That this Equation may be the eafier managed, fuppofe aa; then the foregoing Equation will become nearly 8 a320 a2 equal to a+ 15a+.5=0. a= And for the firft Suppofition, let g --x be = a, and 3=1; then a4 8a3 +20a2 - 15 a +-.5 3 =+ 1+ 4x+6xx+4x3-004 2400 24xx-8x3 :+201-40x + 20xx 15- 15x 汁.5 -1.5-|- 5x-|- 2xx. ·4x3-- x04 That is cx + dxx +ƒx3 + bx+ =0 b + Whence 'tis manifeft, that 1.26 or 1.27 is nearly a; and confequently 12.6 or 12.7 nearly =a. Now let us fuppofe g+xa, and g 12.7; then ·at + 26014.4641 + 8193·532x+7 967.74x2+50.8 x3-|-x+ -80a3 163870.64 298.6559 = 0564407; therefore 24567-434334 5296.132 5296.132 g+x=12.75644, which is nearly a. Or, by Dr. Halley's Irrational Theorem, =.05644080331... which really is lefs than the true Value of x: But, in order to correct it, fubftract 0026201... 2643.423 1⁄2 fx 3 + bx + √: bd + 4 cc: 00000099117... from it, and you will have .05644179448-x corrected. 00 : And, if you defire yet more Figures of the Root; from the corrected let there be made dx: fx 3 -\- 1x+= -.43105602423...; and√ld+cc-dƒx3-dbox + : d -2648.066 + √ 6987685.67496597577 82.26 .05644179448074402...x; wherefore g+x= 12.75644179448074402 a nearly. The Reafon or Demonftration of the foregoing Corrections is this; Viz. By what hath been already faid and done, any Equation may be reduc'd equal to the following one, viz. b c x + d x2 - \ - ƒ x3 + bx++&c.=0% + And, by Sir If. Newton's Theorem, √: jcc \-bd-dfx3-dbx+-&c: is√4cc+bd: i d ƒ x 3 \ - ž d b x + +&c., &c; therefore 4 ± √: 4 cc \d b : Now, the Value of x being very small, this laft &c. may be rejected, as being abundantly lefs (fecluding the Sign of the Value of x) thanƒx3 + b x+ + &c. ± √ : 4 ec \ - bd: 5 -- and so you have the firft Correction. CHA P. III. Of Logarithms. DEFINITION. Ogarithms are a Series or Set of artificial Numbers accommodated to an other of natural Numbers, in fuch fort that the Sum of the Logarithms (or artificial Numbers) of any two (natural) Numbers is equal to the Logarithm of the Product of the two (Natural) Numbers; and confequently the Remainder of the Logarithms of any two Numbers is equal to the Logarithm of the Quotient of thefe two Numbers; as alfo two, three, four, &c. times, and one half, one third, one fourth, &c, The Logarithm of any Number is equal to the Logarithm of the Square, Cube, Biquadrate, &c. and of the Square Root, Cube Root, Biquadrate Root, &c. of that Number refpectively: Or, as the learned Dr. Halley defines them, Logarithms are the Indexes of the Ratios of Numbers one to an other. LEM LEMMA. If any Indefinite Root be extracted out of each of any two given Numbers, and from each of these two Roots an Unite be fubftracted; I say the Sum of the two indefinitely little Remainders is equal to the indefinitely little Remainder of the faid Indefinite Root of the product of the faid two given Numbers, an Unite being from it fubducted. In order to demonftrate this Lemma, I will fuppofe any indefinite Number = n; as alfo one of the two given Numbers (which I fuppofe are Affirmative) to be fome Binomial or Refidual whofe firft Member is 1, as (fuppose) Ix; that is to fay, if that given Number be 1, then it is = 1+x, but if it be 1, then it is =1x: And let the other given Number be some (finite) Power of 1 ➡x, as (suppose)". Then tho' the Value of Ix were determined, that of 1x will notwithstanding be any Number you please, x being not o, and m being indetermin'd: Thus tho' 1+x were any given Number that is greater than 1, 'tis manifeft 1-xm fhall be Affirmative Number whatsoever, if m that is to say - any be La : Or-x is a, if m be= NB. L stands for Logarithm. Now what we are to demonftrate is, that La LI-X ןןן a, is indefinitely small, and con fequently being added to or taken from any finite Quan 72 tity or Number will not fenfibly, that is, will only by way of increafe or diminish it; wherefore Hence if I be fubftracted from the indefinite-Root of any Number, the Remainder may be taken for the Logarithm of that Number had it not been indefinitely little: But if it be multiplied by 10000 &c. indefinitely (which 10000 &c. indefinitely, in the following Operations, is fup pofed |