# The Gregory-Collins correspondence

Most of the letters we quote from below are between James Gregory in St Andrews and John Collins in London. We do, however, quote from some other correspondence of Gregory's which give other interesting details. The extracts we give are not chosen to illustrate any particular aspect of Gregory's work but rather to give an overview of the range of his interests. Much of the correspondence regards books which Gregory wants Collins to get for him. This makes interesting reading but we have not included this aspect of the correspondence in our extracts. We begin with a quotation from Gregory who was involved in a controversy with Christiaan Huygens. Gregory had written a work attempting to show that π and $e$ are transcendental. The proof is not satisfactory, as Huygens pointed out, but Huygens also accused (unfairly) Gregory of plagiarising some of his theorems. Gregory wrote to Collins on 20 January 1669:

I must have my controversy ended before I publish my 'Optics and Astronomy'; for I have several things in my head, as yet only committed to memory, neither can I dispose of myself to write them in order and method till I have my mind free from other cares.

This is an important quote since it shows a lot about Gregory's personality and explains his later reluctance to publish. On 15 February 1669 Gregory is asking Collins about astronomical instruments:

I would have a quadrant after the enclosed fashion, made of jointed wood, with a brass limb, and a ball and socket (or something else to turn upon) in the centre of gravity, three feet in the radius exactly, and as minutely divided as the quantity can suffer. I do not desire any other scales and divisions on it, save only a needle of a considerable length, for observing the declination of the same upon one of the radiuses or middle, as you think most convenient. I would also have a chain, two good compasses, and a brass sector. I desire to know the price of every one of these by themselves.

Collins, having made detailed enquiries about astronomical instruments, replies on 15 March 1669:

The price of the Instruments mentioned in you letter will be
Ball and socket with three-foot staff £1-8-0
Needle a foot long with graduated box £1-0-0
2 pair of compasses one to have 3 feet £0-10-0
A Brass Sector £1-5-0
If you mean Gunter's sector the contrivance is not good, as wanting chords and tangents from the centre; if you would have another contrivance, then it were well ordered that chords, sines, tangents might all be taken off, at one opening to the same radius, without altering the sector, and as for the quadrant the instrument maker thinks the quadrant better strengthened in this form than in that you sent.

Gregory went to London in the summer in the summer of 1669 and no letters passed between Gregory and Collins between March and November. In a letter to Collins written on 6 January 1670, Gregory writes a little about his duties in St Andrews:

I am now following out, in my public lectures, a full course of mathematics, which, if I publish, either altogether or in part, I will add to it what else of my own I think worthy of public view, and print it in Edinburgh.

On 29 January 1670 Gregory writes to Collins:

Mr Barrow, in his Optics, showed himself a most subtle geometer, so that I think him superior to any that ever I looked upon: I long exceedingly to see his geometrical lectures, especially because I have some notions upon that same subject by me. I entreat you to send them to me presently as they come from the press, for I esteem the author more than you can easily imagine.

On the 9 April 1670 there was an eclipse (predicted by Flamsteed) but Gregory was disappointed not to be able to observe it "it was not seen by any in Scotland, because of a great fall of snow fell out that day." Collins, in chatty mood, writes to Gregory on 10 May 1670:

... I had ere this sent you Slusius, but I am informed the carrier goes no further than to Newcastle upon Tyne, where the books may lie long enough, unless sent to an acquaintance that may pay the carriage there and take care to send them to Edinburgh. Within three weeks or a month Mr Barrow's 'Geometrical Exercises' will be finished, and as soon as it comes out, I intend to send it as before ... Within a fortnight or three weeks after Mr Barrow's book is finished will likewise be completed Dr Wallis's book 'De Calculo Centri Gravitatis' which will be large, viz about 60 or 70 sheets of paper, and I must embrace some subsequent opportunity to send you that likewise. Fermat's Diophantus will not be extant ths year yet, and I am afraid Borelli will be disturbed in Sicily from publishing Maurolico's Archimedes with his own annotations, for Turks having got together a great naval force, it is said 80 sail of galleys and 40 sail of ships besides those of Barbary, and two Greeks were executed as spies for surveying the Plains of Palermo: Alteiri is chosen Pope a man 74 years of age, deaf and dark ...

In fact at this time Scotland was an independent nation and there were customs barriers at Newcastle upon Tyne which caused trouble for Collins sending books from London to Scotland. Gregory received Barrow's book by the end of July 1670 and wrote in a letter of 5 September 1670:

I have read over Mr Barrow's lectures with much pleasure and attention, wherein I find him to have infinitely transcended all that ever wrote before him.

He also enclosed with the letter, or sent shortly after it, details of four of his discoveries. The include integrating $√(1 + 1/x^{2})$, the transcendence of the exponential function, various series expansions for sin x and a general form of the binomial theorem. In December 1670 Collins began discussing the solution of algebraic equations with Gregory. It was on the back of such a letter that Collins sent to Gregory, dated 24 December 1670, that Gregory worked out his method of solving cubic and quartic equations.

See James Gregory on equations at THIS LINK

Notes which Gregory wrote on the blank spaces of Collins' 29 January 1671 letter shows that he was by this time quite expert in computing the Taylor series of functions. Gregory writes to Collins on 15 February 1671:

I can hardy believe, till I see it, that there is any general compendious and geometrical method for summing a harmonic progression ... at present I have not the leisure to examine it, but afterwards I may. I admire much the pocket tube of Mr Newton, being only 6 inches long to magnify 150 times: if it magnifies the diameter of an object so it is incredible ...

