Leonhard Euler

“ Recently an international mathematical society has been organized with the sole purpose of supporting the publication of the complete works of the most prolific mathematical writer, Leonhard Euler, who died in 1783. Each member of this society is expected to pay at least ten francs annually until this publication is completed, which is expected to require about fifteen years. ”

“ At the age of nineteen he communicated to Leonhard Euler his idea of a general method of dealing with “isoperimetrical” problems, known later as the Calculus of Variations. It was eagerly welcomed by the Berlin mathematician, who had the generosity to withhold from publication his own further researches on the subject, until his youthful correspondent should have had time to complete and opportunity to claim the invention. This prosperous opening gave the key-note to Lagrange’s career. ”

“ In 1764 Leonhard Euler employed the functions of both zero and integral orders in an analysis into the vibrations of a stretched membrane; an investigation which has been considerably developed by Lord Rayleigh, who has also shown (1878) that Bessel’s functions are particular cases of Laplace’s functions. There is hardly a branch of mathematical physics which is independent of these functions. ”

“ His mathematical discoveries were extended and overshadowed by his contemporaries. His development of the equation xm + px = q in an infinite series was extended by Leonhard Euler, and particularly by Joseph Louis Lagrange. In 1761 he proved the irrationality of π; a simpler proof was given somewhat later by Legendre. The introduction of hyperbolic functions into trigonometry was also due to him. His geometrical discoveries are of great value, his Die freie Perspective (1759-1774) being a work of great merit. ”

“ Daniel Bernoulli, Leonhard Euler, Colin Maclaurin and Antoine Cavalleri. The first D. Bernoulli and Others. three adopted not only the theory of gravitation, but also Newton's method of the superposition of the two ellipsoids. Bernoulli's essay contained an extended development of the conception of the two ellipsoids, and, under the name of the equilibrium theory, it is commonly associated with his name. Laplace gives an account and critique of the essays of Bernoulli and Euler in the Mécanique céleste. ”

“ This equation is in general of the 2nth order, and the 2n arbitrary constants which are contained in the complete /primitive-must be adjusted to satisfy the conditions that y, y′, y′′, . . . y (n−1) have given values at the limits of integration. If the function y is required also to satisfy the condition that another integral of the same form as the above, but containing a function φ instead of F, may have a prescribed value, Euler achieved his purpose by replacing F in the differential equation by F+λφ, and adjusting the constant λ so that the condition may be satisfied. ”

“ Leibniz and John Bernoulli once argued the question whether a negative number has a logarithm. Bernoulli claimed that since , we have and , and finally . Euler proved that has really an infinite number of logarithms, all of which are imaginary when is negative, and all except one when is positive. He then explained how might equal , and yet not equal .
The subject of infinite series received new life from him. To his researches on series we owe the creation of the theory of definite integrals by the development of the so-called Eulerian integrals. ”

“ Were the character which superstition assumes in different ages an indication of man’s advance or retrogression, assuredly the nineteenth century would have no reason to plume itself, in comparison with the sixth B.C. A more earnest attempt to account for the more perfect consonance of the smaller ratios was made by the celebrated mathematician, Euler, and his explanation, if such it could be called, long silenced, if it did not satisfy, inquirers. Euler analyzes the cause of pleasure. We take delight in order; it is pleasant to us to observe means “co-operant to an end.” ”

“ But when the word has become general, the specialist is not very wise if he refuse to follow the mass, and perfectly foolish if he insist on others following him. There have been a few who demanded that Euler should be pronounced in the German fashion: [611] Euler has long been the property of the world at large; what does it matter how his own countrymen pronounce the letters? Shall we insist on the French pronouncing Newton without that final tong which they never fail to give him? ”

“ A round assertion of the non-existence of anything which stands in the way is the refuge of a certain class of minds: but it succeeds only with things subjective; the objective offers resistance. A philosopher of the appropriative class tried it upon the constable who appropriated him: I deny your existence, said he; Come along all the same, said the unpsychological policeman.
Euler [10] was a believer in God, downright and straightforward. The following story is told by Thiébault, [11] in his Souvenirs de vingt ans de séjour à Berlin, [12] published in his old age, about 1804. ”

“ Euler was probably the most versatile as well as the most prolific of mathematicians of all time. There is scarcely any branch of modern analysis to which he was not a large contributor, and his extraordinary powers of devising and applying methods of calculation were employed by him with great success in each of the existing branches of applied293 mathematics; problems of abstract dynamics, of optics, of the motion of fluids, and of astronomy were all in turn subjected to his analysis and solved. ”

“ Mr. Airy [554] there contends, and proves it both by Leverrier [555] and by Adams, [556] that the limited publication of a private letter is more efficient than the more general publication of a printed memoir. The same may be true of a dead letter, as opposed to a dead book. Our eye was caught by a letter of Oughtred (1629) , containing systematic use of contractions for the words sine, cosine, etc., prefixed to the symbol of the angle. This is so very important a step, simple as it is, that Euler [557] is justly held to have greatly advanced trigonometry by its introduction. ”

“ Perhaps it is to the quiet abstraction of his life that he owed the perpetuity of his tenure of investigation. Many eminent mathematical discoverers have run the brilliant part of their career while comparatively young. Euler “ceased to calculate and to live” at once. But it may be that this was a part of his natural constitution, and a distinct feature of his mind. ”

“ His writings are the first in which the modern analysis is uniformly the instrument of investigation. His predecessors, James and John Bernouilli, had perhaps the largest share in bringing the infinitesimal analysis of Newton and Leibnitz to the state of power required for extensive application. To Euler (besides important extensions) belongs the distinct merit of showing how to apply it to physical investigations, in conjunction with D’Alembert, who ran a splendid and contemporary career of a similar character in this respect. ”

