Thus metaphysics and mathematics are, among all the sciences that belong to reason, those in which imagination has the greatest role. I beg pardon of those delicate spirits who are detractors of mathematics for saying this .... The imagination in a mathematician who creates makes no less difference than in a poet who invents.... Of all the great men of antiquity, Archimedes may be the one who most deserves to be placed beside Homer.
Algebra is generous: she often gives more than is asked for.
Allez en avant, et la foi vous viendra
Push on and faith will catch up with you.
[advice to those who questioned the calculus]
One magnitude is said to be the limit of another magnitude when the second may approach the first within any given magnitude, however small, though the second may never exceed the magnitude it approaches.
Geometry, which should only obey Physics, when united with it sometimes commands it. If it happens that the question which we wish to examine is too complicated for all the elements to be able to enter into the analytical comparison we wish to make, we separate the more inconvenient [elements], we substitute others for them, less troublesome but also less real, and we are surprised to arrive, notwithstanding a painful labour, only at a result contradicted by nature; as if after having disguised it, cut it short or altered it, a purely mechanical combination could give it back to us.
The imagination in a mathematician who creates makes no less
difference than in a poet who invents.... Of all the great men of
antiquity, Archimedes may be the one who most deserves to be
placed beside Homer.
The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this
was long ago; to this day a shape will occasionally appear which will
fit into the garment as if the garment had been made for it. Then there
is no end of surprise and delight.
I am delighted at the contrast between your modesty and the good opinion that other geometers have of themselves, although they have certainly nothing like the same claim. You are a living instance of what you said to me some time ago, that pretensions are ever in an inverse ratio to merit. (letter to Lagrange).
My dear and illustrious friend - they write to me from Berlin that you are about to take what we philosophers call 'le saut perilleux,' and that you have married one of your relations. ... Accept my compliments, for a mathematician ought to have pre-eminent advantages in the calculations of his own happiness, and any calculations of yours are sure to lead to a solution - the solution in your case being marriage" (letter to Lagrange).