This letter contains six examples of functions expanded using Taylor series many years before Brook Taylor was born. Gregory assumes that Newton must be familiar with Taylor series. Gregory writes to Collins on 17 May 1671:

I am now much taken up, and have been so this winter past, both with my public lectures, which I have twice a week, and resolving doubts which some gentlemen and scholars propose to me. This I must comply with, nevertheless that I am often troubled with great impertinencies; all person here being ignorant of these things to admiration. These things do so hinder me, and I have but little time to spend on these studies my genius leads me to; so that it is necessary for me to delay answering you much against my own humour. I resolve, when the college rises, which will be within six weeks, to apply myself seriously to the doctrine of equations ... When I come such a length as to satisfy myself, you may be assured that I will make no secret of it to you. This inclination I find in myself to prosecute these things I had once laid up, hinders my design at present to publish any thing. Yet I cannot express how much I am sensible to your kindness, in exposing yourself freely to so much trouble upon the account of any notions my dullness can produce. ... I have had sufficient experience of the uncertainty of things ... before now, which makes me, since I came to Scotland, however mean and despicable my condition be, to rest contented and satisfy myself with that, I am at home in a settled condition, by which I can live. I have known many learned men, far above me on every account, with whom I would not change my condition.

In the summer of 1672 Gregory attempted to equip his St Andrews observatory with instruments. He made enquiries to Collins in a letter of 23 May about instruments. He went to Aberdeen to try to raise funds to purchase instruments and, on 6 August 1672, he wrote to Collins from Aberdeen:

... several gentlemen have fallen upon a method to gather contributions for mathematical instruments to the University of St Andrews, which probably may take effect. It is like upon this account I may see you within twelve months.

Collins sent some instruments and books to Gregory in December 1672 by sea, arriving in Leith. Gregory received them but doesn't acknowledge this until March 1673. It looks as though it was a slow method of sending things. Much of Gregory's correspondence at this time relates to comparing his design of a telescope with Newton's design. Via Collins, Newton and Gregory discuss the advantages and disadvantages of each. On 10 June 1673 the University of St Andrews commissioned Gregory to set up an observatory in St Andrews:

... proper and necessary instruments and utensils as may serve and conduce for the better, more solemn, and famous profession, teaching, and improving of natural philosophy and the mathematical science, and especially for making such observations on the heavens and other bodies of this universe, (as easily may be by such help, with the great advantage of the pure air and other accommodations of this place) whereby we may be enabled to keep correspondence with learned and inquisitive persons in solid philosophy everywhere ... we have ordered Mt James Gregory, professor of the mathematical sciences here, to go to London, and there to provide, so far as the money already received from our benefactors will reach, such instruments and utensils as he, with the advice of other skilful persons, shall judge must necessary and useful for the above mentioned design. ... And to take notice of the fabric and form of the most competent observatory, that ours here intended may be built with all its advantages.

Gregory went to London and purchased various instruments for his observatory. Several instruments, now on show in the University of St Andrews, may have been purchased by Gregory at this time. Gregory wrote to the Royal Society on 31 December 1673:

... if the lunar eclipse of the 11th of next month be observed by any, I pray you let me have particulars of it; for by it I intend to observe the longitude of this place ...

On 30 April 1674 he wrote to Colin Campbell giving some details of the instruments he had bought:-

It were tedious to write down particularly all the instruments I have brought home, even a larger letter would not contain all their names and sizes, for I have all sorts: our largest quadrant is of oak, covered with brass, 4 foot in radius and actually divided in minutes, of which we can judge $\large\frac{1}{3}\normalsize$ or $\large\frac{1}{4}\normalsize$: we have two semisextans, all of brass, 6 foot in radius, diagonally divided, in which we can judge $\large\frac{1}{6}\normalsize$ or $\large\frac{1}{7}\normalsize$ of a minute. Our largest telescope is 24 foot long; which magnifies one dimension of the object 100 times. ... As for observations or experiments, I dare hardly promise any considerable before the observatory is built: seeing (as the instruments are kept in the library, where I cannot be alone or with my own company and convenience) it is hard if it be at all practical to do anything seriously and with exactness. The latitude here is 56° : 22'. The declination of the needle is 3° : 35' westward.

In the summer of 1674 Gregory went to Edinburgh to become Professor of Mathematics there. In November 1674 a letter to Gregory from the University of St Andrews implies that the St Andrews observatory is now built and the instruments that Gregory bought are now in the observatory. He wrote to Collins on 26 May 1675 concerning the work on equations he had done while in St Andrews:

I have now abundantly satisfied myself in these things I was searching after in the analytics, which are all about reduction and solution of equations. It is possible that I flatter myself too much when I think them of value, and therefore am sufficiently inclined to know others' thoughts, both (as you say) as to the quid and quomodo of them; but that I have no ground to expect till time and leisure allow me to publish them.

In a letter he wrote on 13 July 1675, Gregory explained his reasons for accepting the position in Edinburgh:

... not long after your arrival in Paris, I had a letter from you; to which, the truth is, I was ashamed to answer, the affairs of the Observatory in St Andrews were in such a bad condition; the reason of which was a prejudice which the masters of the University did take against mathematics because some of their scholars, finding their courses and dictats opposed by what they had studied in mathematics, did mock their masters, and deride some of them publicly. After this the servants of the Colleges got orders not to wait on me at my observations; my salary was also kept back from me; and scholars of most eminent rank were violently kept from me, contrary to their own and their parents' wills, the masters persuading them that their brains were not able to endure it. These and many other discouragements obliged me to accept a call here to the College of Edinburgh where my salary is near double and my encouragements otherwise much greater.

Last Updated July 2012