192 ANNUAL EEPOET SMITHSONIAN INSTITUTION, 1912. 



during the century. A great part of this waste would be eliminated 

 if those who would like to test their ability along this line could be 

 induced to read, before they offer their work for publication, the dis- 

 cussion of more than 100 supposed proofs whose errors are pointed 

 out in a German mathematical magazine called "Archiv der Mathe- 

 matik und Physik," pubhshed by B. G. Teubner, of Leipzig. A very 

 useful pamphlet dealing with this question is entitled, " Ueber das 

 letzte Fennatische Theorem, von B. Lind," and was also pubhshed by 

 B. G. Teubner, in 1910. 



A possible good effect of the offered prize is that it may give rise 

 to new developments and to new methods of attack. As the most 

 successful partial solution of the problem was due to the modern 

 theory of algebraic numbers, one would naturally expect that further 

 progi-ess would be most hkely to result from a further extension of this 

 theory, or, possibly, horn a still more powerful future theory of num- 

 bers. If such extensions will result from this offer they will go far 

 to offset the bad effect noted above, and they may leave a decided 

 sm-plus of good. Such a standing problem may also tend to lessen 

 mathematical idolatry, which is one of the most serious barriers to 

 real progress. We should welcome everything which tends to elevate 

 the truth above our idols formed by men, institutions, or books. 



In view of the fact that the offered prize is about $25,000 and that 

 lack of marginal space in his copy of Diophantus was the reason 

 given by Fermat for not communicating his proof, one might be 

 tempted to wish that one could send credit for a dime back through 

 the ages to Fermat and thus secm'e this coveted prize and the won- 

 derful proof, if it actuaUy existed. This might, hov/ever, result more 

 seriously than one would at first suppose; for, if Fermat had bought 

 on credit a dime's worth of paper even dming the year of liis death, 

 1665, and if this bill had been drawing compound interest at the rate 

 of 6 per cent since that time, the bill would now amount to more than 

 seven times as much as the prize. It would therefore require more 

 than $150,000, in addition to the amount of the prize, to settle this 

 bill now. 



While it is very desirable to be familiar with such standmg prob- 

 lems as Format's theorem, they should generally be used by the 

 young investigator as an indirect rather than as a dhect object of 

 research. Unity of purpose can probably not be secured in any 

 better way than by Iceeping in close touch mth the masters of the 

 past,^ and tliis unity of purpose is almost essential to secure real 

 effective work in the immense field of mathematical endeavor. As 

 a class of problems which are much more suitable for dhect objects 

 of research on the part of those who are not in close contact with a 



» Darboux, Bulletm des Sciences Math6inatiques, vol. 32 (1908), p 107. 



