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3. That the Superficies of the Cone DOF is one and
an half of the Superficies of the equilateral Cylinder cir. + Par Corol. cumscrib'd about the same Sphere. For That tis double 17 14 of 10 2T, while this is quadruple to BPM*. Therefore 24, and 26. the conical Superficies will be to the Cylindrical, as *
twice 3 to four times I ; that is, as 8 to 4, or as 3 to 2.]
PROP. XLI. Theorem.
HE Whole Superficies of an equilateral Cone cir
cumscrib'd about a Sphere, is quadruple to the whole Superficies of a Cone inscribed in the same Spkere.
By the foregoing the whole Superficies of the equila teral Cone DOF circumscrib'd, is to the Superficies of the Sphere, as 9 to 4; and the Superficies of the Sphere is the whole Superficies of the inscribed Cone SKT, as 16 to 9 (by 39. of this.) Therefore by Permutation of Equality of Proportion, the whole Superficies of the circumscribed equilateral Cone is to the whole Superficies of the equilateral inscrib'd, as 16 is to 4, or as 4 to 1. 2. F. 2)
PROP. XLII. Theorem.
Sphere hath that Proportion to B Ķ C an equilateral
Cone inscribed in it, which 32 bath to 9. Let the Sphere and Cone be cut by a Plane passing thro’ the common Axis KO, producing in the Sphere the greatest Circle OFKI, and in the Cone the equilateral Triangle BKC. Then a Plane being drawn thro' the Centre A perpendicular to OK, let the Hemisphere FGKI be cut off, in which let the greatest Cone F KI be understood to be inscribed. Now because (by Cor. 5. P.15. I. 4.) the Side B C of the equilateral Triangle cuts off Ó Pa 4th part of the Axis OK, PK will be to AK, as 3 to 2, that is, as 9 to 6. But the Base QT is to the Circle OFKI, that is, to the Base N D, as 3 to 4, that is, as 6 to 8, as appears from what was demonstrated př. 39. Wherefore seeing the Proportion of the Cone BKC to the Cone FKI is (by Schol. 2. pr. 15. 1. 12.) compounded of the Proportion of the Altitude PK to the Altitude AK (that is, of the Proportion of 9 to 6)
and of the Proportion of the Base QT to the Base N D (that is, of the Proportion of 6 to 8) the Cone BKC will be to the Cone F KI, as 9 to 8. Wherefore seeing (by 30. of this) the Sphere CG is quadruple of the Cone FKI, the equilateral Cone BKC will be to the Sphere CG, as 9 to 32. 2. E. D.
PRO P. XLIII. Theorem. N A equilateral Cone circumfcribʼd about a Sphere, Fig.-28.
is eightfold of an equilateral Cone infcribd in the Same Sphere.
Let SKT and DOF be the equilateral Cones inscrib'd and circumscrib'd, and let O KB be the common Axis. Then let as well both the Cones as the Sphere be cut by a Plane passing thro' the Axis; their Sections will be two equilateral Triangles, and the greateft Circle BPM. About the Triangle DOF likewise let there be understood to be describ'd the Circle ND OF, and let the Axis O KB be produc'd unto N. Now because the Side DF of the equilateral Triangle doth (by Corol. 5. pr. 15.1.4.) cut off NB a 4th Part of the Axis ON, it is manifest that ON is double to BK. In like manner, because the Side ST of the other equilateral Triangle cuts off B C a 4th Part of the Axis BK, NO will be to BO, as BK is to CK; and by changing, as N O is to BK, so is BO to CK. But N O is double to BK. Therefore BO is likewise double to CK. Therefore because of the Similitude of the Triangles, DOF, SKT, DF and ST also, to wit, the Diameters of the conical Bases, will (by 4. I. 6.) be in a double Proportion betwixt themselves. Wherefore seeing the Cones DOF, SKT, be like, and consequently (by 12. l. 12.) their Proportion is triplicate to the Proportion of the Diameters D F and ST, which is that of 2 to 1, the Cone DOF will be to the Cone SKT, as 8 to 1. 2. E. D.
PROP. XLIV. Theorem. Asphere hath the same Proportion both in respect of Fig. 28.
Solidity and Surface to the equilateral Cone DOF circumscrib'd about it, which 4 hath to 9.
The Sphere T P is (by 42. of this) to the equilateral Cone SK T inscribid in it, as 32 is to 9. But(by the fore-. going) SKT the equilateral Cone inscribid is to DOF the equilateral Cone circumscribed, as I is to 8, that is, 9 to 72. Therefore by equality of Proportion the Sphere TP is to DOP the equilateral Cone circumfcrib'd, as 32 is to 72, that is, as 4 to 9. But in Prop. 40. we demonstrated that the Superficies of a Sphere is to the whole Superficies of an equilateral Cone circumscribed, as 4 is to 9. Therefore a Sphere both in Solidity and Superficies is to an equilateral Cone circumscrib's about it, as 4 is to 9. 2. E. D.
That therefore which Archimedes was surpriz'd at in a Sphere and Cylinder encompassing it, we have also now demonstrated in a Sphere and an equilateral Cone encompassing it, to wit, that there is the same rational Proportion of the Solidities betwixt themselves, which there is of the Surfaces. For as he found that the Sphere is to the Cylinder about it as well in Solidity as Superficies, as 2 to 3; fo we have now taught, that the Sphere is in respect both of Solidity and Surface to an equilateral Cone encompassing it, as 4 to 9.
But from hence we shall without much labour demonstrate that the very Proportion, to wit, the fesquialteral, which Archimedes shew'd to be betwixt the Sphere and Cylinder, is continued by the equilateral Cone circumscrib'd both in the Solidity and Superficies; and so we shall put an End to the present Work.
PRO P. XLV. Theorem.
See the Fi-' gure prefix= ed to this
N equilateral Cone circumscrib'd about a Sphere,
and a right Cylinder in like manner circumscrib'd Treatise about the Same Sphere, and the Same Sphere it self,
continue the same Proportion ; to wit, the sesquialteral, as well in respect of the Solidity as of the whole Superficies.
For by 32. of this Book, the right Cylinder GK encompassing the Sphere, is to the Sphere, as well in refpect of Solidity as of the whole Superficies, as 3 is to 2, or as 6 to 4. But by the foregoing the equilateral Cone BAD circumscrib'd about the Sphere, is to the Sphere in both the said Respects, as 9 is to 4.