32. of this) But the Spheric is to the Inscrib'd as 8 is to 6, by this present Propofition. Therefore the Circumscrib'd is to the Inscrib'd as 12 is to 6, or 2 to 1. T (as 2 HE Superficies of any Spherical Portion whatever Fig. 26, ILBG) hath the fame Proportion to the Su-25 perficies of the greatest inscribed Cone, which (BG) the Side of the Cone hath to (GO) the Radius of the Bafe. Because (by 25. of this) the Superficies of the Portion IL BG is equal to the Circle of the Radius BG; the Proportion thereof to QT, that is, to the Base of it felf and of the Cone, will be duplicate to the Proportion (by 2. 1. 12.) of BG to GO; that is, (by 14. of this) of the Proportion of the conical Superficies IBG to the same Base QT. Therefore it is manifest (by Def. 10. 1. 5.) that the Superficies ILBG is to the conical Superficies IBG, as the same conical Superficies IBG is to the Base QT. Wherefore seeing the conical Superficies IBG, is to the Base QT, as BG (by 14. of this) is to GO, the Superficies of the Portion will also be to the conical Superficies IBG infcrib'd in it, as BG is to GO. 2. E. D. T PROP. XXXVI. Theorem. HE Superficies of the Hemisphere (EOBD) Fig. 24. hath that Proportion to (EBD) the Superficies of the greatest right infcribed Cone, which in a Square the Diameter hath to a Side; and that Proportion to the Superficies of a like Cone circumscribed, as the Side in a Square hath to the Diameter. I. The Demonstration of the first Part is manifest from the foregoing. For the Superficies of any Portion whatever, and consequently of the Hemisphere, EOBD, is to the conical Superficies inscrib'd, as BD is to DA. But BADK is a Square, whose Diameter is BD and the Side D A. Part II. Let EBC be half of the Square circumscrib- Fig. 6. 1. 4. ed about the Circle (whose Centre is 0); which EBC being turn'd about the Axis OB, let there from thence be produc'd a Cone circumfcribed about the Hemisphere. Now because the Square EC is (by 47. l. 1.) double to the Square E B or GI, the Circle of the Diameter EC also is (by 2. l. 12.) double to the Circle whose Diameter is GI, that is, to the Circle HGDI. But (by 24. of this) the Superficies of the Hemisphere included in the Cone EBC is double to the same Circle. Therefore the Circle of the Diameter EC is equal to the hemispherical Surface. Wherefore seeing the conical Superficies EBC is (by 14. of this) to the Circle of the Diameter EC, to wit, to its own Base, as the Side BE is to EO the Radius of the Base; it will be also to the Shemispherical Superficies inscribed in it, as BE is to EO; that is, as the Diameter in a Square is to a Side. 2. E. D. The fame Figure with Fig. 13.1.5. 13. PROP. XXXVII. Theorem. A Sphere hath the fame Proportion to a Square co nical Rhombus circumscribed about it, both in reSpect of the Solidity and Surface, which in a Square the Side hath to the Diameter. Let the Square EBCF be circumfcrib'd about HG DI, the greatest Circle of a Sphere, from which Square as turn'd round about the Axis B F, let a conical Rhombus encompaffing the Sphere be produc'd. As EB a Side of the Square (see Fig. 6. l. 4.) is to the Diameter EC, even to let S be made to R; (fee Fig. 13. l. 5.) and let this Proportion be continued thro' four Terms, S, R, Q, O; the Proportion then of S to O will be triplicate to the Proportion of S to R; that is, See def. 10. of EB to EC, and the Proportion of Q to R will be 1.5. duplicate to the Proportion of O to Q, or of R to S; that is, of EC to EB; and consequently (by 20. 1. 6.) Ois to R as the Square of EC is to that of EB; from whence (by Schol. Pr. 6. and 7.1.4.) O is double to R. These Things being thus settled, let the Sphere EBCF be understood to be circumscribed about the conical Rhombus. Thus the Sphere HGDI will be to the Sphere EBCF (by 18. 1. 12.) in the triplicate Proportion of the Diameter GI or EB to the Diameter EC; that is, (as I have already shew'd) it will be as S to O. But But the Sphere EBCF is to the conical Rhombus inscrib'd in it (by 30. of this) as 2 is to 1; that is, (as I have shew'd above) as O is to R. Therefore by Equality of Proportion, the Sphere HGDI is to the same Rhombus which is describ'd about it, as S is to R; that is, as in a Square the Side E B is to the Diameter EC. Which was the first Part. Then from the second Part of the foregoing, it appears that the Superficies of the Hemisphere is to the Superficies of the Cone E BC, and consequently the Superficies of the whole Sphere is to the Superficies of the whole Rhombus EBCF, as in a Square the Side is to the Diameter. Therefore the Sphere as well in Solidity as in Superficies is to the square Rhombus EBCF, as in a Square the Side is to the Diameter. 2. E. D. : TH PROP. XXXVIII. Theorem. 