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In introducing negative magnitudes the law of combinations of the signs -- and —, as described in Art. 12, must be observed. The complete symbolical statement of the associative law for addition and subtraction is contained in the formula: + (+ a + b ) = + (+ a ) + (+ 5),
wherein the order of occurrence of the signs -- and — must be the same in the two members of the equation.
16. In Multiplication and Division. Construct upon OA and OM, Fig. 15, the products OL = a X 6, O//= a X b, OE = 5 × c, ON= (a X b) X c, ON. = a X (6 X c). In this construction the lines marked f, f, f, through L, N. and B, E and J. Care, by the rule for constructing products
and as determined by the parallels A, K, f: a ... à : a × 5, (Prop. 13.) f: a ... à X c : (a X 6) X c. (Prop. I.) But the parallels h, h, determine the proportion f: a .: 5 × c : a X (à X c). Therefore W. M. are one and the same point (Prop. 6) and X (X a X 5) × ca X a X (X & X c).
Thus the sign X is distributive over the successive factors of a product; that is, as here follows when a =j,
The same is true of the sign /; for the product of any magnitude by its reciprocal is j (Art. 6), and two products that have their factors respectively equal are obviously themselves equal (Art. 5); that is,
/ a X a = 7, & X / b =j, and (/a X a) X (à X / 5) = j Xj = j. (Art. 7.) Whence, by the associative law in multiplication just proved, and because the means in a proportion between like magnitudes may be interchanged (Prop. 11),
X (X, a X. b) = X (X, a) X (X, 5), wherein the order of occurrence of the signs X and / in the
two members must be the same, and the law of their combination (Art. 13) must be observed.
V. COMMUTATIVE LAW FOR REAL QUANTITIES.
17. In Addition and Subtraction. Let a and 5 represent any two lengths taken in the same sense along a straight line such that a = O.4 and & = AB; then
If A B be negative and equal to —6, lay off a from O to the right, & from the extremity of a to the left. B will fall to the right or left of O according as 5 is less than or
In the former OB will be positive, in the latter negative. If now B' be so taken that B’O = b, then Bo B = a, Hence
Obviously any algebraic sum may have its terms commuted in like manner. Stated symbolically, this law is
wherein each letter carries with it like signs on both sides of the equation.
18. In Multiplication and Division. On two straight lines, meeting at a convenient angle, construct OB, O.B., each equal to b, and OA, , OA, each equal to a. On OB, take O/=j, the real unit, join JA, , JB, draw AM, B, M, parallel respectively to JB, JA, , intersecting OB (produced, if necessary) in M and M, and let OM = m, OM = m. By definition of a product (Art. 5),
a × 6 = m,
and b X a = n.
But the proportion determined by the parallels h and k is
and that determined by the parallels f and g is f : b :: a n,
which by interchange of means (Prop. 11) becomes f : a .: 6: n. ‘. 772 = 72. (Prop. 6.) M and N are hence one and the same point, and 6 × a = a X 6. Inasmuch as the reciprocal of a line-magnitude is itself a line-magnitude, it follows from the above demonstration that & X / a = / a X & ; or, observing the law of signs (Art. 13), this is b/a = | a X & ; and if b carry with it its proper factor sign, the more explicit form of statement contained in this equation is × 5 / a = / a X 5.
The formula for the commutative law in multiplication and division may therefore be written
X & X, a = X, a × 6, and this is easily extended to products of three or more factors. In this formula, the same sign attaches to a or ò on both sides of the equation.
19. Agenda: Theorems in Proportion.
(1). Prove that in a proportion the product of the means is equal to the product of the extremes.
(2). Prove that rectangles are to one another in the same ratio as the products of their bases by their altitudes. Show that the same relation holds for parallelograms and triangles.
(3). If a square whose side is I be taken as the unit of area, the area of any rectangle (or parallelogram) is equal to the product of its base by its altitude.
VI. THE DISTRIBUTIVE LAW FOR REAL QUANTITIES.
20. With the Sign of Multiplication. On OH and OK (Fig. 19), take OC = c, O/2 = a + b, and construct the product (a + 5) × c = OG. Then if O/ be the real unit, laid off on OK, EG and /C are parallel by construction. On OA take OA = a, and construct the product a X c = OF. Then - H A F and EG, being
O ° otó)x6 GT parallel to JC, are Fig. 19. parallel to one another, and //; drawn parallel to OA is equal to AE, that is, A/3 = b, and therefore FG is the product b×c. Hence