Mere Math'ns

His Power

His favorite pursuits

The first theoretical astronomers

The greatest of arithmeticians

The math giant

Greatness of

Lectures to three students

Lectures to three students

His style and method

His estimate of Newton

On the advantage of new calculi

M. and experiment

His

His

M., the queen of sciences

On number theory

On imaginaries

On the notation of sin^2

On infinite magnitude

On non-Euclidean Geometry (part one)

On non-Euclidean Geometry (part two)

On non-Euclidean Geometry (part three)

On non-Euclidean Geometry (part four)

On non-Euclidean Geometry (part five)

On non-Euclidean Geometry (part six)

On the nature of space

On his respect of the female courage

"In other branches of science, where quick publication seems to be so much desired, there may possibility be some excuse for giving to the world slovenly or ill-digested work, but there is no excuse in mathematics. The form ought to be as perfect as the substance, and the demonstrations as rigorous as those of Euclid. The mathematician has to deal with the most exact facts of Nature, and he should spare no effort to render his interpretation worthy of his subject, and to give to his work its highest degree of perfection.

-Glaisher, J. W. L.

"It may be true, that men, who are

-Gauss

"In 1735 the solving of an astronomical problem, proposed by the Academy, for which several eminent mathematicians had demanded several months' time, was achieved in three days by Euler with aid of improved methods of his own.... With still superior methods this same problem was solved by the illustrious Gauss in one hour."

-Cajori, F.

"Astronomy and Pure Mathematics are the magnetic poles toward which the compass of my mind ever turns."

-Gauss to Bolyai

"[Gauss calculated the elements of the planet Ceres] and his analysis proved him to be the first of theoretical astronomers no less than the greatest of "arithmeticians."

-Ball, W. W. R.

"The mathematical giant {Gauss}, who from his lofty heights embraces in one view the stars and the abysses..."

-Bolyai, W.

"Almost everything, which the mathematics of our century has brought forth in the way of original scientific ideas, attaches to the name of Gauss."

-Kronecker, L.

"I am giving this winter two courses of lectures to three students, of which one is only moderately prepared, the other less than moderately, and the third lacks both preparation and ability. Such are the onera of a mathematical profession."

-Gauss to Bessel, 1810

"The great masters of modern analysis are Lagrange, Laplace, and Gauss, who were contemporaries. It is interesting to note the marked contrast in their styles. Lagrange is perfect both in form and matter, he is careful to explain his procedure, and though his arguments are general they are easy to follow. Laplace on the other hand explains nothing, is indifferent to style, and, if satisfied that his results are correct, is correct, is content to leave them either with no proof or with a faulty one. Gauss is as exact and elegant as Lagrange, but even more difficult to follow than Laplace, for he removes every trace of the analysis by which he reached his results, and studies to give a proof which while rigorous shall be as concise and synthetical as possible."

-Ball, W. W. R.

"For other great mathematicians or philosophers, he [Gauss] used the epithets magnus, or clarus, or clarissumus; for Newton alone he kept the prefix summus."

-Ball, W. W. R.

"In general the position as regards all such new calculi is this - That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able - without the unconscious inspiration of genius which no one can command - to solve the respective problems, yea, to solve them mechanically in complicated cases in which, without aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with M�bius's calculus. Such conceptions unite as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius."

-Gauss, C. J.

"In very many cases the most obvious and direct experimental method of investigating a given problem is extremely difficult, or for some reason or other untrustworthy. In such cases the mathematician can often point out some other problem more accessible to experimental treatment, the solution of which involves the solution of the former one. For example, if we try to deduce from direct experiments the law according to which one pole of a magnet attracts or repels a pole of another magnet, the observed action is so much complicated with the effects of the mutual induction of the magnets and of the forces due to the second pole of each magnet, that it is next to impossible to obtain results of any great accuracy. Gauss, however, showed how the law which applied in the case mentioned can be deduced from the deflections undergone by a small suspended magnetic needle when it is acted upon by a small fixed magnet placed successively in two determinate positions relatively to the needle; and being an experimentalist as well as a mathematician, he showed likewise how these deflections can be measured very easily and with great precision."

-Foster G. C.

"The 'Disquisitiones Arithmeticae' that great book with seven seals."

-Mertz J. T.

"It may fairly be said that the germs of the modern algebra of linear substitutions and concomitants are to be found in the fifth section of the

-Matthew G. B.

