296 Cambridge Course of Mathtmalics. 



power of expressing the properties and relations of figures* 

 According to Legendre's plan, the meaning of the term 

 equal, is unnatural. We should have preferred to applj the 

 term coincident 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 eoincidence alone, is comprised in the sixth of 

 his axioms, in which he says, "two magnitudes, whether 

 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. It is remarkable what could have led 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. 



The fourth section treats of regular polygons and the 

 measure of the circle. It is 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- 

 ometry, and has very much occupied the attention of 

 mathematicians. Now we can easily demonstrate, that 

 a circle is equivalent to a rectangle contained by the cir- 

 cumference and half the radius, and by finding a mean pro- 

 portional between the circumference and half the radius, 

 we have the side of the square. The problem of the quad- 

 rature of the circle, is, therefore, reduced to finding the 

 circumference when the radius is given, and to effect this 

 object, it would be sufficient to know the ratio of the circum- 

 ference to the diameter, or to the radius. Mathematicians 

 have not been able to obtain this ratio but by approxima- 



* Laws, B. VI]. 



