in Mathematical Reasoning. 351 
which are in fact comprehended in Euclid’s definition of regular 
polygons. The sum of the interior angles of the first of these 
heptagons is ten right angles, the sum of the interior angles of 
the second is six right angles, and that of the third species is 
two right angles. These new species of polygons were first 
noticed by M. Poinsot, in a highly interesting memoir on subjects 
connected with the Geometry of Situation, read before the 
Institute in 1809, and subsequently printed in the Journal de 
VEcole Polytecnique, 10° Cah. 
Another important business which belongs to this stage of 
the question, is to examine carefully what changes will ensue 
from supposing any peculiar relations amongst the data; or from 
any of the constant quantities becoming infinite or evanescent, 
such circumstances frequently introduce great simplicity, and 
when they refer to geometrical questions, are sometimes the 
means of making us acquainted with general properties, by which 
the construction of the problem is greatly facilitated. 
A careful and laborious attention to all the possible modifications 
of a problem which might result from any relation amongst its 
data, was considered by the ancient Geometers as an indispen- 
sible part of its investigation, and the manner in which this was 
accomplished, was generally little else than a repetition of the 
whole process under the altered circumstances: when the data 
are numerous, the length of such a system of operations becomes 
intolerable, and if more rapid methods had not been contrived, 
Geometry must have become stationary from the accumulation of 
the details with which it was thus encumbered. Many instances 
of the extreme length to which a full investigation of comparatively 
a very simple problem will lead, occur in the treatises De 
Sectione Rationis, &c. The advantage of Algebraic language is 
in this respect very striking: all the data of the questions are 
embodied in the equation in which its solution terminates, and 
