FOUNDATION OF ALGEBRA. 295 



addition of exponents. The combination of a line, not an exponent, 

 with one which is an exponent, by the operation newly learnt, or which 

 might have been learnt, from combining two exponents by the known 

 operation + , is then obviously natural, and its result completes the 

 definitions of algebra in their most comprehensive form. But it is 

 satisfactory to find that the matter is not thus exhausted; and it re- 

 mains a subject of speculation how it arises that a line perpendicular 

 to the unit-line has the same relation to an angle which one on the 

 unit-line has to the logarithm of a length. The following considerations 

 tend, as far as they go, to give some idea of the origin of this circum- 

 stance. 



Let us go on to the generation of quantities by infinite numbers 

 of infinitely small elements, premising that nothing will be said which 

 may not easily be altered into the language of limits by those who 

 object to the infinitesimal phraseology. Every line (r, p) can only be 

 formed by addition of equal* elements in one way, namely, that ex- 

 pressed by f^ {dr, p), p being constant. Let us now consider what takes 



place when (r, p) becomes {r + dr, p + dp). This is obviously equivalent 



to multiplying the first line by (1 + — , dp), or successively by (1-1 — -, 0) 



and (1, dp). The first multiplication alters length only, in the proportion 

 of r + dr to r\ and answers to multiplying by OK, UK being dr : r. 

 Now, OU being the unit-line, make UW = dp in 

 linear units; whence, by the conventions of angular 

 measurement, OW is (], dp), neglecting the diffe- 

 rential of the second order by which OW differs from 

 OU. These ratiunculce (such was the term applied * v v 



to differentials so employed when the theory of loga- ' 

 rithms was first explained), UV and UW, being each used in the 

 multipliers n times in succession, we have resolved the contemporaneous 

 transitions (linear and angular) into distinct and (if we please) successive 

 transitions. If we begin with OU, or (1, 0), and proceed through n 

 multiplications, and make dr : r always the same, and = /n, we have 



* Remember that 'equal' means 'same in length and direction.' 



112 



