Juke 14, 1907] 



SCIENCE 



935 



conscious or unconscious judgment, penetra- 

 tion into the heart of a matter — whatever we 

 like to call it — ^play a large part in the selec- 

 tion of means to an end? It is not neces- 

 sary — is it even advisable? — to stop and con- 

 sider the theoretical possibilities of a new step 

 if one feels certain it is to lead to the desired 

 end, especially in the application of mathe- 

 matics to physical problems. If astronomers 

 had known that the series they were to use 

 were nearly all divergent without at the same 

 time knowing that they could still be used, 

 would they have been inclined to undertake 

 the enormous calculations which have resulted 

 in our present tables for the positions of the 

 moon and planets? 



The centipede was happy till 

 One day the toad in fun 



Said, " Pray, which leg comes after which ? " 



This raised his thoughts to such a pitch, 



He lay distracted in a ditch, 

 Not knowing how to run. 



At the same time, one does not in the least 

 wish to depreciate the value of the labors of 

 those, and above all of Poincare himself, who 

 have rendered such magnificent services to the 

 cause of pure science by placing the methods 

 of the applied mathematicians on a secure 

 foundation. 



In reading through the memoirs there are 

 certain features of Dr. Hill's work which im- 

 press themselves on the mind. His power of 

 dealing with long and complicated expressions 

 with apparent ease is often the secret of his 

 success. Unlike the methods in the two 

 papers just mentioned which possess an ex- 

 cellent symmetry of mathematical form, ex- 

 pressions best adapted for computation are 

 usually least symmetrical. And the reason 

 for this is not difficult to understand. For the 

 symmetry frequently implies some kind of re- 

 lation between the symbols constituting the 

 expression, which relation can often be used 

 for abbreviating the work. In many cases one 

 would shrink from attempting to reduce Dr. 

 Hill's formulas to numbers, but he rarely fails 

 to give one or more numerical examples to 

 show how his methods can be applied. For 

 instance, in memoir 79, which is an attempt 

 to introduce the use of purely periodic terms 



to express the coordinates of the planets in 

 terms of the time, instead of the usual method 

 which involves secular terms, he estimates that 

 some 2,800 special values of a certain ex- 

 pression will have to be computed. He im- 

 mediately sets out the computation and the 

 results for 175 of these to 13 places of 

 decimals. And again, in the last paper on 

 'Dynamic Geodesy' in which he examines 

 methods for computing the effects of the 

 continents and seas in order to obtain a more 

 accurate expression for the value of gravity 

 at any place, there are five suppositions as to 

 the distribution of the earth's mass; in each 

 case the value of g and the deviation of the 

 plumb line is found at several positions on the 

 earth's surface. 



The freshness and originality of Hill's work 

 make it difficult to attach him to any par- 

 ticular school of mathematicians ; if any such 

 attempt is made, he belongs perhaps more 

 closely to that of the mathematical astrono- 

 mers of the latter part of the eighteenth cen- 

 tury, and of their immediate successors. This 

 is not unnatural, for it was to them that he 

 owed his first inspiration. But his methods 

 are essentially his own, even when he is ex- 

 pounding or using the work of his predeces- 

 sors. We perhaps need more such men, lest 

 the font of originality be choked up by the 

 attempt to assimilate the mass of work which 

 is being turned out every year in increasing 

 quantities. That Dr. Hill has by no means 

 ceased to contribute his share is shown by the 

 last five papers contained in over a hundred 

 pages which had not previously been pub- 

 lished. The range of subjects is sufficiently 

 varied. Two are continuations of memoir 79, 

 to which reference has already been made ; one 

 is on the development of the disturbing func- 

 tion; one on the construction of maps, in 

 which he sets forth a method for the better 

 representation of large areas of the earth's 

 surface or of the sky on paper; and the final 

 one, also mentioned above, on geodesy. 



It is to be regretted that the paper used for 

 the reproduction is thick and unsized. In 

 consequence of this the volumes are heavy and 

 those who wish to make notes will find it 

 necessary to use a sharp pencil rather than a 



