336 A CENTURY OF SCIENCE 



Instead of three scalar equations for each of the innumer- 

 ably large number of particles involved, he showed how 

 to reduce the ordinary dynamical equations to a number 

 equal to that of the degrees of freedom of the system. 

 This is made possible by a combination of d'Alembert's 

 principle, which eliminates the forces due to the connec- 

 tions between the particles, and the principle of virtual 

 work, which confines the number of equations to the num- 

 ber of possible independent displacements. The aim of 

 Lagrange was to make dynamics into a branch of 

 analysis, and his success may be inferred from the fact 

 that not a single diagram or geometrical figure is to be 

 found in his great work. 



Celestial Mechanics. Almost simultaneously with the 

 publication of the "Mecanique Analytique" appeared 

 Laplace's "Mecanique Celeste." Laplace's avowed 

 aim was to offer a complete solution of the great 

 dynamical problem involved in the solar system, taking 

 into account, in addition to the effect of the sun's gravi- 

 tational field, those perturbations in the motion of each 

 planet caused by the approach and recession of its 

 neighbors. So successful was his analysis of planetary 

 motions that his contemporaries believed that they were 

 not far from a complete explanation of the world on 

 mechanical principles. Laplace himself was undoubt- 

 edly convinced that nothing was needed beyond a 

 knowledge of the masses, positions, and initial velocities 

 of every material particle in the universe in order to 

 completely predetermine all subsequent motion. 



The greatest triumph of these dynamical methods was 

 to come half a century later. The planet Uranus, dis- 

 covered in 1781 by the elder Herschel, was at that time 

 the farthest known planet from the sun. But the orbit 

 of Uranus was subject to some puzzling variations. 

 After sifting all the known causes of these disturbances, 

 Leverrier in France and Adams in England independ- 

 ently reached the conclusion that another planet still 

 more remote from the sun must be responsible, and com- 

 puted its orbit. Leverrier communicated to Galle of 

 Berlin the results of his calculations, and during the next 

 few days the German astronomer discovered Neptune 

 within one degree of its predicted position ! 



