. Carnegie Institution of Washington publication. 216 HISTORY OF THE THEORY OF NUMBERS. [CHAP. V. On the ratios of the sides to the radius of the inscribed circle see Gerono,150 Ch. XXIII. The following papers were not available for report: C. Klobassa, tjber Pythagoreische u. Heronische Zahlen, Progr., Troppau, 1908. E. Haentzschel, Das Rationale in der algebraischen Geometric [an address], Unterrichtsblatter Math. Naturw., 21, 1915, 1-5. RATIONAL QUADRILATERALS. A rational quadrilateral is one whose sides, diagonals and area are expressed by rational numbers. Brahmegupta1 (§ 38) stated that
. Carnegie Institution of Washington publication. 216 HISTORY OF THE THEORY OF NUMBERS. [CHAP. V. On the ratios of the sides to the radius of the inscribed circle see Gerono,150 Ch. XXIII. The following papers were not available for report: C. Klobassa, tjber Pythagoreische u. Heronische Zahlen, Progr., Troppau, 1908. E. Haentzschel, Das Rationale in der algebraischen Geometric [an address], Unterrichtsblatter Math. Naturw., 21, 1915, 1-5. RATIONAL QUADRILATERALS. A rational quadrilateral is one whose sides, diagonals and area are expressed by rational numbers. Brahmegupta1 (§ 38) stated that " the legs of two right triangles multi- plied reciprocally by the hypotenuses give the four sides of a ; Bhdscara130 (born 1114) illustrated this construction of a rational quadri- lateral by starting with the right triangles (3, 4, 5), (5,12,13). Multiplying the legs of the first by the hypotenuse of the second, we get two opposite sides of the quad- rilateral; multiplying the legs of the second by the hypotenuse of the first, we get the two re- maining sides of the quadrilateral. One diago- nal is the sum 4 • 12 + 3 • 5 = 63 of the products of the legs of one triangle by the corresponding legs of the other. The other diagonal is 4-5 + 3-12 = 56. As the Commentator Gane'sa (1545 ) indicated (p. 81), the quadri- lateral is formed by the juxtaposition of four right triangles obtained by multiplying the sides of each given triangle by the perpendicular and base of the other. Bhaseara noted that if we take the sides of the quadrilateral in the new sequence 25, 39, 52, 60, one diagonal is still 56, but the other is now the product 65 of the two hypotenuses. He noted (§§ 179-184, pp. 76-8) that the quadrilateral with the sides 40, 51, 68, 75 and diagonals 77, 85 has the area 3234. M. Chasles131 made clear the true sense of Brahmegupta's theorem. Let a, 6, c, d, e be integers, such as 3, 4, 5, 12, 13, for which a2 + b2 = c2, c2 -f d? = e2. Construct th
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