LEONARD, professor of mathematics, member of the Imperial Academy of Petersburg, ancient director of the Royal Academy of Berlin, and fellow of the Royal Society of London, as also a corresponding member of the Royal Academy of Sciences at Paris, was born at Basil on the 15th of April 1707, of reputable parents. The years of his infancy were passed in a rural retreat at the village of Richen, of which place his father was minister. Being sent to the university of Basil, he attended regularly the different professors; and as his memory was prodigious, he performed his academical tasks with uncommon rapidity; but all the time he thus gained was consecrated to geometry, which soon became his favourite study. The early progress which he made in this science only gave new ardour to his application; and thus he obtained a distinguished place in the esteem of John Bernoulli, who was at that time one of the first mathematicians in Europe, as well as in the friendship of Daniel and Nicolas Bernoulli, who were already the rivals of their illustrious father. In 1723 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 afterwards 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; but, with his father's consent, he returned to geometry as his principal pursuit. He continued to avail himself of the counsels and instructions of John Bernoulli; he contracted, as already stated, an intimate friendship with his two sons, Daniel and Nicolas; and it was in consequence of these connexions that he became afterwards the principal ornament of the Academy of Petersburg. The project of erecting this academy, which had been formed by Peter the Great, was executed by Catherine I.; and the two young Bernoullis being invited to Petersburg in 1725, promised Euler, who was desirous of following them, that they would use their utmost endeavours to procure for him an advantageous settlement in that city. In the mean time, by their advice, he applied himself with ardour to the study of physiology, to which he made a happy application of his mathematical knowledge; and he also attended the medical lectures of the most eminent professors of Basil. This study, however, did not wholly engross his time, nor even relax the activity of his powerful and comprehensive mind in the cultivation of other branches of natural science. For whilst he was keenly engaged in physiological researches, he composed 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 early embarked in the curious and important study of navigation, which he afterwards enriched with so many valuable discoveries.

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 academical honours had not been there distributed by lot. But chance having decided against him in regard to a certain situation to which he aspired, 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 the commencement of his new career he enriched the academical collection with many memoirs, which excited a noble emulation between him and the Bernoullis; and this emulation 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 a new degree of perfection the integral calculus, invented the calculation of sines, reduced analytical operations to a greater simplicity, and threw new light on all the parts of mathematical science. In 1730 he was promoted to the professorship of natural philosophy; and in 1733 he succeeded his friend Daniel Bernoulli in the mathematical chair. In 1735 a problem was proposed by the academy which required expedition, and for the solution of which several eminent mathematicians had demanded the space of some months. The problem was solved by Euler 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. 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; a problem the solution of which required the most arduous calculation and comprehended the theory of the solar system. Euler's discourse on this question was considered as a masterpiece 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 inferior reputation. Rarely if ever did such a brilliant competition adorn the annals of the academy; and no subject, perhaps, proposed by that learned body, was ever treated with such accuracy of investigation and force of genius, as that which here displayed the philosophical powers of these three extraordinary men.

In the year 1741 Euler was invited to Berlin in order to augment the lustre of the academy which was there rising into fame. He enriched the last volume of the Miscellanea or Miscellanies of Berlin with five memoirs, which make an eminent, perhaps the principal, figure in that collection; and these were followed, with an astonishing rapidity, by a great number of important researches, which are scattered throughout the memoirs of the Prussian academy, of which a volume was regularly published every year. The labours of Euler will appear more astonishing when it is considered, that whilst he was enriching the Academy of Berlin with a prodigious number of memoirs on the deepest parts of mathematical science, containing always some new views, often sublime truths, and sometimes discoveries of great importance, he did not discontinue his philosophical contributions to the Academy of Petersburg, which granted him a pension in 1742, and the memoirs of which display the marvellous fecundity of Euler's genius. It was with much difficulty that this great man obtained, in 1766, permission from the king of Prussia to return to Petersburg, where he desired to pass the remainder of his days. Soon after his return, which was graciously rewarded by the munificence of Catherine II., he was seized with a violent disorder, which terminated in the total loss of his sight. A cataract, having formed in his left eye, which had been essentially damaged by a too ardent 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 tailor's apprentice, who was absolutely devoid of mathematical knowledge, his Elements of Algebra; which, by their intrinsical merit in point of perspicuity and method, and the unhappy circumstances in which they were composed, have excited equal applause and astonishment. This work, though purely elementary, discovers the characteristics of an inventive genius; and it is here alone that we meet with a complete theory of the analysis of Diophantus.

About this time Euler was honoured by the Academy of Sciences at Paris with the place of foreign member of that learned body; and after this, 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 for 1770 and 1772 were designed to obtain from the labours of astronomers a more perfect theory of the moon. Euler, assisted by his eldest son, was a competitor for these prizes, and obtained both. In this last memoir he reserved for further 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. But he had the courage afterwards to review his whole theory, with the assistance of his son and Messrs Kraft and Lexell, and to pursue his researches until 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; and 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 analytical 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 effected. It was when he was totally blind, and consequently obliged to arrange all his computations by the sole powers of his memory and his genius; when he was embarrassed in his domestic circumstances by a dreadful fire, which had consumed the greater part of his substance, and forced him to quit a ruined house, every corner of which was known to him by a habit that in some measure supplied the place of sight;—it was in these circumstances, and under these privations, that Euler composed a work, which alone is sufficient to render his name immortal. The heroic patience and tranquillity of mind which he displayed need no eulogy here; and he derived them not only from the love of science, but from the power of religion. His philosophy was too genuine and sublime to stop its analysis at mechanical causes; it led him to that divine philosophy of religion which enables human nature, and is alone capable of forming a habit of true magnanimity and patience under suffering.

Some time after this the celebrated Wenzell, by couching the cataract, restored Euler's sight; but the satisfaction and joy which this successful operation produced were of short duration. Some instances of negligence on the part of his surgeons, and his own impatience to use an organ the cure of which was not complete, deprived him of vision a second time; and this relapse was accompanied with tormenting pain. With the assistance of his sons, and of Messrs Kraft and Lexell, however, he continued his labours; neither the loss of his sight nor the infirmities of an advanced age being sufficient to damp the ardour of his genius. Having engaged to furnish the Academy of Petersburg with as many memoirs as would be sufficient to complete its acts for twenty years after his death, he in the space of seven years transmitted to the academy above seventy memoirs, and left above two hundred more, which were revised and completed by the author of this notice. Such of the memoirs as were of ancient date were separated from the rest, and form a collection which was published in the year 1783, under the title of Analytical Works.

Euler's knowledge was more universal than could well be expected in one who had pursued with such unremitting ardour mathematics and astronomy as his favourite studies. He had made very considerable progress in medical, botanical, and chemical science; he was an excellent scholar, and possessed what is generally called erudition in a very high degree. He had read, with attention and taste, 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 to have been wholly 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 an uncommon memory, which seemed to retain every idea that was conveyed to it, either from reading or 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 which he used.

But several attacks of vertigo, in the beginning of September 1783, which did not prevent his calculating the motions of the aerostatical globes, proved the forerunners of his mild and happy passage from this scene to a better. His death was sudden; he ceased to calculate and to live at nearly the same instant of time. Whilst he was amusing himself at tea with one of his grandchildren, he was struck with apoplexy, which terminated his illustrious career at the age of seventy-six. Euler, says Condorcet, was one of those men whose genius was equally capable of the greatest efforts and of the most continued labour; who multiplied his productions beyond what might have been expected from human strength, and who notwithstanding was original in each; whose head was always occupied, and his mind always calm. The nature of his pursuits, by withdrawing him from the world, preserved that simplicity of manners for which he was originally indebted to his character and his education; and he employed none of those means to which men of real merit have sometimes recourse in order to enhance the importance of their discoveries. It is true that fecundity such as his renders useless all the little calculations of self-love; but still great lucidity of mind and uprightness of character are necessary to trace, as he has done, the history of his thoughts, even when his investigations have proved fruitless, or the results disappointed the expectations which he had formed. Euler's constitution was uncommonly vigorous; his health was good; and the evening of his long life was serene, being sweetened by the fame which follows genius, the public esteem and respect which are never withheld from exemplary virtue, and several domestic comforts, which he was capable of feeling, and therefore deserved to enjoy. The works which Euler published separately are, 1. Dissertatio physica de Sono, Bâle, 1727, in 4to; 2. Mechanica, sive Motus scientia, analytice exposita, Petersburg, 1736, in 2 vols. 4to; 3. Einleitung in die Arithmetik, or Introduction to Arithmetic, ibid. 1738, 2 vols. 8vo, in German and Russian; 4. Tentamen Novae Theoriae Musicae, ibid. 1739, in 4to; 5. Methodus inventiendi Linæa curvae, maximæ minimæ proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti, Lausanne, 1744, in 4to; 6. Theoria motuum Planetarum et Cometarum, continens methodum faciem ex aliquot observationibus orbitas, etc. determinandi, Berlin, 1744, in 4to; 7. Beantwortung, etc. or Answers to different Questions respecting Comets, ibid. 1744, in 8vo; 8. Neue Grundsatze, etc. or New Principles of Artillery, translated from the English of Benjamin Robins, with notes and illustrations, ibid. 1745, in 8vo; 9. Opuscula varii argumenti, ibid. 1746–1751, in 3 vols. 4to; 10. Nouve et correcte Tabula ad loca Lunæ computanda, ibid. 1746, in 4to; 11. Tabula Astronomica Solis et Lunæ, ibid. 4to; 12. Gedanken, etc. or Thoughts on the Elements of Bodies, ibid. 4to; 13. Retung der Gotlichen Offenbarung, etc. Defence of Divine Revelation against Freethinkers, ibid. 1747, in 4to; 14. Introductio in Analysis Infinitorum, Lausanne, 1748, in 2 vols. 4to; 15. Scientia Nauzialis, seu Tractatus de construendis ac dirigendis Navibus, Petersburg, 1749, in 2 vols. 4to; 16. Theoria motus Lunæ, Berlin, 1753, in 4to; 17. Dissertatio de principio minimo actionis, una cum examine Objectionum cl. prof. Königig, ibid. 1753, in 8vo; 18. Institutiones Calculi Differentialis, cum ejus usu in analysi Infinitorum ac doctrina Serierum, ibid. 1755, in 4to; 19. Constructio Lentiæ Objectiværum, etc. Petersburg, 1762, in 4to; 20. Theoria motus Corporum solidorum sex rigidorum, Rostoch, 1765, in 4to; 21. Institutiones Calculi Integralis, Petersburg, 1768–1770, in 3 vols. 4to; 22. Lettres à une Princesse d'Allemagne sur quelques sujets de Physique et Philosophie, Petersburg, 1768–1772, in 3 vols. 8vo; 23. Anleitung zur Algebra, or Introduction to Algebra, Petersburg, 1770, in 8vo; 24. Dioptrica, Petersburg, 1767–1771, in 3 vols. 4to; 25. Theoria motuum Lunæ nova methodo practicata, ibid. 1772, in 4to; 26. Nouve Tabula Lunares, ibid. in 8vo; 27. Théorie complète de la construction et de la manœuvre des Vaissceaux, ibid. 1773, in 8vo; 28. Éclaircissements sur les établissements en faveur tant des Veneux que des Moris, without a date; 29. Opuscula Analytica, Petersburg, 1783–1785, in 2 vols. 4to.