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PROCEEDINGS 



CAMBRIDGE PHILOSOPHICAL SOCIETY. 



November 27, 1843. 



On the Foundation of Algebra, No. III. By Augustus De Morgan, 

 of Trinity College, Professor of Mathematics in University College, 

 London, &c. 



In the second paper of this series a general definition of the ope- 

 ration A was laid down, A and B being each of them any form of 

 p + q V — 1 . The logarithm (or as Mr. De Morgan calls it, the logo- 

 meter) of a line is thus described : — a line whose projection on the unit- 

 axis is the logarithm of the length, and whose projection on the per- 

 pendicular is the angle made with the unit- axis (or its arc to a radius 

 unity). Thus a line r inclined at an angle 9 has for its logometer a 

 line v (log'V+9-) inclined at an angle whose tangent is 9: log r. 



This being premised, he universal definition of A is the line whose 

 logometer is B x logom. A. 



The object of this third paper is to show that the preceding defi- 

 nition of the logometer is not the most general. Take any two lines 

 whatsoever passing through the origin, and style them the bases of 

 length and direction. Set off on the first a line representing the 

 logarithm of the length in question, and on the second a line repre- 

 senting the angle it makes with the unit-axis, both on any scale of 

 representation. Then the diagonal of the parallelogram described 

 on the lines just set off is a logometer to the length and direction 

 from which it was derived ; and if under this meaning of the word 



logometer the preceding definition of A be employed, the equations 



A B A C =A B+C , A B C B =(AC) B , (A B ) C =A BC 

 are universally true. 



There is no necessity for the introduction of this more general 

 system, since all its results can be expressed in terms of those of the 

 more simple definition in the second paper. This new definition of 

 the logometer is really nothing more than the process answering to 

 the extension of the theory of logarithms from the system constructed 

 on the Napierian base, to that which is on any base whatsoever. 



No. I. — Proceedings of the Cambridge Phil. Soc. 





