186 PROFESSOR DE MORGAN, ON THE 



required than of a correction to already existing formulae : and in this 

 view I perfectly agree. If we define log x, or rather Xx, (reserving log x 

 for the numerical logarithm of the length) to be any legitimate solution 

 of e Xr a x, it is plain that the logarithm of n inclined at an angle v, (or 

 of N) to the base b inclined at an angle (S, (or B) is to be derived (avoid- 

 ing ambiguity) from 



. at logn + v\/ - 1 



or \ B N= = ,-^-r — r, J - • 



logi + ^-l 



This result is real when . ° , = g ; nor is it more surprising that an 



impossible quantity (hitherto so called) should have a possible logarithm, 

 than that exponential operations not containing v - 1, or not inter- 

 changing exponents of length and direction, should in certain cases enable 

 us to pass from one line to another. I need not enter into details of the 

 properties of the preceding equation. If we admit all symbols to be 

 algebraical (in the old sense) which denote lines drawn in the unit 

 line or its continuation, whatever may be the number of complete revolu- 

 tions after which they rest there, we must then admit that the logarithm 

 of a unit which is in its position for the (m + l) ,h time, with respect to 

 e which is in its position for the (« + l) th time is 



1 + 2»7T\/^1 



the form proposed by Mr Graves. 



In a work of M. Cauchy, and perhaps in other writings which I am not 

 acquainted with, mention is made of a singular point in curves which 

 he calls point d 'arret, at which the branch suddenly stops. Such a point 

 has long been admitted in the spiral of Archimedes and other curves, 

 owing to the neglect of making those extensions with regard to the sign of 

 the radius vector which were necessary to complete the connexion* of polar 

 and rectangular co-ordinates ; and from the assumption of the impossibility 



* On this subject I may be allowed to refer to page 341 of my Treatise on the Differential 

 Calculus. 



