Fig. 25.

The Cone whofe Bafis is the hemifpherical Superficies EOBD, and its Altitude the Radius A B, is to the Cone EBD (by II. l. 12.) as Bafe is to Bafe; that is, as the hemifpherical Surface EOBD is to the greatest Circle PT. Therefore feeing the hemifpherical Superficies EOBD is double to the greatest Circle (by 24. of this), the Cone alfo which hath the Superficies EO BD for its Bafe, and the Radius AB for its Altitude, is double to the Cone E BD. But (by 28. of this) the Hemisphere is equal to a Cone which hath the Radius for its Altitude, and the hemifpherical Superficies for its Bafe. Therefore the Hemifphere is alfo double to the Cone E BD. Q. E. D.

i.

L'

PROP. XXXI. Theorem.

ET a Sphere be divided into two Segments IL BG, ISK G, by the Plane IQGT which doth not pass thro' the Centre A; and let the Diameter BOK be perpendicular to the cutting Plane.

As the Altitude O B of the Segment IL BG, is to the Radius of the Sphere A B: So let OK, the Altitude of the other Segment, be made to another Line KN.

In like manner, As OK, the Altitude of the Segment ISK G, is to the Radius AK or AB, So let the Altitude O B of the other Segment be made to another Line B D. Which Things being fuppos'd, I say,

1. The Cones ING and IDG, whofe Altitudes are ON, OD, and IQGT their common Bafe, are equal to the Spherical Segments.

2. There is the fame Proportion of the Segments as there is of the right Lines DO, NO. S

3. The Segment ISKG is to the greatest Cone IK.G infcrib'd in it, as NO is to KO; and the Segment ILBG is to the greatest Cone IBG infcrib'd in it, as DO is to BO.

Part

Part I. Let the Sphere and Cones be cut by a Plane thro' the Diameter BK. There will be produced in the Sphere the greateft Circle BLK G, and in the Cones the Triangles BIG, IKG. And because BOK the Diameter is (by the Hypothefis) perpendicular to the Circle QT, IOB (by Def. 3.1.11.) will be a right Angle. The Angle BIK in the Semicircle is also a right one (by 31. 1. 3.) Because therefore in the Triangle BIK, there is drawn from the right Angle, IO perpendicular to the Bafe BK; BI will be to IO, as (by 8. l. 6.) BK to KI. Therefore the duplicate Proportion of BI to IO is equal to the duplicate Proportion of B K to KI, that is, (because B K, KI, KO [by Corol. 2. Pr. 8. 1. 6.] are three Proportionals) equal to the Proportion of BK to KO.

Then because OB is (by the Hypothefis) to BD, as OK is to the Radius A B; by Inverfion it will be always thus, DB is to BO, as AB to OK; and by Permutation thus, DB is to BA, as BO to OK; and by Compounding thus, DA is to BA, as BK is to OK. Because therefore I have already fhew'd the Proportion of BK to OK to be duplicate to the Proportion of BI to IO, and confequently (by 2. l. 12) equal to the Proportion betwixt the Circles defcrib'd by the Radius's BI, IO; DA will also be to BA, as the Circle of the Radius BI, to the Circle of the Radius IO. Therefore the Cone under the Altitude D'A, and for the Bafe, the Circle of the Radius IO, that is, the Circle QT, is equal to the Cone under the Altitude BA, (by 15. l. 12) which hath for its Bafe the Circle of the Radius BI; that is, (by Corol. Pr. 29. of this) the spherical Sector AIBG. Wherefore if the fame Cone IAG be added. as well to the Sector AIB G, as to the Cone under DA, and the Circle QT, the Wholes will be equal; to wit, the fpherical Segment IL BG will be equal to two Cones, whereof one is that which is under the Base QT and the Altitude DA, and the other I A G is under the fame Base QT, and the Altitude OA. But these two Cones (by 14. l. 12.) make up the Cone IDG. Therefore the Segment ILBG will be equal to the Cone IDG. 2.E.D.

By the fame Reasoning, the Segment ISKG will be equal to the Cone ING, with this only Change,

Fig. 24.

that the Cone IA G, which before was added, be now taken away.

Part II. This is manifeft from the firft. For the Cones IDG and ING are betwixt themfelves (by p. 14. l. 12.) as are DO and NO. Therefore the Segments alfo I LBG, ISK G, equal to thofe Cones, are betwixt themselves, as the right Lines, DO, NO.

Part III. This likewife is manifeft from the firft. For the Cone IDG is to the Cone IBG, (by the fame) as DO is to BO. Therefore the Segment alfo ILBG, which is equal to the Cone ID G, is to the Cone IBG, as DO is to BO.

FR

Scholium.

Rom the first Part of this Propofition there arifes another Way of measuring spherical Segments, and that a very eafy one; if, to wit, the Cones I DG, ING, be measured; which will be done if the third Parts of the right Lines DO, NO, be drawn into the Circle QT.

A

PROP. XXXII. Theorem.

Right Cylinder (GK) is both in Solidity and the whole Superficies to the Sphere about which it is circumfcrib'd as 3 to 2.