"Mathematics is the queen of sciences and arithmetic the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank."

-Gauss.

"The higher arithmetic presents us with an inexhaustible store of interesting truths, - of thruths too, which are not isolated, but stand in a close internal connection, and between which, as our knowledge increases, we are continually discovering new and sometimes wholly unexpected ties. A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity upon them, are often easily discoverable by induction, and yet are of so profound a character that we cannot find their demonstration till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simpler methods may long remained concealed.

-Gauss C. F.

"That this subject [of imaginary magnitudes] has hitherto been considered from the wrong point of view and surrounded by a mysterious obscurity, is to be attributed largely to an ill-adapted notation. If for instance, +1, -1, �1 had been called direct, inverse, and lateral units, instead of positive, negative and imaginary (or even impossible) such an obscurity would have been out of question."

-Gauss C. F.

"I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a

-Gauss C. F.

"I am exceedingly sorry that I have failed to avail myself of out former greater proximity to learn more of your work on the foundations of geometry; it surely would have saved me much useless effort and given me more peace, than one of my disposition can enjoy so long as so much is left to consider in a metter of this kind. I have myself made much process in this matter (though my other heterogeneous occupations have left me but little time for this purpose); though the course which you assure me you have reached, as much to the desired end, which you assure me you have reached as to the questioning of the truth of geometry. It is true that I have found much which many would accept as proof, but which in my estimation proves

-Gauss C. F.

"On the supposition that Euclidean geometry is not valid, it is easy to show that similar figures do not exist; in that case, the angles of an equilateral triangle vary with the side in which I see no absurdity at all. The angle is a function of the side and the sides are functions of the angle, a functions of the angle, a function which, of course, at the same time involves a constant length. It seems somewhat of a paradox to say that a constant length could be given a priori as it were, but in this again I see nothing inconsistent. Indeed it would be desirable that Euclidean geometry were not valid, for then we should possess a general a priori standard of measure."

-Gauss C. F.

"I am convinced more and more that the necessary truth of our geometry cannot be demonstrated, at least not

-Gauss C. F.

"There is no doubt that it can be rigorously established that the sum of the angles of a rectilinear triangle cannot exceed 180�. But it is otherwise with the statement that the sum of the angles cannot be less than 180�; this is the real Gordian knot, the rocks which cause the wreck of all.... I have been occupied with the problem over thirty years and I doubt if anyone has given it more serious attention, though I have never published anything concerning it. The assumption that the angle sum is less than 180� leads to a peculiar geometry, entirely different from Euclidean, but throughout consistent with itself. I have developed this geometry to my own satisfaction so that I can solve every problem that arises in it with the exception of the determination of a certain constant which cannot be determined a priori. The larger one assumes this constant the more nearly one approaches the Euclidean geometry, an infinitely large value makes the two coincide. The theorems of this geometry seem in part paradoxial, and to the unpracticed absurd; but on a closer and calm reflection it is found that in themselves they contain nothing impossible.... All my efforts to discover some contradiction, some inconsistency in this Non-Euclidean geometry have been fruitless, the one thing in it that seems contrary to reason is that space would have to contain a

-Gauss C. F.

"there is also another subject, which with me is nearly forty years old, to which I have again given some thought during leisure hours, I mean the foundations of geometry.... Here, too, I have consolidated many things, and my convictions has, if possible become more firm that geometry cannot be completely established on a priori grounds. in the mean time I shall probably not for along time yet put my

-Gauss C. F.

"I will add that I have recently received from Hungary a little paper on non-Euclidean geometry in which I rediscover all

-Gauss C. F.

"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 B�otians. 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."

-Sartorius, W. V. Waltershausen

"But how to describe to you my admiration and astonishment at seeing my esteemed correspondent Monsieur LeBlanc metamorphose himself into this illustrious personage who gives such a brilliant example of what I would find it difficult to believe. The enchanting charms of this sublime science reveal only to those who have the courage to go deeply into it. But when a woman, who because of her sex and our prejudices encounters infinitely more obstacles that an man in familiarizing herself with complicated problems, succeeds nevertheless in surmounting these obstacles and penetrating the most obscure parts of them, without doubt she must have the noblest courage, quite extraordinary talents and superior genius."

-Carl Friedrich Gauss