since BC and AC are slightly > BE and AE, respectively. х Now, when n is taken indefinitely great, and become n n Q.E.D. (236) being the limits of variables always equal. 318. COR. 1. Rectangles with equal bases are to each other as their altitudes. For (317) the equal bases may be taken as altitudes, and the altitudes as bases. 319. SCHOLIUM. Since a rectangle is determined by its base and altitude (144), that is, by any two adjacent sides, as AB, BC, we employ the expression, the rectangle contained by AB and BC, or more briefly, the rectangle AB, BC, to denote the rectangle determined by AB and BC; and use the symbol rect. AB AC, or simply AB BC, for either of these expressions. Since a square, again, is determined by its base, i.e., by a side, we employ the expression, the square of AB, or the symbol AB2, an abbreviation for AB · AB, to denote the square whose base is AB. 320. DEFINITION. The symbol, to be read, is equivalent to, is the symbol of equivalence. 321. COR. 2. If four lines, A, B, C, D, are in proportion, the rectangle contained by the extremes is equivalent to that contained by the means. For since rect. AD: rect. BD = A: B, and rect. B. C: rect. B.D=CD=A: B,} (317) rect. AD: rect. B. C. = rect. BD rect. BD, (232'') .. rect. A Drect. B. C. 322. COR. 3. If A, B, C, are lines such that A: B=B: C, then B2 -2 =rect. A. C. (321) That is, the square of a mean proportional between two lines is equivalent to the rectangle contained by those lines. 323. The unit of area is the square having as base the linear unit. Thus if the base AB of the square AC is equal to the linear unit, then the square AC is the unit with which all areas are compared. Α B 324. As already defined, the numerical measure of a quantity is the number that shows how many times the quantity contains its unit; in other words, it is the ratio of the quantity to its unit. As regards triangles, it is customary to denote the numerical measure of a side by means of the small letter corresponding to the capital designating the opposite angle. Thus in ABC, B we employ a, b, c, to denote the numerical E measures of BC, AC, AB, respectively. As regards polygons of more sides than three, there is no such convention, but we specify AB = a, BC = b, CD = c, and so on. Wherever it may occur, henceforth, such an expression as the product of A and B is to be understood as a convenient abbreviation for the product of the numerical value of A by that of B. Great care should be taken, however, not to forget the real meaning of such abbreviations. Beginners are often confused by the careless use of such expressions as, length multiplied by breadth gives area, forgetting that what is meant is: the numerical measure of the length multipled by that of the breadth gives as result the numerical measure of the area, as we find explained in Art. 326. A B PROPOSITION II. THEOREM. 325. Rectangles are to each other as the products of the numerical measures of their altitudes and bases. Given: Two rectangles AC, A'C', with altitudes AB, A'B', and bases BC, B'C', respectively, whose numerical measures are, respectively, a, a', and b, b'; To Prove: Rectangle AC: rectangle A'c' = a × b: a' × b'. Construct a rectangle EG with altitude EF = A'B', and base FG = BC, and let the numerical measures of AC, A'C', EG, be x, y, z, respectively. Since AC and EG are rectangles with equal bases, (Const.) AC: EG = AB: EF, or x: z = a : a' ; (318, 232') EG and A'C' are rectangles with equal altitudes, (Const.) EG: A'C' FG: B'C', or z: y = b: b'. (317, 232') From these numerical proportions we obtain (242) xy=ax b: a' x b', = .. rect. AC: rect. A'c' a x b: a' x b' Q.E.D. (232") 326. COR. The area of a rectangle is measured by the product of its base and altitude. For if AC be any rectangle, and s the unit-square, then ... area AC: Sax b:1 x 1, area ACS × ab; i.e., the area of AC is ab A S B D times the unit-square s. If, for example, the numerical measures of AB and BC are 5 and 7, respectively, then the numerical measure of the area of AC is 35; that is, the area of AC is equal to 35 unit-squares. In the enunciation of this corollary, as elsewhere, the term area is for brevity used for numerical measure of the area; that is, the number of unit-squares to which the surface in question is equivalent. PROPOSITION III. THEOREM. 327. Any parallelogram is equivalent to the rectangle having the same base and altitude. Given: A parallelogram AC and a rectangle EC, with the same base and altitude BC, EB; To Prove: Parallelogram AC is equivalent to rectangle EC. Since AC and EC are parallelograms, (Hyp.) (136) (72) .. par'm AC rect. EC. Q.E.D. (Ax. 3) 328. COR. 1. Parallelograms with equal bases and equal altitudes are equivalent. For each is equivalent to the same rectangle. (327) 329. COR. 2. Parallelograms with equal altitudes are to each other as their bases; and those with equal bases to each other as their altitudes. For they are as the rectangles having those bases or altitudes. 330. COR. 3. The area of any parallelogram is equal to the product of its base and altitude. PROPOSITION IV. THEOREM. 331. Any triangle is equivalent to one half the rectangle contained by its base and altitude. D B E Given A triangle ABC, having a base BC and altitude AD; To Prove Triangle ABC is equivalent torect. AD · BC. Complete the parallelogram ABCE. Then since AC is a diagonal of par'm BE, ΔΑΒC= Δ ACE, .. ▲ ABC par'm BE, (140) .. ▲ ABC≈rect. AD · BC. Q.E.D. (327) 332. COR. 1. Triangles with equal bases and equal altitudes are equivalent. 333. COR. 2. Triangles with equal altitudes are to each other as their bases; and those with equal bases, as their altitudes. 334. COR. 3. Triangles are to each other as the products of their bases and altitudes. EXERCISE 461. Prove Prop. III., when the upper base of the rectangle lies with- E 462. Prove the same proposition when the upper and lower bases lie without each other, though in the same lines. B FA |