1 HE Superficies of the Portion (BGKD) which Fig. 27. contains an equilateral Cone (BKD) is double to the Superficies of the Same Cone. : This is manifest from 35. For the Superficies of the Portion BGKD is to the infcrib'd conic Superficies (by 35. of this) as BK is to BA. But because the Cone BKD is fuppos'd to be equilateral, KB is equal to BD, and confequently double to BA. Therefore the Superficies BGKD is also double to the inscribed conical Superficies BKD. 2. E.D. PROP. XXXIX. Theorem. THE Superf HE Superficies of a Sphere is to the whole Super- Fig. 27. equilateral Cone infcrib'd in it, as 16 to 9. Let Z be the Center of the Sphere, and BKD the equilateral Cone inscribed, and KZAO the Ax Mere the Diameter of the Basis of the Cone QT. And because causethe Axis of the Cone K A is perpendicular to the Base QT, BAK (Def. 3. 1. 11.) will be a right Angle. Therefore the Square of B A is equal to the Rectangle KAO. (Corol. 1. Pr. 17.1.6.) Now because the Side of the equilateral Triangle cuts off (Corol. 5. Pr. 15. 1. 4.) a 4th Part of the Axis AO, the Rectangle KA O, that is, the Square of BA, will be triple to the Square of A O (by 1. 1. 6.) Wherefore seeing the Square of the Radius ZO is (Corol. 3. Pr. 4. 1. 2.) quadruple of the Square of AO, the Square of the Radius ZO will be to the Square of the Radius BA, as 4 is to 3. Therefore the Circle OBKD is also (by 2.1. 12.) to the Circle QT, as 4 is to 3. Therefore four Circles OBKD, that is (by 24. of this) the whole spherical Superficies DG is to the Circle QT, as 16 is to 3. But (Corol. 1. Pr. 14. of this) the Superficies of the equilateral Cone BKD is to the Circle QT, to wit, its own Base, as 2 is to 1; and confequently the whole Superficies of the Cone BKD, including its Base, is to the Base, to wit the Circle QT, as 3 is to r, or 9 to 3. Therefore feeing I have shew'd that the Superficies of a Sphere is to the same Circle, as 16 is to 3, the Superficies of the Sphere DG will be to the whole Superficies of the equilateral Cone, as 16 is to 9. 2. E. D. Or other wife thus: BEcause (by Corol. 5. Pr. 15.1.4.) the Side BD of the equilateral Triangle cuts off a 4th Part of the Axis AO, the spherical Superficies BOD will be a 4th Part by 27. of this, and consequently the Superficies BGKD, three 4th Parts of the Superficies of the whole Sphere. Wherefore if the whole Superficies be suppos'd to be 16, the Superficies BGKD will be 12. But (by the foregoing) the Superficies BGKD is double to the conical Superficies BKD, and consequently is to it, as 12 to 6. Therefore the whole Superficies of the Sphere is to the conical BKD, as 16 is to 6. Then because the Superficies of the Cone BKD (as being equilateral) is (by Corol. 1. Pr. 14. of this) double to the Bafe QT, it is manifeft that the conical Superficies BKD (to wit, without the Base) is to the whole Superficies of the Cone, as 2 is to 3; that is, as 6 to 9. Therefore by equality of Proportion the whole Superficies of the Sphere Sphere is to the whole Superficies of the equilateral S T PROP. XL. Theorem. こ HE Superficies of a Sphere bears that Propor-Fig. 28. tion to the whole Superficies of an equilateral Cone circumfcrib'd about it, which 4 doth to 9. Let there be circumfcrib'd about the greatest Circle of a Sphere BPM, the equilateral Triangle DOF; by which, as turn'd round about the Axis OAB, let there be produc'd an equilateral Cone, circumfcrib'd about the Sphere. And let there also be circumfcrib'd about the equilateral Triangle DOF the Circle NDLOF, which, as is manifest, is concentrical to the former; and let the Axis O A B be produc'd to N. Because BN is a 4th Part of the Axis ON, (as is manifest from Corol. 5. Pr. 15. l. 4.) ON is double to BK. Wherefore the Proportion betwixt Circles being duplicate (by 2. 1. 12.) of the Proportion of the Diameters, the Circle BPM will be to the Circle NDLOF, as I to 4. But it hath already been shew'd in the first foregoing Demonstration, that the Circle NDLOF is to the Circle QT, the Base of the equilateral Cone infcrib'd in the Sphere F L, as 4 is to 3. Therefore by equality of Proportion the Circle BPM is to the Circle QT, as I is to 3. But the whole Surface of the Cone DOF is (by Cor. 1. Pr. 14. of this) triple to QT. Therefore the whole Superficies of the Cone is ninefold of the Circle BPM. Wherefore seeing the Superficies of the Sphere TP is quadruple (by 24. of this) of the same Circle BPM, the whoe Superficies of the equilateral Cone DOF is to the Superficies of the Sphere to which it is circumscrib'd, as 9 is to 4. 2. E. D. Coroll. 1. From this Demonstration it is manifest that the Axis BO of an equilateral Cone circumscrib'd about a Sphere, is one and a half of the Diameter of the Sphere BK, or as 3 to 2. 2. That QT the Base of the Cone DOF is also one and an half of both Bases of the Cylinder circumscrib'd about the fame Sphere. For QT is to BPM, as 3 to 1. Therefore 2T is to BPM twice, as 3 is to 2. 3. That : |