* Ex. 2. If A, P, B, Q form an harmonic range, and M is the middle point of AB, prove that MA or MB is the mean proportional between MP and MQ. THEOR. II. A straight line which divides the sides of a triangle proportionally is parallel to the base of the triangle. Let the straight line DE divide the sides AB, AC of the triangle ABC, so that AD: DB :: AE : EC, D B . then shall DE be parallel to BC. For the parallel to BC through D divides AC in the ratio AD: DB IV. 9, Cor. 2. and therefore must pass through E, the only point in which AC is divided in this ratio; IV. 10. therefore DE is parallel to BC. Q.E.D. THEOR. 12. Rectangles of equal altitude are to one another in the same ratio as their bases. Let AC, DF be two rectangles of equal altitude on the bases AB, DE: then shall the rectangle AC be to the rectangle DF as the base AB to the base DE. Suppose 2AB = 3DE. On AB produced take BB' equal to AB, so that AB'=2AB; and on DE produced take EE', E'E' each equal to DE, so that DE"=3DE. Complete the rectangles BC', EF, E'F". Then because AB=BB', therefore the rectangle AC=the rectangle BC', I. 30, Cor. and therefore the rectangle AC'=twice the rectangle AC. In like manner the rectangle DF"=three times the rectangle DF. But, by supposition, AB'=DE", therefore the rectangle AC'=the rectangle DF", 1. 30, Cor. or, twice the rectangle AC=three times the rectangle DF. If m times AB=n times DE, we can prove by like reasoning that m times the rectangle AC=n times the rectangle DF. Hence the rectangle AC is to the rectangle DF as the base AB to the base DE. Q.E.D. Cor. Parallelograms or triangles of the same altitude are to one another as their bases. For a parallelogram is equal to a rectangle whose base and altitude are equal to those of the parallelogram. II. 1, Cor. 1. Also a triangle is equal to half a rectangle whose base and altitude are equal to those of the triangle (11. 2), and the ratio of two magnitudes is equal to that of their doubles. IV. 3. *Ex. 3. Parallelograms or triangles on equal bases are to one another as their altitudes. THEOR. 13. In the same circle or in equal circles angles at the centre and sectors are to one another as the arcs on which they stand. [NOTE.-In Book III. it was only necessary to consider arcs less than the whole circumference, and angles less than four right angles; but Theors. 4 and 5, Book III., are equally true for arcs greater than one or any number of circumferences, and the corresponding angles greater than four right angles.] In the equal circles DEF, HKL, of which A and B are the respective centres, let the angles DAE, HBK stand on the arcs DE, HK: then shall the angle DAE be to the angle HBK as the arc DE is to the arc HK. Suppose twice the arc DE=three times the arc HK. On the circumference DEF take the arc EE' equal to the arc DE, so that the arc DE'=twice the arc DE ; and on the circumference HKL take the arcs KK', K'K" each equal to the arc HK, so that the arc HK"=three times the arc HK. Join AE', BK', BK". III. 5. Then because the arc DE=the arc EE', therefore the angle DAE=the angle EAE', and therefore the angle DAE'=twice the angle DAE. In like manner the angle HBK"=three times the angle HBK. But, by supposition, the arc DE'=the arc HK", therefore the angle DAE'=the angle HBK", or, twice the angle DAE=three times the angle HBK. If m times the arc DE=n times the arc HK, we can prove by like reasoning that m times the angle DAE=n times the angle НВК. = Hence the angle DAE is to the angle HBK as the arc DE is to the arc HK. Q.E.D. DEFINITIONS OF BOOK IV. PART I. DEF. I. DEF. 2. DEF. One magnitude is said to be a multiple of another magnitude when the former contains the latter an exact number of times. According as the number of times is 1, 2, 3...m, so is the multiple said to be the 1st, 2nd, 3rd,...mth. One magnitude is said to be a measure or part of another magnitude when the former is contained an exact number of times in the latter. DEF. 3. If two or more magnitudes have a common multiple or measure they are said to be commensurable. 4. The ratio of one magnitude to another of the same kind is a certain relation of the former to the latter in respect of quantity, the comparison being made by considering what multiples of the two magnitudes are equal to one another. The ratio of A to B is denoted thus, A:B, and A is called the antecedent, B the consequent. Thus the ratio of a half-crown to a florin is the relation expressed by stating that 4 half-crowns = 5 florins. The ratio A:B is said to be equal to the ratio P: when like multiples of A and P (mA and mP) are equal respectively to like multiples of B and Q (nB and nQ), and the four magnitudes are said to be proportionals, or to form a proportion. The equality of the ratios is denoted by the symbol :: ; and the pro a DEF. 5. |