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:

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:

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

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:

On 29 January 1670 Gregory writes to Collins:

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:

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:

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.

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:

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:

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:

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:

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:

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

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:

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