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Sine LABDs, its Co-fine = p, A Bb,⋅
BDd,

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

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10 reduc'd 11432b2 + 4p2b2 xx4 (after fubfti

tuting

dd — xx | 3

for y)

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+p2b2c2+2rdc2pb—4d2s2b2'

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N: B. Part III. begins Page 417, Signature H h h

+ s2b2 d2 c2 d4s2 b2

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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.

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But

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bx

But altho'd had been lefs than

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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

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Then I fay the Sum (z) of 1+1, 2+1, 3p+1, 4p+1, &c.

continu'd to n Terms, will be=

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n

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pa

Pt.2

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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. +

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the nth Rank; alfo 1+ 2

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the n Ith Rank; alfo 1

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n

+ 3 + 4 + &c.

P D P + 2 + 3 + 4 + &c.

the n2th Rank, &c. of the faid leffer Series:

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+an, the Sum of the said

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Sum of the faid - 1th Rank will be=

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ip +

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+axn-31%, &c. Wherefore the Sum of the said leffer Se

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+ a x n − 1)2 + n − 2+2 3+ &c. to n — 1 Terms

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