October 15, 1914] 



NATURE 



185 



has been maintained to prove that the moon, like the 

 other bodies of the solar system, obeyed this law to 

 its farthest consequences. While the theory was being 

 advanced, the observers were continually improving 

 their instruments and their methods of observing, 

 with the additional advantage that their efforts had 

 a cumulative effect ; the longer the time covered by 

 their observations, the more exact was the knowledge 

 obtained. The theorist lacked the latter advantage ; 

 if he started anew he could only use the better instru- 

 ments for analysis provided by the mathematician. 

 He was always trj'ing to forge a plate of armour 

 which the observer with a gun the power of which 

 was increasing with the time could not penetrate. 

 In the struggle the victory rarely failed to rest with 

 the observer. Within the last decade we theorists 

 have made another attempt to forge a new plate out 

 of the old materials ; whether we have substantially 

 gained the victory must rest partly on the evidence I 

 have to place before you to-day and partly on what 

 the observer can produce in the near future. 



There are three well-defined periods in the history 

 of the subject so far as a complete development of the 

 moon's motion is concerned. From the publication 

 of the " Principia " in 1687, when Newton laid down 

 the broad outlines, until the middle of the eighteenth 

 century, but little progress was made. It seems to 

 have required more than half a century for analysis by 

 symbols to advance sufficiently far for extensive appli- 

 cations to the problems of celestial mechanics. 

 Clairaut and d'Alembert both succeeded in rescuing 

 the problem from the geometrical form into which 

 Newton had cast it and in reducing it to analysis by 

 the methods of the calculus. They were followed by 

 Leonard Euler, who in my opinion is the greatest of 

 all the successors of Isaac Newton as a lunar theorist. 

 He initiated practically every method which has been 

 used since his time, and his criticisms show that he 

 had a good insight into their relative advantages. A 

 long roll of names follows in this period. It was 

 closed by the publication of the theories of Delaunay 

 and Hansen and the tables of the latter, shortly after 

 the middle of the nineteenth century. From then to 

 the end of the century the published memoirs deal 

 with special parts of the theory or with its more 

 general aspects, but no complete development appeared 

 which could supersede the results of Hansen. 



My own theory, which was completed a few years 

 ago, is rather the fulfilment to the utmost of the ideas 

 of others than a new mode of finding the moon's 

 motion. Its object was severely practical — to find in 

 the most accurate way and by the shortest path the 

 complete effect of the law of gravitation applied to 

 the moon. It is a development of Hill's classic 

 memoir of 1877. Hill in his turn was indebted to 

 some extent to Euler. His indebtedness would have 

 been greater had he been aware of a little-known 

 paper of the latter, " Sur la Variation de la Lune," 

 in which the orbit, now called the variation orbit, is 

 obtained, and its advantages set forth in the words : 

 "Quelque chim^rique cette question j'ose assurer que, 

 kI I'on r^ussissoit k en trouver une solution parfaite on 

 ne trouveroit presque plus de difficult^ pour determiner 

 le vrai mouvement de la Lune r^elle. Cette question 

 est done de la derni^re Importance et il sera toujours 

 bon d'en approfondir toutes les difiicult^s, avant qu'on 

 en puisse esp^rer une solution complete." 



In the final results of my work the development 

 aims to Include the gravitational action of every par- 

 ticle of matter which can have a sensible effect on the 

 moon's motion, so that any differences which appear 

 between theory and observation may not be set down 

 to want of accuracy in the completeness with which 



NO. 2346, VOL. 94] 



the theory is carried out. Every known force capable 

 of calculation is included. 



So much for the theory. Gravitation, however, is 

 only a law of force ; we need the initial position, speed, 

 and direction of motion. To get this with sufficient 

 accuracy no single set of observations will serve; the 

 new theorj' must be compared with as great a number 

 of these as possible. To do this directly from the 

 theory is far too long a task, and, moreover, it is not 

 necessary. In the past every observation has been 

 compared with the place shown in the " Nautical 

 Almanac," and the small differences between them have 

 been recorded from day to day. By taking many of 

 these differences and reducing them so as to corre- 

 spond with differences at one date, the position of the 

 moon at that date can be found with far greater 

 accuracy than could be obtained through any one 

 observation. At the Greenwich Observatorj- the moon 

 has been observed and recorded regularly since 1750. 

 With some 120 observations a year, there are about 

 20,000 available for comparison, quite apart from 

 shorter series at other observatories. Unfortunately 

 these observations are compared with incorrect 

 theories, and, in the early days, the observers were 

 not able to find out, with the accuracy required to-day, 

 the errors of their instruments or the places of the 

 stars with which the moon was compared. But we 

 have means of correcting the observations, so that 

 they can be freed from many of the errors present in 

 the results which were published at the time the 

 observations were made. We can also correct the 

 older theories. They can be compared with the new 

 theory and the differences calculated ; these differences 

 need not even be applied to the separate observations, 

 but only to the observations combined into properly 

 chosen groups. Thus the labour involved in making 

 use of the earlier observations is much less than might 

 appear at first sight. 



For the past eighteen months I have been engaged 

 in this work of finding the differences between the old 

 theories and my own, as well as in correcting those 

 observations which were made at times before the 

 resources of the astronomer had reached their present 

 stage of perfection. I have not dealt with the observa- 

 tions from the start; other workers, notably Air\', in 

 the last century, and Cowell in this, have done the 

 greater part of the labour. My share was mainly to 

 carry theirs a stage further by adopting the latest 

 theory and the best modern practice for the reduction 

 of the observations. In this way a much closer agree- 

 ment between theory and observation has been ob- 

 tained, and the initial position and velocity of the 

 moon at a given date are now known with an accuracy 

 comparable with that of the theory. I shall shortly 

 return to this problem and exhibit this degree of 

 accuracy by means of some diagrams which will be 

 thrown on the screen. 



I have spoken of the determination of these initial 

 values as If it constituted a problem separate from the 

 theory. Theoretically It is so, but practically the two 

 must go together. The Increase In accuracy of the 

 theory has gone on successively with increase in 

 accuracy of the determination of these constants. We 

 do not find, with a new theory, the new constants 

 from the start, but corrections to the previously 

 adopted values of these constants. In fact, all the 

 problems of which I am talking are so much Inter- 

 related that It is only justifiable to separate them for 

 the purposes of exposition. 



Let us suppose that the theory and these constants 

 have been found in numerical form, so that the posi- 

 tion of the moon is shown by means of expressions 

 which contain nothing unknown but the time. To 



