A treatise of algebra, in three parts. Containing

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that arises by substitutim a the differe ebet x x and soris, in the pro sed inu

Νoin the values of Wares me of the divisorsos , hic is the term lest hen o sim sex mi and the values of the fcare ibine of the

rical progressio increasing by the common disiseretice unit, ecause x - , are in that progression Andrit is obvious the semereasening may be extende to any equatio os haleve degree. O that this gives a generalmethod for the resolutionis equations hostmota are commensuratae.

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mbos common disserence is unit; and the values

when the arithmetica progressio increases, burnegative When it decreases.

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s o i l

in mo

ia o

cra d

Os theses ur arithmetical progression having their common distaence equa torimit, thesilast sives

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ma discove it root asterWard hy the resolutionis these quadrati equations ori fisei ther these simple quation nor thel quadratic

equations a be Bund, et is findire aiine or Aquadrati stat is a diviser of the proposed

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present the quotient arising by dividim the pri posed equatio by that divisor, then

equation.

9 63. Not me arcto observe, stat, supposing the quation has a simple diviser, et is, is, e substitute in me quation E , in place os, an quantity, a the me quantity that in result rom his substitution ill

Is, substitute successivel si x an arithmetical progression, 'si c. thequantities that, ill result rom these substitutions Will have among their divisors

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ma -- - - 16 whicli are ali in arithmetical progression, having their common disserence equa to me. Is sor example, M substitute oris the terms of this progression et, o, - , he quantities that result have among thei divi r the arith

Where the difference of the term is, and thetem elongin to the supposition o x iis n.

6 64. It is manifest thereiore that when an equation has an simple diviser, Dyou substitute soro the progression , o, a there millhes und among the divisors of the sum that

resuli fio these substitutionis, ne arithmeticalprogressionis least, hos common difference

Willie unico a divisor m of the coeffcient os the hi est term and whicli milite the coefficient of . in the simple diviso required i and whos term Mising roin the suppositio os mi, willi ththiother member of the simple From,hicli this Rule is deduce fur disco-vering suc a simple divisor, When there is

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RULE. Sabiniste foris in the pro eis quam suo

m dira, and the divisor uill eis is, and the, rute, ill coincidi illi that sive in the en of the last hapter, hich, demonstrates aster a. different maniter the diviser eing m,tbe value o x ill e is, the ea in f the

progressio that is a divisor of the sum that arist stom supposingis of this case we

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gave example in the last chapter and thoughit is eas to reduce an quation hole ighest term has a coessicient different Dominit, to ne

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. among thei divisera, respectively. Thele term are notiso' as in thecia sese, in arithnietica progression , ut is ou subtractinem stom ille 'vares of the te s. a is, Hae a tae, c. muhiplied by m a divisor of the ighest term of the proposed equation, that is stom

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.subtractedi ' respectivesn

dissere est ' η and Whos ter arisini mine sissistitution os o sor His κει- hic it sollows that by this omniation is the proposed quatio has a quadraticdivisor, o Will in an arithmeticu progres