736 



MR EDWARD SANG ON THE THEORY OF COMMENSURABLES. 



39. To construct a tetragon which may have the co-ordinates of its corners 

 and its sides all rational; subject to the condition that its sides have, nearlj', 

 given inclinations to the axes. 



Since the inclinations of the sides of the actual polygon to the axes must be 

 muarif, we must select such muarif angles as may be suitable ; and, moreover, 

 we may always transform the co-ordinates, so that one of the axes may be 

 parallel to one of the proposed sides. 



If, then, l^, 4, 4, l^ be the lengths, and 6^, 6 ^, ^3, 6^ the bearings of the four 

 sides taken in order, we must have — 



1, sin ^1 + ^2 ^^^ ^2 + ^3 sin ^3 + ^4 sin ^4 = 

 Ij^ cos 6j^ + Zj cos $2 + Z3 cos ^3 + l^ cos ^^ = 



in which, if we put the appropriate rational fractions for the sines and cosines, 

 we shall form two indeterminate equations, involving four unknown quantities. 



By placing one of the axes of co-ordinates parallel to (say) the fourth side, 

 we cause the equations to take the form — 



Zj sin 0j + l^ sin 0^ + l^ sin 6^ = l^ 

 Zj cos 6^ + Zg cos B^ + /g cos ^3 = 



in which we have only to consider the latter. 



As an example, let it be proposed to construct a tetragon ABCD, such that 

 AB may have about the direction 22"^, BC about 74°, CD about 168°, and DA 

 270°. The muarif angles corresponding to these directions are, — 



22 . 



. 37 . 



. 11 



13 



12 



5 



73 . 



. 44 . 



. 23 



25 



7 



24 



167 . 



. 19 . 



. 21 



41 



-40 



9 



270 . 



. 00 . 



. 00 



1 







-1 



and thus if 13a, 25b, 41c, and d be assumed as the four sides, we must have, — 



12a + 76-40c=0; 5a + 2Ab + 9c=d 



in which, if the values of a, b, c, be obtained in integers, that of d must also be 

 integer, so that we have only to consider the indeterminate equation 12a -f- 7^— 

 40c = ; this may be put under the form, — 



12a + 76 



that is to say, we have to obtain such values of a and b as may make 12 a + 7b 

 divisible by 40. The lowest solution of this indeterminate problem is a = l, 

 J = 4, c=l ; whence the sides are 13, 100, 41, 110. 



By an extension of the same process we may obtain polygons of any number 

 of sides, having all their sides and ordinates commensurable. The area of such 

 a polygon is also commensurable with the square of the linear unit ; but it does 

 not follow that the diagonals drawn from one corner to another are rational. 



