SCIENCE. 



[N. S. Vol. XIV. No. 340. 



field was already occupied by the philos- 

 ophy of Descartes. The glittering gener- 

 alities found here in place of the severe 

 mathematical reasoning of ISTewton proved 

 too attractive to be at once overthrown. 



Fifty years elapsed before anything ma- 

 terial was added to the science of Newton, 

 then came a galaxy of distinguished men, 

 including Euler, Clairaut, D'Alembert, La- 

 grange and Laplace, who, by a series of 

 most brilliant and exhaustive researches, 

 made possible by the development of much 

 more powerful mathematical instruments 

 than those possessed by Newton, practically 

 disposed of every objection which the op- 

 ponents of Newton's theory could discover. 



Their last stronghold and one which 

 proved extremely difficult to carry was the 

 explanation of the secular acceleration of 

 the moon's motion. Halley, one of New- 

 ton's disciples, had found by a comparison 

 of ancient and medieval eclipses with those 

 of modern times, that the lunar month is 

 now shorter than was the case 2,000 years 

 ago. How this could be if the motion of 

 this body were governed by the law of 

 gravity only proved an extremely difficult 

 question. It was finally shown by La- 

 place to be one of the consequences of this 

 law, that what is now an acceleration will 

 in the future become a retardation, thus 

 preserving the system essentially as we see 

 it to-day. This took place near the begin- 

 ning of the nineteenth century. We may, 

 therefore, say that this century began its 

 course with the law of universal gravitation 

 firmly established. In fact no one was 

 hereafter found to seriously call it in ques- 

 tion whose opinion was worthy of notice. 



It is not to be supposed, however, that 

 the science of celestial mechanics was now 

 finished. In fact we can hardly say that 

 any branch of physical science ever has been 

 or ever will be finished. Results depend- 

 ing for their value upon observations of 

 any kind must share the inevitable imper- 



fections of the observer and of his instru- 

 ment. 



The problems of celestial mechanics have 

 accordingly occupied the attention of a 

 large number of distinguished men during 

 the century past. With the discovery of 

 new planets and comets and stellar sys- 

 tems, new applications are constantly ap- 

 pearing calling for the most refined skill 

 and active perseverance of its votaries. 

 Refinements in observation and improve- 

 ments in method also call for frequent 

 revision of the old investigations. This is 

 particularly true of the lunar theory. 



The study of the moon's motions was one 

 of the first problems to attract the atten- 

 tion of the ancient astronomers. Since 

 their day we are probably safe in saying 

 that no problem in the entire range of sci- 

 ence has called out anything approaching 

 the labor and ingenuity bestowed upon this 

 one. Yet the theory is hardly in a satis- 

 factory condition to- day. Such a state- 

 ment may seem to those who have never 

 looked into the intricacies of the problem 

 like a confession of failure. 



So far as it concerns a purely mathe- 

 matical statement of the conditions the 

 problem offers no difficulties. The rela- 

 tions between the coordinates of a body 

 like the moon, acted on by any system of 

 forces, are very readily expressed by a series 

 of three differential equations of the second 

 order. The forces here are the mutual at- 

 tractions exerted by the sun, moon and 

 planets. If only two bodies are present 

 the solution offers no difficulty. If there is 

 a third, we have the famous problem of the 

 three bodies. In spite of all that has been 

 done during the century in the way of 

 mathematical advancement this problem 

 still defies its most powerful resources. 

 Recourse must be had to methods of ap- 

 proximation, expansion in the form of 

 series being one of the most obvious. No 

 one can form an idea of the intricacy and 



