Archimedis Opera cum Apollonii Conicorum Libri IIII

Prop. XVI.

Fig. 19.
20.

The entire surface (S) of a right cylinder, excluding the base, is equal to a circle whose radius (A) is the mean proportional between the side of the cylinder (L) and the diameter of the base (2 R).

a 6 of this.
b hyp.
c 15. 5.
d cor. 20. 6.
e 1. 12.
f 1. 6.
g
h 14. 5.
k const.
l 8. 5.
m 10. 5.
n before

If you deny it, let S first be greater than circle A; and let a figure (called C) be circumscribed about circle A, and another similar one (called I) be inscribed, such that C less than I less than S. less than circle A. Then, let a figure similar to C be conceived as circumscribed about the base of the cylinder, which is named K; and let its perimeter be called P. Now, because 2 R : A hyp.

) 2 A : 2 L; or by halving the antecedents, R : A R : 2 L 1. 12. RP/2 : LP; therefore, since K = RP/2 14. 5., it will be C = LP. const. Therefore LP : I less than S. less than circle A. Yet LP greater than circle A. 8. 5. greater than LP : I. Therefore, more so, LP : circle A less than S. circle A. 10. 5. Whence LP less than S; that is, the surface of the inscribed prism is less than the surface of the cylinder, contrary to the 4th Corollary of the preceding. Therefore S is not greater than circle A.

If it is said that S less than circle A, a 6 of this let C : I less than circle A : S; and let a figure similar to I be conceived as inscribed in the base of the cylinder, which is called Y, and its perimeter P. Then, because Y : I

RP/2 : LP, and Y less than RP/2, h 8. 5. it will be I less than LP. But C : I b hyp. less than circle A : S; l 8. 5. greater than C : S. m 10. 5. And so S greater than I. Therefore, even more so, S greater than LP; that is, the surface of the cylinder is less than the surface of the inscribed prism, contrary to the 3rd Corollary of the preceding. Therefore S is not less than A. It remains, therefore, that S = circle A. Q. E. D.

Corollaries.

1. Cylindrical surfaces established upon equal bases are to one another as their sides, or heights.
2. Cylindrical surfaces of equal height are to one another as the diameters of their bases.
3. Cylindrical surfaces have a ratio composed of the ratios of the sides and the diameters.
4. Similar cylindrical surfaces have the squared ratio of their sides or diameters.
5. The sides and diameters of equal cylindrical surfaces are reciprocally proportional; and conversely, if these are reciprocally proportional, those surfaces are equal.