# Letter from Newton to John Collins, dated 16 July 1670

July 16^{th} 1670.

{illeg}|W|orthy S^{r}

I sometimes thought to have altered & enlarged Kinkhuysed|n| his discourse upon surds but judging those examples I added would in some measure supply his defects I contented my selfe w^{th} doing that onely. But since you would have it more fully done, if the booke goe not immediately into y^{e} presse I desire you'le send it back w^{th} those notes I have made (since you are resolved to print them also) & I will doe something more to it or if you please to send all but the first sheete or two, while y^{t} other is {illeg}|in| the press \is printing/, Ile reveiw the rest & not only supply y^{e} wants about surds but that about Æquations soluble by trisection, & somthing more I would say in the chapter [Quomodò quæstio aliqua ad æquationem redigatur.] that being the most requisite & desirable doct{illeg}|r|ine to a Tyro & scarce touched upon by any writer unles in generall circumstances bidding them onely Nota ab ignotis non discernere & adhibere debitum ratiocionium.

As to Fergusons rendering the roots of Æquations soluble by trisection, his defect will appeare by example. Let us take his 2^{d} ${\mathrm{x}}^{3}=6\mathrm{x}+4$, in pag 12. In order to solve this hee bidds extract y^{e} cubick root of these binomiums $2+\sqrt{}-4$, & $2-\sqrt{}-4$ To doe this his rule pag 4 is: " Multiply y^{e} binomium by 1000, put in pure numbers &c: Now $2+\sqrt{}-4in1000$ makes $2000+\sqrt{}-4000000$, but to put this in pure numbers is impossible for $\sqrt{}-4000000$ is an impossibe|l|e quantity & hath noe pure number answering to it. \His rule therefore failes &/ The like difficulty is in his 3^{d} example & in all other such cases. In generall I see not w^{t} hee hath done more then in Cardans rules. For in this instance Cardans rule will give you $\mathrm{x}=$$=\sqrt{}c:\stackrel{\u203e}{2+\sqrt{}-4}+\sqrt{}c:\stackrel{\u203e}{2-\sqrt{}-4}$. in w^{ch} y^{e} only difficulty as before is to extract y^{e} rootes of y^{e} binomiums $2+\sqrt{}-4$ & $2-\sqrt{}-4$. Which roots indeed are $-1+\sqrt{}-1$ & $-1-\sqrt{}-1$, as he assignes them, but tells not how to extract them. Nor doe I see w^{t} hee hath done more then Descartes in his Solution of biquadratick Equations: for both goe y^{e} same way to worke in reducing them first to Cubick & then to quadratick æquation. Lastly I see not in what case his rules will render ro|th|e roots of cubick or biquadratick Æquations in proprio genere where those of Cardan or Descartes will not. But in hast I must take my leave remaining

Yo^{r} most obliged s^{r}vant

I. Newton

< insertion from lower down >I thank you for yo^{r} two last bookes.