represented by /a, /b, etc.

Since the means, in any pro

portion between like magnitudes, may be interchanged, if b = /a, then also a

and vice versa.

| b,

A reciprocal, being itself a line-magnitude, may enter a product as one of its factors.

7. Idemfactor: Real Unit.

If in the proportion

jabm we write aj, that is, make AM in Fig. 10 coincide with JB, then

jjbm, that is, mb..

But by the definition of a product m =j X. b;.

(Prop. 2.)

An operator which, likej, as factor in a product leaves the other part of the product unchanged, is called an idemfactor. This particular real idemfactor is what was defined in Art. 5 as the real unit. In arithmetic it is denoted by the numerical symbol 1.

8. Quotient. The product defined by the proportion jcam is cX/a and is called the quotient of c by a. The sign before may be omitted without ambiguity and this quotient be denoted by the simpler notation ca, in which c is called the dividend and a the divisor.

The proportion ja :: bj defines

b = |a, a = | b, and a × b=j;

hence

*Benjamin Peirce: Linear Associative Algebra (1870), p. 16, or American Journal of Mathematics, Vol. IV (1881), p. 104.

and so for any magnitude whatever. Hence we may describe the real unit as the quotient of any real magnitude by itself.

The quotient ca is also represented by ca, or by The latter notation will be frequently employed in the sequel.

9. Agenda. Problems in Construction.

(1). From the definitions of Arts. 5, 6 and 8 prove that the following construction for the quotient a/b is correct: On one of two straight lines, making any convenient angle with one another, lay off OA=a, OB=b, on the other 0J=j; join B and Jand draw AM parallel 'to BJ to intersect OJ in M. OM is the quotient sought. (2). Given a Xa = m, construct a.

(3). Given a, b and c, construct (a X b) X c.

(4). Prove that a X b is > = or <b according as a is >

= or < I

(5). Draw OX and OY making any convenient angle with each other; on O Ylay off OJ=j, OA=a, OB=b, and on OX take 0]=j. A straight line through / parallel to OX will be cut by JA and JB in two points P and Q. Show that if A and B are on the same side of O, the distance between P and Q is PQ/a/b, where

means difference between, but if A and B are on opposite sides of O, then PQ= a+/b. In this construction a and b are supposed to be positive magnitudes.

10. Infinity is defined as the reciprocal of zero; in symbols

When a magnitude decreases and becomes zero, its reciprocal obviously increases and becomes infinite. Since zero

is not a magnitude, neither is infinity as here defined.
In Art. 6 it was shown that b =
hence, from the definition / o = ∞

/a implies also a = | b ; follows

The construction for a product (Art. 5) shows that when one of its factors becomes o or ∞, the other remaining finite, the product itself is also o or, so that for all finite values of a

From the definition of addition (Art. 3) it is also obvious that

II. Indeterminate Algebraic Forms. When a sum or product assumes one of the forms+∞∞, o×∞,0/0, ∞∞, it is said to be indeterminate, by which is meant: the form by itself gives no information concerning its own value.

(i). The form + ∞

∞.

On a straight line ABP take at random two points A, B, so that AB is any real finite magnitude whatever. Take P, Q, R, on the same line,

AP=| AQ= |a, BP=|BR= | b,

and let pass into coincidence with A. P then passes out

of finite range, R passes into coincidence with B and the difference

gives no information con

Hence, taken by itself, ∞

cerning its own value and is indeterminate.

(ii). The forms o X ∞,'0/0, ∞ /∞. In the figure of Art. 5, let M and J remain fixed, while MA and JB, being always parallel to one another, turn about M and J until MA coincides with, and JB becomes parallel to OM. At this instant a becomes zero, b infinite, and a Xb assumes the form o × ∞ ; and because the original value of a b is anything we choose to make it, the expression OX∞ gives no information concerning its own value and is therefore an indeterminate form.

Since o ∞ and / ∞ = o, we may replace o/o by OX and ∞ / ∞ by ∞ Xo. The two forms o/o and ∞∞ are therefore also indeterminate.

An expression, such as ab, that gives rise to an indeterminate form, may nevertheless approach a determinate value as it nears its critical stage. To find this value is described as evaluating the indeterminate form. (See Arts. 43 and 52.)

III.

LAW OF SIGNS FOR REAL QUANTITIES.

12. In Addition and Subtraction. The sign+, by definition, indicates that the magnitude following it is to be added algebraically to what precedes, without having its character as a magnitude in any way changed; and the sign indicates that the magnitude immediately

following it is to be reversed in sense (taken in the opposite direction) and then added algebraically to what precedes.

Any symbolic representative of quantity, a letter for example, unattended by either of the signs or —, but still thought of as part of an algebraic sum, is supposed to have the same relation and effect in such a sum as if it had before. it the sign +: This usage necessitates the

following law of signs in addition:

and by a, unattended by any sign, is understood + a.

13. In Multiplication and Division. In Art. 8 it was agreed that a / b shall stand for the product a × /b. This convention requires that the combination shall produce. Looked at from another point of view, the symbol indicates that the letter following it is to be used as a factor with its character as a magnitude unchanged, while gives notice that the reciprocal of the magnitude immediately following it is to be used as a factor.

Any symbolic representative of quantity, a letter for example, unattended by either of the signs X or /, but still thought of as part of (factor in) an algebraic product, is supposed to have the same relation and effect in such product as if it had before it the sign X. The usage here described necessitates the following law of signs for X and /: