and another to the Lord Provost of Edinburgh on the subject, part of each of which we shall transcribe as curiosities. In his letter to Mr. Maclaurin he expresses himself thus: "I am very glad to hear that you have a prospect of being joined to Mr. James Gregory in the Professorship of the Mathematics at Edinburgh; not only because you are my friend, but principally because of your abilities; you being acquainted as well with the new improvements of mathematics as with the former state of those sciences. I heartily wish you good success, and shall be heartily glad to hear of your being elected." In his letter to the Lord Provost he writes thus: "I am glad to understand that Mr. Maclaurin is in good repute among you for his skill in mathematics, for, I think, he deserves it very well; and to satisfy you that I do not flatter you, and also to encourage him to accept the place of assisting Mr. Gregory, in order to succeed him, I am ready, if you please to give me leave, to contribute 201. per annum towards a provision for him till Mr. Gregory's place becomes void, if I live so long, and I will pay it to his order in London." In November 1725, he was introduced into the University of Edinburgh. The number of his pupils amounted to about 100 every year, and being of different standing and proficiency, he was obliged to divide them into four or five classes, in each of which he employed a full hour every day from the first of November to the first of June. In the first class he taught the first six Books of Euclid, plain trigonometry, practical geometry, the elements of fortification, and an introduction to algebra. In the second class he taught algebra, the 11th and 12th Books of Euclid, spherical trigonometry, conic sections, and the general principles of astronomy. The third class were instructed in the principles of astronomy and perspective, in Newton's Principia, and in the elements of fluctions. The fourth class were instructed further in fluctions, in the doctrine of chances, and read the remainder of Newton's Principia. In 1784, when Dr. Berkeley published his Analyst, Mr. Maclaurin, thinking himself included in the charge of infidelity alleged against mathematicians in general, resolved to write an answer to the Bishop's book. But the work accumulating under his hands terminated in his celebrated Treatise on Fluctions, which was published at Edinburgh, in two quarto volumes, in 1742. In this book he demonstrates the whole doctrine of fluctions without having recourse to infinitely small quantities, or any other supposition capable of being contested; but, according to the rigid method of the ancients, following a mode of demonstration often employed by Archimedes in his works. But his demonstrations are often so long and complicated, and require such severe attention to follow them, that we believe they are seldom perused by the mathematicians of the present day, who, having turned almost the whole of their attention to the analytical method, are not so capable as their predecessors of following long synthetical demonstrations. But it will be acknowledged by every person who Methodus Ju peruses the book, that all the objections of Dr. Berkeley against the doctrine of fluctions are completely refuted, and whatever doubts the most captious metaphysicians may think proper hereafter to start about the nature of infinitics, the mathematician has no more concern with them than with the famous sophisms about space and motion. In the year 1740, Mr. Maclaurin gained a prize of the Academy of Sciences, at Paris, for his Explanation of the Motion of the Tides from the Theory of Gravitation. He had only ten days to draw up this paper, and was obliged in consequence to send the first imperfect copy to the Academy. He afterwards revised the whole, and inserted it in his Treatise on Fluctions. His Algebra and his Account of Sir Isaac Newton's Discoveries were published after his death. In 1745, having been very active in fortifying the city of Edinburgh against the rebel army, he was obliged to take refuge in the North of England, where he was invited by Dr. Herring, then Archbishop of York, to reside with him during his stay in that country. During this expedition, being exposed to cold and hardships, he laid the foundation of an illness which soon after put an end to his life. This disease was an ascites, of which he died, on the 14th of June, in the forty-eighth year of his age. During his last moments he requested his friend, Dr. Monro, to account for the flashes of fire which seemed to dart from his eyes, while in the mean time his sight was failing so that he could scarcely distinguish one object from another. This singular request is a proof of the great calmness and serenity of mind which he enjoyed at his last moments. He left behind him two sons and two daughters. One of his sons was a lawyer and an eminent writer. He was appointed one of the Lords of Session, in Edinburgh, when he assumed the name of Lord Dreghorn. X. The Methodus Incrementorum, for the invention of which we are incrementoru debted to Dr. Brook Taylor, ought rather to have preceded than followed the Brook Taylor, invention of fluctions. Brook Taylor was born at Edmonton, in Middlesex, in the year 1685. In 1701, he entered St. John's College, Cambridge, and in 1708, he wrote his Tract on the Centre of Oscillation. This paper was published in the Philosophical Transactions for 1714, so that John Bernoulli's accusation of plagiarism, which he made with so much violence and indecorum, is totally groundless. In 1709, he took the degree of Bachelor of Laws. In 1712, he was elected a Fellow of the Royal Society, and the same year took his degree of LL.D. Dr. Taylor died in the year 1731, when only forty-six years of age. He published a variety of excellent papers in the Transactions, and was one of the chief writers in the memorable dispute between the Bernoullis. and the British mathematicians respecting the discovery of fluctions. His great. works were his Principles of Linear