On Eulerʹs attempt to compute logarithms by interpolation: A commentary to his letter of February 16, 1734 to Daniel Bernoulli

In the letter to Daniel Bernoulli, Euler reports on his attempt to compute the common logarithm log x by interpolation at the successive powers of 10. He notes that for x = 9 the procedure, though converging fast, yields an incorrect answer. The interpolation procedure is analyzed mathematically, and the discrepancy explained on the basis of modern function theory. It turns out that Eulerʹs procedure converges to a q-analogue S q ( x ) of the logarithm, where q = 1 10 . In the case of the logarithm log ω x to base ω > 1 (considered by Euler almost twenty years later), the limit of the analogous procedure (interpolating at the successive powers of ω ) is S q ( x ) with q = 1 / ω . It is shown that by taking ω > 1 sufficiently close to 1 and interpolating at sufficiently many points, the logarithm log x can indeed be approximated arbitrarily closely, although, if x, 1 < x < 10 , is relatively large, extremely high-precision arithmetic is required to overcome severe numerical cancellation. An alternative procedure for computing log x by interpolation at points in [ 1 , 10 ω ] , ω > 0 , accumulating at the lower end point, is shown to converge to the desired limit, but also not without numerical complications.