Benjamin Robins' parents were John Robins (1666-1758) and Sarah Broughton. John and Sarah were Quakers and they married in 1700, according to the Quaker monthly meeting records, "decently without any superfluity either in meats, drink or apparel." Benjamin was his parents' only child. John Robins was a tailor and the family was rather poor financially. It is unclear whether Benjamin attended school in Bath; some biographers claim that he did, but no record of his attendance has been found. It is likely that he studied languages and mathematics without the help of tutors. By whatever means he acquired an education while in Bath, he showed sufficient promise that he was recommended to go to London to be coached in mathematics. James Wilson, in his account of Robin's life and writings in , writes:-
However, some particular friends of Mr Robins being very desirous that he might continue his pursuits, and his merit not be left in obscurity, wished for this purpose that he could be properly recommended to teach in this town [London] the mathematics, which had been one of the principal objects of his studies. With this view therefore they communicated to a gentleman here [in London] a paper written by him, in order to learn what judgement persons of knowledge might make of his abilities. This was shown to Dr Pemberton, who, thence, conceiving a good opinion of the writer, for a further trial of his proficiency sent him some problems, of which the Doctor required elegant solutions, not those founded on algebraical calculations; adding an example of such a solution that the young geometer might the more readily comprehend his meaning. An answer was returned by Mr Robins that gave a very advantageous idea of his taste, as well as invention. Upon this he came to London; where his presence still increased the favourable sentiments that had been entertained of his talents.
In London he was taught by Dr Henry Pemberton, who, at the time, was preparing the third edition of Newton's Principia for publication. Pemberton soon had Robins reading, in English translations, the classic Greek texts on geometry by Apollonius, Archimedes and by Pappus. Robins loved this geometrical approach to mathematics and retained a preference for geometry over algebra or analysis throughout his life. In addition to the Greek texts he read works by Fermat, Huygens, de Witt, Sluze, James Gregory, Barrow, Newton, Taylor and Cotes. He progressed quickly, publishing in the Philosophical Transactions of the Royal Society in 1727, the year he was elected a Fellow of the Royal Society. This paper gave a proof of a result by Isaac Newton on quadratures. He then wrote on Johann Bernoulli's laws of motion and impacts of rigid bodies, refuting Johann Bernoulli's theory of elastic collisions in The Present State of the Republic of Letters in 1728. With this work he achieved considerable fame so that he was able to attract many pupils for mathematics tuition. He gave this as individual tuition to pupils who were aiming to enter the University of Cambridge, but he never taught a class. Wilson writes that at :-
... about this time he quitted the peculiar garb and profession of a Quaker; for not having the least tincture of obstinacy, superstition or enthusiasm in his nature; he soon got over the prejudices of education, and had an utter aversion to act a feigned part. However, he continued to cultivate a friendship with several deserving persons of that persuasion ...
Gradually Robins gave up teaching to become an engineer. He went on to construct bridges, mills and harbours. He also worked at making rivers navigable and draining fen land. In addition to this he began to study gunnery and fortifications. To gain experience he travelled through Flanders studying fortification work there; there was certainly much to see. On his return to England he published A discourse concerning the nature and certainty of Sir Isaac Newton's method of fluxions and prime and ultimate ratios. This was written to support the differential calculus against attacks by George Berkeley and James Jurin. In Robins own words from the article he published about his tract in The Present State of the Republick of Letters (October 1735):-
Some doubts having lately arisen concerning Sir Isaac Newton's doctrines of fluxions, and of prime and ultimate ratios; this treatise was written with design to give such an idea of both these subjects, as might clear them from uncertainty, without entering into the discussion of any particular objections.
Florian Cajori writes  (or see ):-
Robin's tract is remarkable for clearness and soundness of exposition; it is a marked advance in that respect. The use of infinitely small quantities is rigidly excluded.
To avoid the objections relating to infinitesimals, in "A discourse" Robins had defined a limit as a number:-
... to which a varying magnitude can approach within any degree of nearness whatever, though it can never be made absolutely equal to it.
Jurin, who wrote under the pseudonym of Philalethes Cantabrigiensis, did not find it convincing, however. He replied in the November part of The Present State of the Republick of Letters arguing that Robins had not faithfully represented the concept of limit as used by Newton. For two years the controversy raged covering in total over 700 pages. Cajori writes  (or see ):-
The debate is the most thorough discussion of the theory of limits carried on in England during the eighteenth century. It constitutes a refinement of previous conceptions.
