A treatise of algebra, in three parts. Containing

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GENERA PROPERTIES

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conic seruons, the ancient an modern geometem have initten very sulis concerning the figures,liichare referre to the superior ordere, lines, litue hasbeen deli vered hesore NEWTO M. That most illustri-Dus an in his tra concerning the numeration VLines of the Third Order, has revived this su edi, whichhad long Iam neglected, and has strewn it tote Worthyof the geometer 's notice For the generat properties Ostheis lines, whichae has la id down, are so consonant totheanown properties of the coni festions that theystem to e conformabie to the fame law, and rom his eAample many other have been since induce to mahethis subjed thei study, and bave clearly comprehendedan explaine the analog whic there is belwee figures of such very different Ends. Thelains,hichtho have been at in the illustration an surther investi gationis these matters have deserviat met wit applause, since there is nothing in pure mathematics ita ut be calle more beavisses, in that is more

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436 ENERA PROPERTIE orapirio delio a min desimus o inviestigatin truth, than the agreement an harmon o disserent inires, and the admirabie connection os ste succeeding Mincte preceding, where the more simple alWay open theway tocinose Which are ni ne dissiciat. Most of the genera properties of lines of the ihild der, desivere is N -' relate in segments of p restas and aiymptotes some other os mei assinion os a disserent itid, I have bries potnted ut in myTreatis os Fluxions lates publimeri Art. 3a , and o I. . The amous cius sermeri discovete a m bbeautila properi os geometrica lines, liberto un- published whic has been communicatexto me by the

Rev. r. Oler Smith, master o Trinit College, Cambridge, a 'enilem an not m remetirhahlerior his learnin and wolhq than or his fidelit and regardfor his stle s. Whill G ad these unde consideration, ome ther genera theorem offered themselves ;which, as the seem to conduce to the augmentationand illustration of this dissiculi par of geometry, I havdithought ficto throw together, an esita in e cpound nocter, and demonstrate

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referred to the superior ordem es lines is different. Todesine and dra out thei properties, genera equations inust be applied expressing the relatio of the co- Ordinates Let x represent the abscissa Ariis the or Fig. i.

dinate PM of the figure M H, and letis, Adenote an invariabie coemients an havin thaangle AP gion, is the relation of the c ordinates ea b be definia by an equation whicli, besides the co- ordinates the elves involvo ni invariabie coeff- cients the line ΜΗ is calle a geomestica one; whicli indeed by seme author is calle an algebraicalline, by ther a r tiona line. But in order of the line depellas pon the highest index of xor in the

right line AD, o PN, scio bestaken on the other sideo the abscissa AP solo the contra ituation os right lines a iners to the contrar sigiis os the coemienis.. Is me assirmative values of x denote rightarnes di a nsrom A, thes ginning of the abscissa, to the righi ha , in negative values ill denote right line drawn stomine sume besti inito the lest. an in lita mai ne is

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38 ENERA PROPERTIEsthe assirmative values of ri, Irrelent the ordinates constitute above the abscissa, the negative ones, ill denote in ordinates eis in abscisa, Mawn the o posite Way-I'he generi equation soria sine of the seconia ordo is es his form,

cal lines os superior ordercare defined. La. in geometrica line may meet a right in inas many potnis a there a re unita in the numbe whichdenotes the order of the equatio ordine, an neve in more. The number of times that an curve Mi meetit abscisi A is determine by ultio, mi in whicli case there remains Only the last terni os the equa , tion into whicli, Oes no enter. For example, a lineo the thita ordo meris ille abscissa AP hen δε ga' ' of hic 'equatio is stere bethree rea mots, in three potnis. In lita manne in ille genera equation o any orde the ighes index os in abscissa, is equa to the number Whicli denores the order of the line, but neve greater, and of course expresses the number os times that he curve in meet

the abscissa o any othor right line. mutiunce one motos a cubi equation is gways eat, and that the sameis true o an quation of the fifth or an od order hecause vernil rui est naryo Oot lias necesiarii iis et lowJ, it sollows that a line of the thir or an otherod orde cui an ris hi line, o parallel to the symptote

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stippine domi in the sim plane, in one piant at γαλ Butarii, right tu be parallel in the asymptote, in his eas it is commoni meet the curve at . an infinite istance siue heresere os annota orderii necessi ril tWo branches dic maybe producta in Asinitum. But os a quadratio, o an other equa tion o an even number os risu, an the number os mois maybe somelimes imaginary, thereis it may bethat a right sine minii in the plane os a curve es an

inde is semetimes compounded of so many simple ones,

freta froni surd in fractions multi,ille into e ch thecas olim a the pro sed dimensons os that equation ex eis; in hiis case the figure pMwis no cur, vilinear, but is made u es se many right sines orare destribed by the simple equations thus determined, as in 1. In lita manne is a cubi equation be compouo ed of two equalion multiplied into rach other, ne ofv hiemis a quati tie and the ossier a simple one, inc Mus wil not be a line of the inita ortar, properlyso called, ut a conic section Oined with a right line.

No the properties xv hic are generali demonstrates of geometrica lines of higher order are tote affirmedallo os lines of ii terior oria ers, i the number denotingtheir ordeis, ahen together mali o the numberwhic denotes the order of the sat superior line. Those, hich, for ex ample, a re generali demonstrat edo lines os the thir order, re also to heri Trmed of three right line drax an in the sitne plane or of a coniesed ion ogether init one right lini describe in thesa mea lanc. O the otheriand there an carce any

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cienti genera to hic so me assection os lines opsu. perior Order does no correspond. Bucto derive these Domahose, it is no et ery one that caestahe thetains. This doctrine in a great measure depend upo the properties of generat equati s Whic it is here oesypropera mention

6 4. In e ver equation the coefficient os the secondier is equat to the excessos the sum of the affirmativeroot above the sum es the negative ones an is that term he wanting, it is an indicatio that in sum ofthe assirmative an negative more, o the sum of the

ordinates constitute uidisserent sides of the abscissa,vre eques. et in genera equation esse a line of

latae substitute ita value foris and in the

transsorme equatio the secon term willae Manting as appears sto the calculation, o sto the doctrine of equations, very where delivered: and rom merit alis appears, stati hypothesis very value of

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α -- so that Q may mis right sine on both sidet me polat in terminates a the curve, ili malae m. the me sum. Now the locus of the mini 2 is the

isom hende it appears that a right line manat ways bedrawn whic stati Q cut an number of paralleis, meetinia geometrica line in a many potnis a the ditiaenis sions of the figure express that the sunt of the segments of every pararet, terminated a the curve o one fide of

the cultim line, may twaysie eques o be sum of the segmenta of the semeson the other side the cutura line. mincit is manifest stat a right lines hicli cure any two parallela in this manne is necessarii that whla walcut allisther parallel in the fame manneri And stomhence appeus the truth os the Newimila theorem, in

ita is containta the genera propert of geometrices lines, analogous to that wel known propert of theconi sections. For in thes a right line hic bisecta anni ci paralleis terminated at the section, is a diame ter, and hi sem allisther parallel to these, an terminate at the section. And, in like manner a right line, Whicli ut an t o paralleis, meetinia geometri calline imas many poliat ascit has dimension' so that ille sum es thiparis standinion onestae of the cultire lineand