period, the demonstration, or analysis, however complicated it might be. In addition to the works before-mentioned, he published one well calculated to promote the study of ancient geometry, entitled Propositiones Geo metrica more Veterum Demonstratæ. The method here employed constituted an important part in the analysis of the ancient geometers, and a few examples of it have been preserved: those in the Propositiones are particularly valuable. 1718. JOHN CANTON. JOHN CANTON, the son of a broadcloth-weaver, was born at Stroud, in Gloucestershire, on the 31st of July, He was educated at a school in his native place, and there apprenticed to his father, who soon became alarmed for his health, in consequence of the ardour with which he devoted his leisure time to astronomical studies. Without any other aid than the Caroline tables annexed to Wing's Astronomy, he computed eclipses of the moon and other phenomena, and constructed various kinds of sun-dials. His father, to prevent his labours at night, had forbidden him the use of a candle, but this he found means to secrete in his chamber, till the family had retired to rest. The prohibition only seems to have stimulated his exertions; as, during this period, he cut out, with a knife, upon common stone, the lines of a large upright sun-dial, on which was indicated, besides the rising of the sun, his place in the ecliptic, and some other particulars. So gratifying a specimen of his son's abilities, Mr. Canton felt too much pride to conceal, and its exhibition on the front of his house caused great adiniration, and the introduction of the maker to several gentlemen in the neighbourhood. The books from some private libraries to which he thus gained access, greatly facilitated his progress in mathematics, and, at the same time, gave him a taste for natural philosophy. He was introduced to the metropolis, by Dr. Miles, of Tooting, with whom he resided from March, 1737, till the following May, when he articled himself, for five years, as clerk to Mr. Watkins, master of the academy in Spital Square. At the exDiration of this term, he was taken into partnership, for three years, and on the retirement of Mr. Watkins, he became his successor, and continued in Spital Square for the remainder of his life. He married, in December, 1744, a Miss Colebrooke, the niece of a banker in London. Electricity appears to have been the first subject in science to which Mr. Canton gave his attention, after his arrival in London. The discovery of the Leyden phial having just taken place, he made an experiment to determine the quantity of electricity accumulated in it. He effected this by ascertaining the number of sparks it would give to an isolated conductor, and communicated the result, in a paper to the Royal Society, in 1746. In the following year, he published two electrical problems in The Gentleman's Magazine. In 1749, he was occupied, with Mr. Benjamin Robins, in ascertaining the height to which rockets may be thrown; they, in general, rose four hundred, but some reached one thousand yards, the latter of which were seen at a distance of nearly forty miles. In 1750, a communication to the Royal Society, of a method of making artificial magnets, without the use of, and yet far superior to, any natural ones, procured his election as a member of that body, and their gold medal for that year. About the same time, he was created M.A. by the University of Aberdeen; and, in November, 1751, was elected one of the council of the Royal Society. On the changing of the style, in 1752, he sent to the Earl of Macclesfield several memorial canons for finding leap year, the epoch, &c., to be inserted in the Common-prayer Book; but, being too late for this purpose, they were given a place in Dr. Jenning's Introduction to the Use of the Globes. It was in this year also that Mr. Canton was the first to verify Dr. Franklin's hypothesis, by drawing lightning from the clouds during a thunder-storm. In this experiment he obtained sparks half an inch long, and of the duration of two minutes. From his Electrical Experiments, &c., read before the Royal Society, in December, 1753, he also appears to have made a discovery contemporaneous with that of Franklin, that some clouds possess the positive, and some the negative, state of electricity. In a paper which he communicated, in November, 1754, he proved the dependence of the plus and minus electricities to be on a rubber or electric, and not on the nature of the substance which was rubbed, as commonly received. He also showed, by an admirable apparatus, well known by the name of Canton's electrometer, that air is capable of receiving and retaining electricity. In 1756, he solved a prize question in the Ladies' Diary, respecting the nature and concomitant circuni stances of shooting stars; and, in September, 1759, he inserted in The Gentleman's Magazine, an account of the laws, by which the tourmalin exhibits its electric states during the