A treatise of algebra, in three parts. Containing

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tahen istiirmatimet to thei sum illi thi ei propersigm, the willisive dotibi the sum of the cubes of the assirmative oots. An is ou subtrae the secon .sum mira the issi, there will re in douiae the sum os the cubes of the negative roots. Whencerit folio rus, that hal the sum os the mea proportiona belwix thesum of the square and the sum os the biquadrates, and of the sum os the cubes of the more it thei proper os excred the simo the cubes of the amrmative opis and hal thei difference exceed the sum os thecubes of the negative oots.' An by extracting the cube rom of that sum an differetice, ou ill obtain limit that mali Meeditu ciunt of the assirmative and of the negative iovis. An sipo it a Ur, rom ha hasbeen

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been atready explainta, to diministi the ootao an quation O that the allina hecome negative ut one, it appears o by this means yo ma approximit Ver no to that motia But this Ges no serve when there are impossibie

severat other Rules like these might beii vensor limiting the mota os equations. e mallgive one not mentioned by other Authors.

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L hecia term os an equation is the productis it mota stom hicli it talows,

that the oots of an equation, hen commensu- rabie quantities, ill e Mund among the di

ubere in rust ternari eta'. Whos simple literat

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rat divisors areis, sta 2b, ach of whic maybestae either positivel or negativel : ut ashere,e fuid there are variations os sens in theequation, me nee oni tari them positively. Suppose, the firmo the divisors, and substitutinia sis, the equationiecomes

so that the whole vanishing, it sellam that ais one of the mois os the equation. Aster the same manner, is yo substitute Linplace of the equation is whicli vanisti istews , t be another mot sine equation. Again, Dyo substitute i. sor Milfin Lal the term destro one another o as Omine ille sum est. For it Will theni Whence we in stat et is the third mo os the equation hicli, after the firlt two by had been Bund might have been collected sto this that the last terna Ming the produci: of the three motu insono .

a the

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The quotient stat ove a quadrati equison Ἀ- 3 o whicli must be the product of the othe two simple equations rom, hicli thecubi is generared, and whose oota therelare morbe two of the risis of that cubici No the more os that quadratic equation areeasit flaundi Chap. 3. Par I tote, Land

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and findingahat s one of the divisors of o, is ne of the oore, o divide stat ubi byx , and the quotient oves me quadratic

ys . his Rule suppost that ou can indan the diviser of the last terna hicli ou V

b italos diviser, prauedint thus illi Ioviavea quotieni ibat is mi divisebis number greater ita iunii ' his quotient, illi thesedivisors, are thera si or simple divisors of the quantity And the producis of the multiplication o an et, a, , of them are therem und divisors. As, o finit the divisors os 6o frst I divideri et, and the quotient ro again by et the the

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nex quotient Lisci anxii quotient of thia division scis no sarther divistin is an integer a veilut so that Theimsile divisere Me a, a, a, Thesprodum o tWo, IO, IS.

The producta of three, . . . . . Ia, O, 3

Thesproducti os ali lauti The divisors of so are found aster the lame

Simple divisors, a 3 S- Thesprodest os Wo, . . . . , s , o 1 The products of three, . . . . 18, 3o, S-

Thesproduc O ali Our ........ O.

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greatest negative coessicientis unit is to beneglectes.' Thus in resolving the quation: rix' - 2s a6x o et mi as scis the notest negative coemient, e conclude stat the diviser of Izo that excee et may bebe neglected.

vili give in them allis positive result so that numhe will e greater than the greate root, an est the divisors sciet that exceta it imay be neglectes. Τhat this investigation a b easteri eought to egi alWays With ahat expression, Where the negatio mota seem to prevat most; as here in me quadrati expression---- as; where finding that 6 substitute foris gives that expression positive, and ives ali the ther expression a the iam time positive, I conclude that 6 is greater than anyin the moth and that

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ing the greates negative te . ut is omwe require What number substitute seris,aigive ali these expression positive:

sitive result, hoc ali the divisors os et stat exceta are iste rejectei and thus is a se.

trials we find that in a is the positive mot fine equation. The divid in the quatio hyx - 2, and resolving the quadrati equatio thatis the quotient os the division, mussita the othermo more tote - , and S s. ut there is another method that reduces the divisbrs of the las term stat canteusifill, stilicio more narro limita. Supposse the cubic equation, ix rm oris p oposed to e resolved Transform itto an quation hole oois hal be est inanthe values os, b unity, assumitigari: - . And the la term os the transsernaeis quationwillae I - - - - ru hich is oundi substitutin unit, the disserence os, and seris, in