Mr. DE morgan, ON THE FOUNDATION OF ALGEBRA. 141 



show that the signification of ordinary exponentials involving -y/— 1 is completely changed: thus 

 el,„^' signifies a line of the length e ''* inclined at the angle ad to the unit-line. 



Without going further into details we may see that, as before remarked, it is not necessary to 

 retain this extended notion of A^, since the consequences of the extension can be expressed by the 

 particular case in general use; which cannot be said of JB as compared with A + B, or of A" 

 as compared with AB. This rejection is a generalization of the rejection of all logai-ithniic bases in 

 favor of e, and the extended definition of A" is itself a substitution of logarithmic bases in their most 

 general form. For whereas, in the common system, e and e^"' are the logarithmic bases* employed 

 for ordinary and periodic magnitude, we have, in the system above described, employed 



^,. + „V-ir and e*----^-')-. 



Great care will be necessary, in verifying the conclusions, not to confound the meanings of A^„ 

 and A"^, or the operations performed upon them. Thus the function whose WM^iz-logometer is 1, 

 may be represented by 



' V-i 





e»i + W-i or by e""'" 



and €"*"J^'' = ■X'. AVithout such care, the inquirer will infallibly be led to equations of con- 

 dition between m, n, fi, and v, which he will find are satisfied by m= i, n = Q, n = 0, v - I : 

 that is, he will imagine he has proved the system of my last paper to be necessary. 



From the expression of X in terms of its logometer, we derive the following, e meaning 

 (f , 0) ; 



^V-l, 





On this it is to be observed, that the notion formed from the ordinary modes of expression, 

 namely, that in ep + i^-^ there is a peculiar reference to length in p, and to direction in q, is not 

 altogether correct. Tlie imaginary part (it may perhaps be allowed to retain the nominal dis- 

 tinction of real and imaginary) determines the direction, but the length depends upon both parts. 

 The interpretation of c^*^',/"' is, that it represents a line of the length e''"'"' inclined to the unit- 

 line at an angle aq\ or (e''"*"', aq). One case, and one only is indefinite, when (fi + v^/- 1) -^ 

 (m + w-^— 1) is real, that is, when m = 0, /^ = 0, or when « = 0, u = 0, or when m : n :: ix : v, 

 which last includes the others. In this case the line takes the form (0, co ) or (m , eo ) the inde- 

 finite character of the result arising from the coincidence of the bases of length and direction ; 

 it resembles the attempt in common algebra to form a system of logarithms to the base unity. 

 But when {ij. + r ^y - ]) -H (»« + n %/- = ~ \/- 1, which gives 6 = 0, a = - I, we find («, ^) 

 represented by le"'^,,"'. Here the bases of length and direction are at right angles to one another, 

 but that of length is in advance of that of direction. This case requires that n = n, v — — m, 

 and the logometer is (m 4- n '^— 1) (log ,r — ^ -y/— 1). 



There would be little use in entering into more detail than is necessary to illustrate the 

 general meaning of the symbol A". But it must be considered necessary, in all future explana- 

 tions of the elements of algebra, to point out the complete meaning of this symbol, not only 

 to avoid defective reasoning, but to prevent the student from attaching an undue weight to the 

 connexion of -\/- 1 with the representation of direction. It is a strong corroboration of what 

 seems to have been pointed out by the course of the complete science up to the present time, 

 namely, that we must not expect any new imaginary or impossible quantities. I must own that 



* Ar far an 1 know the basCH actually employed are four, « and e*^'' in analysis as above described, 10 iu tlie facilitation of com- 

 putatlona, and ^2 in the numerical coHHideration of the musical scale. 



t2 



