Leonhard Euler

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Although to penetrate into the intimate mysteries of nature and thence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena.

Leonhard Euler (15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician, and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory. He is considered to be one of the greatest mathematicians of all time.

See also:
Euler's identity


Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.
To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be.
  • Madam, I have come from a country where people are hanged if they talk.
    • In Berlin, to the Queen Mother of Prussia, on his lack of conversation in his meeting with her, on his return from Russia; as quoted in Science in Russian Culture : A History to 1860 (1963) Alexander Vucinich
    • Variant: Madame... I have come from a country where one can be hanged for what one says.
  • Now I will have less distraction.
    • Upon losing the use of his right eye; as quoted in In Mathematical Circles (1969) by H. Eves
  • Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.
    • As quoted in Calculus Gems (1992) by G. Simmons
  • All the greatest mathematicians have long since recognized that the method presented in this book is not only extremely useful in analysis, but that it also contributes greatly to the solution of physical problems. For since the fabric of the universe is most perfect, and is the work of a most wise Creator, nothing whatsoever takes place in the universe in which some relation of maximum and minimum does not appear. Wherefore there is absolutely no doubt that every effect in the universe can be explained as satisfactorily from final causes, by the aid of the method of maxima and minima, as it can from the effective causes themselves. Now there exist on every hand such notable instances of this fact, that, in order to prove its truth, we have no need at all of a number of examples; nay rather one's task should be this, namely, in any field of Natural Science whatsoever to study that quantity which takes on a maximum or a minimum value, an occupation that seems to belong to philosophy rather than to mathematics. Since, therefore, two methods of studying effects in Nature lie open to us, one by means of effective causes, which is commonly called the direct method, the other by means of final causes, the mathematician uses each with equal success. Of course, when the effective causes are too obscure, but the final causes are more readily ascertained, the problem is commonly solved by the indirect method; on the contrary, however, the direct method is employed whenever it is possible to determine the effect from the effective causes. But one ought to make a special effort to see that both ways of approach to the solution of the problem be laid open; for thus not only is one solution greatly strengthened by the other, but, more than that, from the agreement between the two solutions we secure the very highest satisfaction.
    • introduction to De Curvis Elasticis, Additamentum I to his Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes 1744; translated on pg10-11, "Leonhard Euler's Elastic Curves", Oldfather et al 1933
  • To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be.
    • As quoted in Fundamentals of Teaching Mathematics at University Level (2000) by Benjamin Baumslag, p. 214
  • La construction d'une machine propre à exprimer tous les sons de nos paroles , avec toutes les articulations , seroit sans-doute une découverte bien importante. … La chose ne me paroît pas impossible.
    • It would be a considerable invention indeed, that of a machine able to mimic speech, with its sounds and articulations. … I think it is not impossible.
    • Letter to Friederike Charlotte of Brandenburg-Schwedt (16 June 1761)
    • Lettres à une Princesse d'Allemagne sur différentes questions de physique et de philosophie, Royer, 1788, p. 265
    • As quoted in An Introduction to Text-to-Speech Synthesis (2001) by Thierry Dutoit, p. 27; also in Fabian Brackhane and Jürgen Trouvain "Zur heutigen Bedeutung der Sprechmaschine Wolfgang von Kempelens" (in: Bernd J. Kröger (ed.): Elektronische Sprachsignalverarbeitung 2009, Band 2 der Tagungsbände der 20. Konferenz "Elektronische Sprachsignalverarbeitung" (ESSV), Dresden: TUDpress, 2009, pp. 97–107)
  • It will seem a little paradoxical to ascribe a great importance to observations even in that part of the mathematical sciences which is usually called Pure Mathematics, since the current opinion is that observations are restricted to physical objects that make impression on the senses. As we must refer the numbers to the pure intellect alone, we can hardly understand how observations and quasi-experiments can be of use in investigating the nature of numbers. Yet, in fact, as I shall show here with very good reasons, the properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. There are many properties of the numbers with which we are well acquainted, but which we are not yet able to prove; only observations have led us to their knowledge. Hence we see that in the theory of numbers, which is still very imperfect, we can place our highest hopes in observations; they will lead us continually to new properties which we shall endeavor to prove afterwards. The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful.
    • Opera Omnia, ser. 1, vol. 2, p. 459 Spcimen de usu observationum in mathesi pura, as quoted by George Pólya, Induction and Analogy in Mathematics Vol. 1, Mathematics and Plausible Reasoning (1954)
Original title: Introductio in analysin infinitorum. Translated as Introduction to Analysis of the Infinite (1988–89) by John Blanton (Book I ISBN 0387968245; Book II ISBN 0387971327 (online version).
  • A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities.
    • §4

A conjecture about the nature of air (1780)

