Found a great quote while reading through Lagarias’s recent paper on Euler’s constant,
Jeffrey C. Lagarias. Euler’s constant: Euler’s work and modern developments, Bull. Amer. Math. Soc. (N.S.), 50 (4), (2013), 527–628. MR3090422.
Lagarias quotes Truesdell,
Clifford Ambrose Truesdell, III. An idiot’s fugitive essays on science, Methods, criticism, training, circumstances. Springer-Verlag, New York, 1984. MR769106 (86g:01060); and Great scientists of old as heretics in “the scientific method”. University Press of Virginia, Charlottesville, VA, 1987. MR915762 (88m:01038).
C. Truesdell [303, Essay 10], [304, pp. 91–92], makes the following observations about the methods used by Euler, his teacher Johann Bernoulli, and Johann’s teacher and brother Jacob Bernoulli:
Always attack a special problem. If possible solve the special problem in a way that leads to a general method.
Read and digest every earlier attempt at a theory of the phenomenon in question.
Let a key problem solved be a father to a key problem posed. The new problem finds its place on the structure provided by the solution of the old; its solution in turn will provide further structure.
If two special problems solved seem cognate, try to unite them in a general scheme. To do so, set aside the differences, and try to build a structure on the common features.
Never rest content with an imperfect or incomplete argument. If you cannot complete and perfect it yourself, lay bare its flaws for others to see.
Never abandon a problem you have solved. There are always better ways. Keep searching for them, for they lead to a fuller understanding. While broadening, deepen and simplify.