Euler, Leonard

, a very eminent mathematician, was born at Basil, on the 14th of April, 1707: he was the son of Paul Euler and of Margaret Brucker (of a family illustrious in literature), and spent the first year of his life at the village of Richen, of which place his father was protestant minister. Being intended for the church, his father, who had himself studied under James Bernoulli!, taught him mathematics, as a ground-work of his other studies, or at least a noble and useful secondary occupation. But Euler, assisted and perhaps secretly encouraged by John Bernoulli, who easily discovered that he would be the greatest scholar he should ever educate, soon declared his intention of devoting his life to that pursuit. This intention the wise father did not thwart, but the son did not so blindly adhere to it, as not to connect with it a more than common improvement in every other kind of useful learn-, ing, insomuch that in his latter days men often wondered how with such a superiority in one branch, he could have been so near to eminence in all the rest. Upon the foundation of the academy of sciences at St. Petersburgh, in, 1723, by Catherine I. the two younger Bernouillis, NichoJas and Daniel, had gone thither, promising, when they set out, to endeavour to procure Euler a place in it: they accordingly wrote to him soon after, to apply his mathetics to physiology, which he did, and studied under the best naturalists at Basil, but at the same time, i. e. in 1727, published a dissertation on the nature and propagation of sound; and an answer to the question on the masting of ships, which the academy of sciences at Paris judged worthy of the accessit. Soon after this, he was called to St. Petersburgh, and declared adjutant to the mathematical class in the academy, a class, in which, from the circumstances of the times (Newton, Leibnitz, and so many other | eminent scholars being just dead), no easy laurels were to be gathered. Nature, however, who had organized so many mathematical heads at one time, was not yet tired of her miracles and she added Euler to the number. He indeed was much wanted the science of the calculus integralis, hardly come out of the hands of its creators, was still too near the stage of its infancy not to want to be made more perfect. Mechanics, dynamics, and especially hydrodynamics, and the science of the motion of the heavenly bodies, felt the imperfection. The application of the differential calculus, to them, had been sufficiently successful; but there were difficulties whenever it was necessary to go from the fluxional quantity to the fluent. With regard to the nature and properties of numbers, the writings of Fermat (who had been so successful in them), and together with these all his profound researches, were lost. Engineering and navigation were reduced to vague principles, and were founded on a heap of often contradictory observations, rather than a regular theory. The irregularities in the motions of the celestial bodies, and especially the complication of forces whitfh influence that of the moon, were still the disgrace of geometers. Practical astronomy had jet to wrestle with the imperfection of telescopes, insomuch, that it could hardly be said that any rule for making them existed. Euler turned his eyes to all these objects he perfected the calculus integralis he was the inventor of a new kind of calculus, that of sines he simplified analytical operations and, aided by these powerful help-mates, and the astonishing facility with which he knew how to subdue expressions the most intractable, he threw a new light on all the branches of the mathematics. But at Catherine’s death the academy was threatened with extinction, by men who knew not the connection which arts and sciences have with the happiness of a people. Euler was offered and accepted a lieutenancy on board one of the empress’s ships, with the promise of speedy advancement. Luckily things changed, and the learned captain again found his own element, and was named Professor of Natural Philosophy in 1733, in the room of his friend John Bernouilli. The number of memoirs which Euler produced, prior to this period, is astonishing,*

