This is a Renaissance book by a Renaissance man. Artmann gives a full summary of the "Elements", using considerable modern notation. It is accurate and detailed, and the various themes he traces (such as Symmetry, or Incommensurables) let him include a wide range of topics: architecture, design, sculpture, myth, history -- even philology and poetry. Some may think he limits himself too narrowly to the classical Greeks, does too little digging in the Babylonian or Egyptian parts of the story.
To Artmann's credit, his book disregards the smallscale disputes amongst superspecialists ("all modern translations of Elements are satisfactory"). He overturns the fashionable idea that the "Two Cultures" cannot communicate. So, Rilke has something to say -- perhaps not to Hilbert, but to the widely cultured mathematician, or to the general reader -- about Contradiction, or Widerspruch.
About the pre-Euclidean origins of mathematics in Greece, he overmodestly disclaims specialist knowledge. An example: he traces the earliest technical work on the dodecahedron and the icosahedron via pre-Euclideans such as Theaetetus (Plato's friend), and up to the highly abstract Group Theory work on isomorphisms of the 1990s A.D. -- and does this well and surefootedly. Too bad his modesty barred him ("I leave that to the specialists") from analyzing the pre-history of Euclid's Book XII, the classical ancestor of our integral calculus. The fact is that he knows a great deal about Eudoxus (another friend of Plato's). Perhaps more detail in a Second Edition?
His work on the so-called Euclidean Algorithm (finding a greatest common factor) is another valuable contribution. Its autobiographical flavor is reminiscent of Archimedes in "Sand Reckoner". It allows him to stake out a clear and non-partisan position on the "where is the algebra?" question, on which scholarly debates often produce more heat than light.
So multi-faceted a book, one could wish an Index fuller than a mere 2 pages. Typos are too frequent for a good house like Springer, including two I found in names of authors or book titles. But the book's cultural sweep is admirable throughout, its bibliography good.
TL Heath's 1933 report about the Cambridge undergraduate, so struck by Euclid ("a book to be read in bed or on a holiday") may have been exaggerated, making him over into a Young Werther. But Artmann's charming and learned book really is hard to put down, on or off holiday.
[note: this is a lightly revised version of a review I submitted a few days ago. -Malcolm Brown]