in Mathematical Reasoning. 363 
the difference is apparently trivial, but the convenience or incon- 
- venience of notation frequently depends on differences as trifling, 
It may be observed, that in the first equation, a denotes the tangent 
of an angle, and 4 an absolute line; two things which have no 
relation to each other, and which are therefore justly represented 
by dissimilar signs. In the second equation, the line and the 
angle, are both represented by the same letter of different alpha- 
bets; a circumstance, which will infallibly suggest some idea of 
a relation that does not exist. When two straight lines enter 
into the question, other reasons present themselves: they may 
be represented in any of these four ways; 
y=ar+hb y=ar+ta y=ar+b y=ar+h 
y=ar4+t y=br+By y=axr+B y=cur+d 
and if we seek the ordinate of their point of intersection it will be 
ab’—a'b aBp—ba aB —ab ad —ch 
Shit laletigl eo Weep te Be. fm a? ae ee 
the latter of these expressions is quite devoid of all symmetry in 
regard to its letters, and the larger the number of lines about 
which we reason, the more confused will such a mode of expressing 
them render the result. In the first and third mode, it is sufticient 
to remember that the letter a, under all its forms, represents the 
tangent of an angle, and that the letter >, in every form, always 
represents a particular ordinate: with this principle in our mind, 
we can see at a glance, however numerous the lines introduced, 
to what property of them each individual letter refers; whereas 
in the last method, we must, in order to discover the meaning of 
any letter, refer back for each individual one, to the original trans- 
lation into algebraic language. The third plan will suffice, where 
only a few different lines are concerned, but its application is 
limited by the smallness of the number of different alphabets we 
can command. The second method may be defended on the 
ground, that the tangents are denoted by one class of letters; 
Vol. Il, Part Il. 3A 
