cal discoveries. H. S. E. ISAACUS NEWTON, EQUES AURATUS, QUI ANIMI VI PROPE DIVINA COMETARUM SEMITAS OCEANIQUE ESTUS, PRIMUS DEMONSTRAVIT, RADIORUM LUCIS DISSIMILITUDINES, COLORUMQUE INDE NASCENTIUM PROPRIETATES, SEDULUS, SAGAX, FIDUS INTERPRES, DEI OPT. MAX. MAJESTATEM PHILOSOPHIA Asseruit, SIBI GRATULENTUR MORTALES, TALE TANTUMQUE EXTITISSE NATUS XXV. DECEMB. MDCXLII. OBIIT XX. MART. His mathemati- The mathematical discoveries of Sir Isaac Newton were so numerous and so important, that it is no easy task to give an idea of them. As his geometrical studies were conducted, in a great measure, without a master; and as the first books, to which he paid particular attention, were the Geometry of Descartes, and the Arithmetic of Infinites of Dr. Wallis, he never possessed any intimate acquaintance with the methods of the ancient mathematicians; a circumstance which, as we are informed by Dr. Pemberton, he afterwards regretted; but which, probably, contributed to render his invention so fertile and so happy. He made a great many discoveries while perusing the two works above-mentioned; and we have complete evidence that he was in possession of all his inventions before the age of 24. A complete collection of his works was published in 1779, by Dr. Horsley, in five quarto volumes, accompanied by a commentary, which, however, is any thing but complete. It is to be regretted that the mathematical world is yet destitute of a good commentary on the works of this consummate mathematician. Some of his books, indeed, have been fully commented on. Thus the Jesuits' copy of the Principia, if it has any fault, abounds too much with notes; and Stirling's Commentary on Newton's Treatise respecting Lines of the Third Order, is excellent. We have also a very elaborate Commentary on his Universal Arithmetic; and, perhaps his Optics stands in need of no other commentary than the few optical discoveries which have been made since he wrote, and which enable us to rectify one or two of his opinions on that difficult subject. Newton communicated many of his original discoveries to Dr. Barrow, at that time Professor of Mathematics, at Cambridge; and by him they were made known to various other British mathematicians. He likewise entered into a correspondence with Collins and Oldenburg, and by them was induced to x 1 a number could be interpolated between r and v r3 in the second series, corresponding to the interpolation of in the first series between 0 and 1, that this number would represent the quadrature of the circle. But Wallis could not succeed in making this interpolation; it was left for one of the first steps of Newton, in his mathematical career. Newton arranged the terms of the second series given above, under each other in order, and examined them as follows: On considering this table, Newton observed, that the first terms are all r; that the signs are alternately positive and negative; that the powers of x increase by the odd numbers; that the co-efficient of the first term is 1; that the co-efficient of all the other terms are fractions; that the denominators of these fractions are always the indices of r, in the respective terms; that the numerators in the second terms are the ordinary numbers; in the third terms, the triangular numbers; in the fourth terms, the pyramidal numbers; &c. These observations made him master of the laws that regulated the whole of the series. Hence he concluded, that having to develope in general (1 − x2)m, the series of numerators for the respective fractions in the different terms must be &c., for these are the expressions which represent the natural, triangular, and pyramidal numbers. Now this will hold good whether m be a whole number or a fraction. In the case which occasioned the investigation, namely, (122); m = 1⁄2, and, consequently, the numerators deduced from the preceding formulas are 1, †, TT, TT, &C. These, multiplied into the terms of the series, namely, r 1; m; m.m 1 M. M --- 1.m- - 2 ; 1.2 1.2.3 + Quadrature of x7 + 9 &c., & C., the ofrcle, Binomial theorem. Fluctionary calculus. x3 1 5 give us the following series: r — 8.5.23-17 — 128.9oo, &c., a series which obviously represents the area of the circular segment, corresponding to the abscissa r. This investigation led him likewise to the discovery of the binomial theorem, so celebrated in algebra, and of so much importance in an infinite number of investigations. curves. Newton had already made these discoveries, and many others, when the Logarithmotechnia of Mercator was published; which contains only a particular case of the theory just explained. But, from an excess of modesty and of diffidence, he made no attempt to publish his discoveries, expressing his conviction, that mathematicians would discover them all before he was of an age sufficiently mature to appear, with propriety, before the mathematical world. But Dr. Barrow having contracted an acquaintance with him soon after, speedily understood his value, and exhorted him not to conceal so many treasures from men of science: he even prevailed upon him to allow him to transmit to some of his friends in London a paper containing a summary view of some of his discoveries. This paper was afterwards published under the title of Analysis per Equationes Numero Terminorum Infinitas. Besides the method of extracting the roots of all equations, and of reducing fractional and irrational expressions into infinite series, it contains the application of all these discoveries to the quadrature, and the rectification of curves; together with different series for the circle and hyperbola. He does not confine himself to geometrical curves, but gives some examples of the quadrature of mechanical He speaks of a method of tangents, of which he was in possession, in which he was not stopped by surd quantities, and which applied equally well to mechanical and geometrical curves. Finally, we find in this extraordinary paper the method of fluctions and of fluents, explained and demonstrated with sufficient clearness; from which it follows, irresistibly, that before that period he was in possession of that admirable calculus: for the editors of this paper, which was published in the Commercium Epistolicum, attest that it was faithfully taken from the copy which Collins had transcribed, from the manuscript sent by Barrow. At the request of Dr. Barrow, he drew up a full account of this method, which was only described in the first tract with great conciseness. This new work he entitled Methodus Flucionum, et Serierum Infinitarum. This last book he meant to publish at the end of an English Translation of the Algebra of Kinckuysens, which he had enriched