TENDENCIES IN MATHEMATICAL SCIENCE 333 



The theory of probabilities is at bottom nothing but common 

 sense reduced to calculus ; it enables us to appreciate with exactness 

 that which accurate minds feel with a sort of instinct for which oft- 

 times they are unable to account. If we consider the analytical 

 methods to which this theory has given birth, the truth of the prin- 

 ciples on which it is based, the fine and delicate logic which their em- 

 ployment in the solution of problems requires, the public utilities 

 whose establishment rests upon it, the extension which it has received 

 and which it may still receive through its application to the most 

 important problems of natural philosophy and the moral sciences; 

 if again we observe that, even in matters which cannot be submitted 

 to the calculus, it gives us the surest suggestions for the guidance of 

 our judgments, and that it teaches us to avoid the illusions which 

 often mislead us, then we shall see that there is no science more worthy 

 of our contemplations nor a more useful one for admission to our 

 system of public education. 



MODERN ASTRONOMY. TELESCOPIC DISCOVERIES. The im- 

 mense impetus given to astronomy by the revolutionary discov- 

 eries of Copernicus, Tycho Brahe, Galileo, Kepler, and Newton, 

 followed in the eighteenth century by the complete working out of 

 the mathematical consequences of the gravitation theory by Laplace 

 and others, placed the science in advance of all its rivals and 

 seemed to make it a model for their imitation. 



A different and most far-reaching tendency appears with the 

 work of the Herschels. Friedrich Wilhelm Herschel (1738-1822), 

 a poor German musician emigrating to England and devoting his 

 spare time unremittingly to astronomy, with the help of his capable 

 sister laid the foundations of modern physical astronomy. In 1781 

 he amazed himself as well as the scientific world by discovering be- 

 yond Saturn a new planet, Uranus, taking it at first for a 

 comet. Constructing more and more powerful telescopes he dis- 

 covered several satellites of Uranus and two of Saturn. He also 

 determined a motion of the solar system as a whole, towards a 

 point in the constellation Hercules. He catalogued more than 

 800 double stars and more than 2000 nebulas, recognizing among 

 the latter, as he believed, different stages of the evolution of other 

 planetary systems. He observes : 



