292 PROFESSOR DE MORGAN, ON THE 



If we look for that system in which the newly determined line is 

 the product, RS, or * (r, p) x («, a) or (rs, p + a), we find that we must 

 have cp (r, p) = log r, \J/ (r, p) = p. The system of logarithms is meant 

 to be the purely arithmetical one, and for reasons of numerical con- 

 venience, the base e is taken, and the angle is measured by the ratio 

 of the arc to the radius. This species of determinant is what should 

 be called the logarithm of (r, p) ; but, considering it desirable to retain 

 the word logarithm for purposes purely numerical, I should prefer to 



call (\/(logr) 2 + p 2 , tan - ' p^ — J the logometer of (r, p). All this would 



throw no light upon the general meaning of the sign x , but it leads 

 immediately to that comprehensive definition of R s or (r, p) (* "\ which 

 Mr. Warren might have adopted, to the introduction of e 9 ^ -1 , without 

 creating the smallest flaw in his well-secured title to be considered 

 as having most strictly adhered to explained definitions only : and which 

 Dr. Peacock might have regarded as the complete interpretation of every 

 symbolical result in which an exponent occurs that cannot be laid down 

 on one side or the other of the unit-line. That definition is as follows : 

 R s means the line of which the logometer is obtained by multiplying 

 together S and the logometer of R. Thus, OU being the unit-line, 

 let it be required to lay down OR 0S . Let OL be the logarithm of 

 OR, and ML the arc of z ROU (rad. OU): then OM is the logometer 

 of OR. Take OT a fourth proportional to OU, OM, OS, and z TOU 

 = l SOU + iMOU; then TO is the logometer of the result required. 

 Place a line of which the logarithm is TV 

 at an angle whose arc is OV, and that line, ▼ 



OW, is the one represented by OR os . The 

 fundamental laws of operation are so readily 

 established that I do not feel it necessary 

 to enter upon them ; and the equation 

 e 9V-! = cos 6 + J — 1 sin 6 is a mere corollary of the definition; for the 



logometer of e, or (e, 0) is (1, 0), and (l,0)x^-l or (1, 0) x (fl.^) 



* It is to be understood that all operations upon the small letters are those of common 

 arithmetic. 