A conjecture about the nature of air, by which are to be explained the phenomenon which have been observed in the atmosphere (Conjectura circa naturam aeris, pro explicandis phaenomenis in atmosphaera observatis) (1870) (online version).
  • Quanquam nobis in intima naturae mysteria penetrare, indeque veras caussas Phaenomenorum agnoscere neutiquam est concessum: tamen evenire potest, ut hypothesis quaedam ficta pluribus phaenomenis explicandis aeque satisfaciat, ac si vera caussa nobis esset perspecta.
    • Although to penetrate into the intimate mysteries of nature and thence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena.
    • §1

Quotes about Euler

Alphabetized by author
I discovered the works of Euler and my perception of the nature of mathematics underwent a dramatic transformation. ~ Alexander Stepanov
  • He calculated without any apparent effort, just as men breathe, as eagles sustain themselves in the air.
    • François Arago; Variant: Euler calculated without apparent effort, as men breathe, or as eagles sustain themselves in the wind.
  • The most influential mathematics textbook of ancient times is easily named, for the Elements of Euclid has set the pattern in elementary geometry ever since. The most effective textbook of the medieval age is less easily designated; but a good case can be made out for the Al-jabr of Al-Khwarizmi, from which algebra arose and took its name. Is it possible to indicate a modern textbook of comparable influence and prestige? Some would mention the Géométrie of Descartes or the Principia of Newton or the Disquisitiones of Gauss; but in pedagogical significance these classics fell short of a work by Euler titled Introductio in analysin infinitorum.
    • Carl B. Boyer on Euler's Introduction to the Analysis of the Infinite in "The Foremost Textbook of Modern Times" (1950)
  • The Introductio does not boast an impressive number of editions, yet its influence was pervasive. In originality and in the richness of its scope it ranks among the greatest of textbooks; but it is outstanding also for clarity of exposition. Published two hundred and two years ago, it nevertheless possesses a remarkable modernity of terminology and notation, as well as of viewpoint. Imitation is indeed the sincerest form of flattery.
  • Of no little importance are Euler's labors in analytical mechanics. ...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. Thus on reading Virgil's lines. "The anchor drops, the rushing keel is staid," he could not help inquiring what would be the ship's motion in such a case. About the same time as Daniel Bernoulli he published the Principle of the Conservation of Areas and defended the principle of "least action," advanced by P. Maupertius. He wrote also on tides and on sound.
  • Somebody said "Talent is doing what others find difficult. Genius is doing easily what others find impossible." ...by that definition, Euler was a genius. He could do the seemingly impossible, and he did it throughout his long and illustrious life. ...Way to Go, Uncle Leonhard!
  • Euler calculated the force of the wheels necessary to raise the water in a reservoir … My mill was carried out geometrically and could not raise a drop of water fifty yards from the reservoir. Vanity of vanities! Vanity of geometry!
  • The study of Euler's works will remain the best school for the different fields of mathematics and nothing else can replace it.
    • Carl Friedrich Gauss, as quoted by Louise Grinstein, Sally I. Lipsey, Encyclopedia of Mathematics Education (2001) p. 235.
  • It is customary to consider Chebyshev, Gauss, Jacobi, and Legendre as the main creators of the theory of orthogonal polynomials. However, their contributions were directly influenced by Brouncker and Wallis who, in March of 1655, made discoveries which influenced the development of analysis for the next hundred years. Namely, Wallis found an infinite product of rational numbers converging to 4/π and Brouncker gave a remarkable continued fraction for this quantity. ...The only mathematician who understood the importance of these discoveries was Euler. ...he felt that the recovery of the original Brouncker's proof could open up new perspectives for analysis. As usual, Euler was right.
    • Fritz Gesztesy, Spectral Theory and Mathematical Physics in Proceedngs of Symposia in Pure Mathematics (2007) Vol.76, Part 2
  • Following a suggestion by Daniel Bernoulli, Euler gave the first treatment of elastic lines by means of the calculus of variations in the Additamentum I to his Methodus inveniendi (1744...) which carries the title De curvis elasticus. Euler characterized the equilibrium position of an elastic line by the following variational principle: Among all curves of equal length, joining two points where they have prescribed tangents, to determine that which minimizes the value of the expression [where is the radius of curvature]. In other words, Euler interpreted an elastic line as an inextensible curve with a "potential energy" of ,   [i.e., ] being the curvature function of , whose positions of (stable) equilibrium are characterized by the minima of the potential energy, i.e., by Johann Bernoulli's principle of virtual work. Thus the problem of the elastic line leads to the isoperimetric problem
    • Mariano Giaquinta, Stefan Hildebrandt, Calculus of Variations I (2004) Grundlehren der mathematischen WissenSchaften Vol. 310.
  • Galileo does not attempt any theory to account for the flexure of the beam. This theory, supplied by Hooke's law, was applied by Mariotte, Leibnitz, De Lahire, and Varignon, but they neglect compression of the fibres, and so place the neutral in the lower face of Galileo's beam. The true position of the neutral plane was assigned by James Bernoulli 1695, who in his investigation of the simplest case of bent beam, was led to the consideration of the curve called the "elastica." This "elastica" curve speedily attracted the attention of the great Euler (1744), and must be considered to have directed his attention to the elliptic integrals. Probably the extraordinary divination which led Euler to the formula connecting the sum of two elliptic integrals, thus giving the fundamental theorem of the addition equation of elliptic functions, was due to mechanical considerations concerning the "elastica" curve; a good illustration of the general principle that the pure mathematician will find the best materials for his work in the problems presented to him by natural and physical questions.
