in several articles in this Journal, of the principles of the method as well as the proofs of its utility, are sufficient for removing all doubts on this head, and that it will now be regarded as a powerful instrument in the hands of mathematicians.

ON THE IMPOSSIBLE LOGARITHMS OF

QUANTITIES.*

IN a Paper printed in the fourteenth volume of the Transactions of the Royal Society of Edinburgh,† I gave a short sketch of what I conceive to be the true nature of Algebra, considered in its greatest generality; that it is the science of symbols, defined not by their nature, but by the laws of combination to which they are subject. In that paper I limited myself to a statement of the general view, without pretending to follow out all the conclusions to which such views would lead us: such an undertaking would be too extended for the limits of a memoir, and would involve a complete treatise on Algebra. It will not, however, be attempting too much to trace out, in one or two cases, some of the more important elucidations which this theory affords of several disputed and obscure points in Algebra, and therefore in the following pages I shall endeavour to point out the deductions which may be derived from the definition of the operation +, given in the paper above alluded to. I there stated, that we must not consider it merely as an affection of other symbols, which we call symbols of quantity, but as a distinct operation possessing certain properties peculiar to itself, and subject, like the more ordinary symbols, to be acted on by any other opera

* Cambridge Mathematical Journal, Vol. I., p. 226.
See page 1 of this volume.

tions, such as the raising to powers, &c. The definition of the operation represented by this symbol is, that

+ + = +,

which leads to the equation

(+)” = +,

r being any integer. And this peculiarity—that the operation repeated any number of times gives the same result as when only performed once is the origin of certain analytical anomalies, which do not at first sight appear to be connected.

The first of these is the fact of the existence of a plurality of roots of a quantity, when the corresponding powers have only one value. It seems a fair question, to ask the cause of so great a difference between two operations so analogous in their nature, but it is one which I have not seen anywhere discussed. The distinction is, I conceive, to be traced to the nature of the operation +, according to the definition of it which I have given; and much of the obscurity connected with the subject is due to an oversight, by which the existence of this is wholly overlooked. For it is not a, but+a, which has a plurality of roots: and though these quantities are usually reckoned to be the same, this idea is founded on an illegitimate extension of a supposed relation in the science of number. I say supposed, because I hold, that even in Arithmetic a and +a are different, and ought not to be confounded-the former being an absolute operation, the other always a relative one, and consequently incapable of existing by itself. But however this may be, there is no doubt that it is entirely illegitimate to suppose that in all cases a and +a are the same, since generally we know not even what their meanings may be. Indeed, in Geometry the distinction is pretty broadly marked, since a represents a line considered with reference to magnitude only, +a with reference both to magnitude and direction.

I therefore maintain, that in general symbolical Algebra we must never consider these quantities as identical; and if at any time we conceive the existence of the +, we must take cognizance of its existence throughout all our processes, subjecting it to the operations we may perform on the compound quantity. Now, that in the usual theory of the plurality of roots the existence of is supposed, though not always expressed, is easily shown from the very first case of plurality of values which occurs. It is argued that, since a×a = a2 and ах a=+a2 also, we have two values, a and -a, for (a). But this, it will be seen, depends. on the supposition that + a2= a2, since in the case of the product -a --a the + is exhibited. If, instead of saying a×a=a2, we were to say that +ax+a=+a2, we should have undoubtedly +a as the result in both cases, and we are therefore entitled to say that (+a2) has two values, +a and a. The reason for this plurality is now very plain, for

(+ a2)1 = + 1 (a2) * = + * a.

But from the definition of + it appears that + will be different according as we suppose the + to be equivalent to the operation repeated an even or an odd number of times. In the former case it will be equal to +, in the latter to And generally, if we raise + a to any power m, whether whole or fractional, we have

Now, as from the definition of it appears that +”=+, → being any integer, it is indeterminate which power of + it may represent in any case, and therefore we must substitute +* for +, and then, assigning all integer values to r, discover how many values +ma" will acquire. So long as m is an integer, rm is an integer, and +mam has only one „P p p +7 will acquire

rm

value; but if m be a fraction of the form

rm

different values, according as we assign different values to r.

It will not, however, acquire an infinite number of values, since after r receives the value q, the values will recur in the same order. Hence the number of values of a quantity raised to a fractional power, is equal to the number of digits in the denominator of the index. It is to be observed, that we must never make r=0, since that assumption is equivalent to supposing that the operation + is not performed at all, which is contrary to our original supposition. From this we see, that the reason why there is a plurality of values for the roots of a quantity, is to be found in the nature of the operation +; and that it is only the compound operation + a, which admits of this plurality, a itself having only one value for each root. This view serves to explain an apparent difficulty which is noticed by various writers on Algebra. Since by the rule of signs -×

gives +, we ought to have

√(− a) × √(− a) = √(+ a2) = ± a ;

whereas we know that it must be only -a.

Now this fallacy arises from the sign of the root not being made to affect the + as well as the a. The process is really this,

√(− a) × √(− a) = √(+ a2) = √(+) √ (a2) = — a ;

+ or

2

for in this case we know how the + has been derived, namely, from the product +, which of course gives us +*=-, there being here nothing indeterminate about the +.

It was in consequence of sometimes tacitly assuming the existence of +, and at another time neglecting it, that the errors in various trigonometrical expressions arose; and it was by the introduction of the factor cos2r+− 1 sin2rπ (which is equivalent to +") that Poinsot established the formulæ in a more correct and general shape. Thus the theorem of Demoivre that

(cos +− 1 sin0)TM = cosm✪ + − ± sinme