52 THE POPULAR SCIENCE MONTHLY. 



by the unaided eye (this I say for the benefit of those who otherwise 

 might not understand what " pernickity " creatures astronomers are) ; 

 but they are large enough to be easily observable, and even obtrusive, 

 amounting to several seconds of arc, or miles of space. As we have 

 seen, the attempt to account for them by the irregularity of the earth's 

 rotation has apparently failed ; and we are driven to the conclusion, 

 either that other forces than gravitation are operative upon the lunar 

 motions, or else (what is far more probable, considering the past his- 

 tory of theoretical astronomy) that the mathematical theory is some- 

 where at fault. 



To one looking at the matter a little from the outside it seems as 

 if that which is most needed just now, in order to secure the advance 

 of science in many directions, is a new, more comprehensive, and more 

 manageable solution of the fundamental equations of motion under 

 attraction. Far be it from me to cry out against those mathemati- 

 cians who delight themselves in transcendental and n-dimensional 

 space, and revel in the theory of numbers — we all know how unex- 

 pectedly discoveries and new ideas belonging to one field of science 

 find use and application in widely different regions — but I own I feel 

 much more interest in the study of the theory of functions and differ- 

 ential equations, and expect more aid for astronomy from it. 



The problem of any number of bodies, moving under their mutual 

 attraction, according to the Newtonian laws, stands, from a physical 

 point of view, on precisely the same footing as that of tioo bodies. 

 Given the masses, and the positions and velocities corresponding to 

 any moment of time, then the whole configuration of the system for 

 all time, past and future (abstracting outside forces, of course), is ab- 

 solutely determinate, and amenable to calculation. But while, in the 

 case of two bodies, the calculation is easy and feasible, by methods 

 known for two hundred years, our analysis has not yet mastered the 

 general jaroblem for more than two. In special instances, by compu- 

 tations, tedious, indirect, and approximate, we can, indeed, carry our 

 predictions forward over long periods, or indicate past conditions with 

 any required degree of accuracy ; but a general and universally prac- 

 ticable solution is yet wanting. The difiiculties in the way are purely 

 mathematical ; a step needs to be taken, corresponding in importance 

 to the introduction of the circular functions into trigonometry, the 

 invention of logarithms, or the discovery of the calculus. The problem 

 confronts the astronomer on a hundred different roads ; and, until it 

 is overcome, progress in these directions must be slow and painful. 

 One could not truly say, perhaps, that the lunar theory must, in the 

 mean while, remain quite at a stand-still : labor expended in the old 

 ways, upon the extension and development of existing methods, may 

 not be fruitless, and may, perhaps, after a while, effect the reconcile- 

 ment of prediction and observation far beyond the present limits of 

 accuracy. But if we only had the mathematical powers we long for, 



