=629.= A mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution.--HILBERT, D. _Mathematical Problems; Bulletin American Mathematical Society, Vol. 8, p. 438._ =630.= The great mathematicians have acted on the principle “_Divinez avant de demontrer_,” and it is certainly true that almost all important discoveries are made in this fashion. --KASNER, EDWARD. _The Present Problems in Geometry; Bulletin American Mathematical Society, Vol. 11, p. 285._ =631.= “Divide _et impera_” is as true in algebra as in statecraft; but no less true and even more fertile is the maxim “auge _et impera_.” The more to do or to prove, the easier the doing or the proof.--SYLVESTER, J. J. _Proof of the Fundamental Theorem of Invariants; Philosophic Magazine (1878), p. 186; Collected Mathematical Papers, Vol. 3, p. 126._ =632.= As in the domains of practical life so likewise in science there has come about a division of labor. The individual can no longer control the whole field of mathematics: it is only possible for him to master separate parts of it in such a manner as to enable him to extend the boundaries of knowledge by creative research.--LAMPE, E. _Die reine Mathematik in den Jahren 1884-1899, p. 10._ =633.= With the extension of mathematical knowledge will it not finally become impossible for the single investigator to embrace all departments of this knowledge? In answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which at the same time assist in understanding earlier theories and to cast aside some more complicated developments. It is therefore possible for the individual investigator, when he makes these sharper tools and simpler methods his own, to find his way more easily in the various branches of mathematics than is possible in any other science.--HILBERT, D. _Mathematical Problems; Bulletin American Mathematical Society, Vol. 8, p. 479._