On the Simplification of Mathematical Analyses. 179 
even to exclude the conchoid, a curve full as easy to describe as a 
circle. 
From the very nature of things it follows, that some curves are 
more fitted to the solution of some problems than others, and 
for that reason it would seem equally rational, to require a ma- 
thematician to give solutions to the purely circular loci and pro- 
blems, by means of the conie sections, or higher curves, as to 
fasten him down to the use of a circle and right line, in problems 
whose solution would be much simplified by the use of conic 
sections, or curves of the higher genera. It is impossible with 
the aid of all the known curves, excluding the circle, to apply a 
line of a given length, from a given point, to a curve of an 
order : therefore, in this case it plainly appears that the place of 
the circle cannot be supplied by any of the other curves; whence 
it follows that, strictly speaking, the circle is here a curve of the 
highest order, since it overcomes difficulties the others are unequal 
to. Whether this rejection of the other curves be well or ill 
founded, I may hereafter discuss, but shall for the present confine 
myself to problems requiring only the right line and circle. Now 
as it was before shown, that one part of the art of analysis con- 
sists in diminishing the number of data; therefore it follows, 
that we should first of all know how to apply the properties of 
our tools, which are here only two, a right line and circle. ‘The 
application of those known properties, in a manne: most suitable 
to the attainment of the desired end, in a great mvasure depends 
on having a perfect knowledge and experience of ‘the nature of 
the geometrical loci, by means of whose proper application, one 
of the conditions of the given problem may be always so fulfilled, 
that there will not be any necessity of that condition occupying 
the attention of the geometrician any more; and so, if two loci 
be discovered, the problem is still further simplified, 
Rules for Analysis, 
The great art of obtaining geometrical solutions depends on a 
proper application of the geometric loci; therefore our first rule 
will direct their substitution whenever the nature of the data ad- 
mits of such a simplification, 
Rule \. If you have either the sum or the difference of any 
two lines given, to determine them with the further assistance of 
some other given data,—A line must be taken equal to the given 
sum or difference, and the problem will be reduced to finding in 
that line a point, such, that the lines determined by that required 
point, and the two already known, may satisfy the conditions 
specified in the given data. But in case of the two lines pro- 
ceeding from two given points to intersect under conditions that 
M do 
