Archimedis Opera cum Apollonii Conicorum Libri IIII (1675) - Page 31

Prop. XVII.

Fig. 21.

The surface (S) of any isosceles cone, excluding the base, is equal to a circle whose radius (A) is the mean proportional between the side of the cone (L) and the radius (R) of the circular base.

a 6 of this.
b hyp.
c cor. 20. 5.
d 1. 12.
e 1. 6.
f 14. 5.
g hyp.
h 8. 5.
k 10. 5.
l 11. of this.
m 10 of this.

If you deny it, let S first be greater than circle A; and let a figure C be circumscribed about circle A, and another, I, be inscribed, such that C : I hyp.

S : circle A; then let a figure similar to C be circumscribed about the base of the cone, which is called K, and let its semi-perimeter be called P. Now, because R : A hyp.

A : L. cor. 20. 5. It will be Rq : Aq 1. 12. (that is, K : C)

R : L 1. 6.

RP : LP; therefore, since K a 6 of this = RP, it will be C = LP. g hyp. Wherefore LP : I greater than S : circle A. h 8. 5. But LP : circle A greater than LP : I. Therefore, even more so, LP : circle A greater than S : circle A. Whence LP greater than S; k 10. 5. that is, the surface of the pyramid circumscribed about the cone is less than the conical surface, contrary to what was shown before in the 2nd Corollary of Prop. 15 of this.

But secondly, let S less than circle A. a 6 of this And let C : I greater than circle A : S; and let a figure Y, similar to I, be inscribed in the base of the cone, whose semi-perimeter is called pi $\varpi$. Now Y : I d 1. 12.

Rq : Aq c cor. 20. 5.

R : L :: l 11. of this Rpi : Lpi, and Y greater than Rpi. Whence I greater than Lpi. But C : I hyp. greater than circle A : S greater than C : S. k 10. 5. And consequently S greater than I greater than Lpi. m 10 of this That is, the surface of the cone is less than the surface of the pyramid inscribed in the cone, likewise contrary to what was demonstrated in the 1st Corollary of Prop. 15 of this.

Corollaries.

1. Conical surfaces placed upon equal bases are to one another as the diameters of their bases.
2. Conical surfaces of equal height, or having equal sides, are to one another as the diameters of their bases.
3. Conical surfaces have a ratio composed of the ratios of the sides and the diameters.
4. Similar conical surfaces have the squared ratio of their sides or diameters.
5. The sides and diameters of equal conical surfaces are reciprocally proportional, and conversely.