with notes. But, in consequence of the disagreeable disputes into which he had been dragged, by his discoveries respecting the different refrangibility of the rays of light, he altered his intention, and the treatise, to the great injury of mathematics, and ultimately, likewise, to the diminution of his own peace, lay unpublished till after his death. About the time that this paper of Newton's was sent to London, or about the year 1668, James Gregory published his Exercitationes, a book which con tained several important facts connected with the discoveries which Newton had made. In particular there is a new demonstration of Mercator's Series for the Hyperbola. Collins communicated Newton's discoveries to various mathematicians, and among others to Gregory. He first sent him Newton's Series for the Circle, concerning the accuracy of which Gregory at first had his doubts; but he soon discovered his mistake, and by pondering over the subject for about a year, there appears sufficient evidence from his letters in the Commercium Epis- Discovered tolicum, that he divined Newton's method, and consequently had the merit of gory. discovering the fluctionary calculus at least in part. But he declined publishing any thing on the subject, as he states in one of his letters, that he might not interfere with the rights of the original inventor. of also by Gre James Gregory, having been one of the most eminent mathematicians of his James Gregory. time, and a Fellow of the Royal Society, we must not omit this opportunity of giving a few particulars respecting his life. He was born at Aberdeen in 1639, and made great proficiency in classical learning; but still greater in mathematics and philosophy. He invented his reflecting telescope when only 24 years age, and came to London, in 1664, in order to get it constructed. But not being able to find any person capable of grinding a speculum in the truly parabolic figure, he was obliged to lay aside his design for the present, on account of the difficulty of putting it in execution. Next year he went to Italy, at that time considered as the great school for mathematics and science in general; and while in that country printed his book, entitled, Vera Circuli et Hyberbola Quadratura, a book which raised his reputation to a great height; though some of his positions were afterwards attacked by Huygens, and a controversy ensued in which Mr. Gregory was considered by mathematicians as in the right. Mr. Gregory was Professor of Mathematics in Aberdeen, and died of a fever in 1675, when only thirty-six years of age. Besides his work above-mentioned, he published likewise a Treatise on Optics; and the Commercium Epistolicum contains many egregious specimens of his analytical skill. Newton had made all his discoveries before his future rival Leibnitz had en- Leibnitz. tered upon his mathematical career. Godfrey William Leibnitz was born at Leipsic on the 23d of June, old style, 1646. His father was Professor of Ethics, and Secretary of the University of Leipsic, and dying when his son William was very young, the care of his education devolved upon his mother, who spared no expence to make it as complete as possible. His passion for knowledge was excessive, and at the age of 15 he began to embrace, with incredible ardour, every species of learning. Poetry, history, antiquities, philosophy, mathematics, jurisprudence, and law, engaged him in succession, and with equal cagerness. His poetical career did not go far, and seems to have terminated about the 15th year of his age. After this he studied philosophy and mathematics at Jena and at Leipsic, till the year 1663, when he was made Master of Arts. He next applied his mind with great assiduity to the study of the Greek philosophers, and made many efforts to reconcile the opinions of Plato and Aristotle, and those of Aristotle and Descartes. But finding his endeavours fruitless, he abandoned these barren pursuits for the law, in which he made such progress that, in the year 1667, he was made Doctor of Laws by the University of Altorf, and was offered a Professorship extraordinary in the law, which however he declined accepting. About this time Baron de Boinebourg, meeting him at an entertainment, was so much pleased with his conversation, that he procured him the patronage of the Elector of Mentz, and, in 1672, he went to Paris to manage some affairs of that Elector. Here he became acquainted with Huygens, and with some other Members of the Academy of Sciences, and began to turn his attention to mathematical investigations, but his knowledge of the subject was still very imperfect. In 1673, he went to London, where he was introduced to Oldenburg and Collins, and by their means was made acquainted with some of the Newtonian discoveries. With these gentlemen he ever after kept up a correspondence, and procured from them a great many important mathematical facts, particularly different series of Newton and Gregory; and two remarkable letters were sent him by Newton himself at the request of Mr. Collins, in which that illustrious philosopher gives a detail of the steps by which he was led to his various discoveries, and announces his possession of the fluctionary calculus, though he does not give any explanation of it. From London he returned again to France, and, in the year 1676, he came a second time to England, and there is complete evidence that, during his residence in London, he had an opportunity of perusing all the different documents afterwards published in the Commercium Epistolicum, among which there is a concise account of the fluctionary method of Newton. From London he returned to Germany by the way of Holland, having attached himself to the service of the Elector of Hanover. Here he was so much engrossed in business that he had no longer leisure to prosecute his philosophical ideas in detail; though he continued to enrich the Leipsic Acts with a great variety of writings, partly on mathematics and partly on natural philosophy. In the year 1700, when the Elector of Brandenburg, afterwards King of Prussia, instituted an Academy of Sciences at Berlin, Leibnitz, whose reputation by this time had reached the highest point, was made President, and he was continued in the office, though he could only be occasionally present at the meetings of the Academy. In the Memoires of this Academy he published a number of papers both on mathematical and other subjects. The same year was admitted a Member of the Academy of Sciences at Paris. His reputation by this time was so high that honours and pensions accumulated upon him from all quarters. He was Privy Counsellor of Justice to the Elector of Hanover, he |