Bernoullis being invited to Petersburgh 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 math. matical knowled, e, 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 embark ed in the curious and useful study of naval architecture, which he afterwards 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 academical honours had not been there distributed by lot. The lot being against him in a certain promotion, he left his country, set out for Petersburgh, and was made joint professor with his countrymen, Herman 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 astonish

ment 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 afterwards brought on a total blindness.

The Academy of Sciences at Paris, which in 1768 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 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 less magnituse. Seldom, if ever, did such a brilliant competition adorn the annals of the Academy; and perhaps no subject, proposed by that learn. ed body, was ever treated with such force of genius and accuracy of investigation, as that which here displayed the philosophical powers of that extraordinary triumvirate.

In the year 1741, M. Euler was invited to Berlin, to direct and assist the Acade. my 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 profusion 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 Petersburgh Academy, whose memoirs display the surprising fecundity of his genius, and which granted him a pension in 1742.

It was with great difficulty that this extraordinary man, 1766, obtained permission from the King of Prussia to return to Petersburgh, where he wished to pass the remainder of his days. Soon

after his return, which was graciously rewarded by the munificence of Catharine the Second, 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 tailor'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, for 1770 and 1772, were designed to obtain from the labours of astronomers a more perfect theory of the moon. M. Euler, assisted by his eldest son, was a competitor for these prizes, and obtain ed them both. In this last memoir, he reserved for farther consideration several inequalities of the moon's motion, which he would not determine in his first theory, on acccount 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, 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 would 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 effected. It was when our author was totally blind, and consequently obliged to arrange all his computations by the sole powers of his memory, and of his genius: it was when he was embarrassed in his domestic affairs by a dreadful fire, that had consumed great part of his property, 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 occulist Wenzell, by couching the cataract, restored our author to sight; 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 finished, 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 engag ed to furnish the academy of Petersburgh with as many memoirs as would be suffi cient to complete its acts for twenty years after his death. In the space of seven years he transmitted to the academy above seventy memoirs, and above two hundred 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 of 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 would repeat the Eneid 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 grand children, he was struck with an apoplexy, which terminated his illustrious career at seventysix 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 fifty pages, fourteen 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 thirtyeight volumes of the Petersburgh acts, (from six to ten papers in each volume ;) in several volumes of the Paris acts; in twenty-six volumes of the Berlin acts, (about five papers to each volume ;) in the Acta Eruditorum, in two volumes; in the Miscellanea Taurinensia, in vol. ix. of the Society of Ulyssingue; in the Ephemerides of Berlin; in the Memoires de la Société Oeconomique, for 1766.

EVOLUTE, in the higher geometry, a curve, which, by being gradually opened, describes another curve. Such is the curve BC F; (Plate V. Miscel. fig. 7) for if a thread, FC M. be wrapped about, or applied to the said curve, and then un

wound again, the point M, thereof, will describe another curve, A M M, called by M. Huygens, a curve described from evolution. The part of the thread, M C, is called the radius of the evolute, or of the osculatory circle described on the centre, C, with the radius, M C.

Hence, 1. When the point, B, falls in A, the radius of the evolute, M C, is equal to the arch, B C; but if not, to A B, and the arch B C. 2. The radius of the evolute, C M, is perpendicular to the curve, A M. 3. Because the radius, M C, of the evolute continually touches it, it is evident, from its generation, that it may be described through innumerable points, if the tangents in the parts of the evolute are produced until they become equal to their corresponding arches. 4. The evolute of the common parabola is a parabola of the second kind, whose parameter is 27 of the common one. 5. The evolute of a cycloid is another cycloid equal and similar to it. 6. All the arches of evolute curves are rectifiable, if the radii of the evolute can be expressed geometrically.

EVOLUTION. See ALGEBRA.

EVOLUTION, in the art of war, the motion made by a body of troops, when they are obliged to change their form and disposition, in order to preserve a post, or occupy another, to attack an enemy with more advantage, or to be in a condition of defending themselves the better. It consists in doublings, counter-marches, conversions, &c. A battalion doubles the ranks, when attacked in front or rear, to prevent its being flanked, or surrounded; for then a battalion fights with a larger front. The files are doubled, either to accommodate themselves to the necessity of a narrow ground, or to resist an enemy which attacks them in flank; but if the ground will allow it, conversion is much preferable, because, after conversion, the battallion is in its first form, and opposes the file leaders, which are generally the best men, to the enemy; and likewise, because doubling the files in a new or not well disciplined regiment, they may happen to fall into disorder.

