The author offers a pleasantly discursive presentation of the origins of the concept of orthogonality in mathematics. He begins by introducing the continued fraction which is written here as \[ \{ 1 / \{ 1 \;+ {\mathbf K}[ ( 2k-1 )^{2} / 2 + \| k \geq 1 ] \}\tag{\(*\)} \] and suggests that Brouncker, to whom this expansion is due, was one of the originators of orthogonal polynomial theory. He also suggests that Euler had some difficulty in proving that expansion (\(*\)) converges to \( \pi / 4 \). However, there is a simple recipe for constructing continued fractions whose convergents are expressible in closed form: the successive partial sums of the series \[ \{ 1 / v(1) \} - \{ 1 / v(2) \} + \{ 1 / v(3) \} - \ldots \] are the successive convergents of the continued fraction \[ 1 / \{ v(1) + {\mathbf K}[ v(k)^{2} / v(k+1) - v(k) + \| k \geq 1 ] \} \] The first expansion to be derived in this way results from setting \( v(k) = k \) and is \( 1 / \{ 1 + {\mathbf K}[ k^{2} / 1 + \| k \geq 1 ] \} \) converging to \( \log 2 \). The second, namely (\(*\)), is derived from Gregory’s series \( 1 - (1/3) + (1/5) - \ldots \) for \( \pi / 4 \) by setting \( v(k) = 2k - 1 \), and it is hard to believe that Euler had much trouble with it.
An analogue of Stirling’s formula is derived for an extension of (\(*\)).
For the entire collection see [Zbl 1110.00015].