Perspective, in which he first established the true practice of the art on principles which have been followed ever since. This book was published in 1715. The same year he published his Methadus Incrementorum Directa et Inversa. A book in which the calculus is explained so concisely, and with so little developement, that Bernoulli had reason on his side when he complained of it as excessively obscure. It can scarcely be understood by any one, who is not as well acquainted with the method before he begins to read as the author himself. The subject was afterwards explained by Euler, in the first chapters of his Institutiones Calculi Differentialis, with that remarkable order and perspicuity which distinguishes all his writings. To that book therefore we refer those who wish to make themselves acquainted with the Methodus Incrementorum with the greatest facility. The subject was likewise treated by Emerson in his Method of Increments, published in 1763. This book, like all those written by Emerson, contains much valuable matter, but so very inelegantly put together, that it is scarcely possible to read it with patience. The Methodus Incrementorum, of Taylor, contains the famous theorem which The Tayloriau retains his name. It consists in a particular series to express what any function of a variable quantity becomes, when a acquires any increment whatever. Let Y be the function, and let r become Ar,* then, according to Taylor, F becomes Y + &c. If from this series, Axd Y 1.dx + Ax2dd Y + Ax3d3Y 1.2.3.dx3 + Ax1d + Y 1.2.3.4dr13 the law of which is sufficiently evident, we take the first term Y, the remainder theorem. partial dif XI. We have now brought our history of mathematics as low down as is requisite for the mathematical papers contained in the Philosophical Transactions. But several additional improvements have been made, chiefly by foreign mathematicians, which it will be sufficient merely to name here. The first is called Calculus of the calculus of partial differences, and is of great importance in many philoso- ferences. phical discussions, such as the vibration of cords, the propagation of sound, &c. It was invented by Euler in the year 1734, and afterwards by D'Alembert, who was ignorant of what Euler had done, and who first had the merit of applying it to the solution of physico-mathematical questions. Euler has represented this method under a much simpler form than D'Alembert, and on that account mathematicians have followed the notation of Euler. The method of variations is another improvement in the fluctionary calculus Method of of considerable importance. By means of it alone some of the most difficult * We use the foreign notation in this formula, the Newtonian not being fitted for expressing increments. Ar means the increment of x; dY, the fluctions of Y, &c. variations, Papers in the problems de maximis et minimis can be solved. Suppose we have a function of 1. The number of papers to which the title of Arithmetical is given in the Transactions is 14. There is an account of Dr. Wallis's edition of Archimedes' dissertation, entitled, Yaμurns, in which there is an account of the Greek arithmetic lately explained at some length by Legendre.† The next paper * We have no history of mathematics in the English language. Several have been published in Germany and France. The most detailed is that of Montucla, published in four quarto volumes. The first two volumes of that work, which bring down the history to the invention of fluctions, are excellent. The history of mathematics during the 18th century, which occupies the first half of the 3d volume, is more defective. Montucla died while it was printing, and the last part of the 3d volume, and the whole of the 4th, were supplied by La Lande, and are greatly inferior to the rest of the work. +Phil. Trans. 1676. Vol. XI. p. 567. consists of twelve problems on compound interest and annuities by Adam Martindale.*. The next is entitled, On Infinitely Finite Fractions, by Dr. Wood.† It consists of a number of infinite series of fractions with the corresponding numbers to which they are equal. The following table will give an idea of these: In these series it is evident that the numerator is always 1, and the denominators the powers of the first denominator. The next paper is entitled, An Arithmetical Paradox concerning the Chances of Lotteries. By the Hon. Francis Roberts, Esq.‡ The parodox is this. There are two lotteries, at either of which a gamester pays a shilling for a lot or throw. The first lottery, on a just computation of the odds, has 3 to 1 of the gamester; the second lottery but 2 to 1, nevertheless the gamester has the very same disadvantage, and no more, in playing at the first lottery as the second. The next paper is entitled, The Doctrine of Combinations and Alternations, improved and completed. By Major Edward Thornycroft.§ This is an elaborate and complete treatise, though the subject has been explained with more elegance by modern writers, especially by Euler. The next paper is by De Moivre, and is entitled, De Mensura Sortis, seu de Probabilitate Eventuum in Ludis a casu fortuito pendentibus. This is the first sketch of a doctrine which was afterwards fully detailed in our author's celebrated treatise on the subject, published in 1718. I The next paper is entitled, A Short Account of Negativo-affirmative Arithmetic. By Mr. John Colson. This is a contrivance by means of which all numbers larger than five are excluded, and thus the common operations of arithmetic are performed with greater facility. But as the method has not come into practice, and never probably will, it is needless to attempt a particular description of it. The next paper is The Description and Use of an Arithmetical Machine, invented by Christian Ludovicus Gersten, Professor of Mathematics, at Griefsen.** The first arithmetical machine was contrived by Pascal; another was in * Phil. Collections 1681. No. I. p. 34. + Phil. Col. 1681. No. III. p. 415. $ Phil. Trans. 1705. Vol. XXIV. p. 1961. |