"Long algebraic calculations were at first hardly necessary for progress in Mathematics; the very simple theorems hardly gained from being translated into the language of analysis. It is only since Euler that this briefer language has become indispensable to the new extensions which this great mathematician has given to science. Since Euler calculations have become more and more necessary but more and more difficult, at least insofar as they are applied to the most advanced objects of science. Since the beginning of this century computational procedures have attained such a degree of complication that any progress had become impossible by these means, except with the elegance with which new modern mathematicians have believed they should bring to their research, and by means of which the mind promptly and with a single glance comprehends a large number of operations." (Évariste Galois, "Two memoirs in pure analysis", 1831)
"It is a remarkable fact that complaints of the want of clearness and rigour in that part of Mathematics which respects calculation, - whether it be called 'Arithmetic', Universal Arithmetic', 'Mathematical Analysis', or aught else, - recur from time to time, now uttered by subordinate writers, now repeated by the most distinguished of the learned. One finds contradictions of the theory of 'opposed magnitudes'; - another is merely disquieted by 'imaginary quantities'; - a third finally meets difficulties in 'infinite series', either because Euler and other distinguished mathematicians have applied them with success in a divergent form, while the complainant thinks himself convinced that their convergence is a fundamental condition, - or because in general investigations general series occur, which, precisely because they are general, can be neither accounted divergent nor convergent. ... The [...] question: how may the paradoxes of calculation be most securely avoided? - obliges us to submit to a very exact examination of the subject of mathematical analysis, its first and simplest ideas, as also the methods of reasoning which are applied to it." (Martin Ohm, "The Spirit of Mathematical Analysis and its Relation to a Logical System", 1842)
"I have been able to solve a few problems of mathematical physics on which the greatest mathematicians since Euler have struggled in vain. [...] But the pride I might have held in my conclusions was perceptibly lessened by the fact that I knew that the solution of these problems had almost always come to me as the gradual generalization of favorable examples, by a series of fortunate conjectures, after many errors. I am fain to compare myself with a wanderer on the mountains who, not knowing the path, climbs slowly and painfully upwards and often has to retrace his steps because he can go no further - then, whether by taking thought or from luck, discovers a new track that leads him on a little till at length when he reaches the summit he finds to his shame that there is a royal road by which he might have ascended, had he only the wits to find the right approach to it. In my works, I naturally said nothing about my mistake to the reader, but only described the made track by which he may now reach the same heights without difficulty." (Hermann von Helmholtz, 1891)
"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, in conformity with the ideal of logical completeness; but every great mathematician possesses the rarer faculty of constructive imagination." (Ernest W Hobson, "Presidential Address British Association for the Advancement of Science", Nature, 1910)
"Having found a method differing from that of Ferrari for reducing the solution of the general biquadratic equation to that of a cubic equation, Euler had the idea that he could reduce the problem of the quintic equation to that of solving a biquadratic, and Lagrange made the same attempt. The failures of such able mathematicians led to the belief that such a reduction might be impossible. The first noteworthy attempt to prove that an equation of the fifth degree could not be solved by algebraic methods is due to Ruffini (1803, 1805), although it had already been considered by Gauss. [...] The modern theory of equations is commonly said to date from Abel and Galois. [...] Abel showed that the roots of a general quintic equation cannot be expressed in terms of its coefficients by means of radicals." (David E Smith, "History of Mathematics", 1925)
