CHAP. V] RATIONAL PYKAMIDS; RATIONAL TKIHEDRAL ANGLES. 221 



with diagonals crossing at right angles at 7, and having E, F, G, H as the 

 projections of / on the sides and K, L, M, N as its projections on EF, FG, 

 GH, HE, there are integral values for the distances from / to these 12 points, 

 for the 8 segments on the sides, and for the 8 segments on EF, FG, GH, HE. 



E. Turriere 146 gave known results on inscriptible quadrilaterals with 

 rational sides and diagonals. 



W. F. Beard stated and G. N. Watson 147 proved that if two circles with 

 centers and 0', and radii R and R', are such that quadrilaterals can be 

 inscribed in the first and circumscribed about the second circle, then the 

 least integral values for R, R', c = 00' are 35, 24, 5, while general solutions 

 follow from 



(R 2 - c 2 ) 2 = [R'(R + c)} 2 + (R'(R - c)} 2 . 



For rational quadrilaterals, see Turriere, 61 " 62 Euler 148 and Schwering 160 ; 

 also, Euler 32 of Ch. XV. Cf. Berton 49 of Ch. XXIII. 



RATIONAL INSCRIBED POLYGONS. 



L. Euler 148 gave a construction to find a polygon with any number n 

 of sides, inscribed in a circle with center and radius unity, such that the 

 sides, all diagonals, and the area are rational. Employ n 1 arbitrary 

 angles 2A, 2B, , and take as the nth angle one whose sine and cosine 

 equal the sine and negative of the cosine of the sum of those n 1 angles. 

 Take arc AB = 2A, arc BC = 2B, arc CD = 2C, etc. Hence side AB is 

 2 sin A, side BC is 2 sin B, -, diagonal AC is 2 sin (A + B), . To 

 make all the sines and cosines rational, take sin A 2ab/(a z + 6 2 ), etc. 

 Since triangle AOB equals sin A cos A, the area is rational. He gave 

 complicated expressions which serve as rational sides and diagonals of 

 an inscribed quadrilateral, but do not make the area rational. 



H. Schubert 47 (pp. 28-38, or Schubert, 88 pp. 55-67) considered an in- 

 scribed polygon with the sides a\, , a n . Let Ion be the arc subtended by 

 at. Let n 1 of the a's (and hence all) be Heron angles. 47 Let 



n 1 



tan \ai = qtlpi, 4r = H (P\ + ff*), 



i=l 



so that r is the radius of the circumscribed circle. Then the sides 

 a,i = 2r sin a,- are rational, also all diagonals since the ratio of any one to 2r 

 is the sine of a sum of certain a's. The area (sin 2i + + sin 2a n )r 2 /2 

 is rational. 



J. Cunliffe 67 found rational inscribed pentagons. 



RATIONAL PYRAMIDS; RATIONAL TRIHEDRAL ANGLES. 



A rational pyramid is one whose edges and volume V are rational. 

 R. Hoppe 149 considered a rational trihedral angle (one having rational 

 sines and cosines of the face and dihedral angles). Let a, b, c be the tangents 



146 L'enseignement math., 18, 1916, 408-410. 



147 Math. Quest, and Solutions, 4, 1917, 31-2. 



148 Opera postuma, 1, 1862, 229 (about 1781). 

 " 9 Archiv Math. u. Phys., 61, 1877, 86-98. 



