A treatise of algebra, in three parts. Containing

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I. Uthe two firstiaris,e considered Aly-1 bra asi independent of Geometry and demonstrared iis operations soni iis own princi-ples. It remalas stat, no explain the use of

Algebra in the resesurion os geometri a problema;

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o reasoning abnut geometrica figures and theis of geometrica lines an figures in theresia iocos minitions. Tli mutuat intem eourse of these sciences lias produces mstre ex tensive an beavilibi Theoris, the hie ofwhic ine mal enneavou to explain heginaing With the relatio i etWix curve line and theis equations. 64. - are no in conside quantities asrepresente by fines Hown quantit by a give line, and an uη nown by an Meremimed

linea

But ascit is lassicient that it be indet linedon one tae, Wima supposione extremit: to be

Thusiae line AB, in extremities, and

Bare both determined ma represent a gi ven quantity: bile AP, hose extremit His un- determinia, a represerit an undetermines quantit' a ri indetermine quantii maybe representes by AP, aini neare in A. and is o luppof to move OKard A thenmill ΑΡ, successivesy, represent ali quantities lasi than the iis AR an asteris has coi

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oninio stae re the geη ess of the quan ity, and thelmuis Mithin whic it is possibie. In in e geometrica resolutionis a question, the ining requires is exhibite oes in those cases Khen the question admits of a rea solutioni and heyond thos limita, o solution ap- peus. So in finding the intersections os a givencirese and a stratot line, is ou determine themis an quation, o mill finx o generat expression sor the distances of theloints of inter- stelion Do the perpendicula drawndrom thecentre o the ove line. ut geometricalinthos intersections illis exhibite oesy whenthe distance of the straight line fiom the centreis test than the radius of the ive circle. s. 'hen in an quation there are two undetermine quantities, Canda, then sor achparticula value of there a be a many values of as it has dimension in in t equa,

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So iliat, i A par of the indelisite line AE reprelent and the perpenaicular mre Mnesthe corre Onadire valites os thenthere Mille a mina prim the extremities of the perpendicularcor ordinaus, as thereare dimentions iis in thesequation. And the

6 6. When an equatio involving tW un known quantities x and 3 is propostri thensubstitutinisor, an particular valueia' is theequatio that ara se has ali iis mois positive, thepoinrs , ill ite o one fide of AEa ut is any of them are Bund negative, then these area bese ost o the otheride of AE tomes Pin. Is foris, hicli is suppost undetermined, yo substitute a negative quantity, as Ap, then

correspondinito est the possibie values of x. Is in an case, ne os the values os 3 vanissi,

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comptate to the curve, or Ouches it at an in

Mue distance,' is AP is itfel finite. Is, limis is suppost ii sim great a, alue es, sanissumen the curate an aches toAE producta as an inmptote.'' I an values of 3 ecome impossibis, thens many pol nisi vanish. 's'. rom What hac en salo it appears that When an equation is proposed involving two un-ψα--ω quantillas Minddi 'th- may beas many int resections os the curve that is thecus of the equation, and of the line Pinas thereare dimension ob in the equ*tion, and as many inters mons os se curve and the line AE asst ere are dimensioni os, in the equinon.'

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vit arist, z-d, wil nou liave more dinae si s than the hi est dimensionis Handis m the proposed quation, o the ighes sum ofthei dimension talien together in the termswhere thenare both seund an consequently, in dram in Where in the plane of the curve illisol meet it in more potnis stan there are unit in the hi est dimensionis, ora, o in the highest sum of thei dimensions in heterms,her both are seu .' NoW the dimensionis the equation or curve Ming denominate isto the hioest dimensionis cor a m it, orsroin the sum os diei dimensiona here thesare

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molli e conclude that the number os potnis in hic the curve an meet Mith an sua hesine, is eques to the number that expresses the inmension of the curve.' Ι appears also rom his article, hom hena equation o a curve sciri ven expressing themation os the ordinate mand abscisse Arimu may transformat, solas in expressalie res tion etween an other ordinate M and theabaeisse Aia by substitutiniser , iis valueata in x iis valueri .

6 p. hos curve lines that cante described by the resolutionis equations the relation of isos ordinates mand abscisses AP ea be

expressed by an equatio involvin nothingiui

They are dividia into orderi accordinito ste, dimensions of thei equations o number of potnis in Whicli the can intersect a straight line. Tbes metuet ius the elve 3 sqnstitute thesis ordo es lines , and when the equation ex-scessing the relation os Lando is of one dimension