Cafe 2. If the Value of g be taken lefs than that of a; then, I say the Value of the 24g will be greater than that of N-gr But, fince the Value of g (which is fuppos'd to be affirmative) is less than that of a, the true Value of x will be affirmative; confequently (n being suppos'd not 2) a, Cafe 3. If the Value of any g be greater than that of a ; I fay the Value of the next following g will ftill be the foregoing g; for but But, fince g isa, the true Value of x is Negative; let us therefore defign it here by a negative Symbol, viz −z; then it is evident that g gn 2 gn-3x3, &c. is gang 12: Whence 22x I n gn 3 N g.n ng" (= following 8) is 8+ gn &c. (a): That is 2 to fay, if the Value of the foregoing g bea, the Value of the following g will be alfoa. lowing g) isg: That is to fay each following g is lefs inValue than the foregoing g: But (by what has been abovefaid) ftill greater than the Value of a: Whence (fince the 2d Cafe is reduced to the 3d, as it is by one only Renewal) the Approximations in the 2d and 34 Cafes are demonftrated. i.e. The Value of x From the foregoing Demonftration it is obvious that asx, fecluding its Sign, is fmall in refpect of g, the Series must converge accord produces an Affirmative; and this multiplied by the next preceeding Term, which is Negative will produce a Negative. ingly swift; for the Difference between g and a being only the faid, that of the next g and a will be only. 22 I 2 8-xx+&c. which, if the faid x be but very fmall in refpect of g, is but very little more than 72 I 2 g 30 30. Now this Quantity must be much less than the faid x if the faid * be much less than g. This Series converges fo fwift, not only in the foregoing fimple Equations; but alfo in all adfected Equations, when you have got fo many of the Figures of the Root you are about extracting, as to free it from the Intanglements and Incumbrances of the other Roots of the Equation, as intirely or almoft to double the true Figures in the affumed g by each Renewal. In the first Edition of this Book I have demonftrated that g being affumed greater than the greatest Value of the Root a in any Equation, the faid g would converge to that greatest Value of a provided it was not imaginary. But, because of that Provifo, I will not infert here what I have already done upon this Subject, but leave it to be im proved by fuch as pleafe to do it. That Defect may be remedied in feveral Cafes by particular Methods; as by * changing the Negative Roots into Affirmatives, and the affirmative Roots into Negatives; or by fuppofing a Fraction whofe Numerator is 1, as the Root a, &c. But all thofe Methods do not extend the Demonftration to Univerfality; and withal the Root fought may not be the Greatest, nor any of them that may be reduc'd to fuch by the abovementioned Methods: Wherefore I y The fureft Way of extracting the Roots of adfected Equations is to begin the Operation with or by the numeral Exegefis, and fo go on with it till the Divifor takes Place, and then purfue it by this Method, or by any of Dr. Halley's Theorems, either of which will ferve to find as many Figures of the Root you feek (it being, I prefume, not imaginary) as are requifite. * See p. 113. and 114. Note, Note, It will, for the moft Part, be fufficient to find the firft Member of the Root you seek, for to apply this Theorem thereto. I think it needlefs to infert here the Univerfal Theorem from which Mr. Raphson's 2 Set of Theorems are deducible; and therefore will only deduce from the foregoing Univerfal Theorem fome particular ones, by fome of which I fhall extract fome of the Roots of fuch Equations as I fhall chufe for Examples. Example I. If aa2; what is a equal to? N-gg 2 Suppose g=1, then (N=88) = = = 54 2 g Therefore g-x=1+.5=1.5=248. 2 2.834 -.002783 th Therefore 1.417.0027831.44217=48: Then x (or 4th x)=2—2.000009723089 2.828434 ·.000003437622.....; Therefore 1. 414217 1. 000003437622 2 1.414213562378g the 5th a fought nearly. Answer. 13824; 'tis required to find one of the affirmative Values of a. First I fuppofe a=10; then a4a3 (10000-4000) 6000; but 600013824; therefore a 10. Again I fuppofe a=20; then at 4a3 (= 160000— 32000) 128000; but 128000 13824; therefore a 20; but 10, and nearer to 10 than to 20: Therefore fuppofe g=10 N = 3 + + / 82) — 13824—10000+4000 48-3188 2800 =2...; therefore g+x=10-1-2 = 12=8 the 2d a fought. 3 Dr. Edmund Halley found two other Univerfal Theorems for Evolving Equations, the one Rational, and the other Irrational, which are better than Mr. Raphjon's; for every Renewal of each of Dr. Halley's Theorems Trebles the true Figures in the Value of g. The Method of finding which two Theorems is as follows. Let |