C 209 ] 



XXXVIII. Defence of Professor Leslie against th» 

 Edinburgh Review. 



To Mr. TiUoch. 



Sir, It appears tVom the last Edinburgh Review, No. sgf, 

 pp. 89, 90, 91,) that Mr. Leslie has impugned a certain 

 mode of reasoning used by Legendre, which on the other 

 hand the reviewer strenuously defends ; and those who 

 admire the most what they understand the least (a nume- 

 rous class certainly) may perhaps think him in the right. 



He who denies that a line can be a function of an 

 angle (because, according to him, an angle is of the nature 

 of number, and of no dmiension.) must surely be ignorant 

 of a curve called the spiral of Archimedes, of which it is 

 the very nature that the polar ordinate is directly as an 

 angle. Perhaps, however, this curve must be classed with 

 impossible quantities. 



I confess, sir, the postulate of the reviewer appears to 

 me to be mere verbiage, and gratuitous silliness. 



" The quantities A, B, C, are angles 5 they are of the 

 same nature with numbers, or mere expressions of ratio, 

 and, according to the language of algebra, are of no di- 

 mension.'* 



The strange and unsupported assertion that angles are of 

 the same nature with numbers, can only be accounted for 

 from his confoimding* (in an inquiry in which, more than 

 in any other, such things must not be confounded) an 

 angle, with the cosine divided by the radius. 



That an angle is of no dimension, if it simply mean that 

 an angle is not a line, nor a plane, will hardly be disputed ; 

 but then, on the other hand, a line is not an angle, nor 

 angular space of any kind ; so that the argument applies as 

 much on one side as the other; 



If the assertion mean more than this, it is not easy to 

 perceive its truth ; for, as a line is considered of one di- 

 mension, when compared with a plane; so may an angle 

 be considered of one dimension, when compared with 

 '* the angular space subtended by a surface," this having 

 two dimensions. If, therefore, there are different dimen- 

 sions of space, angular space has also its different dimen- 

 sions ; and the heterogeneity is reciprocal. Nor can an 

 ansile be of the nature of number any more than a linej 

 there must be an arbitrary choice of an unit in both cases. 



. r I • COf.C n' + b'^ — ^'' ,. , . „ . , 



• In the equation =• ; which he calU an equation be- 



rad. a«fl 



twe<2Q C and a, b,c^ 



Leaving: 



