than BZ: But GU is greater than BK; much more then is GU greater than BZ, and the square of GU than the square of BZ; and AU is equal to AB; therefore the square of AU, that is, the squares of AG, GU, are equal to the square of AB, that is, to the squares of AZ, ZB: but the square of BZ is less than the square of GU; therefore the square of AZ is greater than the square of AG, and the straight line AZ consequently greater than the straight line AG.

Cor. And if in the lesser sphere there be described a solid polyhedron, by drawing straight lines betwixt the points in which the straight lines from the centre of the sphere. drawn to all the angles of the solid polyhedron in the greater sphere meet the superficies of the lesser; in the same order in which are joined the points in which the same lines from the centre meet the superficies of the greater sphere; the solid polyhedron in the sphere BCDE has to this other solid polyhedron the triplicate ratio of that which the diameter of the sphere BCDE has to the diameter of the other sphere : For if these two solids be divided into the same number of pyramids, and in the same order, the pyramids shall be similar to one another, each to each: Because they have the solid angles at their common vertex, the centre of the sphere, the same in each pyramid, and their other solid angle at the bases equal to one another, each to eacha, be- · B. 11. cause they are contained by three plane angles, each equal to each; and the pyramids are contained by the same number of similar planes; and are therefore similarb to one an- b 11 Def.ll. other, each to each: But similar pyramids have to one another the triplicate c ratio of their homologous sides. There- Cor. 8.12 fore the pyramid of which the base is the quadrilateral KBOS, and vertex A, has to the pyramid in the other sphere of the same order, the triplicate ratio of their hom mologous sides, that is, of that ratio which AB from the centre of the greater sphere has to the straight line from

Book XII. the same centre to the superficies of the lesser sphere.

And in like manner, each pyramid in the greater sphere has to cach of the same order in the lesser, the triplicate ratio of that which AB has to the semidiameter of the lesser sphere. And as one antecedent is to its consequent, so are all the antecedents to all the consequents. Wherefore the whole solid polyhedron in the greater sphere has to the whole solid polyhedron in the other, the triplicate ratio of that which AB the semidiameter of the first has to the semidiameter of the other; that is, which the diameter BD of the greater has to the diameter of the other sphere. ?

PROP. XVIII. THEOR. SPHERES have to one another the triplicate ratio ✓ of that which their diameters have.

Let ABC, DEF, be two spheres, of which the diameters are BC, EF. The sphere ABC has to the sphere DEF the triplicate ratio of that which BC has to EF. - For, if it has not, the sphere ABC shall have to a sphere either less or greater than DEF, the triplicate ratio of that which BC has to EF. First, let it have that ratio to a less, viz, to the sphere GHK; and let the sphere DEF have

the same centre with GHK: and in the greater sphere * 17.12. DEF describea a solid polyhedron, the superficies of which

KF M

does not meet tbe lesser sphere GHK; and in the sphere ABC describe another similar to that in the sphere DEF:

Therefore the solid polyhedron in the sphere ABC has to Cor. 17

the solid polyhedron in the sphere DEF, the triplicate ra"'* tio b of that which BC has to EF. But the sphere ABC

has to the sphere GHK, the triplicate ratio of that which BC has to EF; therefore as the sphere ABC to the sphere GHK, so is the said polyhedron in the sphere A BC to the solid polyhedron in the sphere DEF: But the sphere ABC

12

is greater than the solid polyhedron in it: therefore c also Boon XIT, the sphere GHK is greater than the solid polyhedron io then sphere DEF: But it is also less, because it is contained 5%. within it, which is impossible: Therefore the sphere ABC has not to any sphere less than DEF, the triplicate ratio of that which BC has to EF. In the same manner, it may be demonstrated, that the sphere DEF has not to any sphere less than ABC, the triplicate ratio of that which EF has to BC. Nor can the sphere ABC have to any sphere greater than DEF, the triplicate ratio of that which BC bas to EF: For, if it can, let it have that ratio to a greater sphere LMN: Therefore, by inversion, the sphere LMN has to the sphere ABC, the triplicate ratio of that which the diameter EF has to the diameter BC. But as the sphere LMN to ABC, so is the sphere DEF to some sphere, which must be less than the sphere ABC, because the sphere LMN is greater than the sphere DEF, therefore the sphere DEF has to a sphere less than ABC the triplicate ratio of that which EF has to BC; which was shown to be impossible: Therefore the sphere ABC has not to any sphere greater than DEF the triplicate ratio of that which BC has to EF: and it was demonstrated, that nei. ther has it that ratio to any sphere less than DEF. Therefore the sphere ABC has to the sphere DEF, the triplicate ratio of that which BC has to EF. Q. E. D.

the sphere Do Edt which Elore the sphatiplicate ea, that