422 Dr. Rankine on Rational Approximations to the Circle. 



series of approximations of any degree of closeness to the cir- 

 cumference of a circle may be obtained, consisting wholly of 

 rational quantities, computed by the operations of common arit - 

 metic only. 



3. This process is not to be recommended for practical use, 

 because of the extreme length of the calculations required by it ; 

 but it appears to me to be worthy of some attention on account 

 of its great simplicity in principle. 



4. The method of this paper is so obvious that I was at first 

 unwilling to believe that it could be new ; and I now publish it 

 only because I have not been able to find or to hear of any pre- 

 vious publication of it. 



5. The following lemmata, or preliminary propositions, are 

 already well known. " ■ 



Lemma I. Problem. — To construct an angle whose sine 

 and cosine (and therefore its tangent also) shall be rational frac- 

 tions, and whose sine and tangent shall not differ by more than 



given fraction ( — \ of the tangent. 



It is well known that if a and b be any two whole numbers, 

 a 2 + b' 2 ) cft — b 2 , and 2ab represent the sides of a rational right- 

 angled triangle ; and therefore either of the acute angles of that 

 triangle fulfils the conditions of the problem. Thus, let 6 be 

 the angle opposite the side 2ab ; then 



a 2ab a a*-b* . a 2ab 



sm0= 5 -, cos 0= 3 g ; UnU=~^ — m < . (1) 

 a*-\-b^ a lJ rb z ar—b 2 v ' 



To fulfil the condition that tan 0— sin 6< , we must have 



— m 



M* 1 



a z -\-b' i — m 

 and consequently 



a*>(2»i-l)5 2 (2) 



Having, then, in the first place assumed any arbitrary value 

 for b } take for a? any integer square which is not less than 

 (2m — 1)6 2 , and find the required sine, cosine, and tangent by 

 the equations (1). Q. E. F. 



6. Lemma II. Theorem. — If the sines and cosines (and 

 therefore the tangents) of two or more angles are rational frac- 

 tions, so also are the sine, cosine, and tangent of any angle 

 formed by adding or subtracting those angles ; because these 

 quantities are computed from the sines and cosines of the given 

 angles by addition, subtraction, multiplication, and division. 

 For example, in the case of two angles 6 and </>, 



