Carl Friedrich Gauss

"Bei Gegenstdnden mit denen ich mich noch nicht lange beschaftigt habe, bin ich gegen meine eigenen Ansichten, zumal wenn sie einem Laplace widersprechen, misstrauisch und nehme gern die von anderen entgegen. ["I am suspicious of my own views on subjects with which I have not long occupied myself, and gladly accept those of others, especially when my views contradict one of Laplace."]"

"Some of the discoveries of Abel and Jacobi were anticipated by Gauss. In the Disquisitiones Arithmeticæ he observed that the principles which he used in the division of the circle were applicable to many other functions, besides the circular, and particularly to the transcendents dependent on the integral \displaystyle \int \frac{\,dx}{\sqrt{1-x^4}}. From this Jacobi concluded that Gauss had thirty years earlier considered the nature and properties of elliptic functions and had discovered their double periodicity. The papers in the collected works of Gauss confirm this conclusion."

"He is like the fox, who effaces his tracks in the sand with his tail."

"Everything Gauss writes is abomination, as it is so obscure that it is almost impossible to understand it."

"Not only could nobody but Gauss have produced it, but it would never have occurred to anyone but Gauss that such a formula was possible."

"There is no doubt... that mathematicians are generally overzealous about conciseness, and in their passion for brevity indulge in symbols even where these seem no better than a familiar English word or phrase. A faulty judgement has caused mathematicians to equate elegance and conciseness at the cost of intelligibility. Gauss himself wrote elegant, but highly compact, carefully polished papers with no hint of the motivation, meaning, or details of the steps. When criticized, he said that no architect leaves the scaffolding after completing the building. But the fact is that even excellent mathematicians found the reading of Gauss's papers very difficult, and the same is true of many other mathematicians."

"Toward the ends of their lives, Euler, D'Alembert, and Lagrange agreed that the realm of mathematical ideas had been practically exhausted and that no new great minds were appearing on the mathematical horizons. Of course, these men had grown old and their vision was already dimmed, for Laplace, Legendre, and Fourier were in young manhood. In one respect, however, these elder statesmen were correct... their immediate successors continued to explore and polish the very same ideas which the mid-eighteenth century had pursued. But history shows that the human mind is fertile, ingenious, and creative beyond all possible anticipations. ...even the richest vein of thought is ultimately exhausted, and then, indeed, a period of stagnation may ensue. Inevitably, however, there arise new conceptions and new periods of feverish and rewarding research. Euler and his contemporaries failed to reckon with history. ... The man who was to change the course of mathematics was but six years old when Euler and D'Alembert died in 1783... Gauss is commonly ranked with Archimedes and Newton. ...all three of these men were as much devoted to physical research as to mathematics."

"On demandait à Laplace quel était selon lui le plus grand mathématicien de l'Allemagne. C'est Pfaff, répondit-il. - Je croyais, reprit l'interlocuteur, que Gauss lui était supérieur. - Mais, s'écria Laplace, vous me demandez quel est le plus grand mathématicien de l'Allemagne, et Gauss est le plus grand mathématicien de l'Europe."

"It is to Gauss, to the Magnetic Union, and to magnetic observers in general, that we owe our deliverance from that absurd method of estimating forces by a variable standard which prevailed so long even among men of science. It was Gauss who first based the practical measurement of magnetic force (and therefore of every other force) on those long established principles, which, though they are embodied in every dynamical equation, have been so generally set aside, that these very equations, though correctly given... are usually explained... by assuming, in addition to the variable standard of force, a variable, and therefore illegal, standard of mass."

"If explaining minds seems harder than explaining songs, we should remember that sometimes enlarging problems makes them simpler! The theory of the roots of equations seemed hard for centuries within its little world of real numbers, but it suddenly seemed simple once Gauss exposed the larger world of so-called complex numbers. Similarly, music should make more sense once seen through listeners' minds."

"According to his frequently expressed view, Gauss considered the three dimensions of space as specific peculiarities of the human soul; people, which are unable to comprehend this, he designated in his humorous mood by the name Bœotians. We could imagine ourselves, he said, as beings which are conscious of but two dimensions; higher beings might look at us in a like manner, and continuing jokingly, he said that he had laid aside certain problems which, when in a higher state of being, he hoped to investigate geometrically."

"Gauss liked to call [number theory] 'the Queen of Mathematics'. For Gauss, the jewels in the crown were the primes, numbers which had fascinated and teased generations of mathematicians."

"Armed with his prime number tables, Gauss began his quest. As he looked at the proportion of numbers that were prime, he found that when he counted higher and higher a pattern started to emerge. Despite the randomness of these numbers, a stunning regularity seemed to be looming out of the mist."

"The revelation that the graph appears to climb so smoothly, even though the primes themselves are so unpredictable, is one of the most miraculous in mathematics and represents one of the high points in the story of the primes. On the back page of his book of logarithms, Gauss recorded the discovery of his formula for the number of primes up to N in terms of the logarithm function. Yet despite the importance of the discovery, Gauss told no one what he had found. The most the world heard of his revelation were the cryptic words, 'You have no idea how much poetry there is in a table of logarithms.'"

"Maybe we have become so hung up on looking at the primes from Gauss's and Riemann's perspective that what we are missing is simply a different way to understand these enigmatic numbers. Gauss gave an estimate for the number of primes, Riemann predicted that the guess is at worst the square root of N off its mark, Littlewood showed that you can't do better than this. Maybe there is an alternative viewpoint that no one has found because we have become so culturally attached to the house that Gauss built."