Lemma 18 ([Decimal enclosures]). For b=√3/2 , these rationals satisfy p_*<π<p_*+ε, b_*<b<b_*+ε.

Put A(x)=∑_ j=0 ^ 65 (-1)^jx^ 2j+1 2j+1 . For 0<x<1 , integrate the finite geometric expansion of 1/(1+x^2) . Its remainder has the sign and magnitude giving A(x)< x<A(x)+x^ 133 /133 . The tangent addition formula gives (4(1/5))=120/119 ; subtracting an angle with tangent 1/239 gives tangent one, and the resulting angle is in (0,π/2) . Therefore π/4=4(1/5)-(1/239) . The two required rational comparisons are p_*, <16A(1/5)-4(A(1/239)+239^ -133 /133),; 16(A(1/5)+5^ -133 /133)-4A(1/239), <p_*+ε. They follow by clearing positive denominators. Likewise, the second bracket is exactly the pair of integer-square comparisons b_*^2<3/4<(b_*+ε)^2 .