Sine LABDs, its Co-fine = p, A Bb,⋅ BC=" FDx fought. Co-finep, AB 3b, Given. Produce BC to F, and from the Points D and A let fall DF and AG both Perpendiculars to BF; and suppose BF =y, then yy Jy= dd 10 reduc'd 11432b2 + 4p2b2 xx4 (after fubfti tuting dd — xx | 3 for y) +4pb2sc - 4sb2pc -4sbrdc xx3 +r2d2c2 — 4p2b2d2 + s2b2c2) Xx2 jube N: B. Part III. begins Page 417, Signature H h h + s2b2 d2 c2 d4s2 b2 If by adding a Quantity, be it ever fo little, that is, a Quantity indefinitely little, to either of two given Quantities, the Sum thereby becomes greater than the other given Quantity; I fay, the faid two given Quantities must be equal. Expofition. If b and c be the two given Quantities finite, and a an indefinitely little Quantity: And if bac, and cab; I fay that b Demonftration. For if you fuppofe that either of them as b could be the other c; it must be by fome Quantity, which fuppofe d, then b would be + d; And the Confequence would be that da. Ꮋ Ꮒ Ꮒ But bx But altho'd had been lefs than I 999999 &c. till you have Octillions of 9's writ one after another Or less than bx, the Square of that Number; or less than bx by its Cube, &c. Yet b x by fome Power of the said Number muft bed (by Poft. Lib. 10. Eucl. El.) which Power x b let a (for it is evident that it may) be equal to, or less than; then a will bed, which is contrary to the foregoing Confequence; and therefore by fuppofing that b ise, a Contradiction enfues, which proves that b is not c; and for the fame Reasons c is notb, confequently b isc. Q. D. E. That part of Exhauftions which treats of Quadratures is likewife founded upon the following Lemma; that is, upon its Scholia and Coroll. Lemma II. If n be an indefinite Number, and p = fuch a Number as that the Sum (s) of 1o, 2o, 3o, 4", &c. continued to Then I fay the Sum (z) of 1+1, 2+1, 3p+1, 4p+1, &c. continu'd to n Terms, will be= n pa Pt.2 Again, in order to find the Sum of the leffer Series; confider, that fince o is the firft Rank thereof; 1 the 2d, 1P + 2 the 3d, &c. that 1+ 2+ 3+ 4+ &c. + the nth Rank; alfo 1+ 2 7 the n Ith Rank; alfo 1 n + 3 + 4 + &c. P D P + 2 + 3 + 4 + &c. the n2th Rank, &c. of the faid leffer Series: -- +an, the Sum of the said Sum of the faid - 1th Rank will be= ip + +axn-31%, &c. Wherefore the Sum of the said leffer Se + a x n − 1)2 + n − 2+2 3+ &c. to n — 1 Terms |