Page:Philosophical Transactions - Volume 014.djvu/188

Rh If such Equation be cncombred with fractions they are all removed at once, by multiplying most conveniently, b the least; number that is divisible by the Denominators of fnc fractions, hence also the infinite Series before mentioned (and others) are reduce able to Lo arithmg. 26 Where IE nations have all tieir terms adfecfted with the Eimefign boili 'tor -) Mn Nwvtffn and Mr- Gregor] deceased have atlirmed they are all reduce able to tome pure high power, which is of singular use in the infinite Series. And a Learned ivrman where this cannot be done, hath afiertcd ierted that they may be reduced to a higher power, with a variable Coellicient. which is the root fought with ¥lCOIll° adi lend or fubd- icend. And even this would renderan ealy tentative l, g;l, l'l€lllIlCflCl1l way for attaining the root.

27 ll' but o in Rootof an [Equation can be found ata time, then queitioulefs a better Method is not yet attained, then what is mentioned in the printed proposal about Printing Mr. lf»¢»€¢'rf 'I'reatii7:s therein mentioned

28 Lastly, as to Conltrnctions tor Illquations, the following Problelne seenis to be universal.

Any two analytic ='iuf'1/ef (vi#—) such as wherein the Habitude between the Bafe and Ordinate may be expressed by an Equation being given in Magnitude and Pohtion, and from the points of theirinterlection ordinates let tall to the Am of either figure, or upon parallels to the said ./bm, the inquiry is of what [Equation those ordinates are the roots P Dr Barre." liked the proposition as well grounded, and left a discourse about doing itin the conick Sections, in which there .ire 3 eaies, either the axes are parallel or being produced concur, beyond the vertexes of the figures without; or other wile intellect: within the figures. .i4r, (1-/guy entred on the Rune contemplation, but death deprived us of the benefit of his thong; its

Of Analytick (alias Geometrick) Curves there are innumerable forts. ofwhiffh I shall mention one or two kinds. Between an Arithmetical l'rogreHion and its iiluares, or between its filuares and its cubes. or its cubes and Biquadratics, there may be inter poled as many Arithmetical or Geometrical means as you pleale I and thence /.im or <;'m-z/ei ¢l¢ 1'lViLl, which some call i'f1m§ 1c/ni'/f OF l'¢¢r.fzbc//1//<rf, /ICQ G7'0g/)}'iC$ Ucometrzae, mrs mln/cr/aléf P1-jntgd in Img/ in Q, iQ11'[.

Postscript explaining Section the 9th. After you have obtained the Cube roots of Cmlwf Binomials, according to Vw Svbn-fm, in D f tin, i or Kei;/w, if you change the Sines of the rational parts of those roots, as also the