Mr. De Morgan on the relative Signs of Coordi?iates. 253 



But in my remarks, the doubt arose from my not under- 

 standing that an extension was proposed, but imagining that 

 it was asserted that the ordinarij formulse, under ordinary de- 

 finitions, were incomplete. And this appeared Hkely from the 

 title of the paper in the Philosophical Transactions, " An 

 attempt to rectify the inaccuracy of some logarithmic formulfe;" 

 from the comparison of the residts with the corrections of 

 trigonometrical formulae by MM. Poisson and Poinsot, those 

 corrections being of absolute errors caused by the processes 

 not being as general as the definitions; and other things of 

 the same kind. But supposing it explicitly admitted that 

 an extension is proposed, I see no further objection to Mr. 

 Graves's system, but quite the contrary ; and should even 

 suggest that the extension should include, not only the posi- 

 tive forms of the base s, contained in g i + Switt V— i ^ but 

 also the negative ^oxms contained in s i + (2OT + i)7rA/-ij which 



are equally the representations of the number e, affected by 

 the remainder of those symbolic relations, the consideration 

 of which constitutes one principal difference between algebra 

 and arithmetic. The subject is not without analogy to that 

 treated in the former part of this paper. So long as s is ab- 

 solutely considered, the logarithms of y are contained solely 

 in the form x-\-2'ir m\^ —1; but simultaneously with each of 

 the algebraical forms under which e can be exhibited, exists 

 a new class, which, I'elatively to the particular form in ques- 

 tion, are logarithms to the base e, if so defined, but under 

 the common meaning of the term logarithm, to one of the 

 algebraical forms of s. I should myself prefer the latter me- 

 thod of expression. But under either definition, properly un- 

 derstood, the extension may be highly useful, and 1 am happy 

 to bear testimony to the ingenuity which suggested it, and the 

 talent with which it has been carried out. 



In page 287, I think Mr. Graves has not kept in memory 

 a distinction which I have drawn in the Caladiis of Ftmctiojis 

 already alluded to. The Alps upon Alps of solutions which 

 I have shown to be possible in functional equations (§ 119. 

 212.) belong only to those in which there are no independent 

 functional subjects. In the case where there are independent 

 variables (§ 97. in which among others $ x x <Py = <f> {x+y) is 

 treated,) I have made the following remark : " The use of the 

 Calculus of Functions in regard to these is to show that the 

 obvious algebraical I'orm which satisfies them is also the 7)iost 

 gctieial oj continnous J'anctionsy It would surprise me not a 

 little to sec any otlier solution of the last-mentioned equation 



