0. XII. 
8 
[812 
812] 
ON ARCHIMEDES THEOREM FOR THE SURFACE OF A CYLINDER. 
57 
OF A 
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Lrchimedes in 
d'Archimède, 
t the surface 
i proportional 
(2a. h)} 2 ]. 
ribed in and 
ding on these 
sight and the 
em the same 
It is in the preceding propositions (by means of an axiom as to curve lines) shown 
that 
S X >S>S°, B X >B>B°- 
and it is further shown that 
S X = B X , S° = B°. 
It is moreover shown that, by taking the number of sides sufficiently large, the 
ratio B x : B°, or say the fraction B x /B° (which is greater than 1) may be made less 
than any given quantity 1 + e. 
It is then to be shown that S = B. 
If not, then 
either 
B<S. 
This being so, it is possible to make 
B x /B° < S/B, 
that is, 
S x /B° < S/B, 
or 
S x /S <B°/B, 
which is absurd, since 
S x /S > 1 ; B°/B < 1 ; 
or else 
B>S. 
This being so, it is possible to make 
B x /B° < B/S, 
that is, 
B x /S° < B/S, 
B x /B < s°/s, 
which is absurd, since 
B X /B>1; S°/S < 1 ; 
and consequently S= B, the theorem in question. 
I take the opportunity of referring to two theorems by Archimedes, Lemmas, 
Prop. Y. and vi., Peyrard, pp. 429—435, which relate to the contacts of circles. We 
have in each of them the figure which he calls the Arbelon, viz. if A, C, B are 
points in this order on the same straight line, then the figure consists of the three 
semicircles on the diameters AC, CB, and AB respectively, and the Arbelon is the 
space included between the three semi-circumferences. 
In Prop, v., we have also the common tangent at C to the two semicircles 
AC, CB; this divides the Arbelon into two mixtilinear triangles (each bounded by 
the common tangent, one of the smaller semicircles, and a portion of the larger semi 
circle), and inscribing each of these a circle, the theorem is that the two inscribed 
circles are of equal magnitude. 
In Prop, vi., the theorem is that the radii of the smaller semicircles being as 
3 : 2, then the radius of the circle inscribed in the Arbelon is to the diameter of 
the larger semicircle as 6 to 19. But it is noticed that the demonstration would 
apply to any other value of the ratio; and, in fact, if the radii of the two smaller 
circles are as a : h, then the radius of the inscribed circle is to the diameter of the 
larger semicircle as ab to a 2 + ab + b 2 , which is the general form of the theorem.