Niels Henrik Abel

I have found that the general solution of algebraic equations of degree higher than four is impossible.

The mathematicians of our time are too much concerned with the solution of problems, and too little with the development of theories.

The mathematicians have been very much absorbed with finding the general solution of algebraic equations, and several of them have tried to prove the impossibility of it. However, if I am not mistaken, they have not as yet succeeded.

I shall apply all my strength to bring more light into the immense obscurity that today reigns in Analysis. It so lacks any plan or system that one is really astonished that so many people devote themselves to it—and, still worse, it is absolutely devoid of any rigor.

With the exception of the geometrical series, there does not exist in all of mathematics a single infinite series whose sum has been determined rigorously.

It is readily seen that any algebraical solution whatsoever must be a rational function of the roots.

My work in the future must be devoted entirely to pure mathematics in its abstract meaning. I shall apply all my strength to bring more light into the immense obscurity that today reigns in Analysis.

By studying the masters, not the pupils.

I have had to overcome tremendous difficulties; but I have the consolation that I have opened a new path in the analysis of transcendental functions.

I have found a curious theorem...

It is a common fault of mathematicians to assume too easily that the conditions under which a theorem is proved are necessary, and not merely sufficient.

The solution of problems is the most characteristic and peculiar sort of voluntary thinking.

I am working very hard, and I hope that my efforts will not be in vain.

Abel's Theorem: 'The sum of an integrand of an algebraic function, taken between given limits, can be expressed by a finite number of such functions.'

I have now finished a large memoir on a certain class of transcendental functions... to present it to the Institute.

My life is a succession of hardships and struggles.

I have, to be sure, sometimes been lucky enough to have an idea, but most of the time it has been the result of long and painful reflection.

In mathematics, one must be careful not to mistake the clear for the true.

The theory of elliptic functions, which I have had the good fortune to discover, is a vast field for research.

I have had the good fortune to find in Holmboe a friend and teacher who recognized my abilities and encouraged me.