Being a poor minister's son born within three miles of Oxford, and a while instructed in the Grammar school I went out betimes, my parents being dead, apprentice to a bookseller in Oxford, who failing I lived three years at Court and in that space forgot the Latin I had...Hutton tells us in  that because of Collins brilliance:-
... for the study of the mechanical and mathematical sciences that he was taken under the tuition of Mr Marr, who drew several dials which were placed in different positions in the King's garden, under whom Collins made no small progress in the mathematics...In 1642 the Civil War had broken out and the King, together with the Court, had moved to Christ's College Oxford. It was the King's gardens in Oxford that are referred to above. Collins seems to have decided to leave England because of the Civil War and he became a seaman. For seven years he served, and during this time he continued to study mathematics while at sea. His own account of this is quoted in :-
.. the wars here breaking out, I went seven years to sea, most of it in the Venetian service against the Turk, wherein my hazard was almost as various as those of Dr Johnson set out in his sermon in print; and enjoying some leisure I recovered so much Latin that in 1646 I translated some books into English.In 1649 he became a mathematics teacher in London, a post he held until 1660. From 1660 onwards Collins worked at a number of jobs but most often as an accountant for various different organisations. From 1667 he worked as librarian for the Royal Society in London, in addition to his other jobs. He was elected to the Royal Society on 17 October 1667 where :-
... he seems to have been a Barnabas among those mathematical apostles, his tact and devotion in calming the headstrong and drawing out the reticent being above all praise.Collins describes his jobs, other than the Royal Society post, in his own words as follows :-
... I have been employed for a livelihood (being born to no estate nor any left me which, by reason of the wars, my relations either lost or spent) in clerkship and keeping of accounts, particularly those of the Allom farmers whose shipping of Allom to many parts beyond seas have furnished me with a correspondence abroad and a means of procuring books from most parts. I lately [until Christmas 1670] was employed at Brookhouse under the Commissioner of Accounts but at present  I am one of the clerks attending in His Majesty's Council of trades and plantations.Collins also held a position as an accountant in the Excise Office from 1668 to 1670. However times were not easy and Collins only received a small fraction of his proper salary from the Council of Plantations. He therefore resigned in September 1672 and was given a job in the Farthing Office. The Farthing Office was a part of the Mint and Charles II had introduced, in 1672, the copper half-penny and farthing with the Britannia type.
In 1671 Collins had moved to a house in Westminster close to that of William Austin who was the King's chef. Austin's daughter Bellona Austin was the Queen's laundress. Collins married Bellona.
Collins's importance is, as Barrow said, being "the English Mersenne" . He corresponded with Barrow, David Gregory, James Gregory, Newton, Wallis, Borelli, Huygens, Leibniz, Tschirnhaus and Sluze.
Some extracts from his letters to Gregory are at THIS LINK and some letters from Gregory to Collins are at THIS LINK.
Collins published books by Barrow and Wallis and left a collection of 2000 books and an uncounted number of manuscripts.
He did publish works of his own, however. For instance he published works on sundials, trigonometry for navigation and the use of the quadrant. He had a paper on cartography published and also wrote on accounting, compound interest and annuities. His major works were An introduction to merchant's accounts (1652), The sector on a quadrant (1658), Geometrical dialling (1659), The mariner's plain scale new plained (1659) and, in 1664, he published Doctrine of Decimal Arithmetick.
Collins's character and interests are described in :-
Collins was sociable and musical, and played the viol da gamba. Besides tracts, highly acceptable to the public - a plea for bringing over Irish cattle, and keeping out fish caught by foreigners; for the promotion of the English fishery; for the working of tin mines, and the like. He was frequently consulted in nice and critical cases of accounts...A canal was proposed to join the river Isis (the name given to the upper part of the river Thames in Oxford) and the river Avon (which flows west). This project had earlier been considered by Henry Briggs. Collins went to Oxford in 1683 to survey the proposed route of the canal. On this trip, however, he became ill :-
... contracted a disorder by drinking cider when he was too warm...and although he returned to London, he never recovered from the illness.
Article by: J J O'Connor and E F Robertson