FOUNDATION OF ALGEBRA. 175 



more of B attained by motion from A, than of the quantity of length in 

 AB. All three would use, perhaps, the same modes of expression : and I 

 suspect* that there could be detected, among persons who think about first 

 principles, a very considerable degree of variety in the points of view under 

 which fundamental words suggest their objects ; while as much exists, but 

 could not as easily be found, among those who have studied the exact 

 sciences, without paying particular attention to their foundations. 



A symbol may thus denote either magnitude, operation, by which mag- 

 nitude is attained, or the conception of one extreme arrived at, the other 

 having been the previous object of contemplation. The earlier f algebraists 

 most certainly dwelt on the first notion ; a + b is with them the result of 

 an operation, in which the method of obtaining it is so completely for- 

 gotten, that the result a + b is actually obtained by a distinct operation. 



It seems to me that Sir William Hamilton, in his very original and 

 methodical memoir on algebra as the science of pure time, has adopted a 

 view of the third kind. I cannot see why the whole paper might not be as 

 easily applied to succession of points in a line, as to succession of epochs in 

 time. Succession, that is to say continuous succession, might be made the 

 fundamental conception in both cases ; and if such were the author's inten- 

 tion in the use of the word time, I should be very glad to maintain after him 

 that one of the explanations which suffice to convert technical into logical 

 algebra, has been fully established in his memoir. But, if any thing more 

 physical\ be intended by the distinguished author, and if some of his 

 phrases are to be interpreted as of his asserting algebra to be the science of 



* In a short biographical account (which I have before me, in a private communication) of 

 the late Mile Sophie Germain, whose papers on the theory of elastic surfaces are well known, 

 it is asserted that she could never form the conception of space, except by the means of time : 

 this was her own mode of expressing, to the writer of the notice, a state of mind by which 

 he accounts for another fact, namely, that she had very little aptitude for pure geometry, and 

 a great attachment to the theory of numbers. 



t See my Calculus of Functions, sect. 245. 



{ This word is here improperly used ; but I refer to the notion of those who would have 

 made geometry a part of mixed mathematics : that is, if the algebra of Sir W. Hamilton would, 

 in the opinion of those just alluded to, also have been a part of their mixed mathematics, and 

 if Sir W. Hamilton should admit that they have as much reason, his terms being understood in 

 his own sense, for their location of his algebra as for that of geometry, I should then say that 

 the word used in the text is allowable. 



