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 E B R A. 117 A L G by the fird, and the third by the fecomd, and place and that we bare already found, and proceeding as a- the quotients and -f, over the middle terms in this bove you may correct the value of g. It is not only one root of an equation that can be ob- manner, tained by this method, but, by making ufe of the other T T ?=o limits, you may difcover the other roots in the fame manner. The equation of p. 116. col. 2. par. 1. x3~ + — + + —50=0, has for its limits o, 3, 7, 50. We have already found the leaf! root to be nearly 1.02803<5. Then becaufe the fquare of the fecond term multiplied If it is required to find the middle root, you proceed in into the.fradlion that (lands over it, that is, Xp2x ',■ is the fame manner to determine its neared limits to be lefs than 3j»1*4 the redtangle under the firft and third the fecond term the fign —: but as 6 and 7; for 6 fubftituted for x gives a pofitive, and 7 terms, I place under (=:3p4x>) the fquare of the third term multia negative refult. Therefore you may fuppofe and by fubdituting 1 this value for x in that equation, plied into its fradtion is greater thanz nothing, and conyou find p+zf —9/+4=o, fo that /=£ nearly. Or, fequently much greater than —pqx, the negative proof the adjoining terms, I write under the third term fince f— 9—3/^1-/1 , it is (by fubftituting £ for f) dudt the fign +. I write + likewife under x3 and —q the fird and lad terms ; and finding in the figns, thus markf— iwhence nearly. Which two changes, one from + to —, and another from It be corroded as in the preceding articles. ed, — to I conclude the equation has two impoffible value’9 mayT dill After the fame manner you may approximate to the value roots. When two or more terms are. wanting in the equation, of the highed root of the equation. under the fird of fuch terms place the fign —, under the H “ In all thefe operations, you will approximate fooner fecond the third —,’and fo on alternately; onto the value of the root, if you take the three lad ly when+,theunder two terms to the right and left of the defi“ terms of the equation, and extraft: the root of the cient terms have figns, you are always to write “ quadratic equation confiding of thefe three terms.” the fign under contrary the lad deficient term. Then, in p. 116. col. 2. par. 2. indead of the two lad terms of the equation p — i2/'*+36y"—1=0, if As in the equations you take the three lad, and extract the root of the xs+ax* * * * +a*=o quadrartic 12/^—36/H-i=o» you will find y=.028o3i, +s + — + — +s which is much nearer the true value than what you difand x +ax* * * * —a =zo cover by fuppofing 36/^—1=0. + +--++ + It is obvious that this method extends to all equations. “ By affuming equations affedted with general coeffi- the fird of which has four impofilhle roots, and the 0“ cients, you may, by this method, deduce general ther t'M.'o. “ rules or theorems for approximating to the roots of Hence too we may difcover if the imaginary roots lie “ propofed equations of whatever degree.” hid among the affirmative, or among the negative roots. For the figns of the terms which dand over the figns below that change from -j- to —, and — to +, (hew, by Chap. XXII. Of the Rules fer finding the the number of their variations, how many of the impofiiNumber of Impojfible Roots in an Equation. ble roots are to be reckoned affirmative; and that there are as many negative imaginary rdots as there are repetifame fign. The number of impojjible roots in an equation may, tionsAsofin the theJ equation for mod part, be found by this x —4x4+4*s—2X*—5.*—4=0 Rule. Write down a feries of fractions whofe deno+ + — + + +4 3 minators are the numbers in this progreflion, 1, 2, 3, 4, 5. <bc. continued to the number which expref- the figns ( 1 ) of the terms —4X + 4X —2x* fes the dimenfion o«f the equation. Divide every which over the figns -| )- pointing out two affirfraftion in the feries by that which precedes it, and mative dand we infer that two impoflible roots lie aplace the quotients in order over the middle terms of mong theroots,affirmative
 * and the three changes of the

the equation. And, if the fquare of any term mul- figns in the equation (-j—*— three tiplied into the fraction that (lands over it gives a affirmative roots and two negative, the—)fivegiving roots will product greater than the redtangle of the two adjacent be one real affirmative, two negative, and two imagi. terms, write under the term the fign +, but if that nary affirmatives. If the equation had been, product fs not greater than the redtangle, write —; x5—4X4—4X3—2X1—5X—4=0 and the- figns under the extreme terms being +, there will be as many imaginary roots + + — - -b + an as there are changes of the figns from + to —> d from3 — tox +• the terms—4X4—4X3 that dand over the frd vacaThus, the given equation being x +/>x + 3/dx — tion -1 , (hew by the repetition of the fign —, that q — o, divide the fecond fraction of the feries -f, one imaginary root is to be reckoned negative, and the. Gg te ms Vol. I. No. 5. 3