A

a

n

fed of pyramids, the bafes of which are the aforefaid qua- Book X!!.
drilateral figures, and the triangle YRX, and thofe formed in
the like manner in the rest of the fphere, the common vertex
of them all being the point A: And the fuperficies of this fo-
lid polyhedron does not meet the leffer fphere in which is the
circle FGH: For, from the point A draw AZ perpendicular a II. st.
to the plane of the quadrilateral KBOS meeting it in Z, and
join BZ, ZK: And because AZ is perpendicular to the plane
KBOS, it makes right angles with every traight line meeting
it in that plane; therefore AZ is perpendicular to BZ and ZK:
And because AB is equal to AK, and that the fquares of of AZ,
ZB, are equal to the fquare of AB; and the fquares of AZ,
ZK to the fquare of AK; therefore the fquares of AZ, ZB b 47. I.
are equal to the fquares of AZ, ZK: Take from thefe equals
the fquare of AZ, the remaining fquare of BZ is equal to the
remaining fquare of ZK; and therefore the straight line BZ
is equal to ZK: In the like manner it may be demonftrated,
that the ftraight lines drawn from the point Z to the points O,
S are equal to BZ or ZK: Therefore the circle defcribed from
the centre Z, and distance ZB fhall pats through the points K, O,
S, and KBO> fhall be a quadrilateral figure in the circle: And
becaule KB is greater than QV, and QV equal to SO, there-
fore KB is greater than 50: But KB is equal to each of the
ftraight lines BO, KS; wherefore each of the circumferences
cut off by KB, BO, KS is greater than that cut off by OS; and
thefe three circumferences, together with a fourth equal to one
of them, are greater than the fame three together with that cut
off by OS; that is, than the whole circumference of the cir-
cle; therefore the circumference fubtended by KB is greater
than the fourth part of the whole circumference of the circle
KBOS, and confequently the angle BZK at the centre is great-
ter than a right angle: And because the angle BZK is obtuse,
the fquare of BK is greater than the fquares of BZ, ZK ; c 12. 2.
that is, greater than twice the fquare of BZ. Join KV, and
because in the triangles KBV, OBV, KB, BV are equal to OB,
BV, and that they contain equal angles; the angle KVB is e-
qual to the angle OVB: And OVB is a right angle; there- d 4. I.
fore alfo KVB is a right angle: And becaufe BD is less than
twice DV, the rectangle contained by DB, BV is less than
twice the rectangle DVB; that is, the fquare of KB is lefs
than twice the fquare of KV: But the fquare of KB is greater
than twice the fquare of BZ; therefore the fquare of KV is

e 8. 6.

Book XII. greater than the fquare of BZ: And because BA is equal to AK, and that the fquares of BZ, ZA are equal together to the fquare of BA, and the fquares of KV, VA to the fquare of AK; therefore the fquares of BZ, ZA are equal to the squares of KV, VA; and of these the fquare of KV is greater than the fquare of BZ; therefore the fquare of VA is less than the fquare of ZA, and the ftraight line AZ greater than VA: Much more then is AZ greater than AG; because, in the preceding propofition, it was shown that KV falls without the circle FGH: And AZ is perpendicular to the plane KBOS, and is therefore the fhorteft of all the ftraight lines that can be drawn from A, the centre of the fphere to that plane. Therefore the plane KBOS does not meet the leffer fphere.

12.

And that the other planes between the quadrants BX, KX fall without the leffer fphere, is thus demonftrated: From the point A draw Al perpendicular to the plane of the quadrilateral SOPI, and join IO; and, as was demonftrated of the plane KBOS and the point Z, in the fame way it may be shown that the point I is the centre of a circle defcribed about SOPT: and that OS is greater than PT; and PT was shown to be parallel to OS: Therefore, because the two trapeziums KBÓS, SOPT infcribed in circles have their fides BK, OS parallel, as alfo OS, PT; and their other fides BO, KS, OP, ST all equal to one another, and that BK is greater than OS, and OS a 2. Lem, greater than PT, therefore the ftraight line ZB is greater than IO. Join AO which will be equal to AB; and because AIO, AZB are right angles, the fquares of Al, IO are equal to the fquare of AO or of AB; that is, to the fquares of AZ, ZB; and the square of ZB is greater than the fquare of 10, therefore the square of AZ is lefs than the fquare of AI; and the ftraight line AZ lefs than the ftraight line AI: And it was proved that AZ is greater than AG; much more then is AI greater than AG: Therefore the plane SOPT falls wholly without the leffer sphere: In the fame manner it may be demonftrated that the plane TPRY falls without the fame fphere, as alfo the triangle YRX, viz. by the Cor. of 2d Lemma. And after the fame way it may be demonftrated that all the planes which contain the folid polyhedron, fall without the leffer fphere. Therefore in the greater of two fpheres which have the fame centre, a folid polyhedron is described, the superficies of which does not meet the leffer fphere. Which was to be done.

