backwards and rotation through a right angle. The three kinds of quantities will be considered in order; the distinction between them, here roughly outlined, will be made clearer by a study of their properties. 2. Nature of Real Quantities. It is evident that all real quantities may be made concretely cognizable by laying them off (in the imagination) as lengths, in the positive or negative sense, upon one straight line. In this representation every straight line suffices to embody in itself all real quantities, having its own positive and negative sense, that is, its direction forwards and backwards. In particular, all the numbers of common arithmetic, both integral and fractional, are accurately represented by distances laid off from a fixed origin in the positive sense upon a straight line, and in the same way all so-called irrational numbers, though only approximately realizable as true numbers in arithmetic, are accurately represented. Hence the following proposition, which is postulated as self-evident: The laws of algebraic operation that obtain with geometrical real magnitudes, that is, lengths laid off upon a straight line, are IPSO FACTO true when applied to arithmetical quantities, or numbers. But the converse of this proposition is not equally selfevident. For inasmuch as so-called irrational number, that is, quantity in general, is not realizable as true number (integer or fraction) in arithmetic, the proof that the laws of algebraic operation obtain for integers and fractions constitutes not a proof, but only a presumption, that they obtain also for so-called irrational number. In the following pages magnitudes will be represented by straight lines of finite length. II. DEFINITIONS OF ALGEBRAIC OPERATIONS. 3. Algebraic Addition. Simple addition is here defined as the putting together, end to end, different linesegments, or links, in such a way as to form a one dimensional continuum, that is, a continuous straight line. This kind of addition corresponds to the addition of positive numbers in arithmetic. Algebraic addition takes account of negative magnitudes, that is, of lines taken in the negative sense (from right to left, if positive lines extend from left to right), and to add to any line-segment a negative magnitude is to cut off from its positive extremity a portion equal in length to the negative magnitude. This kind of addition includes the addition and subtraction of positive numbers in arithmetic and introduces the new rule that larger positive magnitudes may be subtracted from smaller, producing thereby negative magnitudes. We then extend the idea of negativeness also to number and produce negative number, prefixing the sign to positive number as a mark of the new quality. The result of adding together algebraically several magnitudes is called a sum. In a sum the constituent parts are terms. 4. Zero is defined as the sum of a positive and an equally large negative magnitude; in symbols, +a-a= 0. It is not a magnitude but indicates the absence of magnitude. 5. Algebraic Multiplication. On two straight lines making any convenient angle with one another at O, Fig. 10, lay off OA= a, OB = b, and on OA in the same direction as OA lay off OJ=j, which shall be of fixed length in all constructions belonging to algebra and shall be called the real unit. Join / and B and draw from A b B m Fig. 10. a straight line parallel to jab: m. The length m, thus determined, is defined as the algebraic product, or simply the product, of the real magnitude a by the real magnitude b, and is denoted by ab, or by ab, or more simply still by a b.* In a product the constituent parts are factors. The product a X b may also be a factor in another product, consisting therefore of three factors, as (a X b) X c, and this may in turn be a factor in a product of four factors, and so on. 6. Reciprocals. If m be equal to j, then etc.; or more simply, since j remains unchanged throughout all algebraic operations, they may be conveniently * Descartes: la Géométrie, reprint of 1886, p. 2. represented by /a, /b, etc. Since the means, in any pro portion between like magnitudes, may be interchanged, and vice versa. if b = | a, then also a /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 a =j, that is, make AM in Fig. 10 coincide with JB, then jjbm, that is, m = b. But by the definition of a product m =jXb; (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 X 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 j : a :: b:j defines * 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 c/a is also represented by c÷a, 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 OJ=j; join B and Jand draw AM parallel to BJ to intersect OJ in M. OM is the quotient sought. (2). Given a X a =m, construct a. (3). Given a, b and c, construct (a × b) × c. = (4). Prove that a X b is > or <b according as a is> or < I = (5). Draw OX and OYmaking any convenient angle with each other; on Ylay off OJ=j, OA=a, OB=b, and on OX take OJ=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 |