VOL. XVlIl.] PHILOSOPHICAL TRANSACTIONS. §(1^ 



parallel to the plane ACBD, and these lines will pass through the points S and 

 Q, and being produced as far as the cylindrical surface circumscribing the hemi- 

 sphere, will cut off from the sides of the cylinder right lines, which will like- 

 wise be equal to HS or HA, and they will contain arcs equal and corresponding" 

 to the arcs MN and VP. Now if another plane, parallel and very near to this, 

 be conceived in like manner to be drawn, it appears, by what is above shown, 

 that these two will design a portion of a ring in the cylindrical surface, equal to 

 the portion between the same planes, which is taken away from the hemispheri- 

 cal surface by its perforation. Now if the same construction be supposed to be 

 made at every point in the periphery AHE, all the portions in the cylindrical 

 surface circumscribing the hemisphere, drawn and designed as before, will be 

 equal to the spherical surface taken away by the perforation. Therefore the 

 remaining hemispherical surface will be equal to the remaining cylindrical sur- 

 face, composed of all the right lines HA, erected at the respective points 

 M, N, V, and P, or to the figure of the sines of the semiperipheries ACB,- 

 ADB, that is, by what has been long known to geometricians, to 4 times the 

 square of the radius AE, or to the square of the diameter AB. And since the 

 two whole figures contained by the common section of the said perforating 

 cylindrical surface with the spherical surface, are equal, the remaining hemi- 

 spherical surface, after taking away the 4 bilinear spaces as in the construction, 

 is equal to the square of the diameter A B. QED. 



If the semiperiphery AHE be so inflected, that it may coincide with the 

 equal quadrantal periphery ARC ; the point H will fall on the point M, be- 

 cause of the equal arcs AH, AM; and HS the altitude at H, of the cylindri- 

 cal surface insisting on AHE, will coincide with its equal HA, the altitude at 

 M of the figure of sines erected on AMC; and the same thing holds good in 

 all other points: hence the curve which is the common intersection of the 

 spherical surface with the cylindrical surface on the base AHE, although not 

 inflected in the same plane, in the manner before said, yet will coincide with 

 and therefore be equal to the curve terminating the figure of the sines ; that is, 

 to the common section of the cylindric surface erected on the quadrantal arc\ 

 ARC with the plane cutting the plane of the base erected on AB at half a right 

 angle; or to a quadrant of the elliptic curve, whose less axis is AB, and its 

 greater axis double in power of the same. Therefore the perimeter of the 

 quadrable Florentine sail, consisting of 4 such arches, is equal to the perimeter 

 of the said ellipse. 



It may also be added, that the surface of two perforating cylinders in the 

 sphere, are equal to the spherical surface remaining after the perforation, or to 



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