ON THE UNCERTAINTY OP CONCLUSIONS. 
7 
The mathematician has the great advantage over the 
physicist, the chemist, or the geologist that he not only can, 
but almost necessarily must, completely define the elements 
with which he has to deal. If he deals with matter, before 
he can put it into his equations he must needs restrict it as 
to form and dimensions and endow it with definite physical 
properties, the relations of which are capable of analytical 
expression. If, after this, his power of analysis is sufficiently 
great, the conclusions which he reaches can have no element 
of uncertainty in them, provided always they are considered 
as referring only to the supposititious material with which 
the investigation was begun. That the conclusions are not 
in harmony with known phenomena is evidence only of the 
fact that the material of nature is not the material which is 
symbolized in the formula, and that certain properties which 
are common to both are modified in the former by the pres¬ 
ence of others which are not attributed to the latter. When 
MacCullagh, Neuman, Stokes, Sir William Thomson, or Max¬ 
well each evolves a dynamical or mechanical theory of light, 
a lack of agreement among them or with known principles 
of optics can generally be traced to the fact that the medium 
in which they suppose the action to take place has not been 
endowed with the same common properties by all, and that 
in every case it falls short of an exact representation of the 
real ether itself. With this important restriction upon mathe¬ 
matical reasoning kept continually in mind, mathematics 
may be safely set aside as the “ one science of precision.’’ 
What, now, are the characteristics of the so-called “ exact 
sciences ” other than pure mathematics ? Without attempt¬ 
ing a rigorous definition or a precise classification, it is suffi¬ 
cient for the purpose at hand to declare that the exact 
sciences are those whose conclusions are capable of being, 
and for the most part are, established by experiment and 
verified prediction. 
Among these exact sciences the most notable, in degree of 
exactness, is the science of astronomy. Although the con¬ 
clusions reached in the study of astronomy may not in gen- 
