Euclid's Elements. Vol. II.; Containing the seventh, eighth, ninth, tenth, thirteenth, fourteenth and fifteenth books with the data being the ... Keil. Now first translated from Dr. Gregory

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1745 Excerpt: ...EF, making the Breadth EL. Whence by 23. 10.EL is rational, and Incommenfurable in Length to E F. Again, becaufe the Rectangle contain'd under AC, BC is medial; twice the Rectangle contain'd under AC, CB will be medial. But it is equal to the Parallelogram G H: Wherefore G H too is medial, and is apply'd to the rational Line EF, making the Breadth HL. Wherefore HL is rational, and Incommenfurable to £F in Length. And becaufe from Hyp. AC,CB are Commenfurable in Power only, A C mail be Incommenfurable to CB in Length. But as A C to CB, fo by i. 6. is the Square of AC to the Rectangle under AC, C B: Therefore the Square of A C is Incommenfurable to the Rectangle contain'd under A C, C B, But by 16. i o. the Squares of AC, CB are Commenfurable to the Square of AC; and twice the Rectangle under AC, CB is Commenfurable to that under AC, CB: Therefore the Squares ©f A C, C B are Incommenfurable to twice the Rectangle contain'd under AC, C B. And by Comjlr. the Parallelogram E G is equal to the Squares of AC, C B; and H G equal to twice the Rectangle under A C, C B: Therefore E G is Incommenfurable to H G. But as E G is to H G, fo by i. 6. is the right Line EL to HL: Wherefore EL 'is Incommenfurable toHL in Length. And they are both Rationals. Wherefore EL, HL are Rationals Commenfurable only in Power; and fo by 74. to.J EH is an ApoTome, and H L congruous thereto. After the fame Manfter we demonftrate that HM alfo is congruous to it: Therefore more than one right Line, being Commenfurable only in Power to the whole, is congruous to an Apo tome, which by 80. io. is impoflible. Therefore to a fecond medial Apotome there is congruous but one medial right Line, Commenfurable only in Power to the whole, and containing a medial Rectangle with the whole. W. W....