... the substitution of proper characters to denote the different kinds and velocities of musical time, instead of those vague, indefinite ones which are now in use. ... Every composer of musical airs would be of real service to the practitioner if he would point out the absolute rate at which his music is to be performed; this would be no difficult task as he would only have to mention the length of a pendulum which would make one complete vibration in the time that part of a bar called a beat was performing.At this time Gregory was still being taught by Richard Weston who suggested to him that he contribute, as Weston did himself, to the mathematical problems in the Ladies' Diary. Here are two of the questions Gregory proposed for 1793-94:
The sum of all the chords drawn in a semicircle from one end of the diameter to those points which divide the circumference into a certain number of equal parts (together with the diameter) is 429.7877. Now if the diameter be 100, what is the number of chords and the length of each?
There is an inflexible rod, void of gravity, 26 inches long, at one end of which is suspended 1 cwt. 1 qr. 23 1/2 lbs. of sugar, in a barrel that weighs 21 lb,; at 1 foot distance from this end hangs a weight of 8 1/2 lbs. and 4 inches further a weight of 6 1/2 lbs.; also at 7 inches from the other end weight of 5 lb., and at this other end a weight of 4 1/2 lbs. Query the point of the rod, which being made a fulcrum, these weights etc will remain in equilibrio?
In 1794 he composed a treatise on the slide rule. This was never published but it brought him to the attention of Charles Hutton. Gregory, perhaps encouraged by Hutton, began to think of an academic career. However, while studying Christianity deeply, he went though various different beliefs but felt he could not subscribe to the Thirty-Nine Articles of religion which was required before a student could matriculate. Nevertheless, he went to Cambridge around 1798 where he taught mathematics, ran a bookshop for about a year, and worked as an editor of the Cambridge Intelligencer for about six months. This radical newspaper began publishing in 1793 and ended publication in 1803. Gregory still appears as "Teacher of Mathematics, Cambridge" on the title page of his book A Treatise on Astronomy published in 1802. He dedicated the work to Charles Hutton:-
In testimony of admiration and esteem for the eminent talents and genius he has so frequently and usefully displayed in his public capacity; and for those generous and amiable qualities which shed a lustre over the path of private life ...Gregory explains in the Preface of A Treatise on Astronomy that he has religious reason for studying the subject. While making such a study, he writes:-
... is it possible that we should not be impressed with a sense of the unlimited power, unbounded wisdom, and infinite goodness, of the adorable Creator and Governor? The great excellence of Astronomy then, even as a promoter of morality and devotion, must be admitted if it appear that it furnishes us with stronger arguments to prove the existence of a supreme, intelligent First Cause - shows more effectually his power and wisdom - or gives us more clear and just notions of his other attributes and perfections ...His friendship with Hutton certainly stood him in good stead for Hutton recommended him for the new position of second mathematical master at the Royal Military Academy at Woolwich, a post he took up in December 1802. He married Anne Beddome (born 1789), a daughter of Boswell Brandon Beddome and Ann Wilkins, on 20 December 1809; they had three children, Annie (born 1811), Boswell Robert (born 1813) who sadly drowned in the Thames, and Charles Hutton (1817-1898) who became a well-known engineer. In May 1821 Gregory was appointed to the chair of mathematics at the Royal Military Academy, Woolwich, and held this post until he retired in June 1838. For the last ten years he had suffered from an illness which became progressively more severe. He died at his home on Woolwich Common.
In addition to his duties at Woolwich, Gregory undertook several other tasks. He edited the Gentleman's Diary from 1802 until 1819. From 1818, he edited the Ladies' Diary, was involved in the production of White's Ephemeris (second only to the Nautical Almanac as an astronomical almanac), and was in charge of acquiring the astronomical data necessary for all the almanacs published by the Worshipful Company of Stationers and Newspaper Makers of London. He was an important member of the Philosophical Society of London which grew up in the early 1800s and was officially founded in 1811 with its own rooms in Fleet Street, London. He was elected vice-president of the Society and lectured to it on ballistics. However, the Society only survived until 1815. He then became a founder member of the Royal Astronomical Society, being one of fourteen to attend the first meeting of that Society on Wednesday 12 January 1820. Of these fourteen, Gregory was elected to serve on the committee along with John Herschel, Charles Babbage, Henry Colebrooke, Thomas Colby, Daniel Moore, William Pearson, and Francis Baily. Gregory served as secretary to the Royal Astronomical Society from 1824 to 1828 and as vice-president in 1829-30. Perhaps it is worth noting at this point that Herschel considered Gregory :-
... as rather too coarse in his strictures and behaviour.Proposals to set up a secular university in London were made in 1825 and Gregory soon added his weight to the movement. By the end of the year Gregory was serving on a ten-man committee which had been set up to interview and select the university's teaching staff. His name was inscribed on the foundation stone of the new University, laid in Gower Street on 30 April 1827.
