A treatise of algebra, in three parts. Containing

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CHAR IL

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Dio tine AB and o AB describe ara semia

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semicircle then resse AE perpendicular to ΑΒ, and on ictahe AP, Qq. that is, a mean pr

the oot of the quatio hecome imagina raso demon atta, in another manno, in

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An ali quadratic equatium Ming reducitae

to thes seu forins,

this and the lalt two articles. et s. y these geometrica constructions thelocus o an equatio of two dimensions may edescribed since, by their means the valuosos that correspon to any givere . value may be determined But is, demonstrate that these loci re alWays conis sections thenthe ma more asil be describe by the method that are freta k Wnio describing these

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whos ordinates are parallel o M. DraWAH parallel o CF meeting ΡΜ in N; and AD parallela ΡΜ meeting CF in D. Because

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Whence, i an equation is proposed, and such values of a, c, d, e, an e assime a tomata that equation and this coincide, the theticus of that equation illis a parabola The constructio of Whichisa be deducta smin this article. 3 7. In this genera equatio sor the para-hola, the coessicient of at is the quare fatis the coessicient of su and when an equation is proposed that has his property the locuso it is a parariis.' For, Whateve coemients assed the three last terms, the may be made to avre,ith the coemienti os the last term os the genera equation, b assuming prope values usandis. I appears also that is the locu te a parabola, and the term antire, in termai must also e anting ' any

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wm both in rems, x andis it ma bealmys accommodates to a paralesa' The generes quation se thesellias is

deduce sto the Propert of the ordinates of an diameter, in the fame manner the conis

structio of the figure bring the semesa in vet6. Only, in place of the parabbis, Let M be ais ellipse ullos diameter is KL, haviniit ordinates parallescio Μ, anulet C e the eentre of the ellipse. Suppose

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M A TREAetis of PAR III. An i an equation is proposed that an bemadesto agree With this genera equation, by aD suming prope values os a b d plani ei then the locus os that equation,illae an in se.

6 29. In the genera equation se thesellipse, the term A and γ' have the a me ges anuthe coefficient of is alWay greater than the

terna be anting, et the termis mustissi in iis coefficient, in that case, ein Ε, whicli must e Mways rea an positive. Onthe other and is an quation is pro sed in wnich the coessicient os excetas the 'nare ofhais the coe cient os su or, an quation that wants xy, ut has v an iis of the same sign, iis locus mustie ancillus.' ao. In the hyperbola, as Μ CGL CLD:: pQ ut Μben tris a st diameter, theequation that arises Will dissertim the equationo the eis die ni in the signs of the values of CG and I , an consequently ill ave