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 A, L O E B II A. I next try 6, and fjbftituting it for x we find —i6x-J~ refult muft be greater than the root: and thus proceed$5=36—96+5.5 =>—? 5 which refillt. being negative, I find the root to be betwixt 158 and 159 : froxh conclude that 6 is greater than the root required, which ing,you which you infer, that the leaft root of the propofe-d equatherefore is limited now between^ and 6. And fub/li- tion x1—6x+7=:o is betwixt 1.58 and 1.59, being the tuting 5, the mean between them, in place ofi x, I find hundredth part of the root of x1—6cox'+70oCio=o. Xs'—i6x+55—25—r&o+j5=o; aftd confequently 5 is If the cubic equation x3—1 jxi+63x—50=0 is prothe leaf! root* of the equation. After the fame manner pofed tq be r'efolvedy the equation of the limits will be you will difeover 11 to be the greateft root of that equa- (byp. 110. col.2. par.2.) 3v2—^ox+63=o,>or xa—-iox, tion.+21=0, whofe 3roots are 3, 7 ; and’by fuhftitutirig o for Thus by diminifning the greater, or inereafing the x, the value of x -—x —50 is negative; and by leffer limit, you._ma}r difeover the trtje root when it is a fuoftituting 3 for x, that quantity becomes pofitive. commehfurable quantity. But, by proceeding after this x=x gives it negative, and x=2 gives it pofitive, fo that manner, when you have two limits, the one greater than the root is between 1 and 2, 'and therefore incommen* the root, the other lyfier, that differ from one another furable. You may proceed as in the foregoing examples but by unit, then'you .may conclude the root is incom- to.approximate to the root. But there are other meiiietifur-able. ■ • thods by which you may do that more eafily and readiWe may'however, by continoiifg the operation in ly; which we proceed to explain. fraflions, approximate to it. As if the equation propofed When you have difeovered the value of the root to he. is x1—6-v+7=o, if we fuppofe .,v = 2, the refult is lefs than an unit (as, in this.example, you know it is a 4—-12+7=—1', which being negative, and the fuppofi- little above 1),. fuppofe the difference betwixt its real tion x—o giving a pofirive refult, it follows that the root value and the number that you have found nearly equal is between 0 and 2. Next we fuppofe x=i; whence to it, to be reprefented by /: as in this example. , Let x*—6x+7—1—6+7~+2, which being pofitive, we in- x ~ i --pf. Subftitute this value for x in the equation, fer the rodt is betwixt 1 and 2, ahd Aonfequently in- thus, 2 5 commenfurable. In order to .approximate to it, we fup-„ i+3/+3/ +^1 pofe and find x1 — 6x.+ 7=2^ — 9+ 7 — t 5 an£I —1 30/—ij/ this refult being pofitive, we infer the root muft be be+ 63*= 63+65/. twixt 2 and 14-. And therefore we try 1 + and find —50=—50 x1—6x+7-4-§—V+7=3t6— 1 o-V+7=—tV> which 3 J is negative ; fo that we conclude the root to be betwixt x — 1 sx +63 x—50-—1+36/— 12/2 +/3 =0. and ii. And therefore we try next -i-|, which gi- Now becaufe / is fuppofed lefs than unit, its powers. 3 ving alfo a negative refult, we conclude the root is be- /2,/ , may be negledied in this approximation; fo twixt 14- (or 14) and 1^. We try therefore i-nr> and that afiuming the two firft terms, we have — 1 + the refult being pofitive, we conclude that the root muft 36/—6, or fonly = ^*5-~.027 ; fo that x will be nearly be betwixt x^-tand ahd therefore is nearly i44« 1.027. Or you.niay approximate more eafily by transforming You may have a/1 carer value of x by confidering, that the equation propofed into another whofe roots fhall be feeing —1+36/—i2/2+/5=o, it follows that equal to 10, 100, or ioco times the roots of the former, by p. xo6. col. 1. par. 4. and taking the limits
 * = (by . fubftituting -yS for /)

36—i2/i-/ greater in the fame proportion. This transformation is eafy; for you are only to multiply the 2d term by nearly = -, — - ,. —- — = *02803. 10, 100, or xoooj the 3d term by their fquares, the 36 I2X ^+T5-X-r?4th by their cubes, e^c.. The equation of the laft ex- But the value ofT/ may be corredted and determiz ample is thus transformed into x —6oox+7oooo=o, ned more accurately, by fuppofing g to be the difference whofe roots are 100 times the roots of the propofed e- betwixt its real value and that which we laft found quatioa, and whofe limits are x.oo and 200. Proceed- nearly equal to it. So that /=.02803+0-. Then by this value for/ in the equation ing as before, we try 150, and find x4—6oox+7oooo= fubftituting 22500—90000+70000=:2 500, fo that 150 is lefs than /3—12/2+36/—1=0, it will ftand as follows.. the root. You next try 175, which giving a negative /3 — 0.0000220226+ 0.0023 57^ +o.oSqo^1 +^3 —12/* ——.00942816 —0.67272^-—I2gz + 36/ = 1.00908 +36^ =—0.0003261374+35.329637^—IX.9i95^1+i-3=o. Of which the firft oo two terms, negledting the reft, give .x=:i+/=i.o28o3923I27; which is very near the true root of the equation that was propofed. * 35.329637X^=o.o 326x374, andg=.^2231^13.13 = If ftil! a greater degree of exadtnefs is required, fup35-329637 7 ; and pofe h equal to the difference betwixt the true value, ofandg ■0.00000953127. So that /=o.028o3923X2