Let BQ be the common Axis of the Sphere and Cy linder, and EBD the greatest Cone infcrib'd in the Hemifphere EOBD. Because the Cylinder E K (half of GK) is (by 10. l. 12.) triple to the Cone EBD, while the Hemifphere is double to the fame Cone (by 30 of this), it is manifeft that the Cylinder EK is to the Hemifphere as 3 to 2. Therefore alfo the whole Cylinder GK is to the whole Sphere QE BD, as 3 to 2. Which was the firft Part.

Then because the Side of the Cylinder K. N is equal to GN the Diameter of the Bafe, its Superficies without the Bafes will be fourfold (by Corol. Pr. 12. of this) of the Bafe MI, and confequently taken together with the Bafes, that is, the whole Superficies of the Cylinder, will be fixfold of the Bafe MI, which is equal to the greatest Circle of the Sphere. But the Superficies of the Sphere is fourfold of that greatest Circle. Therefore

the

the whole Superficies of the Cylinder GK is to the Superficies of the Sphere, as 6 to 4, or as 3 to 2. Which was the other Part.

Therefore a Cylinder is both in Solidity and the whole Superficies to the Sphere, about which it is circumfcrib'd, as 3 to 2. 2. E.D.

Scholium.

T is an Argument what a great Value Archimedes puts upon this Theorem, that he would have a Sphere infcrib'd in a Cylinder fet upon his Tomb. And perhaps amongst so many other famous Discoveries, this chiefly and above all others pleas'd him, for this Reafon, to wit, because there was one and the fame rational Proportion both of Bodies, and of the Surfaces which contain them. We have demonftrated a like Identity of Affections betwixt Rings, and the Surfaces of Rings, in the 4th Book of our Cylindricks and Annularies, Prop. 13, 14, 15. And another famous Example of the fame hath alfo offer'd it felf to me in the Sphere it felf. For I have found, that like as a Sphere is to a right Cylinder which encompaffeth it (which will neceffarily be equilateral) as 2 is to 3, and this both in refpect of Solidity and Surface; fo likewise the Sphere hath to an equilateral Cone encompaffing it, that Proportion which 4 hath to 9; and this both in regard of Solidity and Superficies. From which this alfo follows, That the fefquialteral Proportion found by Archimedes in the Sphere and Cylinder, is continued in three Solids, a Sphere, Cylinder, and equilateral Cone. The Demonftration of both which Things, withfome other Theorems of my own, in which thewonderful Nature of the Sphere will more appear, I fhall fubjoin in the thirteen following Propofitions.

Sphere.

PROP. XXXIII. Theorem.

HE Superficies of a Sphere is double to the Su-Fig. 26. perficies of a Square Cylinder infcrib'd in the fame

Let A KLD be the Square infcrib'd in the greatest Circle of a Sphere, from which turn'd round, there is

Q3

defcrib'd

Fig. 26.

defcrib'd a fquare Cylinder; and let AL be drawn a
Diameter common to the Square and Sphere. Becau
the Square of AL is (by 47. l. 1.) equal to the eq
Squares of A K, KL, it will be double to one A
Therefore alfo the Circle of the Diameter AL, is (
2. l. 12.) double to the Circle, whofe Diameter is A F
to wit, to the Circle CN. But the Superficies of t
Sphere is (by 24. of this) fourfold to the Circle who
Diameter is AL; for that is the greateft Circle of th
Sphere, feeing AL is the Diameter of the Spher
Therefore the Superficies of the Sphere is eightfold
the Circle CN. But becaufe L K, KA (by the Hyp
thefis) are equal, the cylindrical Superficies A CL
(by Corol. Pr. 12. of this) quadruple of the Circle CN
Therefore fince the Superficies of the Sphere is eight
fold of the fame Circle, it will be double to the cylin
drical Superficies. Q.E. D.

PROP. XXXIV.

THE

Theorem.

HE Superficies of a Sphere hath that Proportion to the whole Superficies of a Square Cylinder inferib'd in it, which 4 bath to 3.

Let the fame Things be fuppos'd which were in the foregoing Demonstration. Becaufe by the Hypothefis LK the Side of the Cylinder, and AK the Diameter of the Bafe thereof are equal, the cylindrical Superficies CL will be quadruple (by Corol. Pr. 12. of this) to the Bafe CN, and confequently the whole Superficies of the Cylinder is to both Bafes CN and $L, as 6 is to 2. But the Superficies of the Sphere is to both Bafes together CN, SL, as 8 is to 2, feeing in the foregoing it was fhew'd that it is to one Bafe as 8 to 1. Therefore the Superficies of the Sphere is to the cylindrical Superficies CL as 8 is to 6, or 4 to 3.2..E. D.

Corollary.

THE whole Superficies of a right Cylinder defcrib'd about a Sphere, is to the whole Superficies of an equilateral Cylinder infcrib'd, as 2 is to 1. For the Circumfcrib'd is to the fpheric Superficies as 12 is to 8 (by

23.