184 ANNUAL REPORT SMITHSONIAN INSTITUTION, 193 3 



The numerical results have been shown to be so nearly the same, 

 that in only one case has it been possible to detect by observations of 

 the positions of the bodies of the solar system a minute difference 

 between the results deduced from the Newtonian laws of motion and 

 ijravitation and those deduced from the theory of relativity. 



Newton was not content with the bare outlines of the subject but 

 desired to fill in details wherever possible. It is reported that the 

 publication of his work was much delayed because of his difficulty 

 in proving that a uniform sphere would attract an external particle 

 exactly as if all its mass were concentrated at its center. And again 

 he was delayed by an erroneous value of the radius of the earth, a 

 proper value of which was needed in the application of his theory to 

 the motion of the moon, if theory and observation were to agree. 

 Further, he dreaded controversy, and had he given liis proofs by 

 the use of calculus, or " fluxions " as he called it, there would prob- 

 ably have been endless debate concerning the validity of this new 

 and unknown method of argumentation. To avoid it, he performed 

 one of his greatest feats, the translation of the whole process into 

 elementary plane geometry — a method of reasoning which was known 

 and acceptable to the scientific men of his time. 



For nearly a century after the publication (1688) of the Prin- 

 cipia of Newton, progress was slow, perhaps because it demanded 

 considerable development of the calculus originated by Newton and 

 Leibnitz. The geometrical form of argument used in the Principia 

 is much too difficult and complicated for investigation, and it was 

 not until Leonard Euler and some of his predecessors had shown 

 how to deal witli tlie problem by analysis tliat progress was possible. 

 Most of Euler's work appeared in the third quarter of the eighteenth 

 century. The end of it is marked by the advent of Laplace, whose 

 Traite de la Mecanique Celeste furnished mathematicians with a 

 storehouse of facts that were deducible from the equations of mo- 

 tion, and who gave principles for the development of the solutions, 

 which furnished starting points for many of the investigations of his 

 successors. 



The mathematical processes start by translating the laws into 

 algebraic symbols and then equating the gravitational forces to their 

 kinetic reactions. This process — a comparatively simple one — in- 

 volves the setting up of certain formulas known as differential equa- 

 tions which give the rate of change of the velocity of each body in 

 any direction in terms of the positions of all of them at the moment. 

 What is usually needed is the deduction of the positions at any other 

 moment. This second step requires what is known as tlie " solution " 

 of the differential equations. It is at this point that the difficulties 

 begin to arise. No method is known for obtaining these solutions 