but what he did in 1735 is almost incredible, | An important calculation was to be made, without loss of time; the other academicians had demanded some months to do it. Euler asked three days—in three days he did it; but the fatigne threw him into a fever, and the fever left him not without the loss of an eye, an admonition which would have made an ordinary man more sparing of the other. The great revolution, produced by the discovery of fluxions, had entirely changed the face of mechanics; still, however, there was no complete work on the science of motion, two or three only excepted, of which Euler felt the insufficiency. He saw, with pain, that the best works on the subject, viz. “Newton’s Principia,” and “Herman’s Phoronomia,” concealed the method by which these great men had come at so many wonderful discoveries, under a synthetic veil. In order to lift this up, Euler employed all the resources of that analysis which had served him so well on so many other occasions; and thus uniting his own discoveries to those of other geometers, had them published by the academy in 1736. To say that clearness, precision, and order, are the characters of this work, would be barely to say, that it is, what without these qualities no work can be, classical of its kind. It placed Euler in the rank of the first geometricians then existing, and this at a time when John Bernouilli was still living. Such labours demanded some relaxation; the only one which Euler admitted was music, but even to this he could not go without the spirit of geometry with him. They produced together the essay on a new theory of music, which was published in 1739, but not very well received, probably, because it contains too much geometry for a musician, and too much music for a geometrician. Independently, however, of the theory, which is built on Pythagorean principles, there are many things in it which may be of service, both to composers, and to makers of instruments. The doctrine, likewise, of the genera and the modes of music is here cleared up with all the clearness and precision which mark the works of Euler. Dr. Burney remarks, that upon the whole, Euler seems not to have invented much in this treatise; and to have done little more than arrange and methodize former discoveries in a scientific and geometric manner. He may, indeed, not | have known what antecedent writers had discovered before; and though not the first, yet to have imagined himself an inventor. In 1740, his genius was again called forth by the academy of Paris (who, in 1738, had adjudged the prize to his paper on the nature and properties of fire) to discuss the nature of the tides, an important question, which demanded a prodigious extent of calculations, aud an entire new system of the world. This prize Euler did not gain alone; but he divided it with Maclaurin and D. Bernouilli, forming with them a triumvirate of candidates, which the realms of science had not often beheld. The agreement of the several memoirs of Euler and Bernouilli, on this occasion, is very remarkable. Though the one philosopher had set out on the principle of admitting vortices, which the other rejected, they not only arrived at the same end of the journey, but met several times on the road; for instance, in the determination of the tides under the frozen zone. Philosophy, indeed, led these two great men by different paths; Bernouilli, who had more patience than his friend, sanctioned every physical hypothesis he was obliged to make, by painful and laborious experiment. These Euler’s impetuous genius scorned; and, though his natural sagacity did not always supply the loss, he made amends by his superiority in analysis, as often as there was any occasion to simplify expressions, to adapt them to practice, and to recognize, by final formulae, the nature of the result. In 1741, Euler received some very advantageous propositions from Frederic the Second (who had just ascended the Prussian throne), to go and assist him in forming an academy of sciences, out of the wrecks of the Royal Society founded by Leibnitz. With these offers the tottering state of the St. Petersburgh academy, under the regency, made it necessary for the philosopher to comply. He accordingly illumined the last volume of the “Melanges de Berlin,” with five essays, which are, perhaps, the best things in it, and contributed largely to the academical volumes, the first of which was published in 1744. No part of his multifarious labours is, perhaps, a more wonderful proof of the extensiveness and facility of his genius, than what he executed at Berlin, at a time when he contrived also that the Petersburgh acts should not suffer from the loss of him. In 1744, Euler published a complete treatise of isoperimetrical curves. The same year beheld the theory of the motions of tb.e planets and | comets; the well-known theory of magnetism, which gained the Paris prize; and the much-amended translation of Robins’ s “Treatise on Gunnery.” In 1746, his “Theory of Light and Colours” overturned Newton’s “System of Emanations;” as did another work, at that time triumphant, the “Monads of Wolfe and Leibnitz.” Navigation was now the only branch of useful knowledge, for which the labours of analysis and geometry had done nothing. The hydrographical part alone, and that which relates to the direction of the course of ships, had been treated by geometricians conjointly with nautical astronomy. Euler was the first who conceived and executed the project of making this a complete science. A memoir on the motion of floating bodies, communicated to the academy of St. Petersburgh, in 1735, by M. le Croix, first gave him this idea. His researches on the equilibrium of ships furnished him with the means of bringing the stability to a determined measure. His success encouraged him to go on, and produced the great work which the academy published in 1749, in which we find, in systematic order, the most sublime notions on the theory of the equilibrium and mo. tion of floating bodies, and on the resistance of fluids. This was followed by a second part, which left nothing to be desired on the subject, except the turning it into a language easy of access, and divesting it of the calculations which prevented its being of general use. Accordingly* in 1773, from a conversation with admiral Knowles, and other assistance, out of the “Scientia Navalis,” 2 vols. 4to, was produced, the “Theorie complette de la Construction et de la Manoeuvre des Vaisseaux.” This work was instantly translated into all languages, and the author received a present of 6000 livres from the French king: he had before had 300l. from the English parliament, for the theorems, by the assistance of which Meyer made his lunar tables .† | And now it was time to collect into one systematical and continued work, all the important discoveries on the infinitesimal analysis, which Euler had been making for thirty years, and which lay dispersed in the memoirs of the different academies. This, accordingly, the professor undertook; but he prepared the way by an elementary work, containing all the previous requisites for this study. This is called “An Introduction to the analysis of Infinitesimals,” and is a work in which the author has exhausted all the doctrine of fractions, whether algebraical or transcendental, by shewing their transformation, their resolution, and their developernent. This introduction was soon, followed by the author’s several lessons on the “calculus integralis, and differentialis.” Having engaged himself to count Orlow, to furnish the academy with papers sufficient to fill their volumes for twenty years after his death, the philosopher is likely to keep his word, having presented seventy papers, through Mr. Golofkin, in the course of his life, and left two hundred and fifty more behind him; nor is there one of these that does not contain a discovery, or something that may lead to one. The most ancient of these memoirs form the collection then published, under the title of “Opuscula Analytica.” Such were Euler’s labours, and these his titles to immortality His memory shall endure till science herself is no more! Few men of letters have written so much as Euler no geometrician, has ever embraced so many objects at one time or has equalled him, either in the variety or magnitude of his discoveries. When we reflect on the good such men do their fellow-creatures, we cannot help indulging a wish (vain, alas as it is) for their illustrious course to be prolonged beyond the term allotted to mankind. Euler’s, though it has had an end, was very long and very honourable; and it affords us some consolation for his loss, to think that he enjoyed it exempt from the ordinary consequences of extraordinary application, and that his last labours abounded in proofs of that vigour of understanding which marked his early days, and which he preserved to | his end. Some swimmings in the head, which seized him on the first days of September, 1783, did not prevent his laying hold of a few facts, which reached him through the channel of the public papers, to calculate the motions of the aerostatical globes; and he even compassed a very difficult integration, in which the calculation had engaged him .* But the decree was gone forth: on the 7th of September he talked with Mr. Lexell, who had come to dine with him, of the new planet, and discoursed with him upon other subjects, with his usual penetration. He was playing with one of his grand-children at tea-time, when he was seized with an apoplectic fit. “I am dying,” said he, before he lost his senses; and he ended his glorious life a few hours after, aged seventy-six years, five months, and three days. His latter days were tranquil and serene. A few infirmities excepted, which are the inevitable lot of an advanced age, he enjoyed a share of health which allowed him to give little time to repose. Euler possessed to a great degree what is commonly called erudition he had read all the Latin classics was perfect master of ancient mathematical literature and had the history of all ages, and all nations, even to the minutest facts, ever present to his mind. Besides this, he knew much more of physic, botany, and chemistry, than could be expected from any man who had not made these sciences his peculiar occupation. “I have seen,” says his biographer, Mr. Fuss, “strangers go from him with a kind of surprise mixed with admiration; they could not conceive how a man, who for half a century had seemed taken up in making and publishing discoveries in natural philosophy and mathematics, could have found means to preserve so much knowledge that seemed useless to himself, and foreign to the studies in which he was engaged. This was the effect of a happy memory, that lost nothing of what had ever been entrusted to it nor was it a wonder that the man who was able to repeat the whole Æneis, and to point out to his hearers the first and last verses of every page of his own edition of it, should not have lost what he had learned, at an age when the impressions made upon us are the | strongest.*