“ For the tempering ratio of the complement of any concord to the octave, is the same with that of the concord itself, differing only in its sign, which does not sensibly affect the harmony or the rate of beating; while the tempering ratio of the compounds with octaves is not only the same, but with the same sign.
Scholium 1.
There is no point in harmonics, concerning which theorists have been more divided in opinion than in regard to the true measure of equal harmony, in consonances of different kinds. Euler maintains, that the more simple a consonance is, the less temperament it will bear ”

“ This was objected to by D’Alembert (1750) and also by Lagrange on the ground that irregular curves are inadmissible. D. Bernoulli (Berlin Memoirs, vol. ix., 1753) based a similar result to that of Euler on physical intuition; his method was criticized by Euler (1753) . ”

“ Fortunately Euler appended to his memoir a supplement truly worthy of his genius. Father Lozeran de Fiesc and the Count of Créqui were rewarded with the high honour of seeing their names inscribed beside that of the illustrious geometer, although it would be impossible in the present day to discern in their memoirs any kind of merit, not even that of politeness, for the courtier said rudely to the Academy: "the question, which you have raised, interests only the curiosity of mankind." ”

“ No law of nature has been better established than that of universal gravitation, and we believe with the utmost confidence that any body capable of affecting the senses will attract other bodies, and fall to the earth if not prevented. Euler remarks238 that, although he had never made trial of the stones which compose the church of Magdeburg, yet he had not the least doubt that all of them were heavy, and would fall if unsupported. ”

“ His first appearance before the learned world was in a controversy on the subject of Newton's law of refraction with Euler, who in the ‘Berlin Memoirs’ for 1747 (p. 274) had endeavoured to substitute for it a hypothetical principle permitting the colour-correction of telescopes by the employment of combined lenses of glass and water. Dollond expressed his objections in a letter to James Short [q. v.] dated 11 March 1752, which Short persuaded him to send to Euler, and communicate, with his reply, to the Royal Society. ”

“ Has not the public a right to be surprised upon reading this Academic declaration: "the question affords no handle to geometry!" In matter of inventions, to attempt to dive into the future, is to prepare for one's self striking mistakes. One of the competitors, the great Euler, took these words in their literal sense; the reveries with which his memoir abounds, are not compensated in this instance by any of those brilliant discoveries in analysis, I had almost said of those sublime inspirations, which were so familiar to him. ”

“ I. p. 702; “Nova Act. Petropol.” for 1779, Vol. III; “Pièces de Prix de l’Acad. des Sc. de Paris,” Vol. V. Mém. II and IX, this last-named publication, containing likewise a joint Memoir of D. Euler, J. Bernoulli and E. F. Dutour upon the mariner’s compass, which appeared in Paris during 1748; Whewell, “History of the Inductive Sciences,” 1859, Vol. I. pp. 225, 367, 370; Vol. II. pp. 32, 40.
His son, Albert Euler, censured Halley’s magnetical hypothesis, and proposed, in 1766, a theory requiring the assumption of only two poles, distinct, however, from those of the terrestrial axis. ”

“ Euler brought about an emancipation of the analytical calculus from geometry and established it as an independent science. Lagrange and Laplace scrupulously adhered to this separation. Building on the broad foundation laid for higher analysis and mechanics by Newton and Leibniz, Euler, with matchless fertility of mind, erected an elaborate structure. ”

“ Of no little importance are Euler's labours in analytical mechanics. Says Whewell: "The person who did most to give to analysis the generality and symmetry which are now its pride, was also the person who made mechanics analytical; I mean Euler." [11] He worked out the theory of the rotation of a body around a fixed point, established the general equations of motion of a free body, and the general equation of hydrodynamics. He solved an immense number and variety of mechanical problems, which arose in his mind on all occasions. ”

“ In the i8th century Russian thought was still unable to state general laws and could only observe phenomena in part; but in course of time experiment produced a greater knowledge of scientific law, above all in the physical and chemical sciences studied at the Imperial Academy. Before Euler published his Dioptrics, Rihmann had already made some experiments in electricity in which Lomonosov had a certain share; this high-spirited Russian man of science wrote also on optics and formulated some fundamental propositions concerning the mechanical theory of heat. ”

“ This, the most valuable result of her labours in this field, was regarded as the best introduction extant to the works of Euler. It was translated into English by Colson of Cambridge, and into French by d'Antelmy, with the notes of Abbé Bossuet. The plane curve, known as versiera, is also called "the Witch of Agnesi". ”

“ While Maupertuis was absent on account of the wars and in search of health, Euler acted as president, and proved himself well qualified for the duties of that office. But the king had no intention of filling it with a German. The man he wanted was d'Alembert, of Paris, to whom he offered a large salary, rooms in the palace and a seat at the royal table if he would come to Berlin. But d'Alembert belonged to the French Academy and did not care to leave Paris. ”

“ The question was then considered from a new point of view by Lagrange, in a memoir on the nature and propagation of sound (Miscellanea Taurensia, 1759; Œuvres, vol. i.) , who, while criticizing Euler’s method, considers a finite number of vibrating particles, and then makes the number of them infinite; he did not, however, quite fully carry out the determination of the coefficients in Bernoulli’s Series. These mathematicians were hampered by the narrow conception of a function, in which it is regarded as necessarily continuous ”

“ Students of elementary geometry are not prepared to appreciate it, but teachers will be interested in the remarkable formula discovered by Euler (1707-1783) , the great Swiss mathematician, namely, 1 + eiπ = 0. In this relation are included the five most interesting quantities in mathematics,—zero, the unit, the base of the so-called Napierian logarithms, i = √ (-1) , and π. It was by means of this relation that the transcendence of [Pg 281] e was proved by the French mathematician Hermite, and the transcendence of π by the German Lindemann. ”