Glenie, James | The Doctrine of Universal Comparison, or General Proportion [Bound together with] A Geometrical Investigation of Some Curious and Interesting Properties of the Circle

A mathematical sammelband containing the first editions of two scarce texts by the soldier and mathematician James Glenie (1750-1817). The second Glenie volume, A Geometrical Investigation of Some Curious and Interesting Properties of the Circle, is inscribed “From the Author”, and contains a long equation and seven small textual corrections in the same ink, but it is unclear if this is an authorial or secretarial hand.

During his education at St. Andrews Glenie showed aptitude for science and mathematics, but on the outbreak of the American War of Independence he enlisted and was sent to North America, becoming second lieutenant in the engineers in 1776.

“In 1774, while in the army, it seems that Glenie discovered the 'antecedental calculus', and wrote 'a small performance' of it in Latin which was printed in July 1776. He sent a paper on this to the Royal Society, which was read in 1777 and published the following year. At much the same time Glenie wrote papers entitled 'The division of right lines, surfaces and solids' and 'The general mathematical laws which regulate and extend proportion universally', printed in the society's Philosophical Transactions in 1776 and 1777. These publications, with his book, The History of Gunnery with a New Method of Deriving the Theory of Projectiles (1776), secured Glenie's election to the Royal Society on 18 March 1779, while he was still in Quebec... In 1794 Glenie published a new booklet on the antecedental calculus. Newton's approach to the calculus had used the notion of limit unclearly, and also drew upon velocity; Glenie wished to avoid all this, so as an alternative he defined the derivative of a function algebraically by using the binomial theorem in order to express the ratio of the increments of two functions as a power series in the incremental variable h, and then blithely deleting terms containing powers of h above the first. A related work was a letter from Glenie to Francis Maseres, containing 'A demonstration of Sir Isaac Newton's binomial theorem'. This, and other papers by Glenie, were published by Maseres in his Scriptores logarithmici (6 vols., 1791–1807).” (Oxford Dictionary of National Biography).