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It seems clear that $\theta$ is can indeed be chosen to be a number independent of $x$ as stated, to get Stirling's formulas for the gamma function when $x$ is large. The wording, at least in English, is not too helpful in this section. But I'm less clear about where in the formula on page 24 there is supposed to be a mistake. Here as in any mathematics book (especially a translation) one has to be wary about misprints or errors. Probably there is no publicly available list of errata for this small monograph published originally in 1931 in German and later republished in 1964 in an English translation by Michael Butler. This English version is included in the 2007 AMS softcover book Exposition by Emil Artin: A Selection edited by Michael Rosen. (There is an older 1965 book *The Collected Papers of Emil Artin" published by Addison-Wesley and edited by Lang & Tate. This contains Artin's research papers, in the original German or English.) As Zavosh observes, the 1964 preface by Edwin Hewitt reprinted here does indicate one formula corrected in the translation: " ... a small error following formula (59) (this edition) was corrected..." However, the formula seems to be the one actually numbered (5.9). Caveat lector.

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It seems clear that $\theta$ is indeed a number independent of $x$ as stated. But I'm less clear about where in the formula on page 24 there is supposed to be a mistake. Here as in any mathematics book (especially a translation) one has to be wary about misprints or errors. Probably there is no publicly available list of errata for this small monograph published originally in 1931 in German and later republished in 1964 in an English translation by Michael Butler. This English version is included in the 2007 AMS softcover book Exposition by Emil Artin: A Selection edited by Michael Rosen. (There is an older 1965 book *The Collected Papers of Emil Artin" published by Addison-Wesley and edited by Lang & Tate. This contains Artin's research papers, in the original German or English.) As Zavosh observes, the 1964 preface by Edwin Hewitt reprinted here does indicate one formula corrected in the translation: " ... a small error following formula (59) (this edition) was corrected..." However, the formula seems to be the one actually numbered (5.9). Caveat lector.