Worthy Sir

Perceiving by your last that you are willing to take some more paines at present with Kinckhuysen I remand the same, but doe not presse your selfe in time, your paines herein will be acceptable to some very eminent Grandees of the Royal Societie who must be made accquainted therewith, and forasmuch as Algebra may receive a further Advancement from your future endeavours () and that you are more likely than any man I know, herein to oblige the Republick of Learning give me leave to remind you of some discourse I had when I first had the happinesse of your accquaintance, I intimated that D P affirmed that he could most exactly limit any Æquation shewing what the Homogeneum must be to make any paire or paires of rootes gaine or loose their possibility, and Secondly that out of that doctrine of Limits, he could fill up (with no great toyle) Columnes containing all those rankes of rootes both negative and affirmative, I send you a Specimen hereof in Cubick Æquations that my meaning may be the better understood his way of doing it was not by depression as here but scandendo, the like in his limits, limiting precisely first quadratick Æquations, then Cubicks then Biquadraticks, then by ayd of these Columnes making the rootes ordinates applyed either to the respective Homogenea, or to the rootes of those Homogenea according to the degree of the first tearme of the Æquation find the Genius of Curves proper to Æquations of each kind passing through the tops of those Ordinates, this I mention not to put you to any trouble to undertake it, or out of a desire to doe it on my account, or when done that you should impart it, but as a thing that will advance the Science of Algebra and cleare up that which other Authors have pretermitted.

Moreover you have happily found out a method of turning any Æquation of two tearmes besides the Homogeneum into an infinite Series, and the totall of some of the tearmes of that Series may be obtained by tables as a helpe Suppose we then that to such an Æquation as this
a5±Ba=N I assume a ranke of rootes to be in Arithmeticall Progression and make up a Series of N or Homogenea Whose 5th differences will be equall first then I thinke it will be hereafter prooved that if barely that rank of Homogenea were proposed to find what Æquations were common to that ranke, there might be found 5 severall Æquations common thereto, and that each of these Æquations hath or at least may have as the Æquation may be put one ranke of rootes in Arithmeticall Progression and consequently that the differences of the rootes of each Æquation are Proportionall one to another

But to returne into the Way, when I had Fergusons Papers, I only viewed his Examples and that cursorily, it seemes he soared but Icari fine to accomplish what Hudden promised page 503 in annexis Geametriæ Cartesianæ, I scrupled his rootes of negative quadratick quantities, and imagined that they expunged one another being affected with contrary Signes but conceited there might be more done in Cubicks then authors yet insist on, because Hugenius in libro de magnitudine Circuli <9v> divides a Sphære in a given reason by Trisection, and Lalovera in Elementis Tetragonismicis divides a Parabola (throughout Proportionall to a Sphere) by finding of 2 Meanes are not both wayes the Solution of the Cubick æquation of the same kind and for finding the rootes where trisection is used see especially Du Laurens (which bookes Dr Barrow can shew you) page 205 206 207
that Author as also Lalover and Leotaud are deceased.

Bartholinus in 1657 wrote a small booke de arte Analytica inveniendi omnia Problemata Proportionalium maximè harmonicorum wherein (detecting some of Vietas errors) he treates of Cubick Æquations and of framing of rules for Binomiall rootes, I beleive there is little new, but possible his mode of expressing himselfe may be pleasing, I have not the booke of my owne, but hope erelong to send you those Chapters transcribed.

I[Editorial Note 1]

July 19th 1670 to Mr Newton

[Editorial Note 1] Manuscript abruptly ends here

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