6.
Jul 11th 1670.
Trin.Trinity Coll.College
SrSir
I have here sent your Kink-Huysons Algebra wthwith those notes
wchwhich I have intermixed wthwith the Authors discourse. I know not whither I
have hit your meaning or noe but I have added & altered those things
wchwhich I thought convenient to bee added or altered, & I guesse that was your
desire I should doe. All & every part of what I have written I leave wholly to
your choyse whither it shall bee printed together wthwith your translation or not. what
If you think fit to print any of it yethe directions I have writ in english will
shew you where it is to bee inserted. But if you have a mind not to change
yethe Author soe much, I would not have you recede from yoryour intentions
upon yethe accompt of wtwhat I have done. For I assure you I writ wtwhat I send
you not so much wthwith a designe ytthat they should bee printed ats ytthat yoryour
desires should bee satisfied to have me revise yethe booke. And so soone
as you have read yethe papers I have my end of writing them. In
a letter you hinted somthing to bee supplyed out of Ferguson's Laby
rinthus about yethe extraction of cubick roots; if you meant pure rootes, I
have done ytthat in as breif plaine & full a manner as I can. But if you
meant affected roots, tis already done by Kinck-Huyson pag 91 as well as
by Ferguson. Indeed Ferguson seemes to have done more in so much as to
comprehend all cases of cubick roots equations wthwithin yethe same rules; but that [more] is
inartificiall because it supposes yethe extraction of cubick roots out of imaginary
binomiums, wchwhich how to doe hee hath not taught us, his rule taught in pag
4 not extending to it. Thus his second example, supposeth yethe cubick roote of to bee extracted wchwhich indeed is , but I would know by wtwhat direct method hee teacheth to find it. Not but that it may bee done, & I know how to doe it, but I think it not worth yethe inserting into Kinck huyson, yet if you think it convenient (& indeed it may bee congruently enough inserted into him at pag 91) I will send you it done in my next letter. There remains but one thing more & thats about the Title page if you print these alterations wchwhich I have made in the Author: For it may bee esteemed unhandsom & injurious to Kinck huysen to father a booke upo wholly upon him wchwhich is soe much alter'd from what hee had made it. But I think all will bee safe if after yethe words [nunc e Belgico Latinè versa,] bee added [et ab alio Authore locupletata.] or some other such note.
Somthing I have yet to say & that's about your paper concerning yethe aggregate of the termes of a musicall progression: Namely yoryour way deduced from Mercators squareing of yethe Hyperbola is yethe same wthwith yethe last of those two I had sent you together before. Onely I had taken a greate deale of paines to bring it to such a forme might bee most convenient for practise & soe had made it soe intricate as to other respects that is noe wonder if you did not discerne its fountaine or by wtwhat method I had composed it. I begg you pardon therefore for that obscurity: but I have since committed a greater fault then that; & that's a neglect of writing to you, Yet I doubt not but that you have goodnesse enough to pardon all. In confidence of wchwhich I rest
YorYour most humble Servant Is: Newton.
I had sent yoryour booke immediately upon yethe receipt of yoryour letter but that I staid two or three days expecting to see Mr Pitts. As for yethe coppys of Kinck-huysen you mentioned to send to me, I know tis usually not wthwithout some unwillingnesse that Math:Mathematical books are printed. And I would not soe far discourage yethe printing of it as to have any coppys reserved for mee. I had rather purchase yoryour freinship then bookes. Yet if you please to send mee one coppy I shall acknowledg my selfe your debter for that together wthwith Dr Wallis his Mechanicks & yethe rest, I. N.