If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words,the most important parts of mathematics stand without a foundation.

Quoted in G F Simmons,

[*About Gauss' mathematical writing style*]

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

Quoted in G F Simmons, *Calculus Gems* (New York 1992).

It appears to me that if one wishes to make progress in mathematics, one should study the masters and not the pupils.

Quoted in O Ore, *Niels Abel, Mathematician Extraordinary*

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

Quoted in E Maor, *To Infinity and Beyond: a Cultural History of the Infinite*

The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and so many paradoxes ...

Quoted in E Maor, *To Infinity and Beyond: a Cultural History of the Infinite*

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 therefore dare hope that the mathematicians will receive this memoir with good will, for its purpose is to fill this gap in the theory of algebraic equations.

Opening of *Memoir on algebraic equations, proving the impossibility of a solution of the general equation of the fifth degree* (1824)

JOC/EFR December 2013

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