E. AGENDA: SUPPLEMENTARY PROPOSITIONS. (1). If two geometrical magnitudes A, B, have the same. ratio as two integers m, n, prove that nA (2). If A, B be two geometrical magnitudes and m, n two integers such that n A = m B, prove that A: Bm: n. Hence infer the statement in the first part of Definition 4, page 4, concerning commensurable magnitudes. (3). Given A: B :: P: Q and ʼn A n P = m Q. m B, prove that (4). It is a corollary of (3), that if A: B:: P: Q and A be a multiple, part, or multiple of a part of B, then P is the same multiple, part, or multiple of a part of Q. (5). Given A: B :: P: Q and B: CQ: R, prove that P> or < R according as A> or < C. = (6). Given A: B:: P: Q and B: C:: Q: R, prove that (Ex aequali.) Α C :: P: R. (7). Given A: B :: P: Q and B: C :: Q: R and C: D :: RS and D: E:: S: T, prove that A: E: P: T. State and prove the general theorem of which this is a particular case. (8). Given AB :: Q: R and B : C :: P: Q, prove that A: C: P: R. CALIFOR CHAPTER I. LAWS OF ALGEBRAIC OPERATION. I. QUANTITY. 1. Quantities in General. Quantities, whatever their nature, may be expressed in terms of geometrical magnitudes; in particular they may be thought of as straight lines of definite fixed or variable length. Such mag nitudes, in so far as they represent the quantities of ordinary algebra, are of three kinds: real, imaginary, and complex; real if, when considered by themselves (laid off upon the real axis), they are supposed to involve only the idea of length, positive or negative, without regard to direction, imaginary when they involve not only length, but also turning or rotation through a right angle, that is, length and direction at right angles to the axis of real quantities, finally complex if they embody length and rotation through any angle, that is, length and unrestricted direction in the plane. If we think of the straight line as generated by the motion of a point, we may translate length positive or negative into motion forwards or backwards; and it will sometimes be convenient to use the latter terminology in place of the former. It is at once evident that both reals and imaginaries are particular forms of complex quantities, reals involving motion forwards or backwards and rotation through a zero-angle, imaginaries involving motion forwards or 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, +aa=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 J and B and draw from A B m Fig. 10. a straight line parallel to JB to intersect OB in M. Then by Proposition 13 (p. 14) the intercepts on OA, OM by the parallels MJB, AM are proportionals, and if OM=m, 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.* duct the constituent parts are factors. In a pro The product a × b may also be a factor in another product, consisting therefore of three factors, as (a X b) × 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. |