  • Who has studied the works of such men as Euler, Lagrange, Cauchy, Riemann, Sophus Lie, and Weierstrass, can doubt that a great mathematician is a great artist? The faculties possessed by such men, varying greatly in kind and degree with the individual, are analogous with those requisite for constructive art. Not every mathematician possesses in a specially high degree that critical faculty which finds its employment in the perfection of form, hi conformity with the ideal of logical completeness; but every great mathematician possesses the rarer faculty of constructive imagination.
    • E. W. Hobson, "Presidential Address British Association for the Advancement of Science" (1910) in: Nature, Vol. 84, p. 290. Cited in: Moritz (1914, 182); Mathematics as a fine art
  • To the reader of today much in the conception and mode of expression of that time appears strange and unusual. Between us and the mathematicians of the late seventeenth century stands Leonhard Euler... He is the real founder of our modern conception. However non-rigorous he may be in details: he ends and conquers the previous epoch of direct geometric infinitesimal considerations and introduces the period of mathematical analysis according to form and content. Whatever was written after him on the logarithmic series is necessarily based no longer on the already obscured predecessors in the receding mathematical Renaissance, but on Euler's Introductio in analysin infinitorum... in which the entire seventh chapter [De Quantitabus exponentialibus ac Logarithmis] treats of logarithms.
    • Josef Ehrenfried Hofman, "On the Discovery of the Logarithmic Series and Its Development in England up to Cotes" (Oct., 1939) National Mathematics Magazine, Vol. 14, No. 1, pp. 37-38.
  • Read Euler: he is our master in everything.
    • Pierre-Simon Laplace, as quoted in Calculus Gems (1992). variant: Read Euler, read Euler. He is the master of us all. As quoted by S. H. Hollingdale, "Leonhard Euler (1707-1783): A Bicentennial Tribute", Bulletin (1983) Volumes 19-20, Institute of Mathematics and Its Applications, & by Edwin Joseph Purcell, Dale E. Varberg, Calculus with Analytic Geometry (1987), Vol. 1.
  • He was later to write that he had made some of his best discoveries while holding a baby in his arms surrounded by playing children.
    • Richard Mankiewicz, in The Story of Mathematics (2000), p. 142
  • If we compared the Bernoullis to the Bach family, then Leonhard Euler is unquestionably the Mozart of mathematics, a man whose immense output... is estimated to fill at least seventy volumes. Euler left hardly an area of mathematics untouched, putting his mark on such diverse fields as analysis, number theory, mechanics and hydrodynamics, cartography, topology, and the theory of lunar motion. ...Moreover, we owe to Euler many of the mathematical symbols in use today, among them i, π, e, and f(x). And as if that were not enough, he was a great popularizer of science...
    • Eli Maor, e: The Story of a Number (1994)
  • Euler and Ramanujan are mathematicians of the greatest importance in the history of constants (and of course in the history of Mathematics ...)
  • Euler's step was daring. In strict logic, it was an outright fallacy... Yet it was justified by analogy, by the analogy of the most successful achievements of a rising science that he called... "Analysis of the Infinite." Other mathematicians, before Euler, passed from finite differences to infinitely small differences, from sums with a finite number of terms to sums with an infinity of terms, from finite products to infinite products. And so Euler passed from equations of a finite degree (algebraic equations) to equations of infinite degree, applying the rules made for the finite...
    This analogy... is beset with pitfalls. How did Euler avoid them? ...Euler's reasons are not demonstrative. Euler does not reexamine the grounds for his conjecture... only its consequences. ...He examines also the consequences of closely related analogous conjectures... Euler's reasons are, in fact, inductive.
    • George Pólya, Induction and Analogy in Mathematics (1954) Vol. 1 Of Mathematics and Plausible Reasoning
  • It is the invaluable merit of the great Basle mathematician Leonard Euler, to have freed the analytical calculus from all geometric bounds, and thus to have established analysis as an independent science, which from his time on has maintained an unchallenged leadership in the field of mathematics.
    • Thomas Reid, as quoted in Mathematical Maxims and Minims (1988) by N. Rose
  • Perhaps the most surprising thing about mathematics is that it is so surprising. The rules which we make up at the beginning seem ordinary and inevitable, but it is impossible to foresee their consequences. These have only been found out by long study, extending over many centuries. Much of our knowledge is due to a comparatively few great mathematicians such as Newton, Euler, Gauss, or Riemann; few careers can have been more satisfying than theirs. They have contributed something to human thought even more lasting than great literature, since it is independent of language.
  • As analysis was more cultivated, it gained a predominancy over geometry; being found to be a far more powerful instrument for obtaining results; and possessing a beauty and an evidence, which, though different from those of geometry, had great attractions for minds to which they became familiar. 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.
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