296 Cambridge Course of Mathematics. 
power of expressing the properties and relations of figures. 
According to endre’s plan, the meaning of the term 
equal, is unnatural. We should have preferred to apply the 
term cowncident to those figures which are proved to be equal 
by superposition, and to have designated by the term equiv- 
alent, all the remaining part of the common signification of 
the word equal. Coincident and equivalent figures are 
both equal, but these terms designate different kinds of 
equality, and, we think, the introduction of them would 
contribute to the perfection of the language of geometry. 
Equality by coincidence alone, is comprised in the sixth of 
his axioms, in which he says, “two magnitudes, whetber 
they be lines, surfaces or solids, are equal when being ap- 
plied the one to the other, they coincide with each other 
entirely, that is, when they exactly fill the same space.” 
This section is concluded, by demonstrating, that the diag- 
onal and side of a square, are incommensurable quantities, 
and by an investigation of the approximate ratio of the one 
to the other. Jt is remarkable what could have Jed Plato,* 
to attach such an importance to this principle, as to regard 
as unworthy the name of man, him who was ignorant of it. 
It is demonstrated in prop. CXVII. B. X. of Euclid, and in 
several modern treatises of geometry. It is of no great im- 
portance, either when veiwed by itself, or in connexion 
with other truths. 
ne fourth section treats of regular polygons and the 
measure of the circle. [tis well known, that the problem 
of finding a square equal in surface to a circle whose radius 
is given, or as it is usually termed, the problem of the quad- 
rature of the circle, is much celebrated in the history of ge- 
* Laws, B, vil. 
