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How Johann Jakob Ballmer found his formula in 1885 by analyzing and manipulating the ratios of these data:
r(1) = a(1)/a(1) = 1,
a(2)/a(1) = 1.349961..., rounded: r(2) = 135/100 = 27/20,
a(3)/a(1) = 1.511996..., rounded: r(3) = 1512/1000 = 189/125,
a(4)/a(1) = 1.599996..., rounded: r(4) = 16/10 = 8/5,
a(5)/a(1) = 1.6530647..., r(5) = 81/49 = 2-1/(3-1/(9-1/2)), derived from a(5)/a(1) = 2-1/(3-1/(9-3095/6216)) when replacing 3095/6216 by 1/2;
the multiplication of these fractions by 5/36 is the key trick to get more handy figures to see eventually increasing squares in the denominators by an appropriate expansion:
b(1) = r(1)*5/36 = 5 / 36,
b(2) = r(2)*5/36 = 3 / 16,
b(3) = r(3)*5/36 = 21 / 100,
b(4) = r(4)*5/36 = 2 / 9,
b(5) = r(5)*5/36 = 45 / 196;
... b(1) .|.... b(2) ..|.... b(3) ..|.... b(4) ..|.... b(5),
... 5/36 .|.... 3/16 ..|... 21/100 .|.... 2/9 ...|... 45/196,
... 5/36 .|... 12/64 ..|... 21/100 .|... 32/144 .|... 45/196,
(9-4)/9*4 |(16-4)/16*4 |(25-4)/25*4 |(36-4)/36*4 |(49-4)/49*4,
this last step was the crowning achievement: the discovery of the pattern (x-y)/x*y,
b(n) = ((n+2)^2 - 4)/(4*(n+2)^2) = 1/4 - 1/(n+2)^2;
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