<10r>

Sept 27 1670.

Sr

The receipt of yor last letter staying mee from sending back yor Kinck-Huysens Introduction, I have hitherto deferred writing to you, waiting for D Barrows returne from London that I might consult his Library about what you propounded in your last letter but one to mee concerning ye solutions of Cubick æquations, before I sent you my thoughts upon it. And yesterday veiwing {illeg} Hugenius de Quadratura circuli & Lalovera de Elementis Tetragonismicis I can not by that instance you gave mee out of those two compared together bee convinced yt two meane proportionalls may bee found by trisecting an arch, or contrarily. Hugenius indeed divides a sphære in a given ratio by trisection & Lalover divides a cer{illeg}|t|aine Parabolicall segment in a given ratio by t{illeg}|w|o meane proportionalls prop 8 lib 4, but then \the division of/ that segment & ye sphære are not analogous. For suppose akd ye parabola touched at its vertex by ae & {illeg}|y|t ik, ed \are/ parallell to its diameter, then are aik aed such segments as hee divides by {illeg}|2| meanes, but these segments are as ye cubes of the lines ai, ae whereas ye segments of a sphære are not as ye cubes of their respective axes. If indeed ake (fig 2) bee a Parabola insisting on ye base ae & cut by ik parallely to its axis, the segments of this will be analogous to those of a sp{illeg}||re, but this Lalover divides otherways then by 2 meanes prop 9 lib: 4.

I cannot therefore yet bee convinced that anyone problem can be solved both those ways, wch if it could, it would bee noe hard matter to {illeg} take away {illeg}|b|oth ye two middle termes wh{illeg} of any cubick æquation. Which whoever performes I shall esteem <10v> as a great Apollo & admire as much as if hee had squared ye circle, because I judg both impossible. And my reason is this that æquations to what termes soever they are reduced their reall roots never becom imaginary nor their imaginary roots reall (though indeed their true roots may become false & false ones true). But could {illeg} a cubick æquation wch hath 3 reall roots (and consequently is solvible by trisection) have its two middle termes taken away (& consequently become soluble by 3 meanes) two of its reall roots must bee transformed into imaginary ones, for all simple cubick æquations can have but one root reall & two imaginary.

I thank you for yor intimation about the limits of æquations, & differencing their homogeneall termes, but though the speculation bee pritty I much suspect it will never becom usef{illeg}|u|ll for ye solving of æquations. If I chance to meet wth any thing that may improve it you shall have notice thereof.

Upon the{illeg} receipt of yor last letter I sometimes thought to have set upon writing a compleate introduction to Algebra, being cheifely moved to it by this that some thingls I had inserted into Kinck-Huysen were not so congruous as I could have wished to his manner of writing. Thus having composed somthing pretty largely about reducing æqu problems to an æquation when I came to consider his examples (wch make ye 4th part of his booke) I found most of them solved not by any generall Analyticall method but by particular & contingent inventions, wch though many times more concise then a generall method would allow, yet in my judgment are lesse propper to instruct a learner; as Acrostick's & such kind of artificiall Poetry though never soe excellent would bee but impropper examples to instruct one yt aimes at Ovidian Poetry. But considering that by reason of severall divertisements I should bee so long in doing it as to tire you patience wth expectation, & also that there being severall Inc|t|roductions to Algebra already published I might thereby gain ye esteeme of one ambitious among ye croud ambitious to have my scribbles printed, I have chosen rather to let it passe wthout much altering what I sent you before. Yet because you seeme to bee most sollicitous about the doctrine of surds delivered in it, I desire yt when yor leisure will permit you to write you would intimate wherein the particulars in wch you think it most defective. For at my reveiwing ye papers, I judged it not so imperfect as I thought it had beene \when I sent for them back againe/ & soe have hitherto added two or three examples onely more then was done before.

< insertion from the left margin >

I have sent back yor Humes, Van Ceulen, Fergusons Labyrinthus Algebræ both parts of it, & Kinck-Huysen on ye Con-sections. But his Algebra I presume to keepe by me till you have occasion for it. So thanking you for ye said Books, wth other favours & desiring to bee excused for troubling you thus amongst the midst of your buisinesse I rest

Yor humble Servitor            Is: Newton

< text from f 10v resumes >
<10av>

|Mr Newton a Sphere and a Parabola not divided in ratione data by meane Proportionalls|

These

To Mr John Collins at his
house ye next doore to the
Crowne in Bloomsbury

in

London.

wth a Packet.