To the extent possible under law, the Text Creation Partnership has waived all copyright and related or neighboring rights to this keyboarded and encoded edition of the work described above, according to the terms of the CC0 1.0 Public Domain Dedication (http://creativecommons.org/publicdomain/zero/1.0/). This waiver does not extend to any page images or other supplementary files associated with this work, which may be protected by copyright or other license restrictions. Please go to http://www.textcreationpartnership.org/ for more information.
CIrcles(β) 1.1 are in the same Proportion to one another as the Squares of their Diameters.
Suppose a to be the Diameter of one Circle, and b of another; then by Definit. 31. Consect. 1. the Area of the one will be ¼ eaa, and that of the other ¼ ebb. But as aa to bb so is ¼ eaa to ¼ ebb by Consect. 1. Prop. 19. Q E. D.
THe same will in like manner be manifest of like Sectors Circles, while for the parts of the Periphery you put and ib, as for the wholes we put ea and eb: for thus the A•• of the one will be ¼ iaa, and of the other ¼ ibb.
CYlinders whose Altitudes are equal to the Diameters of th•• Bases, are in proportion to one another as the Cubes their Diameters; for the Cylinders will be ¼ea{powerof3} and ¼eb{powerof3}, Cubes a{powerof3} and b{powerof3}.
HEnce also (whatever the Reason of the Sphere is to the Cylinder of the same Diameter and Heighth; which will hereafter Demonstrate, and which in the mean while will denote by the name of the Reason y) I say, hence Sphe•••• which have the same Proportion to one another as these Cyli••¦ders (viz. as ¼ ea{powerof3} to ¼eb{powerof3}, so ¼ yea{powerof3} to ¼ yeb{powerof3}) will also (by C•••• sect. 1.) be in the same proportion as the Cubes, a{powerof3} to b{powerof3} is also evident from these Terms themselves.
Eucl. Prop. 2. l. 12.