MR EDWAED SANG ON THE THEORY OF COMMENSURABLES. 735 



35. In a given circle to inscribe a polygon having all its sides, diagonals, seg- 

 ments, commensurable with the diameter, and their areas with the square of the 

 diameter; and which may approximate to a given inscribed polygon. 



The general solution of this problem is another variation of No. 29 ; we have 

 only to seek the muarif angle approximating to the angle at the circumference 

 subtended by each of the sides of the given polygon, less one. 



36. If, of a rational trigon, one of the angles be double of a muarif angle, 

 the lines bisecting that angle internally and externally are commensurable with 

 the sides. 



If the angle ABC be double of a ^ 



muarif angle, and if the angle at A 

 also be muarif, the two trigons 

 ABD, DBC, into which ABC is 

 divided by the line BD drawn to 

 bisect the angle ABC, have all their 



angles muarif, and so has CBE formed by drawing BE perpendicular to BD. 

 Hence the line AE is cut harmonically and rationally. 



37. If, of a rational trigon, two of the angles be doubles of two muarif angles, 

 the six lines bisecting internally and externally the three angles intersect each 

 other and the sides of the trigon produced, so 



that the segments are all rational, and the 

 areas commensurable with their squares. 



For, if each of the angles OAC, ACO, be f^ 



muarif, their sum FOA must also be muarif, 

 wherefore its complement FBO has all its trigo- 

 nometrical functions rational ; so that, all the 

 angles in the figure being muarif, all the seg- 

 ments of the lines must be commensurable, and all the areas commensurable with 

 the squares of the lines. 



Section 3. — On Co-ordinates. 



38. If any system of points have their co-ordinates from two rectangular 

 axes, all rational, their co-ordinates referred to another pair of rectangular axes, 

 making a muarif angle with the former, are also all rational. 



If a? and p be the co-ordinates of a point, when referred to one system of 

 axes, u and v its co-ordinates when referred to another system with the same 

 origin, and if be inclination of the one system to the other, we have — 



M=a7 COS — y sin 6 ; v=x sin 6 + y cos 6 ; 



wherefore, if x and ?/ be rational, and the angle 6 muarif, u and v must also be 

 rational. 