Let us now look at a few more of Gregory's publications, in addition to those already mentioned above. In 1806 he published the 2-volume work A Treatise of Mechanics, theoretical, practical and descriptive. The book was dedicated, as his earlier work, to John Joshua Proby, earl of Carysfort. Gregory wrote:-
Although the promotion of the arts and sciences has been ever deemed the proper province of the great and noble, yet it has not always been the good fortune of those who employ their time and exertions in the dissemination of knowledge to meet with a patron who is at once distinguished by his rank, his talents, and his zeal in the encouragement of useful performances. I cannot, therefore but esteem it a high honour and a source of considerable gratification that I am permitted to present a 'Treatise of Mechanics' to the public under you Lordship's protection ...In the Preface Gregory explains why he wrote the book:-
It has of late been a too prevailing opinion in this country that a man may become celebrated as a natural philosopher and be very successful in the application of his knowledge to practical purposes, particularly in the construction of machinery, while he is completely unacquainted with the principles of mathematics. Among a variety of unpleasant consequences which have resulted fro this erroneous opinion may be reckoned the rapid decline of the mathematical sciences in Britain, the dissemination of superficial and vague notions on physical topics and the absolute necessity of having recourse to foreign publications for profound and extensive information on these subjects which we should have supposed had not experience convinced us to the contrary, Englishmen would have been proud to cultivate since they were first placed upon an unshaken basis in the 'Principia' of our own countryman, the illustrious Newton.The first volume was divided into five books: Statics; Dynamics, Hydrostatics, Hydrodynamics, and Pneumatics. The second volume covered Practical Mechanics and Description of Machines (Alphabetically arranged).
In 1811 Gregory published Letters on the Evidences, Doctrines, and Duties of the Christian Religion. He explains in the Preface that the letters were written to a friend:-
... much younger than myself, who had a considerable acquaintance with almost all except religious subjects. He expressed much surprise that a person of my habits and pursuits in other respects should adopt the religious notions I had long entertained ...In this work he states:-
We do not deny that the scheme of revelation has its difficulties: for if the things of nature are often difficult to comprehend, it would be strange indeed if supernatural matters were so simple, and obvious, and suited to finite capacities, as never to startle or puzzle us at all. Origen remarked, with his usual sagacity, that "he who believes the Scripture to have proceeded from him who is the Author of Nature, may well expect to find the same sort of difficulties in it as are found in the constitution of nature:" and this obviously suggests the reflection, that he, who denies the Bible to have come from God on account of these difficulties, may, for exactly the same reason, deny that the world was formed by him.Further books by Gregory include: Elements of Plane and Spherical Trigonometry (1816), Mathematics for Practical Men (1825), Memoirs of ... John Mason Good, M.D. (1828), Memoir of the Rev. Robert Hall (1832), Aids and Incentives to the Acquisition of Knowledge (1838), and Hints to the Teachers of Mathematics (1840). The 1838 work is Gregory's Farewell Address on retiring as Professor of Mathematics, delivered on 7 June of that year. In it he addressed the first year students of the Royal Military Academy:-
The genuine object of all sound education is the development of the intellectual, the moral, and the bodily faculties of man; or, as it has been sometimes more tersely expressed, the improvement and application of head, heart, and limb. The system of education in the institution in which you have the honour to receive instruction, embraces all this. The blame will be your own, and it will through life be the subject of regret, if any of you quit this Academy without having acquired the manners of a gentleman, the principles of a man of honour and high and pure morality, the ornamental facilities of an artist, and a competent store of literary and philosophical knowledge.We should not come away with the impression that all of Gregory's contributions came from his books. He also conducted important experiments. For example he carried out experiments with a pendulum in Shetland and also made astronomical observations there, reporting on these experiments in an article in the Philosophical Magazine in 1818. In 1823 he carried out experiments at Woolwich to determine the velocity of sound using mortars, guns and muskets which were fired at various distances from observers. He came up with the result of 1100 feet per second which is a good result (the speed of sound in dry air at 20° C is 1125 feet per second).
Gregory received many honours for his achievements. In the 1826 edition of his Mechanics text he lists himself as: Corresponding Associate of the Academy of Dijon; Honorary Member of the Literary and Philosophical Society of New York; Honorary Member of the New York Historical Society; Honorary Member of the Literary and Philosophical Society of Newcastle-upon-Tyne; Honorary Member of the Antiquarian Society of Newcastle-upon-Tyne; Honorary Member of the Cambridge Philosophical Society; Honorary Member of the Institution of Civil Engineers; etc. The 'etc.' here includes honorary memberships of societies in Bristol and Paris. He was awarded an honorary M.A. by the University of Aberdeen in 1806 and an honorary LL.D. two years later. He was elected president of the Woolwich Institution for the Advancement of Literary, Scientific and Technical Knowledge from its foundation in 1838. Gregory, together with his friend Alexander John Scott, had been the founders of the Institution.
In  Gregory's character is recorded as follows:-
... all he did and said was dictated by benevolence of feeling, and he was man of unbounded charity. As a Christian, he was moral and devout, and as a scholar he merited and obtained consideration of the first mathematicians of the day; his great zeal in his vocation, his parental kindness, his earnest and impressive admonitions, his entertaining, improving, and philosophical conversation, and his ever readiness to assist, will be gratefully remembered by many. He took a warm interest in the cultivation of mathematics, to which he may be said to have devoted, with indefatigable perseverance, nearly the whole of his valuable life.
Article by: J J O'Connor and E F Robertson