where a is perfectly general (a mere supporter), but ß and ช are any real whole numbers subject to the condition (in No. 2.) that 6-y also represent a real whole number.

SECTION 13.

Hereupon we define the difference-product ab as a symbol of this form (ab), in which a is considered as perfectly general, but in which b denotes any difference ẞ-y of two whole numbers, and which represents the difference a. B-a.y. This definition is chosen with reference to (sect. 12, No. 2) in order that the difference-product may include the whole product defined in (sect. 12) as a particular case.

In these definitions are included the equations

(1) a.00 and (2) a (-7)=-a. r,

where -y denotes a negative whole number: and we, therefore, now know what is the meaning of multiplying a perfectly general expression a by zero, or a negative whole number, i. e. what we may, agreeably to the laws of operation, substitute for a.0 and a (y).

SECTION 14.

We next propose the three theorems,

(1) ab=ba; (2) (ab) c = (ac)b = a(bc) ;
(3) (a+b) c = ac+bc = c(a+b),

and prove
them once,
for the case that all the letters a, b, с
represent real whole numbers, and then once again for the case
that all the letters a, b, c represent any differences of two whole
numbers, as e. g. the differences a - ß, y-d, μ — v. (Compare
System of Mathematics, Vol. 1. sects. 90, 91.)

After this has been done, we can propose the (perfectly) general product ab, as a symbol of this form (ab), endowed with the property that ab is interchangeable with ba, as also abc with acb and with a (bc), and finally also (a + b) c with ac+bc, without having to fear that we shall, by so doing, contradict the laws of operation; just because we have as yet nothing but differences of whole numbers, while the permissibility of these interchanges has just been proved for those differences.

If we now continue to mean by "equal expressions," with the same generality as in (sect. 3), such as may be unconditionally substituted for one another, without contradicting the laws of operation, then the equations (1-3) will still hold, although a, b, c are conceived with perfect generality as mere supporters of the operations, so that we can no longer speak of their particular signification; and these equations (1-3) merely express the relation in which the abstractly conceived operations per se of "multiplication" and "addition" stand to one another.

SECTION 15.

We next explain the difference-quotient, or a : b as a symbol

of this form (4), or (a : b), in which it is presupposed that a

and b represent any differences of whole numbers, and which represents that difference of whole numbers which when multiplied (by sects. 11 and 13) by the " divisor" b produces the "dividend" Hence

a.

a

Ъ

(1) That the quotient has not always such a signification, i. e. that there does not always exist a difference of whole numbers which, when multiplied by the divisor b, produces the dividend a; and

(2) That in the particular case when a and b are equal to zero, there are an infinite number of differences of whole numbers, which may be represented by the quotient; finally

a

(3) That, with the exception of the cases in (Nos. 1 and 2), the difference quotient always represents one single determinate difference of whole numbers.

(4) The assertions of (Nos. 2 and 3) may be also enunciated as follows: If each of two differences A and B of whole numbers, be multiplied by C (by sects. 11 and 13), and we find that A.CB. C, then we must necessarily have A=B, provided that C is not zero.

This last result is of extreme importance for the whole of analysis. For it follows from it, that:

(5) If we multiply or divide equal expressions, representing the same difference of whole numbers, by other equal expressions, the result will always be equal expressions; provided that we never divide by zero.

(6) Whenever then a general divisor becomes equal to zero in particular cases, we must no longer consider equations which contain such a divisor as necessarily correct: i. e. never divide by zero.

The form, whose dividend a is any expression, and whose divisor is zero, is consequently inadmissible in any calculation. Whenever it appears it is always a sign that, for the particular case which occasioned its appearance, the general calculation suffers an exception; that we are consequently unable to retain that general calculation in this particular case; that we must

therefore institute a particular calculation for this particular case, at any rate from the point where we first divided by zero*.

SECTION 17.

Moreover, upon the supposition that all the letters denote differences of whole numbers, we immediately deduce from the equations of (sects. 14 and 15) viz. from

(O) a.b=b.a;

(I.) % . b = a ;

(1) (a.b). c = (a.c). b, and (4) (a + b) c = ac+bc; any number of new equations, and among others the equations,

(11.) ab

(2)

= α

*For example, required the point of intersection of two straight lines repre sented by the equations y=ax+b and y=a'x+b'. If such a point of intersection exist, its two coordinates will, when substituted for x and y, render the two equations (1) y=ax+b, and (2) y=a'x+b' simultaneously identical; and we shall therefore find them by solving the equations algebraically. Now, by eliminating y we obtain

(3) (a-a') x=b'¬b;

