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P, T, R, Y, to the Point A, there will be conftituted a certain solid polyhedrous Figure within the Circumferences B X, K'X, composed of Pyramids, whofe Bases are the quadrilateral Figures KBOS, SOPT, TPRY, and the Triangle YRX; and Vertices the Point A. And if there be made the Same Coustruction on each of the Sides KL, LM, ME, like as we have done on the Side KB; and also in the other three Quadrants, and the other He. misphere, there will be conftituted a polyhedrous Fi. gure described in the Sphere, composed of Pyramids whose Bases will be equal and similar to the aforesaid quadrilateral Figures, and Triangle YR X, and Vertices the Point A. Now, I say, the said Polyhedron

does not touch the Superficies of the Sphere, wherein 113. II. the Circle F G H is. Let A Z be drawn I from the

Point A, perpendicular to the Plane of the quadrilateral
Figure KBS O, meeting in the Point Z ; and join
BŽ, ZK: Then, fince AZ is perpendicular to the

Plane of the quadrilateral Figure K BS(, it shall also *Def. 3.115. be * perpendicular to all Right Lines that touch its : and are in the same Plane: Wherefore A Z is

perpendicular to B Z and Z K. And because AB is equal to

AK, the Square of AB shall be also equal to the 74. 17. Square of A K; and the Square of A Z, Z B, are +

equal to the Square of A B; for the Angle at Z is a
Right Angle; and the Squares of AŽ, ZK, are
equal to the Square of A K: Therefore the Square of
A Z, Z B, arc equal to the Squares of A Z, ZK.
Let the common Square of AZ be taken away, and
ther the Square of BZ, remaining, is equal to the
Square of Z K, remaining; and to the Right Line
B'Z is equal to the Right Line ZK. After the fame
manner we demonstrate, that Right Lines drawn from
the Point Z to the Point OS, are each equal to BZ,
ZK. Therefore a Circle described about the Centre
z, with either of the Distances, ZB, Z K, will also
pass thro' the Points 0, S. And, because BKS O is
a quadrilateral Figure in a Circle, and OB, BK,
KS, are equal ; and OS is less than BK; the An:
gles B Z K Thall be obtuse; and fo B K greater than
B Z. But G L also is greater than BK; therefore
GL is much greater than B Z; and the Square of
GL is greater than the Square of B Z. And since

AL

A L is equal to AB, the Square of A L shall be equal to the Square of AB. But the Squares of AG, GL, together, are equal to the Square of AL; and the Squares of B Z, Z A, together, equal to the Square of AB: Therefore the Squares of A G, GL, together,' are equal to the Squares of B Z, ZA, together. But the Square of B Z is less than the Square o:GL; therefore the Square of Z A is greater than the Square of A G; and to the Right Line Z A will be greater than the Right Line A G. But A Z is perpendicular to one Base of the Polyhedron, and A G reaches to the Superficies of the lefser Sphere: Wherefore the Polyhedron does not touch the Superficies of the leffer Spitere. Therefore, there is described a solid Polyhedron in the greater of two Spheres ; having the same Centre, which doth not touch the Superficies of the lefjer Sphere; which was to be done.

Goroll. Also, if a solid Polyhedron be described in some

other Sphere, similar to that which is described in the Sphere BCDE; the solid Polyhedron described in the Sphere BCDE, to the folid Polyhedron defcribed in the other Sphere, shall have a triplicate Proportion of that which the Diameter of the Sphere BCD E hath to the Diameter of that other Sphere. For, the Solids being divided into Pyramids equal in Number, and of the fame Order, the same Pyramids shall be fimilar. But fimilar Pyramids are to each other in a triplicate Proportion of their homologous Sides; therefore the Pyramid, whose Base is the quadrilateral Figure KOBS, and Verrex the Pwint A to the Pyramid of the fame Order in the other Sphere, has a triplicate Proportion of that which the honologous Side of the one has to the homologous Side of the other : that is, which A 3, drawn from the Centre A of the Sphere, to thar Line which is drawn from the Centre of the other S; here. In like manner, every one of the Pyramids, that are in the Sphere whore Centre is A, to every one of the Pyramids of the same Order in the other Sphere, hath a triplicate Preportion of that which AB has to that Line drawn from the Centre of the other Sphere: And as one of the Antecedents is to one of the Consequents, so are all r

the

the Antecedents to all the Confequents. Wherefore the whole folid Polyhedron, which is in the Sphere described about the Centre A, to the whole solid Polyhedron that is in the other Sphere, hath a triplicate Proportion of that which A B hath to the Line drawn from the Centre of the other Sphere; that is, which ine Diameter B D has to the Diameter of the other Sphere.

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PROPOSITION XVIII.

THEOREM.
Spheres are to one another in a triplicate Proper-

tion of their Diameters.

Suppose A B C D E F, are two Spheres, whose

Diameters are BC, EF, I say, the Sphere A B C to the Sphere D E F, has a triplicate Proportion of that which B C has to EF.

For, if it be not so, the Sphere A B C to a Sphers either lefser or greater than D E F will have a triplicate Proportion of that which BC has to EF. Firft, let it be to a lesser, as GHK ; and suppose the

Sphere DEF to be described about the Sphere GHK; * 17 of tbis. and let there be described * a solid Polyhedron in the

greater Sphere DEF, not touching the Superficies of the lesser Sphere GH K; also, let a solid Polyhedron he described in the Sphere A B C, fimilar to that which is described in the Sphere DEF; then the so

lid Polyhedron in the Sphere. A B C, to the solid Cor. to tbc. Polyhedron in the Sphere DEF, will have a + ish Prope

triplicate Proportion of chat which B C bas to EF: But the Sphere A B C to the Sphere G HK, hath a triplicate Proportion of that which B C hath to EF; therefore, as the Sphere A B C is to the Sphere G HK, lo is the solid Polyhedron in the Sphere A B C, to the folid Polyhedron in the Sphere DEF; and (by Inversion) as the Sphere A B C is to the solid Polyhedron that is in it, so is the Sphere G H K to the solid Polyhedron that is in the Sphere DEF. But the Sphere ABC is greater than the solid Polyhedron that is in it; therefore the Sphere G H K is also greater than the solid Polyhedron that is in the Sphere DEF,

and

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