INACCURACY OF SOME LOGARITHMIC FORMULAE. 
185 
maintained in the Encyclopedia Metropolitana, article Algebra, 281. Indeed, when — 2 is admitted to 
be one of the values of 4 T , the extension of the notion “ logarithm ” must be greatly abridged to deny 
that, relatively to the base 4, \ is a logarithm of — 2. 
Note G. — From this theorem it does not follow' that f — 1 9 2 = 2 f — 1 9; an expression that has only 
half as many values as f — 1 9 + f~ 1 9, which admits the addition of any one value of f — 1 9 to any 
other. 
This instance is adapted to give notice of a very insidious species of fallacy, whose intrusion, in 
reasoning on subjects like the present, should be guarded against with vigilance. 
Note H. — As 2 in comprises exactly the same values as 2(i — i)n, and serves as well to show 
that the integer in the numerator of [26] may be chosen without reference to that in the denominator, 
it is preferred for briefness and concinnity in a general formula. 
Note I. — The solution of this problem assists in constructing the figure whose equation is 
■r^ + ■v /r_ 1 B = y. 
M. Vincent has inserted in the commencement of the 15th volume of the “ Annales de Matlie- 
matiques,” &c. published at Nismes in 1824 and 1825, and edited by M. J. D. Gergonne, an in- 
genious paper on the construction of some discontinuous transcendental curves. His paper is entitled 
“ Considerations nouvelles sur la nature des courbes logarithmiques et exponentielles. Par M. Vin- 
cent, Professeur de Mathematiques au College Royal de Reims, ancien eleve de l’ecole normale.” 
His general principles appear to me to be correct ; but, in my opinion, he has occasionally fallen into 
error. For instance, he seems to take it for granted when a is positive, that whatever value of a x be 
considered, da 1 = la a I dac; whereas, wdien the i th value of a 1 is considered (see Appendix § 7.) 
da 1 = (a/ — l Ziit -\-\a) a x &.X. 
To obviate some objections to my general theory, I may here observe incidentally that M. Stein, 
who has occasionally written on the subject of logarithms in the same journal, would introduce a very 
confused and inconvenient notation by supposing a x to vary its signification according to the form in 
which the value of x is expressed — by supposing, for instance, that, while a 1 = a, a* would = (a 2 )* 
i 
or + a. Hence, by the same analogy a would = a . 1 ^ 2 . According to the usual interpreta- 
tion of a x \ which I have adopted, and by which it is identical with f(a; f “ 1 a), a 1 , a T and a have 
all the same signification. 
The following definition of a x , derived from the characteristic property which led to the extension 
of the exponential notation beyond integral exponents, has been suggested to me by my friend Mr. 
Hamilton, Royal Astronomer of Ireland: 
“ a ~ comprises every successive function <px of x, which, independently of x and y, satisfies the 
conditions <px <py = <p (x + y) <p\ — a” 
From this definition does not follow, in all its generality, the equation a x a y = a x for the pro- 
duct of the i th value of a 1 (which I would designate by a x ) multiplied by the i th value of a y is not 
necessarily among the values of a x + y ; a legitimate consequence of the definition of a x is the equa- 
tion a - x a - y — a x + y '. 
tit 
2 B 
MDCCCXXIX. 
