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) Swiss mathematician and physicist, considered to be one of the greatest mathematicians of all time.

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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
• Since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear ... there is absolutely no doubt that every affect in the universe can be explained satisfactorily from final causes, by the aid of the method of maxima and minima, as it can be from the effective causes themselves ... Of course, when the effective causes are too obscure, but the final causes are readily ascertained, the problem is commonly solved by the indirect method...
• As quoted in The Anthropic Cosmological Principle (1986) by John D. Barrow and Frank J. Tipler, p. 150
• 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)

Introduction to the Analysis of the Infinite (Introductio in analysin infinitorum) (1748)

Translated as Introduction to Analysis of the Infinite (1988-89) by John Blanton (Book I ISBN 0387968245; Book II ISBN 0387971327).
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• 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.

Conjecture regarding the nature of the air, so as to explain the phenomena observed in the atmosphere (Conjectura circa naturam aeris, pro explicandis phaenomenis in atmosphaera observatis) (1780)

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• 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.

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He calculated without any apparent effort, just as men breathe, as eagles sustain themselves in the air. ~ François Arago
Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence. ~ Keith Devlin
Alphabetized by author
• 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.
• Carl B. Boyer in "The Foremost Textbook of Modern Times" (1950)
• 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!
• Read Euler: he is our master in everything.
• Pierre Simon de Laplace, as quoted in Calculus Gems (1992) variant: Read Euler, read Euler. He is the master of us all.
• 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
• Euler and Ramanujan are mathematicians of the greatest importance in the history of constants (and of course in the history of Mathematics ...)
• E. W. Middlemast
• 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.

On "Euler's identity"

In mathematical analysis, Euler's identity is the equation "$e^{i \pi} + 1 = 0. \,\!$"
Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth. ~ Benjamin Peirce
• One of the most frequently mentioned equations was Euler's equation, $e^{i \pi} + 1 = 0. \,\!$ Respondents called it "the most profound mathematical statement ever written"; "uncanny and sublime"; "filled with cosmic beauty"; and "mind-blowing". Another asked: "What could be more mystical than an imaginary number interacting with real numbers to produce nothing?" The equation contains nine basic concepts of mathematics — once and only once — in a single expression. These are: e (the base of natural logarithms); the exponent operation; π; plus (or minus, depending on how you write it); multiplication; imaginary numbers; equals; one; and zero.
• Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence.
• Our jewel ... one of the most remarkable, almost astounding, formulas in all of mathematics.
• There is a famous formula, perhaps the most compact and famous of all formulas — developed by Euler from a discovery of de Moivre: $e^{i \pi} + 1 = 0. \,\!$ It appeals equally to the mystic, the scientist, the philosopher, the mathematician.
• Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth.
• Benjamin Peirce, as quoted in notes by W. E. Byerly, published in Benjamin Peirce, 1809-1880 : Biographical Sketch and Bibliography (1925) by R. C. Archibald; also in Mathematics and the Imagination (1940) by Edward Kasner and James Newman