there is a plus ultra in these affairs, that may deserve the pains of the learned to consider, and hope you will not be displeased for so doing. For when your doctrine of Tangents, of the Infinite Series, and of these matters are explained, certainly every one will think the most invincible difficulties, and greatest toil, in pure mathematics are conquered and removed.

As to mathematical intelligence, I send you a transcript of a part of a letter from Dr. Pell's scholar that wrote the high Dutch Algebra, (translated into English and enlarged by Dr. Pell,) to Mr. Haak, an ancient gentleman of the R. S., that translated the notes of the Dutch divines on the Bible into English.

CCI.

J. GREGORY TO COLLINS.

Worthy Sir,

St. Andrew's, March 7, 1670.

I have lately received yours of the 12th Febr., of which I have answered the greatest part in my last to you. I have spoke to several here concerning Sir

* *u, but I can find none who know him: some only have heard of him, that he is poor and base, in so far that he doth no way regard his credit. However, I have written with this post to one Mr. David Thoires, an advocate in Edinburgh, who is my cousingerman, concerning your business. I have desired him also to give me notice what may be expected of this Sir * * and in what condition he

*

*

is. Ye may write to Mr. David Thoires yourself,

t This extract has not been found.

a The name is suppressed,

being that of a respectable family in Scotland.

directing your letter thus: "For Mr. David Thoires, Advocate in Edinburgh." I have signified to him that ye will do so. Any thing that ye send to me, direct it unto him.

66

I see Mr. Anderson hath dealt very basely with you what I have to say against his book I sent you in my last. last. The truth is, I cannot constrain myself to examine his book accurately, for I find almost in every line some nonsense: as for example, pag. 105, speaking of numbers," the sum shall be a square and its root commensurable;" first, the last words are superfluous, for the root of a square in numbers is always a number, and then they are nonsense, for there is no quantity, which is not commensurable to infinite other quantities: and immediately after, "the product may be AZ;" nay, the product must be AZ: and immediately after," the sum of their squares and product may be," for must be and immediately after, “ equal to a square," for, which may be equal to a square: and then he, speaking in his book De quantitate continua, or of Magnitudes, (as in Prop. 23, Prop. 24,) he citeth the demonstrations of the 7th [book] of Euclid, which, albeit they be true in quantitate continua, can never be demonstrated from the principia of the 7th book of Euclid and then, he giveth the title of his 22nd Prop. in numbers, (and afterwards citeth it, speaking de quantitate continua,) and demonstrateth it in indefinite quantities; and then he citeth the 7th book, which is only proper to numbers, and the 6th, which is only proper to rectilineal figures. In his preface to the reader he saith, that seven years since he resolved these following propositions, and in the mean time, saving two or three, they were all resolved seven hundred years ago. It were easier to find an hundred mistakes of this nature than to read over his book.

I have received all those books, concerning which

ye [wrote], save only Slusius. I shall be very willing ye writ to Dr. Caddenhead in Padua, for some of my books. In the mean time, I desire you to present my service to him, and to inquire of him if my books be suppressed, and the reason thereof. No more at present, but rest

yours to serve you,

J. GREGORY.

Extracts of a letter from J. Gregory to Collins.

Analytically there can be no touch line given to the rhumb spiral. I know not if from this, and the fourth problem in the 124th page of Dr. Barrow's lectures, it follow that there is no analytic proportion betwixt assignable parts in a circle, and an hyperbola, which is of considerable consequence, and in the inquisition of which I have very often concerned myself.

Extract of a letter from J. Gregory to Collins. I suppose these series I send you here enclosed, may have some affinity with those inventions you advertise me that Mr. Newton had discovered. It was upon this account I so often desired you to communicate the same unto me. I shall also give here an approximation for the sines".

x From a paper in Collins's handwriting, on which he has written in his letter of the 5th of Sept. 1670."

y In Collins's handwriting, headed," Mr. Gregory in his letter of 23 Nov. 1670."

z These extracts seem to have been made by Collins for his own convenience, and unfortunately the original letters do not appear. It will be found that the working of some portions is inaccurate, and the detected errors leave a suspicion of

the existence of others; but to introduce corrections with their proofs would needlessly lengthen the foot-notes of the page, and they will be detected by any one, who takes sufficient interest in the subject to examine it throughout. To correct them in the text would not have given a fair view of the letters as they stand. Two have been noticed, but the numerical labour of examining the work throughout is far beyond what could be introduced.

Sit arcus aliquis minor quadrante, sitque defectus = c Item alius quilibet quem superat quadrans excessu=a Sinus totus ...

=b

Et ejus excessus super sinum prioris arcus....... =e Fiant duæ series continue proportionalium, prima nimirum

a2-ac-1×2× c2 a2-ac-2 × 3 × c2

1 × 2 × c2

Et vocetur primus terminus, et productum ex duo

bus primis ?, ex tribus primis 2, ex quatuor

ex " с

Sitque productum ex duobus primis, ex tribus primis

S ex quatuor primis, &c. Arcus ille, quem superat

с

с

quadrans excessu a, habebit sinum =

20e+ae 4pe2 + 2re2 __ 8qe3 + 4se3

+

с

bc

bec

+ &c.:

+

hinc, posito a:c:: 1:7, erit sinus arcus, quem superat quadrans excessu a,=

b.

-&c.

49 24016 11764962 40353607 b3

I have been more large in the approximations to the sines, and numbers of logarithms, than in these to the arches and logarithms, because I suppose the former are more unknown. However these approximations to the arches I hint at are the same that I mentioned in my last answer to Hugenius.

Nam positis, radio

...

...

=1',

Semisse lateris quadrati circulo inscripti =d, Et differentia inter radium et quadrati latus = e, esset semicircumferentia=

quæ series facile ita producitur, ut ab ipsa semicircumferentia minori differat intervallo quam quævis ejus pars assignata, imo nullo negotio infinitæ tales exhibentur.

It were no hard matter to bring from these several approximations for the segments of a circle, but it were to no purpose, seeing I cannot take away the alternate powers, as Mr. Newton doth in his series, (if it be one, for the truth is I cannot reduce it to any of mine,) yet I imagine that my series may be as shortly performed as his, seeing that my continued proportion (cæteris paribus) is much greater than his, et proinde seriei termini multo citius evanescunt. I . could also apply many of the former approaches to the hyperbola, but by that means I should gain nothing more ready than what is already known in it.

I must also give you a series for an arch of a circle;