Letter from John Collins to John Wallis

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S^r

1 To your Apology about Writing in English I have this to adde that we have very few Latin booksellers that trade beyond Sea and such as doe make a more quick and profitable returne of their Stock than to adventure it in printing of Later Mathematiques as you well know by Experience.

2 That tis the Designe of the Royall Societie to promote and encourage the printing of Mathemaicks and other bookes of Art in our owne tounge.
Whereas you have given an Acco^t of the learned paines of Englishmen namely of Harriot, your owne workes M^r Newton, you may yet adde one Stone more to the Monum^t of the fame of English {illeg}|W|riters namely

To Commend the deceased D^r Barrow for setting such a studious \painfull/ and learned Successor as M^r Isaac Newton, and to give an acco^t of the \his/ Optick and Geometrick Lectures, which amongst most knowing Geometers have obtamed great esteeme though here they have mett with hard fate, the booksellers that first undertooke the same failing and what could not cost them the greatest part of the Impression which could not cost them lesse than 4^s for an \Exemplar of/ both Tracts, afterwards sold for 1^{s 6d}, the doing hereof will not only please M^r Newton but likewise the Universitie of Cambridge and possibly revive the Sale of a booke that is slow and by consequence encourage Stationers in future undertakings Concerning Leibnitz, you see he fell into the method of mr M^r Newtons infinite Series which himself the s^d m {sic}^r Newton himselfe graunts, by a new method of transformation of Curves by which the extraction of the rootes of simple Powers in Species is avoyded, whether this was not learnt or may \not/ be derived from D^r Barrows Geometrick Lectures is the question, also the \said Leibnitz/ avoyds the Extraction of the rootes of adfected æquations in Species by an improoved method of Tangents, he is a most learned ingenious man, a Member of the Society, and now lately sent for home and admitted a Member of the Privy Councell to the of the Duke of Hannover: to explaine his doctrine more largely than himselfe could have leisure to doe, doubtlesse will not be unpleasing either to himselfe or to the R. Society who may order it to be printed \with due respect to the author/ without incurring the any censure for publishing what was occasioned by his private Letters, in the interim the Presse same Compositor is employed to pr sett a Booke about Pallaviconos Religion of the Roman Court recommended by your Son and now licensed as also your English Exercises about Harriot &c which shall be sent you sheete by sheete, and if this retards, the other to proceed

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S^r

I am loath to incurre your displeasure, but yet must take liberty to tell you some things concerning your intended Explanation of M^r Newtons Series
If have had been so minded, I could about 9 yeares since namely at the beginning of 1669 have imparted to you a full treatise of his \of/ that Argument but did not, in regard you lye under a hard censure from the World \diverse/ for printing things \discourses/ that come to you in private Letters without permission or consent \as is said/ of the parties concerned, at last {he} \Mr Newton last yeare/ sent up these Letters^*, |* \you have seen/ with particular leave upon my importunity to print the same|, and I seeing you therein mentioned I imparted the first Letter to you (which I thinke if I had not I beleive you had not seen either to this day)

In your narrative you say Mr Newton began to fall into these methods in 1669 or 1670, whereas in the larger Letter he tells you he seemed delighted hisce ventis < insertion from f 2r > namely in Calculating Log^mes and Van Ceulens Numbers < text from f 1v resumes > in his retirem^t from the University in the Plague yeare in 1665, and in 1666 he writt the treatise above mentioned, all the account you can give out of those Letters is but very slender in relation to his performances, he intends a full treatise of Algebra |consisting of these Parts according to the best of my {app^s}|

1 for the an Introductory part he bought a translation of \from/ Kinckhuysen \Introduction/ out of low Dutch \turned by Mercator/ into Latin, r{illeg} \which he bought/ and is so excellent, that it comprehends many of Huddens reductions, and those mentioned by Dary at the end of his tract of Interest & some others to which he \M^r Newton/ added diverse things \much/ of his owne

2 A Collection of diverse Excellent A Problems discourse about bringing Problemes to an Æquation with a Collection of diverse notable ones

3 A Treatise about the Construction of Problemes and Æquations which I have seen, all Solid Problems are by viz those of 4 and 3 Dimensions are solved by ayd of one Con < insertion from f 2r > stant Circle (if so desired) supposed to be intersected by Conick Sections, the description whereof is avoyded by helpe of mooveable angles that give the severall Points of Intersection sought, other Equations betweene the 5 and 9 degree inclusive he performes by ayd of a Cubicall Parabola that being once described in like manner remaines constant, and is to be intersected by a Conick Section as before the description whereof is avoyded as before &c < text from f 1v resumes >
|he hath| With \also/ diverse tentative Constructions \for Cubicks and Biquads/ from Plaine Geometry

4 A Discourse concerning \the/ severall kinds of infinite Series considering which kinds more fitt \are most convincing and fitt/ for Calculation, and which for Construction and Demonstration, of this Argum^t he hath an < insertion from f 2r > d of the whole buisinesse of Series he hath < text from f 1v resumes > |written a new and large treatise since that above mentioned, and hath per| < insertion from f 2r > formed aboundantly more than is either mentioned or can be guessed from the Letters above mentioned < text from f 1v resumes >

5 A Treatise de Locis

6 The same applyed to Dioptriques concerning \the worth of/ both which D^r Barrow affirmed he was so farr su was not only surprized but others would thinke it incredible

Scotus owes debt to Aristotle, yet doth him plainer make
Therefore should he in Scots debt be, if there is no mistake

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Mr Gregory having but{illeg} one of M^r Newtons Series's sent him namely that for the Zone of a Circle after some study fell into the method, and began to be prurient about publishing something concerning it, as being offended I did not publish his Solution of Keplars Probleme, which I would not doe as knowing M^r Newton's Series's were made use of therein and yet he had a good right so to doe for he really advanced the Doctrine, namely after a few tearmes of a Series were attained he had approaches for attaining the Sum of a great many more, moreover he could give a fractionate part of some pure \but/ high Power of the root of a Series equall to the Sum of as many tearmes therein as were desired < insertion from f 2v > These Series of M^r Newton did not hinder M^r Gregory from prosecuting his owne Converging Series, as more proper for Construction though worse for Calculation, by and ass who when he was last here that much after the Same manner as he had streightned an Arch of a Circle \which I formerly accquainted you with/ he could streighten most other Curves, and particularly had streightened the Ellipticall Parabolic all and Hyperbolicall Lines < text from f 2r resumes > which Dr Pell saith he can doe by a Canon of Sines & Tangents, the which indeed will remoove the use but not the necessity of Series from whence to derive such Canons and this being attained addes a topstone to this most excellent doctrine.

After Mercator had published his Logarithmotechnia you mend his Series in the Transactions adde a Series better than his which you now claime, What sa{illeg}|yes| Gregory to this, namely that he emptyed both Mercators and your Series of each other tearme, and what Mercator, \namely/ in an Appendix now at the Presse, namely \he shewes/ by his owne methods without \rejecting/ these advantages he shewes how to make a whole Canon of Logarithmes by Addition, you may therefore omitt such claime, without any Dispendium of renowne

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S^r In sum if you thinke \not/ fitt the tendency of what you have writt, is to show the world that M^r Newton's Series w{illeg}|er|e derived from yours this he graunts, and tis better to sayth so himselfe he attained these methods \his/ Seriess by 3 Methods concerning the first he ha \as I remember for I have not as yet a {illeg}|C|oppy of the/ |larger letter| attained it /was\ by interpoling yours where your \selfe/ gave over, and afterwards \he/ forsooke this method and having m{illeg} {at} \falne into/ two method \ones/ of his owne

|To D^r Wallis|