A treatise of algebra, in three parts. Containing

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GENERA PROPERTiε or Fig. ii and terminate at the curve mayae qua re the sumo the paris of the iam parallel standinio the other

hente, are led tota nother properi os geometricallines, ni li,s genera than that ah ove. et the right line M meet a line of the hird orde in M, m and

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to unit' In like, noercit is demonstriited ibath ving grue in angi APΜ, is the risi, lino AP,m, cui in eometrica lineis anforde in a many minis ascit has dimensons, stat me product of the segments of the irst, terminate by 'and the curve, illat wayche to the produc of the segments of the lalter, terminate by the semes in and the curve, in an Aivariable ratio. 6 6. In the precedin article e have supposed, wit Newton that the right line cui a line of thethird orde in three potiatri I, Κ, butabat this amous theorem ma be reddere more generes, lacussippos that the abscissa AP ut in curve in oesyone potnt, and et statae A. Theresere betaus 1 vanissies let, vanis alis, the las ter of the equation, in this case, Will ef e rx hae ressae c

itiae AP cut the curve in ne polia only sor in hiscas the rom of the quadrati equation-- o hare necessarit imaginary that a b is greater inanu, and the quantity Preal. When there-

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I', willae to the soli unde the abscissa Ariand thesqua re of the distance of the poliat 'fro a give potnt in a constant ratio'. Ab, eing oined is to Aa, as

Theroore in an conic section, is the right line AP does notheet the stmon the angle AP beingitven therectangi containe unde right lines standiniat theptanti and herminalia a the curve is to the quare os the distance of the polii 'stom me gion Dintd in aconstant ratio, hic in a circle is that os equality. No it is manifest stat the fame method may be ap-plied to a line os the ourin ordet hic the abscisa

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6 8. his Ming premis , Pproceia to explain the a vobvisus properties of geometrices lines almost in theiam orta in Whic the occurre in me. Misaused the solio nilemma, derived stom hi doctrine of

Dther, an alis the quantities X, , , V, c. the product of the forme besto the product of the latae la

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Waycrema in the sanae theloi nil remaining and theright line P beingitvenis position. For letis suppos the right lines ABC, D to beearried by motion parallel to themselves, o that their concoursei proceeds in the right line PDgi ven in positio, since AP, PB, P, e is lwaycto is, is, in in Leonstant ratio by Art. s. l. AP reprolent the fluxion o AP, B the fluxion o BP ano CP, ΕΡ c. the fluxions os the right lines C Ρ, ΕΡ, c. respectively that in seles multiplicatio of rm is

line AP is carrie by a motion parallel to iiself, it is weli known inat AP, the fluxion os ille right line AP, is is Eri the fluxion of the right line I las AP in

. . .

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Thino are so wheneve the minis Κ, Μ, M. and l, ni M. Me dron ine sanie side of the poli tand se ineluxions os in right lines AP, BP, CP, M.

op e P, e. have est the iam sign. ut is, therthings emaining the fame, seme potnis 1 and Mali Fig. 7.on me contrar fide of Ρ, the while the res of the ordinates AP, BP Sec increase, the ordinates fanderiare necessari ly dimita istaed, and thei fluxion are tobe accounte subtractive, o negative an si in his onerat, in collectin these sums, in term are tobe affected it the sani o contrar sigm, a the stamenis fali on the fame, contrat fide os ille givenrint P. yio. Is a right line re meet a curve in a manyprinis D, E, I, M. a iis dimentans express the sum

constant or invariabie, ill e qua to the sum praggregate M. i. e. to the sum M the reciprocais to the segments of the right line ΡΕ, given in positioni an determine by the give potntrand the curves in bicri is an semen beson bootherside os heloin P, iis reciproca is iste su tractes.

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a . .

aio. 6 a The synaptotes of geometrices lines are deterinminerseo the gium diretrion of their infinite branches' or leg by this proposition hey may bescoesiderinas tangent to the leg producta in infinitum et theright line A, parallel to the asymptote, meet thecurve in the minis A, B, M. ut the right line PE cut the curve in D f. I, c. Lot M e taeninobis se that in Vae eques to the accestir

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through μὴ ut is these sumsi equat, the curve vill beta parabolii, the asympi ore miniosF in infinitum. Is o determine the curvature o geometricallines by one genera theorem, et CD be a circle ich the right line PR meet in D anda, and the righthnelain C and μ; et the tangent i cui me right

line PD in M, and the right linei remaining fixed latis suppo se the right line Cito be carried by a motion always parallelao iiset tili in minis D, C, coincide, and let the last value of the disserence