A treatise of algebra, in three parts. Containing

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the equation, and f. i, then there Willieno variation of the signs in the quatio of): si it appeam rom . the la articie, that is allthe three p r were eques in one another, and e qua to any one of them increased by Unit, a to the at the term of the

he positive, the negative pari, hic in Volves pandis, einiciminillaed, hile the positive partand the negative involvini remain a besere. q. Aster the fame manne it is demonstrates, that is, is the reatest negative messicient in theequation, and e is suppost α' - , then allthe term of the equationi is di illis positive and consequently mill e greater than any of the values of What we have sud of the cubi equation px' - α o is easit applicatae in

o generat, e conclude stat in greates' negative coefficient in an equation increata Munit, is alway a limit that exceed est the roots

of that equation. 'But it is o be observes at the fame time, that the reare negative coemient increased by unit, is ver seldo the Mars limit statis est discovered by the Rule in the asth

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.hicli is test than the least, and λ ound accordinito 3 39. Whicli surpasses the notest moesos

Αnd sor thisi pose, Wil suppost a tot theleast mosi balte secondiso the inita, and soon it Ming arbitrary. 45. Is yo substitute o in place of the un-knoWn quantity, Puttinixi O, in quantitythat,illistis stom that sipposition is the linter of the equation, ait ille othere, stat involvex vanishing. Is yo substitute ser a quantitycles than theleast roo the quantit resulting iit avetheriame signtas the la term that is, ill epositive or negative accordinias the equatio is

he negative, and thei productisu be positive

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Is you substitute foris a quantit greater hanthe least ut les than ali the thermota, then the sim os in quantit resul ing

illae contrar tu ha it was helore 'ecauseone actor a comes no positive, allthe other remaining negative ac sere. Is you substitute sor, a quantit greater inanthe two lea mota, ut es than es the rest, both the factor x - a W- b, hecome positive, and the res remai as the were. So that the ole product will have the iam signis thena ter of the quation Thus successivelyplacin instea os, quantities that are limitshet ix the Oot of the quation the quantities that result illiave alternatet the agns and . And, eonvreso Dyo fini quantiaties Whicli substituted in place of x in the pro- post equation do give alternates positive and negative resulta, thos quantities are the limita of that equation. It is sem to observe, that in generes, when, by substitutinian tW number sor in an equation the resulis have contra signs, one or more of the rom os the equationmust e betae ix thos numbers. V Thus, in the equation '-- - smo, is o substitute et an a sor the resulta are 4 ;whenc it sellows that the oota re betwix et anu

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and a sor,hen these resulta have disserent signs, one or ther of the factors hic produce theequations raust have changed iis sigi suppos itis, m e then it is plain that e multra betwixtine numbers supposed equa to, 6 Let the cubic equation pin He x

where the last term ae' rape is stomahenature os equations produced of the remaining values of I, or of the excesses of two other valuesos, above Matris suppose eques to e since

i'. Is eie equa to the least value of x, thentho mo excesses eing both positive, theywillii is positive produci, an consequentlyae ope 's il lae, in his case, positive.

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so that is roti nil the limit stat excreta thea greatest

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have demoliti rate of cuhic equations is eastly extende to allisther, 1 that e conclude,

that the last term but one of the trans medequation is the quatio se determining the limita of the proposed equation.' Or, that theequation arising by multiplying each term by the inde of the uni nown quantit incit, is the equa

6 48. For the fame reason, it is plain that the oot of the simple quatio ae i. e. phis the limit bet en the two mors of

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the quadrati in P -- 3pe O. So that we have a complete series of thesesequation arisingstom a simple equation to the propos , eac os whic determines the limita of the mowiis

equation.

6 49. I two more in the proposed quationare equat, the the limit that ought tot he twix them must, in his case, ecome equa toone os the qua root themselves.' Whichperfecti agrees,ith What was demonstrate in the last chapter concerning the Rule se finding the equa roots of quations. And the fame quatio that ives the Iimithiivire alio one of the qua mors, WhentW o more are equat, it appears that is you substitute a limit in place of the unknown quantit in an quation, and instead os a positive o negative eluit, it be ound α , thenyo ma conclude, stat not ni in limit itiet is a root of the equation, but that there arem mota in stat equation eques locitanda one

are in limits of the oot of the quationax - et o, the three mota of the cubicequation, hicli suppo to e b, c, substi

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o M an equation stat is deduce sto it bynlultiplyin iis ternas by an arithmetica progression a b a 2b, a 3b, ιν 4b, C. Andconverso the root of this ne equation iube limita of the proposed equation,

P0sed equata ea is an impossibie expression beseun in hos excesses, then there,in os consequencet mund impossibi expressium in theset Wo value, of x. Anu

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An rom his observation rules may hededuces se discovering hen there are impossitae mota in quations ' Os,hich, malloeat astematas. si Besides the method atready explained, there are iners by Which limit ma be determines, hicli the rom, an quatio cannot

Since the quare os est res quantities areassirmative, it sollows that the sum foue δε res of the risu os an equatis must be

equation.

powers of the oot of the quation, an extrae the biquadrati mot of that sum, it illias exceed ille greatest motis the equation. 6set. Dyo findis mea proportiona be- tween the sum of the quare o any two ootri