"There is a famous formula, perhaps the most compact and famous of all formulas developed by Euler from a discovery of de Moivre: It appeals equally to the mystic, the scientist, the philosopher, the mathematician." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)
"The unfortunate controversy over the discovery of the calculus between Newton and Leibniz and their protagonists occupies more space in the literature than its real importance justifies, but in this case (perhaps the most striking one of this type) the additional space redounds about equally to the advantage of both men. If it were eliminated entirely, Newton's position would still be preeminent, but Leibniz would probably be placed below Lagrange and Euler, rather than above them. As a matter of fact, however, all that we are justified in concluding, considering the probable errors of their positions, is that all three of these men are of practically equal rank [...]" (Walter C Eells, "One hundred eminent mathematicians", The Mathematics Teacher, 1962)
"The two most outstanding mathematicians of the eighteenth century were Euler and Lagrange, and which of the two is to be accorded first place is a matter of debate that often reflects the varying sensitivities of the debaters. Euler certainly published far more than Lagrange, and worked in many more diverse fields of mathematics than Lagrange, but he was largely a formalist or manipulator of formulas. Lagrange, on the other hand, may be considered the first true analyst and, though his collection of publications is a molehill compared with the Vesuvius of Euler's output, his work has a rare perfection, elegance, and exactness about it. Whereas Euler wrote with a profusion of detail and a free employment of intuition, Lagrange wrote concisely and with attempted rigor." (Howard W Eves, "In Mathematical Circles: Quadrants III and IV", 1969)
"[...] the Greeks, Descartes, Newton, Euler, and many others believed mathematics to be the accurate description of real phenomena and that they regarded their work as the uncovering of the mathematical design of the universe. Mathematicians did deal with abstractions, but these were no more than the ideal forms of physical objects or happenings." (Morris Kline, "Mathematical Thought from Ancient to Modern Times", 1972)
"It is a measure of Euler's greatness that when one is studying number theory one has the impression that Euler was primarily interested in number theory, but when one studies divergent series one feels that divergent series were his main interest, when one studies differential equations one imagines that actually differential equations was his favorite subject, and so forth Whether or not number theory was a favorite subject of Euler's, it is one in which he showed a lifelong interest and his contributions to number theoiy alone would suffice to establish a lasting reputation in the annals of mathematics." (Harole M Edwards, "Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory, 1977)
"The difference is that energy is a property of the microstates, and so all observers, whatever macroscopic variables they may choose to define their thermodynamic states, must ascribe the same energy to a system in a given microstate. But they will ascribe different entropies to that microstate, because entropy is not a property of the microstate, but rather of the reference class in which it is embedded. As we learned from Boltzmann, Planck, and Einstein, the entropy of a thermodynamic state is a measure of the number of microstates compatible with the macroscopic quantities that you or I use to define the thermodynamic state." (Edwin T Jaynes, "Papers on Probability, Statistics, and Statistical Physics", 1983)
"The acceptance of complex numbers into the realm of algebra had an impact on analysis as well. The great success of the differential and integral calculus raised the possibility of extending it to functions of complex variables. Formally, we can extend Euler's definition of a function to complex variables without changing a single word; we merely allow the constants and variables to assume complex values. But from a geometric point of view, such a function cannot be plotted as a graph in a two-dimensional coordinate system because each of the variables now requires for its representation a two-dimensional coordinate system, that is, a plane. To interpret such a function geometrically, we must think of it as a mapping, or transformation, from one plane to another."