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“Another proof of the strength of his memory and imagination deserves to be related. Being engaged in teaching his grandchildren geometry and algebra, and obliged, in consequence, to initiate them in the extraction of roots, he was obliged to give them numbers, which should be the powers of other numbers; these he used to make in his head; and one night, not being able to sleep, he calculated the six first powers of all the numbers above twenty, and to our great astonishment, repeated them to us several days after.”

Nothing can equal the ease with which, without expressing the least degree of ill-humour, he could quit his abstruse meditations, and give himself up to the general amusements of society. The art of not appearing wise above one’s fellows, of descending to the level of those with whom one lives, is too rare in these days not to make it a merit in Euler to have possessed it. A temper ever equal, a natural and easy chearfulness, a species of satirical wit, tempered with urbane humanity, the art of telling a story archly, and with simplicity, made his conversation generally sought. The great fund of vivacity which he had at all times possessed, and without which, indeed, the activity we have just been admiring could not have existed, carried him sometimes away, and he was apt to grow warm, but his anger left him as quickly as it came on, and there never has existed a man to whom he bore malice. He possessed a precious fund of rectitude and probity. The sworn enemy of injustice, whenever or by whomsoever committed, he used to censure and attack it, without the least attention to the rank or riches of the offender. Recent examples of this are in the recollection of all who hear me.” As he was filled with respect for religion, his piety was sincere, and his devotion full of fervour. He went through all his Christian duties with the greatest attention. Euler loved all mankind, and if he ever felt a motion of indignation, it was against the enemy of religion, particularly against the declared apostles of infidelity. He was of a very religious turn of mind. He published a New Demonstration of the Existence of God, and of the Spirituality of the Soul, which last has been admitted into several divinity schools as a standard book. With scrupulous exactness he adhered to the religion of his country, that of Calvinism, and, fortified by its principles, he was a good husband, a good father, a good friend, a good citizen, a good member of private society.

“Euler was twice married, and had thirteen children, four of whom only have survived him. The eldest son was | for some time his father’s assistant and successor the second, physician to the empress and the third a lieutenantcolonel of artillery, and director of the armory at Sesterbeck. The daughter married major Bell. From these children he had thirty-eight grand-children, twenty-six of whom are still alive. Never have I been present at a more touching sight than that exhibited by this venerable old man, surrounded, like a patriarch, by his numerous offspring, all attentive to make his old age agreeable, and enliven the remainder of his days, by every species of kind solicitude and care.”

The catalogue of his works in the printed edition makes 50 pages, 14 of which contain the ms works. The printed books consist of works published separately, and others to be found in the several Petcrsburgh acts, in 38 volumes, (from 6 to 10 papers in each volume) in the Paris acts in 26 volumes of the Berlin acts (about 5 papers to each volume) in the “Acta Eruditorum,” in 2 volumes; in the “Miscellanea Taurinensia” in vol. IX. of the society of Ulyssingue in the “Ephemerides de Berlin;” and in the “Memoires de la Societe” CEconomique for 1766.“His” Letters on Physics and Philoophy" were translated by the late Dr. Henry Hunter, and published in 1802, 2 vols. 8vo. ¹