time of heating and cooling. Without going into a detail of Mr. Canton's subsequent experiments, it will be sufficient to state their results as communicated to that the Royal Society. He proved that the attractive power of magnets is less, the higher the temperature; water is not, as before concluded, incompressible; that phosphorus may be produced from common oyster-shells; and that the luminousness of the sea arises from the putrefaction of its animal substances. Several of his papers, besides those mentioned, have been published in Priestley's History of Philosophy. Mr. Canton died on the 22nd of March, 1772, highly respected by his private friends, and deeply regretted by the philosophical world. JOHN LANDEN. JOHN LANDEN was born at Peakirk, near Peterborough, in Northamptonshire, in January, 1719. Though he was one of our most eminent mathematicians, little has been recorded of his life, beyond an account of his writings, which must form, therefore, the chief subject of the present memoir. As early as 1744, we find him a writer in the Ladies' Diary; and he was soon among the principal of those who contributed to the support of that celebrated periodical, in which almost every English mathematician of eminence has, at one time or other of his life, become a candidate for fame. In 1754, he published, in the Philosophical Transactions, an Investigation of some Theorems, which suggest several very remarkable properties of the circle, and are, at the same time, of considerable use in resolving fractions, the denominators of which are certain multinomials, into more simple ones, and by that means, facilitate the computation of fluents. In 1755, he published a volume of about one hundred and eighty pages, entitled Mathematical Lucubrations, containing a variety of tracts relative to the rectification of curve lines, the summation of series, the method of finding fluents, and other branches of the higher mathematics. The title to this publication, we are informed, by Mr. Hutton, was made choice of by the author, as a means of informing the world, that the study of the mathematics was, at that time, rather the pursuit of his leisure hours, than his principal employment, which appears to have been that of a farmer. This business he carried on at the village of Walton, near Peterborough, till the year 1762, when he removed to Milton, the seat of Earl Fitzwilliam, upon his being appointed land-steward to that nobleman. He had, in the meantime, published in the Philosophical Transactions, a new method of computing the sums of certain series; and, by subscription, his Discourse on the Residual Analysis, in which he resolved a great variety of problems, by an entirely new mode of reasoning, and pointed out the seperior elegance of his method to that which had been derived from the fluxionary calculus. In 1764, he published the first book of the Residual Analysis, in which he applies it to the drawing of tangents, and finding the properties of curve lines; to describing their involutes and evolutes; finding the radius of curvature, their greatest and least ordinates, and points of contrary fluxure; and to the determination of their cusps, and the drawing of asymptotes. He proposed, in the second book, to show its application to a great variety of mechanical and physical problems; but he never found leisure to put his papers on this subject in order for the press. In January, 1766, he was elected a fellow of the Royal Society; and, in 1768, published, in their Transactions, his specimen of a new method of comparing curvilineal areas, by which many such areas may be compared, as have not yet appeared to be comparable by any other method; a circumstance of great importance in that part of natural philosophy which relates to the doctrine of motion. In the Transactions for 1770, he gave some new theorems for completing the whole area of curvelines; and, in the same work, for 1771, appeared his Disquisition concerning certain Fluents, which are assignable by the arcs of the conic sections; where are investigated some new and useful theorems for computing such fluents. This subject had previously been considered by Maclaurin and D'Alembert, but some of the theorems of these celebrated mathematicians, being in part expressed by the difference between an hyperbolic arc and its tangent, and that difference not being directly attainable when the arc and its tangent both became infinite, as they will do, when the whole fluent is wanted, though such fluent be finite; these theorems, therefore, fail in such cases, and the computation becomes impracticable without further help. Mr. Landen removed this defect, by assigning the limit of the difference between the hyperbolic arc and its tangent, while the point of contact is supposed to be removed to an infinite distance from the vertex of the curve; and he concludes the paper by stating a curious and remarkable property relating to pendulous bodies, which is deducible