# Frases de Leonhard Euler

## Leonhard Euler

**Data de nascimento:** 15. Abril 1707**Data de falecimento:** 18. Setembro 1783

Leonhard Paul Euler foi um matemático e físico suíço de língua alemã que passou a maior parte de sua vida na Rússia e na Alemanha. Fez importantes descobertas em várias áreas da matemática como o cálculo e a teoria dos grafos. Também introduziu muitas das terminologias da matemática moderna e da notação matemática, particularmente na análise matemática, como também no conceito de função matemática. É também reconhecido por seus trabalhos na mecânica, dinâmica de fluidos, óptica, astronomia e teoria da música.Euler é considerado um dos mais proeminentes matemáticos do século XVIII e também é considerado como um dos grandes matemáticos de todos os tempos, assim como Isaac Newton, Arquimedes e Carl Friedrich Gauss. Foi um dos mais prolíficos matemáticos, calcula-se que toda a sua obra reunida teria entre 60 e 80 volumes de quartos. Viveu a maior parte da vida em São Petersburgo, na Rússia, e em Berlim, que na época era capital da Prússia.

Uma declaração atribuída a Pierre-Simon Laplace manifestada sobre Euler na sua influência sobre a matemática: "Leiam Euler, leiam Euler, ele é o mestre de todos nós".

### Citações Leonhard Euler

### „Na tentativa de mergulhar nos mistérios da natureza, é de grande importância saber se a atração dos corpos celestes uns sobre os outros são por impulso, ou, se uma determinada matéria sutil invisível impele-os uns com os outros, ou se foram dotados de um segredo ou qualidade oculta, pela qual eles são mutuamente atraídos.“

But in attempting to dive into the mysteries of nature, it is of importance to know if the heavenly bodies act upon each other by impulsion, or by attraction; if a certain subtile invisible matter impels them towards each other ; or if they are endowed with a secret or occult quality, by which they are mutually attracted.

"Letters of Euler to a German princess, on different subjects in physics and philosophy", Volume 1 - página 211 http://books.google.com/books?id=_1oIAAAAIAAJ&pg=PA211, de Leonhard Euler, Henry Hunter, Jean-Antoine-Nicolas de Caritat Condorcet, Traduzido por Henry Hunter, Edição 2, Editora Murray and Highley, 1802

### „Quantas vezes os nossos raciocínios são errados! Atrevo-me a afirmar, que somos muito mais freqüentemente enganados por estes do que por nossos sentidos. Mas isso quer dizer que nossos raciocínios são sempre falaciosos, e que não podemos ter nenhuma dependência de qualquer verdade descoberta por nós pela compreensão“

How often are our reasonings erroneous ! I venture to affirm, that we are much more frequently deceived by these than by our senses. But does it follow that our reasonings are always fallacious, and that we can have no dependence on any truth discovered to us by the understanding

"Letters of Euler on different subjects in natural philosophy: Addressed to a German princess. With notes, and a life of Euler", Volume 2 - página 17 http://books.google.com/books?id=IUknAAAAMAAJ&pg=PA17, de Leonhard Euler, Editora J. & J. Harper, 1833

### „Mas você só tem que olhar para a milésima parte de uma polegada através de um bom microscópio, o que amplia, por exemplo, mil vezes, cada partícula aparecerá tão grande como uma polegada a olho nu, e você será convencido da possibilidade de dividir cada uma dessas partículas novamente em mil partes: o mesmo raciocínio pode ser sempre levado avante sem limite e sem fim.“

But you have only to look at this thousandth part of an inch through a good microscope, which magnifies, for example, a thousand times, and each particle will appear as large as an inch to the naked eye ; and you will be convinced of the possibility of dividing each of these particles again into a thousand parts : the same reasoning may always be carried forward without limit and without end.

"Letters of Euler on different subjects in natural philosophy: addressed to a German princess" - Volume 2, página 36 http://books.google.com/books?id=CZLPNtEnFRcC&pg=PA36, de Leonhard Euler, Sir David Brewster, John Griscom, Editora J. & J. Harper, 1833

### „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

Variante: Madame... I have come from a country where one can be hanged for what one says.

### „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“

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" https://www.dropbox.com/s/o09w82abgtftpfr/1933-oldfather.pdf, Oldfather et al 1933

Contexto: 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.

### „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.“

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" https://www.dropbox.com/s/o09w82abgtftpfr/1933-oldfather.pdf, Oldfather et al 1933

Contexto: 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.

### „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.“

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)

Contexto: 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.

### „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

A conjecture about the nature of air (1780)

Original: (la) 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.

### „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

### „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

Introduction to the Analysis of the Infinite (1748)

### „Now I will have less distraction.“

Upon losing the use of his right eye; as quoted in In Mathematical Circles (1969) by H. Eves

### „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.“

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.

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)