312 TRANSACTIONS OF SECTION A. 
all these have to be considered; in fact, almost every one of the departments of 
the astronomy of position must be drawn upon to furnish necessary data. The 
time has now arrived when it may perhaps be possible to repay in some measure 
the debt thus contracted by furnishing to the astronomer, and perhaps also to the 
student of geodesy and, if I may coin a word, of selenodesy, some results which 
can be deduced more accurately from a study of the moon’s motion than in any 
other way. A long-continued exploration with few companions which ultimately 
leads to territories where other workers have already blazed paths gives the 
impression of having emerged from the thick jungle into open country. The 
explorer can once more join forces with his brother astronomers. He can 
judge his own results more justly and have them judged by others. If, 
then, an excuse be needed for overstepping the limits which seem, by silent 
consent, to have been imposed on those who devote themselves to lunar problems, 
it consists in a desire to show that these limits are not necessary and that a study 
of the motion of the moon can be of value and can contribute its share to the 
common funds of astronomy. . 
The history of the motion of the moon has been for more than two centuries 
a struggle between the theorists and the observers. Ever since the publication 
of the ‘Principia’ and the enunciation of the law of gravitation by Isaac 
Newton, a constant effort 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 trying to 
forge a plate of armour which the observer with a gun whose power was increas- 
ing 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 matgrials; 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 as 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 over half-a-century for analysis by symbols to advance sufficiently far 
for extensive applications 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 chimérique cette question j’ose 
assurer que, si l’on réussissoit 4 en trouver une solution parfaite on ne trouveroit 
presque plus de difficulté pour déterminer le vrai mouvement de la Lune réelle. 
