Glenie then qualified as an engineer at the Royal Military Academy, Woolwich. He was an artillery officer when his regiment was sent out to North America in 1775 at the start of the American War of Independence. After reaching the rank of second lieutenant in the artillery in November 1776 he later transferred to the engineers, becoming second lieutenant in February 1779. During his time in North America with the army Glenie worked on mathematics. In fact, even before being sent to North America, he had discovered what he called the antecedental calculus in 1774. The was an attempt to base Newton's fluxional calculus on the binomial theorem rather than on the concept of motion. He published a number of papers on this and other topics; The division of right lines, surfaces and solids being published in the Philosophical Transactions of the Royal Society in 1776 while The general mathematical laws which regulate and extend proportion universally was published in the same journal in the following year. In 1778 the Royal Society published Glenie's paper on the antecedental calculus. In addition to these papers he had also published a book on gunnery entitled The History of Gunnery with a New Method of Deriving the Theory of Projectiles in 1776. For his achievements in mathematics and its applications he was elected a fellow of the Royal Society on 18 March 1779 while he was still based with the army in Quebec.
In 1780 Glenie returned to England and at this time he married Mary Anne Locke from Plymouth. Three years later he became involved in a serious dispute within the Royal Society which is described in detail in  but which we now sketch. In 1779 Hutton had became foreign secretary of the Royal Society but was forced to resign in 1783 by Sir Joseph Banks, the president of the Society. It was an unfortunate affair which led to considerable controversy. Banks claimed that Hutton had failed to carry out his duties efficiently, but many in the Society supported Hutton and felt that it was in fact Banks who had failed to manage the affairs of the Society competently. Hutton was supported by Glenie as well as by Atwood Maseres, Maskelyne, Landen, Hornsby and others. They accusing Banks of using excessive authority and of being "despotic" and Glenie made a strong speech opposing Banks and supporting his fellow mathematicians at a meeting of the Society in February 1784. It was certainly a serious affair with the mathematicians threatening to secede from the Society. In fact there was another element to the argument which reflected the rapid increase in the use of mathematics in physical sciences. Glenie, and the others in the mathematicians' mutiny, strongly supported this.
The "mathematicians' mutiny" in the Royal Society was only one of several disputes that Glenie became involved in. The next involved a dispute with the Duke of Richmond described in :-
The duke of Richmond, appointed master-general of the ordnance in 1783, consulted Glenie about his plans to fortify all naval arsenals and to create lines of defence along the British coast. Glenie's somewhat tactless declaration that these plans were absurd and impracticable was ill received and led to a flurry of pamphleteering on both sides.In many ways Glenie won the argument since the Duke's plans failed to gain the approval of Parliament in 1786. However Glenie chose to resign his commission in the army as a consequence of this dispute and emigrated with his wife and children to New Brunswick, Canada. Glenie still retained his interest in mathematics and he published his ideas in a book entitled the Antecedental Calculus (1793, 1794). He had other interests in Canada, however, serving as a member of the New Brunswick House of Assembly but failing in a business venture as a contractor for ships' timber. Eventually his political career also failed and in 1805 he returned to England with his family. He then taught at the East India Company Royal Military College at Addiscombe from 1805 to 1810. Glenie, sadly, suffered yet another major set-back :-
He was summoned to testify for the crown at the prosecution of G L Wardle MP, but his evidence provoked severe censure from chief justice Lord Ellenborough, and led to his dismissal from his existing posts.Yet again Glenie tried to succeed in another venture. He went to Copenhagen in 1812 in an attempt to purchase a large plantation, acting for a member of parliament. Perhaps, as in other ventures, Glenie was rather naive and had not set up the proper legal protections for himself. When Glenie returned he was not recompensed so he put in a claim for compensation which eventually went to arbitration. Even the arbitrators could not agree and Glenie was left financially ruined. He turned again to his mathematics, attempting to make some money by taking pupils but, like so many of Glenie's ventures, this too was not successful. He died in poverty.
Article by: J J O'Connor and E F Robertson