EULER (Leonard)

, one of the most extraordinary, and even prodigious, mathematical geniuses, that the world ever produced. He was a native of Basil, and was born April 15, 1707. The years of his infancy were passed at Richen, where his father was minister. He was afterwards sent to the university of Basil; and as his memory was prodigious, and his application regular, he performed his academical tasks with great rapidity; and all the time that he saved by this, was consecrated to the study of mathematics, which soon became his favourite science. The early progress he made in this study, added fresh ardour to his application; by which too he obtained a distinguished place in the attention and esteem of professor John Bernoulli, who was then one of the chief mathematicians in Europe.

In 1723, M. Euler took his degree as master of arts; and delivered on that occasion a Latin discourse, in which he drew a comparison between the philosophy of Newton and the Cartesian system, which was received with the greatest applause. At his father's desire, he next applied himself to the study of theology and the oriental languages: and though these studies were foreign to his predominant propensity, his success was considerable even in this line: however, with his father's consent, he afterward returned to mathematics as his principal object. In continuing to avail himself of the counsels and instructions of M. Bernoulli, he contracted an intimate friendship with his two sons Nicholas and Daniel; and it was chiefly in consequence of these connections that he afterwards became the principal ornament of the philosophical world.

The project of erecting an academy at Petersburg, which had been formed by Peter the Great, was executed by Catharine the 1st; and the two young Bernoullis being invited to Petersburg in 1725, promised Euler, who was desirous of following them, that they would use their endeavours to procure for him an advantageous settlement in that city. In the mean time, by their advice, he made close application to the study of philosophy, to which he made happy applications of his mathematical knowledge, in a dissertation on the nature and propagation of sound, and an answer to a prize question concerning the masting of ships; to which the Academy of Sciences adjudged the accessit, or second rank, in the year 1727. From this latter discourse, and other circumstances, it appears that Euler had very early embarked in the curious and useful study of naval architecture, which he afterward enriched with so many valuable discoveries. The study of mathematics and philosophy however did not solely engage his attention, as he in the mean time attended the medical and botanical lectures of the professors at Basil.

Euler's merit would have given him an easy admission to honourable preferment either in the magistracy or university of his native city, if both civil and acade- | mical honours had not been there distributed by lot. The lot being against him in a certain promotion, he left his country, set out for Petersburg, and was made joint professor with his countrymen Hermann and Daniel Bernoulli in the university of that city.

At his first setting out in his new career, he enriched the academical collection with many memoirs, which excited a noble emulation between him and the Bernoullis; an emulation that always continued, without either degenerating into a selfish jealousy, or producing the least alteration in their friendship. It was at this time that he carried to new degrees of perfection the integral calculus, invented the calculation by sines, reduced analytical operations to a greater simplicity, and thus was enabled to throw new light on all the parts of mathematical science.

In 1730, M. Euler was promoted to the professorship of natural philosophy; and in 1733 he succeeded his friend D. Bernoulli in the mathematical chair. In 1735, a problem was proposed by the academy, which required expedition, and for the calculation of which some eminent mathematicians had demanded the space of some months. The problem was undertaken by Euler, who completed the calculation in three days, to the great astonishment of the academy: but the violent and laborious efforts it cost him threw him into a fever, which endangered his life, and deprived him of the use of his right eye, which afterward brought on a total blindness.

The Academy of Sciences at Paris, which in 1738 had adjudged the prize to his memoir Concerning the Nature and Properties of Fire, proposed for the year 1740 the important subject of the Tides of the Sea; a problem whose solution comprehended the theory of the solar system, and required the most arduous calculations. Euler's solution of this question was adjudged a master-piece of analysis and geometry; and it was more honourable for him to share the academical prize with such illustrious competitors as Colin Maclaurin and Daniel Bernoulli, than to have carried it away from rivals of less magnitude. Seldom, if ever, did such a brilliant competition adorn the annals of the academy; and perhaps no subject, proposed by that learned body, was ever treated with such force of genius and accuracy of investigation, as that which here displayed the philosophical powers of this extraordinary triumvirate.

In the year 1741, M. Euler was invited to Berlin to direct and assist the academy that was there rising into fame. On this occasion he enriched the last volume of the Miscellanies (Melanges) of Berlin with five memoirs, which form an eminent, perhaps the principal, figure in that collection. These were followed, with amazing rapidity, by a great number of important researches, which are dispersed through the memoirs of the Prussian academy; a volume of which has been regularly published every year since its establishment in 1744. The labours of Euler will appear more especially astonishing, when it is considered, that while he was enriching the academy of Berlin with a prosusion of memoirs, on the deepest parts of mathematical science, containing always some new points of view, often sublime truths, and sometimes discoveries of great importance; he still continued his philosophical contributions to the Petersburg academy, whose memoirs display the marvellous fecundity of his genius, and which granted him a pension in 1742.

