SECTION 41.

We are now able to introduce the general mth root. By "/a we mean a symbol, i. e. a form, endowed with the property that (a) may be interchanged with a itself, while a is conceived as perfectly general, and m is a positive whole number.

If we now endeavour to determine how many different, unequal, forms exist, represented by x, which have the property denoted by "/a, we find that we must have

xma or x" — α = = 0.

a

But since this is a higher equation of the mth degree it gives m different values, neither more nor less, for x. Therefore is m meaning, and we find all these m different forms, by writing "/a./1 for "/a, in which product we consider the first factor /a as merely single meaning, while "/1 represents the m values a, a2, a3, ... am-1, am or 1, where a has to be still determined*. Hence it follows, that:

=

(A) From A Bm we cannot conclude that A = B, but that A./1=B./1, or at most that A = B."/1, i. e. that we can obtain A by multiplying B by one of the values of "/1, which has to be further determined.

(B) All values of 1 have necessarily the form p+q. i, which form we have termed the general numerical number.

(C) If a is actual or imaginary but of the form p+q.i, all the m values of "/a are actual or imaginary but of the same form P+Q.i.

(D) For these general roots the following laws hold:

́(I.) (~/a)TM = a;

(1) (ab) = a.b;

(3) */(~/a)="/a;

(II.) "/(am) = a."/1;

(2) "/(a: b) = "/a: m/b ;

in such a manner, namely, that exactly as many and exactly the same values stand on both sides of these equations, as by (sect. 3) was required of a correct equationt.

In the same sense the equation

(4) "/(a")=("/a)",

where n is considered as a positive or negative whole number or zero, is not correct, because the expression on the left has m values, while that on the left may have less‡, and will only have as many values when m and n have no common divisor. The

* That if a is a value of "/1, then a2, a3......a", where r is any whole number, must be also (the same or different) values of "/1, follows from the fact that

(a")" = (am)" = 1′ = 1.

+ In (No. I.) there is only one value right and left. In (Nos. II. 1, 2.) there are exactly m values right and left, while in (No. 3) there are exactly mn values right and left.

Thus for example (a2) has the 4 values √√a, −√√a, +√—a, −√—a, while (a) has only the two values √a and √a.

unskilful application of this equation (No. 4.) therefore is to be looked upon as one of the sources of the paradoxes of calculation. (E) We may

not put (p + q)"/a for p."/a+q. "Ja,

nor (a) for Ja. Ja,

nor from /a =a and "a = ẞ conclude that a = ß, unless we have previously ascertained that the same symbol m/a wherever it occurs in the same expression always represents one and the same of its m values, a certainty which, in perfectly general investigations, we are, owing to the nature of the subject, usually unable to obtain. In the same way we may not substitute a for (a), because the latter root denotes one of m different values which has no need to be a exactly, but which is certainly contained in a. "/1.

Generally, therefore, treat such a general mth root as what it is, viz. a symbol which represents, wherever it occurs, one of m different forms, or all the m forms together, but leaves indeterminate which one is intended, and you will never fall into contradictions, whereas the many-meaningness (ambiguity) of roots is otherwise, when not properly regarded, a fruitful source of paradoxes of culculation.

SECTION 42.

The elimination of an unknown expression from two higher algebraical equations, as also the solution of these equations only encounter practical obstacles, but no theoretical obstacles, which have to be logically overcome.

=

Remark 1. So called transcendental determining equations, as for example a*=b, a2√=1 _ a−x√—1_b, where the unknown expression x appears in the exponent, cannot as yet possibly occur, because we have not yet had a power whose exponent is still unknown, and may therefore be also imaginary, nor even such powers as have their dignand general, and their exponent actual or broken.

Remark 2. The binomial theorem for whole exponents, which it is so easy to establish, viz. the proposition x (x − 1) (x − 2) ̧ b3 + . • •

(1 + b)* = 1 + 2 . b

b+

1

x(x-1), b2+

1.2

1.2.3

where x is supposed to be a positive whole number, has only x + 1 terms on the right hand. But the series on the right may be conceived as proceeding ad infinitum because the coefficients of all the terms after the r+1th receive the factor zero, and are therefore themselves equal to zero. Now if we consider this series as really continued in inf. and at the same time the coefficient

and in the same manner all following coefficients arranged according to powers of x, we may arrange the terms on the right according to powers of x, and we obtain

- 1 b2 + 1 ba — 1 b* + . . . in inf.) . æ

2 3

(1 + b)* = 1 + (b −

so that (1 + b) is now transformed into an infinite series proceeding according to powers of x, whose coefficients are also infinite series proceeding according to powers of b. This equation holds for any positive number x; but because this series on the right can be conceived for a general x; it clearly furnishes a means of introducing a general power a* or (1+6)* (for any x), provided we have previously established the "theory of infinite series" in such a manner as to be able to use it as the foundation of rigorous investigations.

