CHAPTER V. 



TRIANGLES, QUADRILATERALS AND TETRAHEDRA 

 WITH RATIONAL SIDES. 



RATIONAL OR HERON TRIANGLES. 



Heron of Alexandria gave the well known formula for the area of a 

 triangle in terms of the sides and noted that when the sides are 13, 14, 15, 

 the area is 84. A triangle with rational sides and rational area is called a 

 rational triangle or Heron triangle. 



Brahmegupta 1 (born 598 A.D.) noted that, if a, b, c are any rational 

 numbers, 



are sides of an oblique triangle [whose 2 altitudes and area are rational and 

 which is formed by the juxtaposition of two right triangles with the common 

 leg a]. 



S. Curtius 20 proposed the following question: Three archers A, B, and 

 C stand at the same distance from a parrot, B being 66 feet from C, B 50 

 feet from A, and A 104 feet from C; if the parrot rises 156 feet from the 

 ground, how far must the archers shoot to reach the parrot? He noted 

 that they stand at the vertices of a triangle the radius of whose circum- 

 scribed circle is 65 feet, while the parrot is 156 feet above its center. Since 

 65 2 + 156 2 = 169 2 , each archer is 169 feet from the parrot. It is stated to 

 be difficult to explain why the radius turns out to be an integer. Cf. 

 Gauss. 14a [The triangle is rational since its area is 2 3 3 5 11 = 1320.] 



C. G. Bachet, 3 in his comments on Diophantus VI, 18, treated several 

 problems, the second of which is to find a triangle with 

 rational sides and a rational altitude (and hence a 

 Heron triangle). Taking a right triangle ADC with 

 the sides 10, 8, 6, he found BD = N such that N 2 

 + 8 2 shall be the square of a rational number (AB). 

 Assuming first that angle B A C is acute, so that DC : 

 AD < AD : BD, we must have 6N < 64, whence 

 N < 32/3. Let N 2 + 8 2 be the square of 8 - xN; then 



10 



1 Brahme-Sphut'a-Sidd'hanta, Ch. 12, Sec. 4, 34 Algebra with Arith. and Mensuration, 



from the Sanscrit of Brahmegupta and Bhascara, transl. by H. T. Colebrooke, London, 

 1807, 306. 



2 E. E. Kummer, Jour, fur Math., 37, 1848, 1. 



2a Tractatus geometricus . . . , Amsterdam, 1617, Quoted by A. G. Kastner, Geschichte der 

 Math., Ill, 294. 



3 Diophanti Alex. Arithmeticorum . . . Commentariis . . . Avctore C. G. Bacheto, 1621, 



416. Diophanti Alex. Arithmeticorum, cum Commentariis C. G. Bacheti & Observa- 

 tionibus D. P. de Fermat (ed., S. Fermat), Tolosae, 1670, 315. 



191 



