PROPOSITION I. Straight lines have to one another the same ratio as rectangles of equal altitude described upon them. Let AG, BH be the given straight lines, and C the given altitude. Then shall AG have to BH the same ratio as rectangle (AG, C) to rectangle (BH, C). On AG, BH construct rectangles having altitudes each equal to C. If now AG be divided into any number whatever of equal parts AM, MN, &c., and lines be drawn through M, N, &c. || to AK: then KG will be divided into the same number of equal parts as AG. If also from BH parts BP, PQ, &c., each equal to AM be cut off until there is no remainder or a remainder less than AM, and lines be drawn through P, Q &c. || to LB: then LH will contain the same integral number of parts, each equal to KM as BH does parts each equal to AM; ... AG has the same ratio to BH as KG has to LH, i.e. as rectangle (AG, C) has to rectangle (BH, C). COR. In a similar manner it may be proved that straight lines have to one another the same ratio as s or ▲s of equal altitude described upon them. DEFINITION. Magnitudes which have the same ratio are called proportionals. The fact of four magnitudes being proportionals is expressed thus: The first is to the second as the third is to the fourth, and written thus the first : the second as the third the fourth. : If two magnitudes are equal one will contain the same integral number of parts, each equal to any third magnitude, as the other. Hence PROPOSITION II. Proportions hold good if for the magnitudes involved others equal to them are substituted. For the test expressed by the definition is satisfied. PROPOSITION III. If four magnitudes are proportional then any others which have the same ratio as the first and second are proportional to any others which have the same ratio as the third and fourth. For the test expressed by the definition is satisfied. PROPOSITION IV. If a magnitude have the same ratio to each of two others then these two must be equal. For if A be the former and B, C the two latter magnitudes, so that A: B as A: C, then B must be equal to C. Otherwise A might be divided into a number of equal parts, each less than the difference between B and C; and B and C could not contain the same integral number of those parts; but they do (by the definition); ... B is equal to C. C. G. II PROPOSITION V. If four straight lines are proportional the rectangle contained by the extremes is equal to that contained by the means. Let the four straight lines a, b, c, d be proportional. On a, b construct rectangles, the altitude of each being equal to c, and on c, d construct rectangles, the altitude of each being equal to a. ...rect. (a, c) rect. (b, c) as rect. (a, c): rect. (a, d); (v. 3) ... rect. (b, c) rect. (a, d). (v. 4) Conversely, If a, b, c, d are four straight lines, such that rect. (a, d) = rect. (b, c), then ab as c: d. For the same construction being made, ..: rect. (a, d) = rect. (b, c); .. rect. (a, c): rect. (b, c) as rect. (a, c) : rect. (a, d); (v. 2) In like manner it may be demonstrated that, if the rectangle contained by two straight lines is equal to the rectangle contained by two other straight lines, then any proportion having the sides of one rectangle for the extremes and the sides of the other for the means is true. Thus if then and rect. (a, d)=rect. (b, c), ba as d c, a c as b : d. COR. If c=b, then rect. (b, c) square on b. Also when = a bas bd, the straight lines a, b, d are then said to be proportional. Hence, if three straight lines are proportional the rectangle contained by the extremes is equal to the square on the mean: and conversely. Hence may be easily established the following Pro positions: PROPOSITION VI If the four straight lines a, b, c, d are proportional, then a cas b : d. For a bas c: d; .: rect. (a, d) = rect. (b, c); ... a: c. as b : d. PROPOSITION VII. If a bas c: d, then ba as d : c. For rect. (a,d) = rect. (b, c), PROPOSITION VIII. If a b as c: d, then a is >= or < b, according as c is >= or < d. ... if be> For rect. (a,d) = rect. (b, c); =ord, a must be accordingly > = or < b. PROPOSITION IX. If a : b as c: d, then a is > = or < c, according as b is > = or < d. This follows in a similar manner to the preceding demonstration. |