Thus we see ourselves, in endeavouring to conceive generally the opposition in triplo between the three last operations (potentiation, radication, and logarithmation), as well as the connection of these three last operations with the four first, which are called the elementary operations, again led, as we had been once or twice before, to infinite series, and we are consequently unable to defer a consideration of them any longer.

MATHEMATICAL ANALYSIS IN ITS RELATION

TO A LOGICAL SYSTEM.

PART IL

THE RELATION OF THE THREE LAST OPERATIONS TO ONE ANOTHER AND TO THE FOUR FIRST.

ANALYSIS FINITORUM.

FIFTH CHAPTER.

SECTION 43.

IN the whole function of x of the nth degree, that is in the form. a + bx + cx2 + dx3 + +px”¬1+qx”,

we may consider the positive whole number n as great as we please; we may therefore consider it as infinitely great, i. e. always greater than any determinate number however great; hence the existence of "infinite series proceeding according to whole powers of x" is necessarily given. But this infinite series cannot be considered as existing, until the law according to which the coefficients are formed, has been determinately enunciated in infinitum.

Now since the infinite series proceeding according to whole powers of x is given simultaneously with the whole function of x of an indeterminate degree, we not only may but we must "calculate" with these forms, whose terms proceed in inf. (according to a determinate law of progression), precisely according to the same laws as hold for whole functions of r of an indeterminate degree.

Now as a whole function of a ceases to be such, and consequently all the laws existing for it as such cease to be applicable, whenever a determinate value in cyphers is substituted for x,in the same way the infinite series ceases to admit of treatment according to the laws of whole functions of x, whenever any determinate value in cyphers is given to the letter of progression x.

Consequently if we wish to "calculate" with infinite series proceeding according to whole powers of x, according to the same rules as hold for whole functions of x, with perfect certainty of success, we must make as the first condition that the letter of progression x remain perfectly general, and that no determinate

value in cyphers whatever be conceived as represented by it; that it should be a mere supporter of the operations*.

And for this reason when we hereafter speak of a general infinite series, we shall not always add the words "proceeding according to whole powers of x or z or 4, &c." since it must necessarily "proceed according to whole powers," and the letter of progression does not always need to be mentioned when it cannot otherwise be mistaken.

SECTION 44.

From this we deduce the following truths, which hold for such general series proceeding according to whole powers of any letter of progression:

(1) Since a whole function of x of the nth degree must have all its coefficients equal to zero, if it is to be equal to zero for all values of x, that is, while x remains perfectly general t,-in the same way a series proceeding according to whole powers of a must have nothing but zeros as its coefficients, if it is to be always equal to zero while x is quite general.

(2) Since two whole functions of x of the nth degree, if they are to be equal to one another for all values of x, that is, while x remains perfectly general, must have all the coefficients of like powers of respectively equal to one another,-similarly, this must be also necessarily the case, when two infinite series proceeding according to whole powers of x are to be equal, while their letter of progression remains perfectly indeterminate, or rather general (a mere supporter of the operations)‡.

*It must here not be overlooked that the whole of analysis, i. e. the so called calculating part of mathematics, has nothing to do with magnitudes, and that we can and may never "calculate" with magnitudes (sect. 6), but only with forms (with symbolized operations), and that consequently all "calculation" with infinite series must necessarily cease, directly the form ceases with which calculation may and can be carried on.

μ

On the other hand, after having introduced the general power, as is done below, we may multiply such a series proceeding according to whole powers of x by x '; so that we obtain a series which begins with negative powers of x, and proceeds according to broken powers of x. Similarly, we may substitute, for example, or 1 for x, so that the series proceeds according to ascending whole powers of 2-1, and therefore may be made by the slightest transformation {of (1) into to proceed according to negative powers of z. But all this can also be done to whole functions of r of the nth degree without in any respect altering their character.

,

For if every coefficient were not zero, we should have an algebraical equation of the nth or a lower degree, which would give at most n or fewer values of therefore would not hold for all values of r.

x, and Different proofs have been given of these two propositions, but all of them have been justly attacked. If from

we concluded that a。=0, because the equation would not otherwise hold for x=0,