Who discovered the logarithmic integral?

Question

This question is about history of logarithmic integral, which is discussed as well here and is the antiderivative of the function $\frac{dx}{\log x}$. Which mathematician discovered that function?

Answer 1

I do not know whether this is the earliest reference where the function is defined, but certainly an early reference is Johann Georg von Soldner's 1809 treatise Théorie et tables d'une nouvelle fonction transcendante (English: Theory and tables of a new transcendental function) [archive.org], which is dedicated to the logarithmic integral function and its properties. (NB the almost-modern notation $\color{#00bf00}{\operatorname{li.} x}$ for the function occurs at the beginning of Chapter 2.)

Obviously, the title suggests the function wasn't considered long before the treatise was written. In a 1811 letter to Bessel (in German), Gauss referred to the function as "his [Soldner's] $\operatorname{li} x = \int \frac{dx}{\log x}$" (Gauss Werke [Gauss' Collected Works], Band VIII [pdf warning], pp. 90-92), which arguably supports that conclusion.

In Chapter 1, Soldner writes that Euler considered the Maclaurin series for the logarithmic integral, but his citations (Tom. I, art. 219 et 228) evidently use a numbering for Euler's papers that differs from the one used by the Euler Archive, so I don't know to which papers he refers and do not know what else Euler achieved along these lines. (Possibly they are among these, and I would be gratified to know to which he referred.)