from those theorems. He also published, in 1771, his Animadversions on Dr. Matthew Stewart's Computation of the Sun's Distance from the Earth, a work written in a most unnecessary and disgraceful style of acrimony. In 1775, he gave the investigation of a general theorem, which he had promised in 1771, for finding the length of any curve of a conic hyperbola, by means of two elliptic arcs; and he observes, that by the theorems there investigated, both the elastic curve and the curve of equable recess from a given point, may be constructed in those cases where Maclaurin's elegant method fails. In 1777, appeared his New Theory of the Motion of Bodies revolving about an axis in free space, when that motion is disturbed by some extraneous force, either percussive or accelerative. He was not aware, at the time, that it had been doubted, whether there is any solid whatever besides the sphere, in which any line passing through the centre of gravity will be a permanent axis of rotation; but, subsequently finding this stated in the Opuscules of D'Alembert, he reconsidered the subject, and succeeded in pointing out several bodies, which, under certain dimensions, have that remarkable property. He published this paper in a volume of memoirs, which appeared in 1780, and which contains also a large appendix, with a complete collection of theorems for the calculation of fluents, principally investigated by himself. In 1781, 1782, and 1783, he published, successively, three small tracts, on the summation of converging sines, in which, with great skill, he explained and extended the theorems of Demoivre, Stirling, and Thomas Simpson. In the beginning of 1782, Mr. Landen had made such improvements in his theory of rotatory motion, that he thought himself able to give a solution of the general problem, mentioned above, namely, to determine "the rotatory motion of a body of any form whatever, revolving, without restraint, upon any axis passing through its centre of gravity." His solution, however, differed so materially from that of D'Alembert, that he began to suspect its correctness, and, for the present, therefore, deferred making it public. He was further confirmed in his doubts on the subject, by finding that Euler's solution of the same problem agreed exactly with that of D'Alembert. The perspicuity of Euler's method enabled him to discover where the difference lay between that and his own, which he now resolved to revise with the greatest attention. He went over his process again and again, with the utmost circumspection; and, being every time more convinced that his own solution was right, and theirs wrong, he at length gave it to the public, in the seventy-fifth volume of the Transactions, for 1785. The solutions, however, of D'Alembert and Euler were still preferred by many eminent mathematicians; and that of Mr. Landen was attacked by the Rev. Charles Wildbore, in a paper on spherical motion, in the Transactions for 1790. He, in consequence, again revised, and greatly extended his own solution, of the truth of which he still remained persuaded, although that of Euler's was further confirmed by Frisi, in his Cosmographia, and by Euler himself, on a revision of the process. Mr. Landen I carried on these and several other investigations, during the intervals of that agonizing disorder, the stone, with which he had, for many years, been afflicted, and which proved fatal to him on the 15th of January, 1790. After his death appeared the second volume of his Memoirs, which contain, among other important papers, a solution of the general problem concerning rotatory motion, the resolution of the problem relative to the motion of a top, and an investigation of the precession of the equinoctial points, in which he had the honour of detecting, for the first time, the miscalculation of Sir Isaac Newton, in his celebrated solution of the same problem. Mr. Landen, undoubtedly, ranks very high as a mathematician; but his character appears to have been, in some respects, far from amiable. He possessed a coarseness of mind, which not only made him treat his inferiors with contempt, but was displayed in his controversies with such men as Euler and Matthew Stewart, in language equally disgraceful to, and unworthy of, a man of genius. From the contrast between his manners, and those of his noble friend, the Earl of Fitzwilliam, the villagers are said to have been in the habit of exclaiming, when they saw them pass together, "There goes Lord Landen and Mr. Fitzwilliam." It is a fact, that his manuscripts were sold for waste paper, to the shopkeepers of | Peterborough. JOHN SMEATON. perpetual screw in brass, a thing little known at that day." THIS eminent mechanic was born at Austhorpe, near Leeds, in Yorkshire, on the 28th of May, 1724. He was the In 1750, he commenced business in son of an attorney, and was educated Holborn, as a mathematical and philowith a view to that profession himself; sophical instrument