Robins also published Remarks on M Euler's Treatise of Motion in 1739. His publications were not restricted to science, however, and around this time he published three famous political pamphlets criticising Robert Walpole's administration. These were entitled: Observations on the Present Convention with Spain; A narrative of what passed in the common hall of the citizens of London assembled for the election of a lord mayor; and An address to the electors and other free subjects of Great Britain occasioned by the late secession; in which is contained a particular account of all our negotiations with Spain and their treatment of us for above ten years past. It was, in part, due to these pamphlets that Walpole was pressured into declaring war with Spain in 1739. Robins' attacks on Walpole meant that after Walpole was removed from office in 1741, Robins was in favour with the Tory Party and he was appointed as secretary to a secret committee set up to investigate Walpole's conduct.
The Royal Military Academy was founded at Woolwich, in south-east London, in 1741 with the purpose of educating "good officers of artillery and perfect engineers." Robins was a candidate for the position of professor of fortification at the Academy but failed to be appointed; the post was given to a Mr Muller. Hutton writes :-
... it has been said that Mr Robins published his discoveries and improvements in gunnery, to show, it was thought, what sort of man had been overlooked on that occasion.
The work Robins published in 1742 was New Principles of Gunnery which formed the basis for all subsequent work on the theory of artillery and projectiles. The work grew out of a course Robins intended to give, but changed his mind on not being appointed to the new Royal Military Academy. He explained in the Preface how the book developed out of this course (see ):-
... I had resolved to render this course as complete as I possibly could, both by large models of different fronts of fortification, and their different attacks, and by an experimental exemplification of the precepts of gunnery with real artillery. I found it necessary to insert under this last head a theory of the force of gunpowder, and certain propositions relating to the resistance of air, which I had discovered, and confirmed by experiments. But these principles being set down in the schemes, which I delivered out as assertions only, without any account of the nature of the experiments made use of for proving them, and being liable to great contestation, on account of their inconsistency with all the received opinions of the writers upon this subject, I thought it incumbent on me to clear up in a more particular manner any difficulties which might have arisen about them, and to evince their certainty by a number of unquestioned experiments. And this has principally given rise to the ensuing treatise, in which the force and varied action of powder is so far determined, that the velocities of all kinds of bullets impelled by its explosion may be thus computed, and the enormous resistance of the air to swift motions (much beyond what any former theories have assigned) is likewise determined.
The New Principles of Gunnery was rated highly by Euler who translated it into German, adding his own commentary, and the translation was published in Berlin in 1745. It was also translated into French by Le Roy at the request of the Académie des Sciences. This, and other work by Robins, would influence not only the British military but also that of other European countries.
Robins invented the ballistic pendulum which allowed precise measurements of the velocity of projectiles fired from guns. As described by Robins, a large wooden block is suspended in front of a gun. When a bullet is fired its momentum is transferred to the bob and can be determined from the amplitude of the pendulum's swing. In 1746 he published Of the resistance of the air; together with the method of computing the motions of bodies projected in that medium. He continued his study of the resistance of air, examining not only spheres moving through the air bur also other shapes such as pyramids. He published results of experiments he carried out in 1746 in the following year in the paper An account of experiments relating to the resistance of the air, exhibited at different times before the Royal Society, in the year 1746. In 1747 Robins received the Copley medal of the Royal Society to recognise his achievement in developing the new science of ballistics.
In his papers Robins often mixed science with a political message. For example in Of the force of fired gunpowder, together with the computation of the velocities thereby communicated to military projectiles (1747) he was critical of military research, while in A Proposal for Increasing the Strength of the British Navy (1747) he suggested that the navy adopt a new gun based on his own design. He experimented with rockets, publishing Rockets and the heights to which they ascend in 1750. He also improved the instruments at the Royal Observatory at Greenwich.
Robins was appointed engineer general for the British East India Company in 1749 sent to India in the following year :-
... having provided himself with a complete set of astronomical and other instruments for making observations and experiments in the Indies, he departed at Christmas in the year 1749 ... In the voyage his ship was very near being cast away; but he arrived at the Indies on the thirteenth of July 1750.
There he prepared the defence of Madras and of Cuddalore. He contracted a fever in India while working on the fortifications for Fort St David in Cuddalore and died, according to :-
... with his pen in his hand while drawing up a report for the board of directors.
Robins father, John Robins, was still living in Bath aged 85 when his son died. John Robins lived for seven more years in Bath :-
... enjoyed a perfect state of health, having had nothing so much to regret as the loss of the only child he ever had; whose reputation in the world, and constant affectionate behaviour towards him, were the chief consolation of his declining years.
Robins never married. In  Wilson gives an indication of his interests outside of science:-
... though he professed teaching mathematics only, he would however sometimes assist particular friends in other parts of knowledge; for he was well qualified to point out the real beauties of writers in all sorts of learning, and also the excellencies in the performances of great artists, as his taste and judgement were not limited to a single subject alone, but extended equally to history, oratory, poetry, music, architecture, sculpture, painting, and works of genius and invention of every kind.
Article by: J J O'Connor and E F Robertson