in Mathematical Reasoning. 347 
usually occur in this first stage; and it cannot be too strongly 
recommended, that every part even of the most difficult problems 
should be fully translated into the language of analysis, before 
any attempts at simplification are made. In the first stage, it is 
scarcely possible to see clearly in what degree the results will 
be affected by a proposed omission; whilst in the second, any 
quantity which it is conjectured will have little influence on the 
result, although it adds greatly to the difficulty of calculation, may 
be kept separate, and the operations to which it is submitted, 
may be indicated rather than performed. In many of the appli- 
cations of analysis, and particularly in its treatment of mechanical 
questions, the principles which regulate the first stage of the 
process are completely known, and little difficulty is experienced 
in translatmg them into the language of signs, the difficulties 
when they occur, usually taking their rise in the solutions of the 
equations thus produced. A_ similar remark is applicable to 
optical questions, and indeed to by far the greater part of those 
which occur in the mixed sciences. 
II. The second stage in the solution of any problem, generally 
begins with the equations into which it has been translated, and 
terminates with their solution. The point at which it commences 
is not always so well defined as that at which it ends, and this 
is more particularly the case when the question relates to geome- 
trical figures, where in some instances, the first and second stages 
are much intermixed. 
The difficulties which now occur are purely analytical, and are 
generally such as have been treated of in works devoted to the sub- 
ject. The solution of one or more algebraic equations is frequently 
the object to be obtained: differential equations, or equations of 
finite differences are another class of analytical expressions to 
which physical problems are often reduced; many of these can 
Vol. Il. Part If. Me 
