Dr. Rankine on Rational Approximations to the Circle. 423 



sin (6 + <j>) = cos cf> sin 6+ cos # sin <p ; 

 cos (# + <£) = cos (£ cos # + sin <£ sin ; 

 tan 6 + tan $ 



tan (0±<£)= ,_ f -. , V 



v r/ 1 + tan <£ tan 



This theorem includes, as a particular case, the proposition that 

 if the sine and cosine of an angle are rational fractions, so also 

 are the sines, cosines, and tangents of all multiples of that 

 angle. 



The known formulae for the sine and cosine of a multiple 

 angle are here given for convenience of reference. 



Let cos 6=c, sin 6=s; then 



= c n —n 



7. Proposition I. Theorem. — If all the sides of a polygon 

 inscribed in a circle, save one, are known to be equal to the sides 

 of rational right-angled triangles of which the diameter of the circle 

 is the hypothenuse, then the remaining side of that polygon also is 

 equal to the side of a triangle of the same sort ; and consequently 

 the perimeter of the polygon is commensurable with the diameter of 

 the circle. 



For the ratios of the given sides of the polygon to the dia- 

 meter are the sines of a set of angles whose sines and cosines 

 are rational fractions ; and the ratio of the remaining side to the 

 diameter is the sine of the sum of those angles, which, by 

 Lemma II., is a rational fraction also, as well as the cosine of 

 that sum ; whence the ratio of the perimeter of the polygon to the 

 diameter of the circle is the sum of a set of rational fractions. 

 Q. E. D. 



8. Proposition II. Theorem. — If tangents be drawn to the 

 circle at the angles of the before-mentioned inscribed polygon, so as 

 to make the corresponding circumscribed polygon, every side of that 

 polygon {and therefore its perimeter) will be commensurable with 

 the diameter of the circle. 



For each of the parts (which may be called half-sides) into 

 which each side of the circumscribed polygon is divided by its 

 point of contact with the circle bears a ratio to the diameter ex- 

 pressed by a rational fraction (being half the tangent of one of 

 the before-mentioned angles) ; and the ratio 3 f the perimeter of 

 the polygon to the diameter of the circle, being the sum of those 

 fractions, is a rational fraction also. Q. E. D. 



