Maboh ao, 1911] 



SCIENCE 



363 



should take. Consider, for example, ana- 

 lytic differential equations. There are five 

 distinct methods of developing their solu- 

 tions — as power series in the independent 

 variable, as power series in parameters, as 

 limits of equations of finite differences, by- 

 successive approximations, and by succes- 

 sive applications of the variation of con- 

 stants — all of which were devised under the 

 pressure of practical astronomical prob- 

 lems and were applied successfully many 

 years before the conditions of their legiti- 

 macy were fully established by mathemat- 

 ical methods. A more recent example is 

 Hill's treatment of the linear differential 

 equation having simply periodic coeffi- 

 cients, the properties of whose solutions 

 were inferred by him from the prop- 

 erties of the motion of the moon. The 

 problems connected with an infinite num- 

 ber of simultaneous homogeneous linear 

 equations also arose in Hill's lunar theory. 

 Poineare's researches in the problem 

 of three bodies led him to the discovery 

 of many new properties of the solutions 

 of differential equations. The question 

 of the legitimacy of the series used in 

 celestial mechanics, particularly when ap- 

 plied for long intervals, has forced a con- 

 sideration of the problem of determining 

 what classes of divergent series may be 

 used and what conclusions may be drawn 

 from them; and the same question has 

 stimulated investigations of other modes of 

 representing solutions, particularly as 

 sums of polynomials in the independent 

 variable, having wider domains of validity. 

 In this direction Painleve has achieved the 

 most important results, and has shown how 

 to construct functions which represent the 

 solution of the general problem of n bodies 

 so long as there are no collisions. If the 

 forces were repulsive instead of attractive 

 the developments would be valid indefi- 

 nitely. But as Laplace said " nature does 



not care for analytical difficulties " ; in fact, 

 it fills the way of the mathematician with 

 them. As a partial recompense for the 

 difficulties it raises it often suggests meth- 

 ods for overcoming them, and these 

 methods being made general in the sym- 

 bolism of mathematics constitute new 

 processes often applicable in many other 

 directions. 



One of the recent movements in mathe- 

 matics is in the application of the postula- 

 tional method. It consists in postulating 

 the existence of certain elements which are 

 wholly without properties except as they 

 are imposed by the postulates and the ex- 

 plicitly stated axioms. The postulates and 

 their implications constitute the theory. It 

 is not to be supposed that the postulates are 

 laid down at random, or even on any simple 

 principle of their individual and separate 

 characteristics. The sole guide is that 

 taken together they shall yield as conse- 

 quences certain relations which are in ad- 

 vance in the consciousness of the investi- 

 gator; the additional implications are the 

 contributions which the developed theory 

 makes. I do not know why there has 

 sprung up the recent interest in this 

 method, but it is fundamentally the method 

 used in natural science. The experiences 

 are the certainties given in advance which 

 must be implications of the theories. The 

 atoms, corpuscles, units of electricity, etc., 

 are the postulated elements. The theories 

 are the postulated relations among the ele- 

 ments. If we let «!, ■■•, an represent the 

 experiences, x-^^, ■■■, Xm the postulated ele- 

 ments, then we shall have 

 U{a!j) =ai, 



where the /» are the theories. If one of 

 these relations fails to hold it is necessary 

 to modify the Xj, or the /,:, or both, and it 

 is easy to cite examples from the history 

 of science illustrating all these possibilities. 

 The recognition of the fundamental iden- 



