A treatise of algebra, in three parts. Containing

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anotheris herar term is alWays bine purei power os, an is positive the seconcis apower sis multiplied by the quantities

-- α - , c. An since these are ali negative, that term must heresere e negative.

ducta made is multiplyingon three of these

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negative.' The Rule is generat, i the impossitile root be allorie to e cister positive or

negative.

, es, In quadrati equations the in moesare either both positive, as in his x cx - ναὶ ἡ . ac ab dio, inere there Me two changes of the rim orsi lares in negative, as in this ere there is no any change os the sens ortitere is one positive and one negative, as in

ere there is necessariis e change of the signfihecause the fies term is positive, and the last negative, and there canielut ne change, M ther the seconeste Or . Theresere the mle give in the isti secti in extenta in ali quadratic equations. 9 2I. In cubi equations the r is may be, i init positive, as in his x-a κος- ιλ -cmi, in Whichahe signflare alternates Handas apperes rom the Tatae. Hiiere are threechanges of the signf. at The mois may besali negative. a in assic

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negative. An there cannot possita be threechanges of the signf. the firmand la te shaving the fame sign. q. There may besone positive oo and wes negative, as in the equation, T κ

signs, since the fir term is positive and thelas negative. And there canielut one change of the signs, since is thes secon term isis nitive, baesi stan . in isties must be

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negative alsi, so that there ill e ut onechange of the signf. si is the secon termis assismative, whiteor the thir term is, there mill be ut ne change os the signf. at amyears heresere, in generat, that in cubic equatioris, there are a many assirmative mois asthere aue changes of the sigm os in term ofine equation. The same a os reasening may be se tende to equations of higher dimensions andine rule desiveres in , 9, extendes in ali Ainciosequations.

Setet There are severat consectaries of wbat has been Hready demonstrates, stat are Misein discovering the oot of quations Buthesore Me proceω to that, it Willi convenientio explain sim transserinations os equatio by whicli the ma osten 'be tendere more simple, and the investigation os their roOt re

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His P. III.

ter My, begiministo the fissi are the sanie in

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in both quations an have the fame sigribein products of an even number of theroois V the product os an imo more having thesime sign4 thei product when both thei signf

But the secondae s and est tinen alternaret, rom them, ecause thei messicient involveat Way the Inducis of an Odd nun ther of theroois, ill have contrar signs in the wo qua-rions. For example the produc of Our, viz.

aberi aving the iam sign in both, an one uation in the fifth term haringabed is

and the other ab ed, e, it solloK that their

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mali bestes stan the root of this quatio bysomeriisen disserence e , that in supposeo, and consequently a m theninstea os, an iis powers, substitute a P ean ita poWers, and there in aris this neπ equation

redire equation by the dictere e o.

Is the mposed quation e in this semet,

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In whic there is no term herebris of two dimensio , and an asteris is place in the momos the second term in me licis,antiis g. 6 26. et the quatio proposed e many number of dimension represented by cn9 and let the messicient of the second term Wissi ita signpretae be then supposing π - - I, an consequently an substitutingiliis value serta in the gi ven equation, here, illaris a neW equation ibat mali an in second