But

But the ftraight line AZ may be demonftrated to be greater Book XII. than AG otherwife, and in a horter 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 he bifected, and its half again bifected, and so on, there will at length be left a circumference lefs than the circumference which is fubtended by a ftraight line equal to GU infcribed in the circle BCDE: Let this be the circumference KB: Therefore the ftraight line KB is lefs than GU: And because the angle. BZK is obtufe, 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 fquare of GU than the fquare of BZ; and AU is equal to AB; therefore the fquare of AU, that is, the fquares of AG, GU are equal to the

of AB, that is, to the fquares of AZ, ZB; but the square of BZ is less than the fquare of GU; therefore the fquare of AZ is greater than the fquare of AG, and the ftraight line AZ confequently greater than the ftraight line AG.

COR. And if in the leffer fphere there be defcribed a folid polyhedron by drawing ftraight lines betwixt the points in which the ftraight lines from the centre of the sphere drawn to all the angles of the folid polyhedron in the greater sphere meet the fuperficies of the leffer; in the fame order in which are joined the points in which the fame lines from the centre meet the fuperficies of the greater fphere; the folid polyhedron in the Iphere BCDE has to this other folid polyhedron the triplicate ratio of that which the diameter of the sphere BCDE has to the diameter of the other fphere: For if thefe two folids be divided into the fame number of pyramids, and in the fame order; the pyramids fhall be fimilar to one another, each to each: Because they have the folid angles at their common vertex, the centre of the fphere, the fame in each pyramid, and their other folid angle at the bafes equal to one another, each to each, becaufe they are contained by three a B. II. plane angles equal each to each; and the pyramids are contained by the fame number of fimilar planes; and are therefore fimilar b 11. def. to one another, each to each: But fimilar pyramids have to one another the triplicate ratio of their homologous fides. c Cor. 12. Therefore the pyramid of which the bafe is the quadrilateral KBOS, and vertex A, has to the pyramid in the other sphere of the fame order, the triplicate ratio of their homologous

II.

Book XII. fides; that is, of that ratio which AB from the centre of the greater fphere has to the ftraight line from the fame centre to the fuperficies of the leffer fphere. And in like manner, each pyramid in the greater fphere has to each of the fame order in the leffer, the triplicate ratio of that which AB has to the femidiameter of the leffer fphere. And as one antecedent is to its confequent, fo are all the antecedents to all the confequents. Wherefore the whole folid polyhedron in the greater sphere hes to the whole folid polyhedron in the other, the triplicate ratio of that which AB the femidiameter of the firft has to the femidiameter of the other; that is, which the diameter BD of the greater has to the diameter of the other fphere.

a 17. 12.

B

SPHE

PHERES have to one another the triplicate ratio of that which their diameters have.

Let ABC, DEF be two fpheres 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 fphere ABC fhall have to a sphere either lefs or greater than DEF, the triplicate ratio of that which BC has to EF. First, let it have that ratio to a lefs, viz. to the fphere GHK; and let the sphere DEF have the fame centre with GHK; and in the greater fphere DEF defcribe

a folid polyhedron, the fuperficies of which does not meet the leffer fphere GHK; and in the fphere ABC defcribe another fimilar to that in the fphere DEF: Therefore the folid polyhedron in the fphere ABC has to the folid polyhedron in the b Cor. 17. fphere DEF, the triplicate ratio of that which BC has to EF. But the sphere ABC has to the sphere GHK, the triplicate ra

12.

N

с

tio of that which BC has to EF; therefore, as the sphere ABC Book XII. to the sphere GHK, fo is the folid polyhedron in the fphere ABC n to the folid polyhedron in the fphere DEF: But the sphere ABC is greater than the folid polyhedron in it; therefore al- © 14. 5. fo the fphere GHK is greater than the folid polyhedron in the sphere DEF: But it is alfo lefs, because it is contained within it, which is impoffible: Therefore the sphere ABC has not to any fphere lefs than DEF, the triplicate ratio of that which BC has to EF. In the fame manner, it may be demonstrated, that the sphere DEF has not to any sphere lefs 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 has to EF: For, if it can, let it have that ratio to a greater sphere LMN: Therefore, by inverfion, the sphere LMN has to the fphere ABC, the triplicate ratio of that which the diameter EF has to the diameter BC. But, as the sphere LMN to ABC, fo is the fphere DEF to fome fphere, which must be less than the sphere ABC, because the fphere LMN is greater than the sphere DEF: Therefore the fphere DEF has to a sphere lefs than ABC the triplicate ratio of that which EF has to BC; which was fhewn to be impoffible: 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 demonftrated, that neither has it that ratio to any sphere Jefs than DEF. Therefore the fphere ABC has to the fphere DEF, the triplicate ratio of that which BC has to EF. Q. E. D.

Q.E.

FIN IS.

T3