A treatise of algebra, in three parts. Containing

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thicar meet the line A in lino potnis , M. the when o the intersectio of the fide. F. SK e me to eliber of these minis, ascit is car

Iarabola, in the third an ellips. The Wjmptotes, hen the curve has ady, re

and alway toxv arcis opposite paris, and through

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An the ω asymptote meet in the centre, constitutinithere an angi m Sn. From aliis constructio it is obvious, fiat When ille circular arci xtouche the lite AE the anse SN Min then, SCN, theline, williecome parallario CS and there- fore CT andra hecome inlata ite, that is, thea sistin Irgoing orto infinity the curve dis 38. There is a ille generat nemo os describing the lines of the Ieconcorderi that deserves ou consideration. Instea os anses in in three russera Din, CN, SP, hicli e suppost to revolvemout the potes D, C, S, and cutisne arinther

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and orderina theamns, it is, In hicli quation, in sim os seme ter mar vax by arring the siluationis the potes

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χνη des*ribes a mist emon. Ont in semeparticular cases the coni sectionisecomes a. straight line. At si example, hen reis undi the straight line CS i so the DF vanishing

linea

Atar the fame manne it appeus that is themines the intersectio os si lines AE, Brisalis in CS, then illi describe a straight line. For in that ras cyanishes, and the quation

be romes,

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o. Fro the secon description, have this solution of the fame problem. Let C S, , Κ, Nie the sive give potnis: dris lines Oining them produce tw of thelines NC, S, ill the meet in D. et threerulere revolve a ut the three potes , S, D, via CP, SQ , DR. et the intersectio osthe rulers CP, DR, e carrie ove the givenline Κ, and the intersectio of the rulers

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o potes hais ver an mali ruter revolvea ut ach of them, an ali the interlections

conete carries alanietvenio lines thatone mali never destribe a lineis veta conicsectio ,' is, linea of rulers, o substitutesve angies hicli ou move o the fame potes, the curve described Mili stillie no morethan a conic section. B caurinione os the intersections necessam in hecie ription ove a rem stimimo, ines of ἡος ordors describessi

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iDensiolis a the quation or by the intersections o any two curves hole indices mulu-plied by each other ive a product eques o steinde of the proposed equation. 'Thus the risis of a biquadratis equatio mayhe determine by the intersections of two conicsectionc sor the equation by Whichahe ordinates stomahe Duritant in hicli these conic sections may cutine another cante determines ill arisem seu dimensions and the conic sections may he assume in such a manner, as O mahe thisequation coincide Mith an proposed biquadratic: so that the ordinates froni thesei r intersections millie equarto the more os the proposed biqua

dratic.

I fine of the inteffections os me conic sectionsalicum the axis, the one of the ordinates vanimes, and the equatio by Whic these ordi .dinates are determine&will thenie of three dimension only, or a cubie,' to Whic an pro- post cubi equatio ma be accommodated.

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So that the three remaining ordinates in besthelhree more of that proposta cubic. 43. hos coni sections oughi tote presera,ed sor his purpos that a re most easit describes. The mus no hoWeve be both ci eos for thes intersections are ni two, and can serve ni sor the realutionis quadratis

equations.

Y et the circle oughi tote one, asieinimost euil describes and the parabola is common assume so the other Thei intersections ara determined in the sollowing manner.

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bola. akem iis axis ille line Ανά is ofit parameter Let te any mint in the planeos mei u abola, and sto illaca centre describe, With an F rassius Cri a rete meeting the par bola in P. et ΡΜ, CD, e perpendiculamon the axis in mandi, and la CN, parallelto the axis, meet Pi I in .

aut, si om themature of the Parabola, M, and substitutin there re these. v laesior, and , it illi be,

i at want the second term since such valuesim he sotmessor a b c alid by corraparing thi With an proposed biquadratic, as to make them coincide. An then the ordinates Dom

this