“It is easier to square the circle than to get round a mathematician.”

“We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact science are mathematics and logic, the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it sees better with one eye than with two.”

“I end with a word on the new symbols which I have employed. Most writers on logic strongly object to all symbols. . . I should advise the reader not to make up his mind on this point until he has well weighed two facts which nobody disputes, both separately and in connexion. First, logic is the only science which has made no progress since the revival of letters; secondly, logic is the only science which has produced no growth of symbols.”

“Every science that has thriven has thriven upon its own symbols: logic, the only science which is admitted to have made no improvements in century after century, is the only one which has grown no symbols.”

“I was x years old in the year x².”

“The moving power of mathematical invention is not reasoning but imagination.”

“Common integration is only the memory of differentiation...”

“This mysterious 3.141592..., which comes in at every door and window, and down every chimney.”

“The imaginary expression √(-a) and the negative expression -b, have this resemblance, that either of them occurring as the solution of a problem indicates some inconsistency or absurdity. As far as real meaning is concerned, both are imaginary, since 0 - a is as inconceivable as √(-a).”

“Imagine a person with a gift of ridicule [He might say] First that a negative quantity has no logarithm; secondly that a negative quantity has no square root; thirdly that the first non-existent is to the second as the circumference of a circle is to the diameter.”

“I don't quite hear what you say, but I beg to differ entirely with you.”

“As to writing another book on geometry [to replace Euclid] the middle ages would have as soon thought of composing another New Testament.”

“The moving power of mathematical invention is not reasoning but imagination.”

“One day at least in every week,
The sects of every kind
Their doctrines here are sure to seek,
And just as sure to find.”

“The gambling reasoner is incorrigible; if he would but take to the squaring of the circle, what a load of misery would be saved.”

“Considerable obstacles generally present themselves to the beginner, in studying the elements of Solid Geometry, from the practice which has hitherto uniformly prevailed in this country, of never submitting to the eye of the student, the figures on whose properties he is reasoning, but of drawing perspective representations of them upon a plane. ...I hope that I shall never be obliged to have recourse to a perspective drawing of any figure whose parts are not in the same plane.”

“Astronomers! What can avail
Those who calumniate us;
Experiment can never fail
With such an apparatus...”

“Great fleas have little fleas upon their backs to bite 'em,
And little fleas have lesser fleas, and so ad infinitum.
And the great fleas themselves, in turn, have greater fleas to go on;
While these again have greater still, and greater still, and so on.”

“Lagrange, in one of the later years of his life, imagined that he had overcome the difficulty (of the parallel axiom). He went so far as to write a paper, which he tool with him to the Institute, and began to read it. But in the first paragraph something struck him that he had not observed: he muttered: 'Il faut que j'y songe encore', and put the paper in his pocket.'” [I must think about it again]