Mascheroni, of Euler-Mascheroni fame

The Euler-Mascheroni constant is arguably the sixth most fundamental complex number (1, 0, i, pi, and e being hard to beat), appearing sometimes unexpectedly in many contexts. It seems to originate in Euler's treatise Institutiones calculi integralis. In his 1790 commentary on Euler's book, Mascheroni attempted to compute this number to 32 decimal places. He succeeded to 19 places, but gave incorrect values for the 20th to 22nd and 31st to 32nd places. Remarkably, he somehow gave the correct values for the eight digits in between. This suggests either a mistake copying his answer or a hidden pattern in whatever method he used to approximate Euler's constant. We will explore this and related curiosities from numerical analysis.