9.
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 RRoyal Societie who must be made accquainted
therewith, and forasmuch as Algebra may receive a further AdvancemtAdvancement
from your future endeavours (at one time or other) 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 ordi rootes ordinates applyed either to the
respective Homogenea, or to the rootes of those Homogenea according
to the highest degree of degree of the high first tearme of the Æquatiōon
find yethe 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 accotaccount, 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.
One thing more 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
I assume a ranke of Numbers rootes to be in Arithmeticall Progression and make up a Series of or Homogenea 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 diffresdifferences 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 ain 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 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 harmonicorūum 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.
IManuscript abruptly ends here
July 19th 1670 to Mr Newton