220 HISTORY OF THE THEORY OF NUMBERS. [CHAP. V 



the sides are 2r sin en, the diagonals are 



e = 2r sin (ai + 2 ), f = 2r sin ( 2 + s). 



The area is %ef sin (o^ + as). In the very special case in which the tangents 

 of %a\, |a 2 , |as are rational, as well as one side or r, the four sides, diagonals 

 and area will be rational. 



A. Ge'rardin 141 juxtaposed two right triangles with a common hypotenuse 

 to obtain a quadrilateral whose sides have the values quoted from Brah- 

 megupta by Kummer; also a second quadrilateral. 



E. N. Barisien 142 noted the inscriptible quadrilateral with the sides 

 AB = 75, BC = 68, CD = 40, DA = 51, segments of diagonals (at right 

 angles) AI = 45, BI = 60, CI = 32, DI = 24, and diameter 85 of circum- 

 scribed circle. 



F. Neiss 142a treated rational triangles and rational quadrilaterals. 



I. Newton 1426 treated the problem to find the diameter x = DA of a 

 circle having an inscribed quadrilateral ABCD, three of whose consecutive 

 sides a = AB, b = BC, c = CD are given, while the fourth side is the diam- 

 eter. We have z 3 (a 2 + 6 2 + c 2 )x 2abc = 0. E. Haentzschel and 

 E. Lampe 142c found rational quadrilaterals of this type by the method of 

 Kummer. 133 



E. Haentzschel 143 treated rational quadrilaterals with perpendicular 

 diagonals by setting c = 0, t = 1, in Kummer's work. Condition (7) is 

 now a 2 (z 2 I) 2 + 4y 2 x 2 = 2 2 ; methods of finding rational solutions are 

 developed. An evident special solution is obtained by taking = 77; then 

 y = x, AB = BC, CD = AD, and the quadrilateral is given by the juxta- 

 position of two congruent rational triangles. Next, taking x = 77 = cfd, 

 y = = a/6, we get Brahmegupta's quadrilateral as quoted by Kummer. 

 More general solutions are found by use of Weierstrass' ^-function. 



Haentzschel 144 noted that the determination of a quadrilateral with 

 rational sides, diagonals, area, and radii of the inscribed and circumscribed 

 circles, depends on the rational solution of 



( M 2 + i)(^ 2 + i) W + i)0 2 + i) + V} = n. 



By use of Weierstrass' ^-function, he found two infinite sets of rational 

 solutions, including the special solutions by O. Schulz 167 (pp. 98-103). 

 Ankum's method to deduce a rational tetrahedron from a rational quadri- 

 lateral is applied to the quadrilaterals found here. 



E. N. Barisien 145 noted that in the quadrilateral with the sides 



AB = 1625, BC = 2535, CD = 3900, DA = 3380, 



141 Sphinx-Oedipe, 6, 1911, 187. 



142 Mathesis, (4), 3, 1913, 263. He noted (p. 14) the quadrilateral with successive sides 15, 



20, 24, 7, diagonals 20, 25, and area 234. 

 1420 Rationale Dreiecke, Vierecke . . . , Diss., Leipzig, 1914. 

 1426 Arithmetica universalis, Amsterdam, 1, 1761, IV, Ch. 1, 140-150. 

 " e Zeitschrift Math. Naturw. Unterricht, 46, 1915, 190-4; 49, 1918, 139-144, 144-5. 



143 Sitzungsber. Berlin Math. Gesell., 14, 1915, 23-31. 

 i Ibid., 14, 1915, 85-94. 



145 L'interme'diaire des math., 23, 1916, 195-6. 



