December 29, 1905.] 



SCIENCE. 



867 



in setting up a fourth necessary condition. 

 These statements apply, however, only when 

 we admit as ' comparison curves ' curves which, 

 although they lie reasonably near the supposed 

 solution, differ greatly from the supposed solu- 

 tion in inclination. For some physical appli- 

 cations it is evident from the nature of the 

 problem that such comparison curves may be 

 neglected. For example, in the famous prob- 

 lem of the ' Surface of Rotation of Least 

 Resistance to a Moving Fluid,' which is due 

 to Newton, there exist no solutions whatever 

 in the sense just explained. That is to say, 

 there exists no surface of revolution whatever 

 which does not offer more resistance than 

 some other surface of rotation, under the 

 Newtonian assumptions regarding resistance. 

 The possibility of such a result was entirely 

 overlooked even within the lifetime of scien- 

 tists now living, it being assumed as an a 

 priori fact that some solution surely existed 

 in this and many other similar problems. On 

 the other hand, it is probable that any physi- 

 cist would hold that the extreme wavyness — ■ 

 the krinklyness, to use a term invented by 

 Professor Moore — which the surfaces of less 

 resistance than the formal ' solution ' must 

 have, is really an objection to their considera- 

 tion from a physicist's standpoint, in that the 

 saw-like edges of the surface would cause a 

 local eruption in the moving fluid which was 

 not contemplated when the original assump- 

 tions regarding the nature and amount of the 

 resistance were made. To meet such reason- 

 able objection, a distinction is made between 

 strong minimizing curves, i. e., curves which 

 render the integral of the problem under con- 

 sideration absolutely less than any other curve 

 whatever in their neighborhood, and, on the 

 other hand, weak minimizing curves, i. e., 

 curves which render the integral less than any 

 other curve which lies in the neighborhood 

 and whose slope differs only slightly from that 

 of the ' solution.' To clinch the point it may 

 be stated that while there exists no solution of 

 Newton's famous problem, there are ' weak [ 

 solutions, and these are precisely the stock 

 solutions of the older books; the difference is 

 that the newer theory demonstrates that there 

 are curves which give a less resistance than 



the stock solution, whereas the older books 

 would have led one to believe that such was 

 not the case. 



While this is merely a typical example, it 

 illustrates in an essential manner the tend- 

 encies of the modern thought. The develop- 

 ment, not of this example, but of the general 

 theory of Weierstrass's condition and that of 

 the sufficient conditions both for ' weak ' and 

 for ' strong ' minimizing (or maximizing) 

 curves forms the subject matter of the third 

 chapter. It may be that such a precise theory 

 is too far advanced for really practical appli- 

 cation, but the writer is among those who be- 

 lieve that mechanics', physics and a few other 

 sciences are upon the eve of the same reforma- 

 tion which has characterized the progress of 

 mathematics during the past century, by which 

 mathematics has become more really an ' exact 

 science.' The fact that errors in conception 

 and errors in proof have been made in that 

 science which has been misnamed ' exact,' and 

 that these errors have been at least partially 

 removed through impartial and accurate in- 

 vestigation may be surprising to many who are 

 not specialists in mathematics, but it would 

 seem to suggest at the same time the possi- 

 bility of similar investigation in sciences 

 which have hitherto been classed as ' inexact,' 

 in that it demonstrates that the difference is, 

 after all, not so fundamental. Especially 

 when the subject treated is as closely allied to 

 mechanics and physics asi is the calculus of 

 variations, the possibility of rendering those 

 subjects just as rigorously exact in certain 

 chapters as are a few of the present branches 

 of mathematics begins to appear less visionary. 

 Possibly a confession in such a general scien- 

 tific organ as is Science, that there are por- 

 tions of mathematics — notably the theory of 

 curves and surfaces — which are still open to 

 fundamental criticism and objection, will tend 

 to further the impression that there is no 

 radical characteristic difference in exactness 

 between mathematics and certain other sci- 

 ences. 



The next chapters will be of especial in- 

 terest chiefly to those who are specialists in 

 mathematics. Suffice it to say here that they 

 treat other less specialized cases than the one 