EVOLVULUS, in botany, a genus of the Pentandria Tetragynia class and order. Natural order of Campanaceæ. Convolvuli, Jussieu. Essential character: calyx five-leaved; corolla five-cleft, rotate; capsule three-celled; seeds solitary. There are seven species, all natives of the East or West Indies.

EUONYMUS, in botany, English spin

le-tree, a genus of the Pentandria Monogynia class and order. Natural order of Dumose. Rhamni, Jussieu. Essential character: calyx five-petalled; capsule five-sided, five-celled, five-valved, coloured, seeds calyptred, or veiled. There are eight species. These are trees or shrubs; the smaller branches or twigs four-cornered; the leaves opposite; peduncles axillary, solitary, opposite, oneflowered, sometimes many flowered, disposed in umbels.

EUPAREA, in botany, a genus of the Pentandria Monogynia class and order. Essential character: calyx five-leaved; corolla five or twelve petalled; berry superior, one-celled; seeds very many, adhering to a free receptacle. There is only one species, viz. E. amoena, a native of New Holland, and Terra del Fuego.

EUPATORIUM, in botany, English hemp agrimony, a genus of the Syngenesia Polygamia Equalis class and order. Natural order of Composite Discoideæ. Corymbiferæ, Jussieu. Essential character: calyx imbricate, oblong; style cloven half way, long; down, plumose; receptacle naked. There are forty-nine species. The species native in the United States are twenty-six in number, of which the E. perfoliatum, or bone set, is a valuable medicine. These are mostly tall, growing, perennial, herbaceous plants. The greater part are natives of North America; many, however, from South America and the West Indies; several are found wild in the East Indies, and one only in Europe.

EUPHEMISM, in rhetoric, a figure which expresses things, in themselves disagreeable and shocking, in terms implying the contrary quality: thus, the Pontus, or Black Sea, having the epithet aževos, i. e. inhospitable, given it, by reason of the savage cruelty of those who inhabited the neighbouring countries, this name, by Euphemism, was changed into that of Euxinus. In which significa tion nobody will deny its being a species of irony but every euphemism is not irony, for we sometimes use improper and soft terms in the same sense with the proper and harsh.

:

EUPHONY, in grammar, an easiness, smoothness, and elegance in pronunciation. Euphony is properly a figure, whereby we suppress a letter, that is too harsh, and convert it into a smoother, contrary to the ordinary rules of this there are abundance of examples, in all languages.

EUPHORBIA, in botany, English euphorbium spurge, a genus of the Dodecandria Trigynia class and order. Natural order of Tricocca. Euphorbiæ. Jussieu. Essential character: corolla four or five petalled, placed on the calyx; calyx one-leafed, bellying; capsule tricoccous. There are ninety-eight species. These are milky plants, mostly herbace ous, a few shrubby, upright for the most part, very few of them creeping; some are leafless; stems angular or tubercled, or more frequently cylindric or columnar; unarmed, or in the angular sorts resembling the upright cactuses; armed with prickles, which are either solitary or in pairs, placed in a single row on the top of the ridges.

EUPHRASIA, in botany, English eyebright, a genus of the Didynamia Angiospermia class and order. Natural order

of Personatæ. Pediculares, Jussieu. Essential character: calyx four-cieft, cylindric; capsule two-celled, ovate, oblong:

lower anthers have a little thorn at the base of one of the lobes. There are nine species.

EURYA, in botany, a genus of the Dodecandria Monogynia class and order. Essential character: calyx five-leaved, calycled; corolla five-petalled; stamina thirteen; capsule five-celled. There is but one species, viz. E. Japonica, a native of Japan.

EURYANDRA, in botany, a genus of the Polyandria Trigynia class and order. Natural order of Coadunatæ. Magnolia, Jussieu. Essential character: calyx fiveleaved; corolla three-petalled; filament much dilated at the tip, with twin disjoined anthers; folicles three. There is only one species, viz. E. scandens, a native of New Caledonia.

EUSTACHIAN tube, in anatomy, begins from the interior extremity of the tympanum, and runs forward and inwards in a bony canal, which terminates with a portion of the temporal bone. See ANA

TOMY.

EUSTEPHIA, in botany, a genus of the Hexandria Monogynia class and order. Corolla superior, tubular, cylindrical, bifid; nectary six cavities in the tube of the corolla; filaments tricuspidate, distinct. There is but a single species, viz. the coccinea.

EUSTYLE, in architecture, a sort of building in which the pillars are placed at the most convenient distance one from another, the intercolumniations being just two diameters and a quarter of the column, except those in the middle of the