<9r>

Worthy Sir

One thing more {illeg} {illeg} \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
${a}^{5}±Ba=N$ I assume a ranke of Numbers rootes to be in Arithmeticall Progression and make up a Series of \N or/ Homogenea {illeg} Whose 5th differences will be equall first then I affirme \thinke it will be hereafter prooved/ that if barely that rank of Homogenea were proposed to find what Æquations were common to the ranke 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 diffres 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 de reduction a|i|n annexis Geametriæ Cartesianæ, I scrupled his rootes of negative |rootes of| quadratick quantities, and imagined he freed himself from them 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 doth by by a {}Drame divides a Parabola (throughout Proportionall to a Sphere) by finding of 2 Meanes both bookes Dr Barrow can shew you \are not both {cases are} \wayes/ the Solution of the Cubick æquation of the same kind/ and for finding the rootes where trisection is used thus saythsee 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 intituled re de arte Analytica inveniendi omnia Problemata Proportionalium maximè harmonicorū wherein \(detecting some of Vietas errors)/ he treates of Cubick Æquations and of framing of rules for Binomiall rootes, wherein 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