_Presidential Address British Association for the Advancement of Science, Section A, (1890); Nature, Vol. 42, p. 467._ =624.= As long as a branch of science offers an abundance of problems, so long it is alive; a lack of problems foreshadows extinction or the cessation of independent development. --HILBERT, D. _Mathematical Problems; Bulletin American Mathematical Society, Vol. 8, p. 438._ =625.= In mathematics as in other fields, to find one self lost in wonder at some manifestation is frequently the half of a new discovery.--DIRICHLET, P. G. L. _Werke, Bd. 2 (Berlin, 1897), p. 233._ =626.= The student of mathematics often finds it hard to throw off the uncomfortable feeling that his science, in the person of his pencil, surpasses him in intelligence,--an impression which the great Euler confessed he often could not get rid of. This feeling finds a sort of justification when we reflect that the majority of the ideas we deal with were conceived by others, often centuries ago. In a great measure it is really the intelligence of other people that confronts us in science. --MACH, ERNST. _Popular Scientific Lectures (Chicago, 1910), p. 196._ =627.= It is probably this fact [referring to the circumstance that the problems of the parallel axiom, the squaring of the circle, the solution of the equation of the fifth degree, have finally found fully satisfactory and rigorous solutions] along with other philosophical reasons that gives rise to the conviction (which every mathematician shares, but which no one has yet supported by proof) that every definite mathematical problem must necessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts.... This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no _ignorabimus_.--HILBERT, D. _Mathematical Problems; Bulletin American Mathematical Society, Vol. 8, pp. 444-445._ =628.= He who seeks for methods without having a definite problem in mind seeks for the most part in vain.--HILBERT, D. _Mathematical Problems; Bulletin American Mathematical Society, Vol. 8, p. 444._