It was with great difficulty that this extraordinary man, in 1766, obtained permission from the king of Prussia to return to Petersburg, where he wished to pass the remainder of his days. Soon after his return, which was graciously rewarded by the munificence of Catharine the 2d, he was seized with a violent disorder, which ended in the total loss of his sight. A cataract, formed in his left eye, which had been essentially damaged by the loss of the other eye, and a too close application to study, deprived him entirely of the use of that organ. It was in this distressing situation that he dictated to his servant, a taylor's apprentice, who was absolutely devoid of mathematical knowledge, his Elements of Algebra; which by their intrinsic merit in point of perspicuity and method, and the unhappy circumstances in which they were composed, have equally excited wonder and applause. This work, though purely elementary, plainly discovers the proofs of an inventive genius; and it is perhaps here alone that we meet with a complete theory of the analysis of Diophantus.

About this time M. Euler was honoured by the Academy of Sciences at Paris with the place of one of the foreign members of that learned body; after which, the academical prize was adjudged to three of his memoirs, Concerning the Inequalities in the Motions of the Planets. The two prize questions proposed by the same Academy sor 1770 and 1772 were designed to obtain from the abours of astronomers a more perfect Theory of the Moon. M. Euler, assisted by his eldest son, was a competitor for these prizes, and obtained them both. In this last memoir, he reserved for farther consideration several inequalities of the moon's motion, which he could not determine in his first theory, on account of the complicated calculations in which the method he then employed had engaged him. He afterward revised his whole theory, with the assistance of his son and Messrs Krafft and Lexell, and pursued his researches till he had constructed the new tables, which appeared, together with the great work, in 1772. Instead of confining himself, as before, to the fruitless integration of three differential equations of the second degree, which are furnished by mathematical principles, he reduced them to the three ordinates, which determine the place of the moon: he divided into classes all the inequalities of that planet, as far as they depend either on the elongation of the sun and moon, or upon the eccentricity, or the parallax, or the inclination of the lunar orbit. All these means of investigation, employed with such art and dexterity as could only be expected from a genius of the first order, were attended with the greatest success; and it is impossible to observe without admiration, such immense calculations on the one hand, and on the other the ingenious methods employed by this great man to abridge them, and to facilitate their application to the real motion of the moon. But this admiration will become astonishment, when we consider at what period and in what circumstances all this was effectuated. It was when our author was totally blind, and consequently obliged to arrange all his computations by the sole powers of his memory and his genius: it was when he was embarrassed in his domestic affairs by a dreadful sire, | that had consumed great part of his substance, and forced him to quit a ruined house, every corner of which was known to him by habit, which in some measure supplied the want of sight. It was in these circumstances that Euler composed a work which alone was sufficient to render his name immortal.

Some time after this, the famous oculist Wentzell, by couching the cataract, restored sight to our author; but the joy produced by this operation was of short duration. Some instances of negligence on the part of his surgeons, and his own impatience to use an organ, whose cure was not completely sinished, deprived him a second time and for ever of his sight: a relapse which was also accompanied with tormenting pain. With the assistance of his sons, however, and of Messrs Krafft and Lexell, he continued his labours: neither the infirmities of old age, nor the loss of his sight, could quell the ardour of his genius. He had engaged to furnish the academy of Petersburg with as many memoirs as would be sufficient to complete its acts for 20 years after his death. In the space of 7 years he transmitted to the Academy above 70 memoirs, and above 200 more, left behind him, were revised and completed by a friend. Such of these memoirs as were of ancient date were separated from the rest, and form a collection that was published in the year 1783, under the title of Analytical Works.

The general knowledge of our author was more extensive than could well be expected in one who had pursued, with such unremitting ardour, mathematics and astronomy as his favourite studies. He had made a very considerable progress in medical, botanical, and chemical science. What was still more extraordinary, he was an excellent scholar, and possessed in a high degree what is generally called erudition. He had attentively read the most eminent writers of ancient Rome; the civil and literary history of all ages and all nations was familiar to him; and foreigners, who were only acquainted with his works, were astonished to find in the conversation of a man, whose long life seemed solely occupied in mathematical and physical researches and discoveries, such an extensive acquaintance with the most interesting branches of literature. In this respect, no doubt, he was much indebted to a very uncommon memory, which seemed to retain every idea that was conveyed to it, either from reading or from meditation. He could repeat the Æneid of Virgil, from the beginning to the end, without hesitation, and indicate the first and last line of every page of the edition he used.

Several attacks of a vertigo, in the beginning of September 1783, which did not prevent his computing the motions of the aerostatic globes, were however the forerunners of his mild passage out of this life. While he was amusing himself at tea with one of his grandchildren, he was struck with an apoplexy, which terminated his illustrious career at 76 years of age.

M. Euler's constitution was uncommonly strong and vigorous. His health was good; and the evening of his long life was calm and serene, sweetened by the fame that follows genius, the public esteem and respect that are never withheld from exemplary virtue, and several domestic comforts which he was capable of feeling, and therefore deserved to enjoy.

The catalogue of his works has been printed in 50 pages, 14 of which contain the manuscript works.— The printed ones consist of works published separately, and works to be found in the memoirs of several Academies, viz, in 38 volumes of the Petersburg Acts, (from 6 to 10 papers in each volume);—in several volumes of 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. 9 of the Society of Ulyssingue; —in the Ephemerides of Berlin;—and in the Memoires de la Société Oeconomique for 1766.