Customer Reviews

Euclid: The Creation of Mathematics

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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.

He largely disregards smallscale battles amongst the superspecialists ("all modern translations of Elements are satisfactory"). He overturns the fashionable idea that the "Two Cultures" cannot communicate. (Rilke has things to say, perhaps not to Hilbert, but to the widely cultured mathematician, about Widerspruch!)

About the pre-Euclidean origins of mathematics, he overmodestly disclaims specialist knowledge. An example: his tracing of the earliest technical work on dodecahedrons and icosahedrons via pre-Euclideans such as Theaetetus (Plato's friend), and on up to the Group Theory work on isomorphisms of the 1990s A.D. is done 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) also contributes importantly. Its autobiographical flavor is reminiscent of that of Archimedes' in "Sand Reckoner". It allows him to stake out a clear and non-partisan position on the question "where is the algebra?" question, on which scholarly debates often produce more heat than light.

So multi-faceted a book, one could wish a fuller Index. But the cultural sweep is admirable throughout. 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 exaggerated, making him over into a Young Werther. But Artmann's charming and learned book really is hard to put down, even at vacationtime.

The material in Euclid's Elements may be divided into four categories of very different degrees of interest for modern readers. (a) Elementary material. To keep us interested when covering tedious proofs of obvious things Artmann discusses foundational issues (as seen by Euclid and contrasted with the modern view), the principles that guide the overall structure of the books, historical topics, etc. (b) Well-known material. This category includes some basic geometry (Pythagoras's theorem, etc.), but primarily it includes all of Euclid's number theory. This is very interesting stuff but less exotic than other parts of the Elements since these pearls have been kept polished and accessible (see, for example, the historically enlightened books by Stillwell, esp. "Elements of Number Theory" and "Numbers and Geometry"). (c) Incomprehensible material. Some parts of the Elements appear mysterious to the modern reader, especially some aspects of "geometric algebra" and of course the theory of incommensurability. A truly faithful guide to the Elements would make it its mission to clarify these things, but Artmann is not that committed, often preferring instead the easy way of looking for agreement with modern mathematical-aestetical principles and commenting on those things instead (e.g. discussions of the role of generalisations and the relation between problems and theories). (d) Constructions. This is the most rewarding part. First there is the remarkable construction of the regular pentagon in book IV. Euclid's construction draws on all previous books, in accordance with his aim to hide his masterplan and unveil it in a flash of brilliance just as we though he was getting lost in a mass of technicalities. Artmann adds helpful commentary on how the principles of construction may be understood through possible earlier constructions that used marked rulers and similar triangles (not developed by Euclid until book VI). The similar triangles proof uses a neat property of the pentagon: a side and a diagonal are in "extreme and mean ratio" (i.e. "golden ratio"), so constructing this ratio is one way to construct the pentagon. Euclid brings this up in connection with the marvellous constructions of the regular polyhedra in book XIII --- the culmination of the entire Elements. "For the construction of the dodecahedron, Euclid starts with a cube and constructs what can be called a 'roof of a house' over each of its faces". The pentagonal faces of the dodecahedron are made up of a quadrilateral piece of one roof and a triangular gable from the roof on the adjacent side. To make this work we must choose the right length for our beams, i.e. we must divide the side of the cube in extreme and mean ratio. The construction topics are not only the most rewarding in themselves but also the starting points of Artmann's most enthusiastic excursions, including the modern algebraic view of constructions as developed by Gauss, the group theoretical view of symmetries and polyhedra, appearances of these figures in art and architecture, etc.

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]