"Why did he [Euler] choose the letter e? There is no consensus. According to one view, Euler chose it because it is the first letter of the word exponential. More likely, the choice came to him naturally, since the letters a, b, c, and d frequently appeared elsewhere in mathematics. It seems unlikely that Euler chose the letter because it is the initial of his own name, as occasionally has been suggested. He was an extremely modest man and often delayed publication of his own work so that a colleague or student of his would get due credit. In any event, his choice of the symbol e, like so many other symbols of his, became universally accepted." (Eli Maor, "e: The Story of a Number", 1994)
"I see some parallels between the shifts of fashion in mathematics and in music. In music, the popular new styles of jazz and rock became fashionable a little earlier than the new mathematical styles of chaos and complexity theory. Jazz and rock were long despised by classical musicians, but have emerged as art-forms more accessible than classical music to a wide section of the public. Jazz and rock are no longer to be despised as passing fads. Neither are chaos and complexity theory. But still, classical music and classical mathematics are not dead. Mozart lives, and so does Euler. When the wheel of fashion turns once more, quantum mechanics and hard analysis will once again be in style." (Freeman J Dyson, "Book Review of ‘Nature’s Numbers’", The American Mathematical Monthly, Vol. 103 (7), 1996)
"[…] and unlike the physics or chemistry or engineering of today, which will almost surely appear archaic to technicians of the far future, Euler’s formula will still appear, to the arbitrarily advanced mathematicians ten thousand years hence, to be beautiful and stunning and untarnished by time." (Paul J Nahin, "Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills", 2006)
"I think e^iπ+1=0 is beautiful because it is true even in the face of enormous potential constraint. The equality is precise; the left-hand side is not 'almost' or 'pretty near' or 'just about' zero, but exactly zero. That five numbers, each with vastly different origins, and each with roles in mathematics that cannot be exaggerated, should be connected by such a simple relationship, is just stunning. It is beautiful. And unlike the physics or chemistry or engineering of today, which will almost surely appear archaic to technicians of the far future, Euler's formula will still appear, to the arbitrarily advanced mathematicians ten thousand years hence, to be beautiful and stunning and untarnished by time." (Paul J Nahin, "Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills", 2006)
"There are many ways to use unique prime factorization, and it is rightly regarded as a powerful idea in number theory. In fact, it is more powerful than Euclid could have imagined. There are complex numbers that behave like 'integers' and 'primes', and unique prime factorization holds for them as well. Complex integers were first used around 1770 by Euler, who found they have almost magical powers to unlock secrets of ordinary integers. For example, by using numbers of the form a + b√ -2. where a and b are integers, he was able to prove a claim of Fermat that 27 is the only cube that exceeds a square by 2. Euler's results were correct, but partly by good luck. He did not really understand complex 'primes' and their behavior." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)
"Euler’s formula is an ideal tour guide because it has access to marvelous rooms that are rarely seen by other visitors. By following Euler’s formula we see some of the most intriguing areas of mathematics - geometry, combinatorics, graph theory, knot theory, differential geometry, dynamical systems, and topology. These are beautiful subjects that a typical student, even an undergraduate mathematics major, may never encounter." (David S Richeson, "Euler’s Gem: The Polyhedron Formula and the Birth of Topology", 2008)
"This equation is considered by some mathematicians and physicists to be the most important equation ever devised. In Euler’s relation, both sides of the equation are expressions for a complex number on the unit circle. The left side emphasizes the magnitude (the 1 multiplying e^iθ ) and direction in the complex plane (θ), while the right side emphasizes the real (cos θ) and imaginary (sin θ) components. Another approach to demonstrating the equivalence of the two sides of Euler’s relation is to write out the power-series representation of each side; [...]" (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)
"At first glance, the number e, known in mathematics as Euler’s number, doesn’t seem like much. It’s about 2.7, a quantity of such modest size that it invites contempt in our age of wretched excess and relentless hype." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)
"Euler’s formula - although deceptively simple - is actually staggeringly conceptually difficult to apprehend in its full glory, which is why so many mathematicians and scientists have failed to see its extraordinary scope, range, and ontology, so powerful and extensive as to render it the master equation of existence, from which the whole of mathematics and science can be derived, including general relativity, quantum mechanics, thermodynamics, electromagnetism and the strong and weak nuclear forces! It’s not called the God Equation for nothing. It is much more mysterious than any theistic God ever proposed." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)
"Graph theory, although it began with Euler, was a fringe mathematical topic until the twentieth century. With the exception of the Euler polyhedron formula, its results had little influence on other branches of mathematics. Indeed, the field of topology, which was largely inspired by the Euler formula, rapidly overtook graph theory in the early twentieth century, reaching a height from which certain topologists could look down on graph theory as 'the slums of topology'." (John Stillwell, "Elements of Mathematics: From Euclid to Gödel", 2016)
"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 J Devlin)
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