NA TURE 



409 



THURSDAY, MARCH 1, 1906. 



MATHEMATICAL ASTRONOMY. 

 The Collected Mathematical Works of George William 

 Hill. Vol. i. Pp. xviii + 363. (Washington: The 

 Carnegie Institution, 1905.) 



IT is a rare mind that can handle the cumbrous 

 developments of practical astronomy and leave 

 uppermost with a reader the impression of variety, 

 ease, and polish; and curiosity will be felt as to the 

 1 in umstances which have developed Hill's remark- 

 able powers. From an interesting introduction to 

 the present volume by M. Po'mcare we learn that 

 he spent three years at Rutgers College, New Jersey, 

 under a certain Dr. Strong. Dr. Strong " etait 1111 

 homme de tradition, un laudator temporis acti; pour 

 lui Euler etait le Dieu des Mathematiques, et apres 

 In", la decadence avait commence; il est vrai que e'est 

 la un dieu que Ton peut adorer avec profit," and if 

 it led Hill to the study of originals, we may overlook 

 the depreciation of the moderns. From New Jersey 

 he went to Cambridge to continue his studies at 

 Harvard; very soon here, by a paper contributed for 

 a prize to a mathematical miscellany, he attracted 

 the notice of Runkle, the editor, who was N'ewcomb's 

 predecessor at the office of the American Ephemeris. 

 Hill became attached to the Ephemeris as computer, 

 and remained in discharge of these duties for thirty- 

 two years. At first he worked at his own home, as 

 was then the custom; but under Newcomb's manage- 

 ment, and in order to complete his theory and tables 

 of Jupiter and Saturn he lived for some years at 

 Washington, incessantly absorbed in his task. "The 

 only defect of his make-up of which I have reason 

 to complain," Newcomb has written, " is the lack of 

 the teaching faculty." In 1892 he withdrew to the 

 little farm where his boyhood was passed, and where 

 he still lives, asking nothing but the liberty to con- 

 tinue his labours. 



The present volume carries us up to 18S1, and 

 includes most, but by no means all, of his best known 

 papers. The essay which attracted Runkle's notice 

 is No. 3, "On the Conformation of the Earth," and 

 was written at the age of twenty-three. It is perhaps 

 not of any permanent importance, yet it is marked by 

 the clearness and the firm hand of his later writings 

 and the same salutary determination that theory 

 should give an account of itself arithmetically. It is 

 natural to compare it with" Stokes's memoir " On the 

 Variation of Gravity," written some twelve years 

 before, when he also was a young man, and the 

 comparison shows strikingly how Stokes is the 

 physicist and Hill the analvst. 



The two great memoirs by which Hill is best 

 known are Xo. 29, " On the Part of the Motion of 

 tin- Lunar Perigee which is a Function of the Mean 

 Motions of the Sun and Moon," and No. 32, " Re- 

 searches in the Lunar Theory." These writings have 

 been greatly praised, but it seems impossible to praise 

 them too highly, whether for their difficulties or the 

 way these are overcome, or the greatness of the 

 NO. 1896, VOL. 73] 



advance which their solution implies. The latter 

 paper was the first which threw any real light upon 

 the general problem of three bodies, and it is well 

 worth notice how large a part arithmetic plays in its 

 success. The analysis is pregnant in the extreme, 

 but it is the actual calculation of a whole sequence 

 of periodic orbits which a moon might occupy that 

 gives it shape and name. 



If this memoir may be said to be the first significant 

 word on the problem of three bodies, the former one, 

 on the motion of the lunar perigee, seems to be 

 almost the last word on a question that had outrun 

 calculation from Newton's day to Delaunay's. It is 

 doubtful whether the more determined effort to 

 calculate this quantity was made by Newton or 

 by Delaunay, but though naturally the degrees of 

 approximation they attained were very different, they 

 had this in common, that they proved the inadequacy 

 of the methods employed. Hill first, with the smooth- 

 ness of a conjurer, gives form to the intractable 

 equations, and then shows how the solution is con- 

 tained in a certain transcendental equation, an infinite 

 determinant. It affords striking evidence of Hill's 

 power to contrast his treatment of this determinant 

 with that of Adams, who followed a similar route, sed 

 longo intervallo, as he said himself. The complexity 

 arising from an infinite sequence of equations might 

 seem to preclude any general conclusions from being 

 drawn, but Hill uses this very feature in the most 

 beautiful manner to derive the eliminant in a trans- 

 cendental form in the shape cos-nr= a known 

 quantity, and from this equation determines c, the re- 

 quired ratio. The secret of the success is now 

 apparent, c is nearly equal to unity; hence it is very 

 much easier to approximate to cos if, where we are 

 in the neighbourhood of a stationary value, than to 

 c directly; and though the difficulty recurs when we 

 seek to find the arc tc from a cosine in the neigh- 

 bourhood of its minimum, it is then an insignificant 

 one, for we are past the true complexities of the 

 problem. 1 



The remaining papers are naturally not of equa! 

 moment with these, but we may be grateful to the 

 Carnegie Institution for making them accessible in 

 the present collection. Several of them arose in con- 

 nection with Hill's duties as computer to the 

 Ephemeris, but even on such hackneyed subjects as 

 eclipse computing and reduction of star places he has 

 something good to say. He is a true artist ; nullum 

 quod tetigit non ornavit. Of considerable general 

 interest are No. 18, " Remarks on the Stability of 

 Planetary Systems," and No. 14, " A Method of Com- 

 puting Absolute Perturbations," which contains a 

 rescension of Hansen. Even the smaller papers, like 

 No. 22, " On the Solution of the Cubic and Biquad- 

 ratic Equations," are usually marked by some 

 analytical felicity that makes one wish that Hill had 

 been able to bring his great powers to bear upon 

 a material not so invariably intractable and over- 

 loaded with tradition, and limited in its problems, as 

 practical astronomy. But if we feel that his hand 



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