maker; and, in but his own taste gave him a preference the following year, made two nautical for scientific pursuits, which his father, voyages, for the purpose of trying a at length, allowed him to follow. He machine he had invented for measuring had developed his mechanical bent of a ship's way at sea. A variety of inmind at a very early period, having, it genious contrivances, communicated by is said, used workmen's tools for play-him to the Royal Society, procured his things, and made machines while in petticoats. His biographer, Mr. Holmes, in describing his occupations at the age of eighteen, observes, "he forged his iron and steel, and melted his metal : he had tools of every sort for working in wood, ivory, and metals: and had made a lathe, by which he had cut a election as a member of that body, in 1753; and he subsequently obtained their gold medal for a paper, entitled An Experimental Inquiry concerning the Natural Powers of Water and Wind to Turn Mills and other Machines depending on Circular Motion. By the experiments alluded to in this paper, he had discovered such improvements as augmented the powers of wind and water, as applied to mechanism, by at least one-third. In 1754, he visited Holland and the Netherlands, for the purpose of making himself acquainted with the mechanical contrivances of those countries; and, on his return, he followed the business of a civil engineer, in which his abilities enabled him to become so famous. These had rendered him so celebrated, that, in the year 1755, on the destruction, by fire, of the Eddystone lighthouse, he was recommended, by the Earl of Macclesfield, president of the Royal Society, as the fittest person in the kingdom to reconstruct it. He was accordingly appointed the architect of this precarious structure, which he completed in 1759, with a durability which the tempests of seventy years have failed to undermine. The reader need not, perhaps, be told, that this edifice is of stone; that the foundations are let into the socket of the rock on which it stands; and that the cement used is the lime of Watchet, from whence Mr. Smeaton contrived to bring it in cyder casks, as the proprietors would not suffer it to be exported in its crude state. His account of the progress of the work, with the history of the two preceding light-houses, is most interesting and curious; particularly as relates to the first light-house, the builder of which, one Winstanley, was swept, with the whole edifice, into the waves, on the night of the 26th of November, 1708. Mr. Smeaton was now considered the most eminent engineer in the kingdom, and there were few public works upon which he was not employed. In 1764, he was chosen one of the receivers of the Derwentwater estate, annexed to Greenwich Hospital, which he greatly benefited by his improve, ments. He rendered the river Calder navigable; gave the first plan and survey for a communication between the Frith and the Clyde; and effected various improvements in Ramsgate harbour, to which he was appointed engineer. The management of the Greenwich and Deptford water-works was also committed to him, which he superintended with his usual ability. He died of a paralytic stroke, at his native place, on the 8th of September, 1792. He Besides the papers contributed by him to the Philosophical Transactions, he wrote several others, in connexion with his professional employment, which appeared posthumously, in three volumes, octavo, under the title of Reports made on Various Occasions, in the course of his employment as an Engineer. Smeaton's character appears to advantage, both in his private and professional relations. was somewhat hasty and peremptory in his disposition, but neither vain nor obstinate; of a sound judgment and ready invention; a lover of science for its own sake, pursuing it rather as a means of becoming useful than famous, of becoming famous than wealthy. He spent much of his time in astronomy, and had fitted up an observatory in his own house, furnished with some curious instruments of his own contriving. The following anecdote is told of Smeaton:-He was a frequent guest at the table of the Duke of Queensbury. and, on one occasion, having, out of complaisance, been induced to join a party at Pope Joan, his attention was only seriously called to the game by finding that the stake which he had, as dealer, to double in Pope, had amounted to almost as much as he possessed. Instead, therefore, of attempting to supply it, he took out a smail piece of paper, and wrote on it an assignment of the chief part of his property. The duke, who requested to peruse it, took the hint in a manner which relieved Smeaton; and, it is said, never afterwards played himself but for the merest trifle. JOHN HOPE. JOHN HOPE, the son of a surgeon, and grandson of Lord Rankeilar, was born at Edinburgh, on the 10th of May, 1725. Having received the rulimets of his education at Dalkeith School, he entered the university of his |