Joël Merker’s main goal in this “modernized translation” is to give English-speaking mathematicians access to Sophus Lie’s famous *Theorie der Transformationsgruppen*, written with the collaboration of Friedrich Engel. It is clearly a labor of love: Merker tells us that he learned German specifically to be able to read Lie, and that he came to the conclusion that “*Lie’s mathematical thought is universal and transhistorical*” (italics in the original). The result is very useful, though it does have some quirks.

In the preface, Merker says that

For several reasons, it was essentially impossible to directly translate the first few chapters in which Lie’s intention was to set up the beginnings of the theory in the highest possible generality, especially in order to eliminate the axiom of inverse, an aspect never dealt with in modern treatises. As a result, I decided in the first four chapters to reorganize the material and reprove the relevant statements, nevertheless retaining all the mathematical content.

Of course, it is clearly *possible* to translate the first few chapters! What Merker means, I guess, is that it is impossible to make sense of those chapters as they stand, and so he has provided a modern account with proofs that are acceptable to us. As a result, these chapters are a mishmash of Merker and Lie-Engel. Some sections are shaded in gray, but even there it is not clear whether we are reading Lie himself or Merker’s modernization. (I may have missed it, but it seems to me that the meaning of the gray shading is never specified.) I would have preferred to see a straight translation next to modern commentary, perhaps on facing pages. Historians interested in what Lie actually said will have to compare the text with the original.

Luckily,

But starting with Chap. 5, Engel and Lie’s exposition is so smooth, so rigorous, so understandable, so systematic, so astonishingly well organized — *so beautiful for thought* — that a pure translation is essential.

Well, almost. There are footnotes, which may or may not be Merker’s. Most chapters end with a horizontal rule, but in some chapters that rule is followed by text which seems to be commentary. And the gray shading reappears at times, mysteriously. There are also some moments when a footnote marker appears, but there is no footnote; instead, a translator’s note appears in parentheses. Like this, from page 261:

In addition, since the mentioned equations are certainly compatible with each other,^{1} (Translator’s note: — and since, furthermore, the lemma on p. 78 insures that, with a suitable choice of generic fixed points \(x_1’,\ldots,x_n’,x_1’’,\ldots, x_n’’,\cdots, x_1^{(q)},\ldots x_n^{(q)}\), the rank of the considered matrix of \(\xi\)’s is maximal equal to \(q\) — ) we obtain…

To summarize: Merker has provided us with a valuable translation of a crucially important text, but Springer could have done a better job of production, and both Merker and Springer should have been more careful to distinguish the commentary from the text.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College and the editor of MAA Reviews.