178 On the Simplification of M4athematical Analyses. 
the case when the number of data is reduced to one ; of less than 
which it is plain no problem can subsist. 
There is also another source of complexity in mathematical 
propositions, and that is the nature of the data when their num- 
ber remains the same; and this will in most cases be found de- 
pendent on the peculiar properties of the subject under consi- 
deration. But in geometry it has always some relation to the 
properties or habitudes of lines considered in a geometrical sense, 
as their sum, differenee, ratios, rectangles, squares, cubes, &e. 
or the position they are in with respect to other lines known or 
unknown. It is not perhaps possible to say decidedly, that any 
one of these data is more complex than the others; only it 
seems natural to infer, that it is a simpler relation of lines, ta 
consider them merely as regarding their length or ratios, than to 
draw the inferences necessary to solution from data concerning 
the squares or rectangles of lines, which latter combination may 
be conceived simpler than one which is connected with the cubes 
or solids. Moreover it is found that some relations of lines, whose 
habitudes or number of dimensions remain the same, are more 
difficult of determination than others, though apparently of equal. 
simplicity: thus it is found that © Given the rectangle and dif- 
Jerence of the squares of two lines to determine them,” is a much 
more difficult problem than “ The rectangle and sum of the 
squares.” But we are not acquainted with any method of reason- 
ing by which this difficulty might have been foretold; and aré 
only able to estimate it by the nature and difficulty of the re- 
quisite analysis. There may however a reason be assigned for 
this difference of difficulty, and this in great measure depends 
upon the almost unaccountable prejudice that most mathema- 
ticians have in favour of the circle above all other curves, which 
prejudice induces them to reject all other modes of solving pro- 
blems, that can by any possible, although intricate, analysis, be 
made to depend only upon the intersections of circles with cir- 
cles, circles with right lines, er right lines with right lines. And 
so it happens, that when in any problem we can obtain the locus 
of a certain point connected with the determimations requisite to 
the solution of the problem supposed to be under consideration, 
that problem is then simplified in a degree proportionate to the 
value of the point whose !ocus is determined: but strict meome- 
ters confine this method of simplification’ to the cases in which 
the determined locus is a circle. And the only reason assigned 
for this rejection of other curves is, the greater relative difficulty of 
their description. This, as a matter merely practica/ly considered, 
is certainly an objection of great weight;, but in the theoretical 
. determination of a problem, I cannot conceive why a prejudice 
of that nature should be indulged, at least to such an extent as 
even 
