Or by the Irrational Theorem, x (= 357. b d + + c c — ÷ ( ) = 57; therefore g+x = = 2. Renew the Theorem, and fuppofeg (or g the 2d) = 357, then, 8+x13 = + 45499293+382347x+1071x2+x3· 438×g+x2 + 55822662-|-312732x+ 438x2 -7825×g+x=2793525 7825x -N98508430 That is +20000+ 687254x+1509x2+x3=0 b+ CX + dx2+ƒx3=0. Then by the Rational Theorem, 2 Therefore g+x=356.9708968182 a very near. Example III. Let it be required to find one of the Affirmative Values of a in this Adfected Biquadratick Æquation; Viz. 14937 a a" — 80 a3 + 1998 a2 3 5000 = 0. That this Equation may be the eafier manag'd, fuppofe aa, then the foregoing Equation will become near And for the first Suppofition, let g=1; That is 1,55 x 2 x 2 4x3- x Then by Dr. Halley's Rational Theorem, 4 Whence 'tis manifeft, that 1.26, or 1.27 is neara, and confequently 12.6, or 12.7 near = a. Now let us fuppofe g = 12.7; 05644; therefore g+x=12.75644, which is near = a; Or by Dr. Halley's Irrational Theorem, = .05644080331, which really is lefs than the true Value fx 3 of x; but in order to Correct it, Subtract ƒ x 3 + { b x + √ b d + ÷ cs 0.0000c099117 from it, and you have .05644179448 x Corre &ted. , 4 b x + will And if you defire yet more Figures of the Root; from the x Corrected, let there be made - dx xfx .43105602423, and c + √ bd + cc - d f x' — db x2 -- d 2648.066 6987685.67496597577 -82.26 05644179448074402 = x. Wherefore g+ x = 12.75644179448074402 = a very near. The Reafon or Demonftration of the foregoing Corrections is this; viz. By what has been already faid and done, any Equation may be reduc'd the following one, viz. b + c x + dx2 +ƒx2 + b·x2 + &c. = o. And by Tranfpofition and Divifion, cc 4 dƒ 3 — dh Therefore 3 c + √ ÷ c c + b d — d f x 3 — dbx1 —&c. 3 — c + √ cc + db _ ¦ ƒ x 3 + ÷ h x + + &c. &c. Now the Value of x being very small, this laft &c. may be rejected, as being abundantly less than See them otherwife defin'd in the latter Part of Ab. de Moivre's Theorem. CHA P. II. Of LOGARITHMS. DEFINITION. Ogarithms are a Sett or Rank of artificial Numbers, accommodated to a Sett 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 of the Product of the faid two (natural) Numbers: And confequently the Remainder of the Logarithms of any two Numbers, is equal to the Logarithm of the Quotient of the faid two Numbers; as alfo twice three times, four times, &c. or one half, one third, one fourth, &c. the Logarithm of any Number is equal to the Logarithm of the Square, Cube, Biquadrat, &c. or of the Square Root, Cube-Root, BiquadratRoot, &c. of the faid Number refpectively. LEMMA. If you extract any Indefinit-Root out of each of two given Numbers, and from each of those two Roots you fubtract 1; I fay 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 Unit being from it fubducted. In Order to Demonftrate this Lemma, I will fuppofe any Indefinite Number to be = n. Either of the two given Numbers must be 1 or not. 1. If either of the given Numbers be 1, and the other (fuppofe) a, then, first thing I propos'd to prove. |