But if a-a=0, that is a=a', then we can no longer admit this result (4), although the result (3) still holds: the equation (3) now becomes 0=b'-b, and contains a contradiction unless b'=b, and this contradiction shews, that the supposition of the existence of a point of intersection in this case involves a contradiction, i.e. that in this case no such point of intersection exists, i. e. that the two straight lines are parallel to one another.

a+bx + cx2+... a'+ bx + c'x2+...

a

Or, we have to transform the quotient into a series proceeding according to whole powers of x, i.e. to find the coefficients Po, P1, P2, &c. of the series Po+ Pix+ P2x2+... which is equal to that quotient. We obtain for the determination of Po the equation a=a'. Po, whence P。= if a' is not zero. But if a'=0, then this general result can no longer be retained, and the direct particular treatment of this case then shews that in this case the series does not begin with such a term as Po but with the term

a

provided that none of the divisors is zero, and also that all the quotients denote differences of whole numbers.

And if these equations are synthetically proposed, they may be proved by multiplying the two expressions to the right and left of the sign of equality in each equation by some one of the divisors which occur in them, and applying the proposition (sect. 15, No. 4), or adding to both sides any expressions which are subtracted, and applying (sect. 3).

SECTION 18.

a

We can now conceive the idea of the quotient quite gene

rally, meaning by it a mere symbol of the form

(*), endowed

with the property that 7.6 may be everywhere interchanged

a

with a. Hence the general quotient represents the simple property, (and consequently every expression which possesses this property), that when it is multiplied by the divisor b, the result is the dividend a.

SECTION 19.

Hence if we still call two expressions which contain (symbolized) divisions, " equal," as in (sect. 3), when they may, agreeably to the laws of operation, be unconditionally interchanged, the question again arises, as in (sect. 3), what is agreeable to the laws of operation? And we must answer this question in a manner analogous to that in (sect. 3). Since we are at present only acquainted with differences of whole numbers, and can therefore only come into contradiction with them, the character of the equation must be such, that two expressions which are recognized as generally equal, shall in all particular cases,-as e. g. when all the letters denote differences of whole numbers,represent one and the same difference of whole numbers. We shall therefore ascertain the equality of two expressions, compounded in any manner whatever by (symbolized) addition, subtraction, multiplication, and division, by shewing that there exists a third expression, which is not equal to zero, and such, that when each of the two first are multiplied by it (sects. 11 and 14), products result that may be made to pass into such expressions as are equal to one another according to the characteristic of (sect. 3), by applying the interchangings permitted by the definitions of product and quotient (sects. 14 and 18), or by substituting two expressions for one another which have already been ascertained to be equal.

SECTION 20.

Now from these definitions, in conjunction with those of (sect. 11), if follows immediately even for expressions which still contain divisions, that:

(1) If two expressions are equal to a third, they are equal to one another.

(2) When such equal expressions are added to, subtracted from, multiplied or divided by equal expressions, the results, provided we never divide by zero, i. e. under the sole condition that no divisor is zero, are also always equal expressions.

And from this, not only does the general correctness follow of all the equations already proposed in the preceding (sects.), but general "calculation" (according to sect. 6) with general forms, is, within the four operations hitherto considered, rigorously established, with this one exception, that general results are no longer to be permitted to hold, as soon as one of the divisors (in any particular case of application) becomes equal to zero. And at the same time we have in no case to regard the signification of the particular letters, so that it is convincingly self-evident, that, to whatever extent we continue the "calculation," (in the meaning of sect. 6), each several equation continues to enunciate nothing more nor less than the relation of these 4 operations or mental acts to one another, each equation in its own peculiar manner. But the relation of these 4 operations to one another is already completely enunciated in the following 7 equations, viz. in the equations

Any further equation (containing no other than these 4 operations) is to be considered as derived from these 7, so that any further equation only appears as a particular case, or as a combination of two or more of these 7 equations.

The two first of these seven equations contain the fundamental property of addition, the third contains the definition of subtraction, the fourth and fifth contain that fundamental property of multiplication which it has in common with addition, the sixth expresses in particular the connection between multiplication and addition, and completes the essence of multiplication, the seventh of these equations, finally, contains the definition of division*.

And whenever two "equal" expressions in any particular case denote differences of whole numbers, they will always denote one and the same difference of whole numbers; and there

We may also say: The first and second of the seven equations contain the character of the sum, the third the definition of the difference; the fourth and fifth enunciate that character of the product which it has in common with the sum, while the sixth shows the connection between the product and the sum, and thus completes the character of the product. Finally, the seventh enunciates the definition of the quotient; all conceived in the most general sense.