5 
by comparison of results derived from them by mathematics 
with certified matters of fact. In so far as the results account 
for the matters of fact, the truth of the hypotheses is estab- 
lished, and an advance is made in physical science. The 
hypotheses of the theory of universal gravitation are, first, 
that the force varies with distance according to the law of the 
inverse square; and, secondly, that it emanates from every 
particle of matter and acts according to that law on all other 
particles; The combination of the reasoning of physical 
astronomy with the data of observational astronomy is con- 
sidered at the present day to have fully established the truth 
of those hypotheses. It is sometimes supposed that Newton 
demonstrated the law of the inverse square. This is true only 
so far as he gave a proof of it a 'posteriori , that is, by deduc- 
ing, mathematically, from the hypothesis of that law, results 
which were found to be verified by facts of observation. It 
is not possible by any such reasoning as that employed for 
demonstrating the propositions of physical astronomy to give 
an a priori demonstration of the law of gravity. I do not say 
that an a priori demonstration is not possible ; but if it be 
possible, it must be effected by theoretical reasoning of a more 
comprehensive order, including, together with the law of 
gravity, the laws of other physical forces. 
7. The department of theoretical science designated above 
as physical astronomy, is only a limited portion of the whole 
domain of science that may be comprehended under the terms 
“ theoretical physics.” It is, however, a part separated from 
the rest by the circumstance that the calculations it requires 
consist in the formation and solution of differential equations 
containing in the ultimate analysis two variables. For assist- 
ing the human intellect in extracting from given relations 
between what is known and what is unknown information 
respecting the latter, no other general method has been 
invented than that of forming equations in accordance with the 
data, and obtaining the desired information by solving the 
equations. Common algebraic equations, as is well known, 
are formed so as to express given relations to which a certain 
number of unknown quantities are subject, and it is proposed 
by treatment of these equations, according to rules of reason- 
ing, to extract from them the values of the unknown quantities. 
In order that this may be done, the number of the equations 
must be equal to the number of the unknown quantities, and 
by known rules they have to be reduced to a single equation 
containing one of the unknown quantities. Then the value of 
this unknown quantity is ascertainable by solving the equation 
according to certain specific rules, and when this is known, all 
