Euclid’s Window by Leonard Mlodinow, PhD (2001, Simon & Schuster)

**My review (out of 5 stars):**

I stumbled across *Euclid’s Window* in a used bookstore a couple of weeks ago, on a particularly auspicious browsing mission in which I also found an old Martin Gardner book and a vintage Isaac Asimov science book that I hadn’t yet read. I vaguely remembered maybe wanting to read *Euclid’s Window* once upon a time. I have a colleague in the math department that I play basketball with regularly, and I’ll usually ask him some off the wall question in the locker room. Recently I asked him to list his top 5 mathematicians of all time. Euclid may well have been on his list, I don’t recall, but Carl Gauss was for sure. Seeing that Moldinow’s book dealt with both, I figured it was worth the flier.

Indeed, it turned out to be a great read, and really went in an unexpected direction. Since I had picked up the book on a lark, I didn’t pay much attention to what it was going to be about, other than, obviously, geometry. I thought I had picked up a book that was carrying Euclid’s ideas into modern times, but that’s not precisely what it was. Rather, it was really more about the modern mathematics behind string theory (from physics) and what ideas led up to string theory.

And that really warms my heart, because that made Mlodinow’s book like any number science books by Isaac Asimov. In explaining any modern idea – like the idea that noble gases like xenon and krypton actually can form compounds, or the notion that stars can be so massive they can collapse to a singularity – Asimov almost always starts at the beginning, which usually means in Ancient Greece (see his books *The Noble Gases* and *The Collapsing Universe*, respectively).

This is an excellent explanatory style, because it starts the reader out with things that can be understood. The Greeks worked out their theories of matter and mathematics without even the benefit of Arabic numbers, place value, and zero, to say nothing of supercomputers. Many of the experiments that took place prior to 1900 used equipment almost anyone is familiar with or can easily visualize (though of course, requiring all the more clever ingenuity in experimental design as a consequence). Following the developments in any field chronologically, then, leads the reader slowly into the more abstruse material. Another advantage is that, even if the modern ideas turn out to be incorrect (string theory, for example, has a lot of work to do to achieve widespread acceptance and validation), 80% of the text will be devoted to ideas that have stood the test of time. (And so the book is worthwhile to get in a used bookstore – even if developments in string theory since 2001 have rendered some of the later chapter material inaccurate.)

Mlodinow divides his narrative up into 5 major sections, each devoted to a giant of mathematics. The first part is given over to Euclid, whom Mlodinow praises for systematizing all of Greek mathematics, but, more importantly, for initiating the mathematical methods of proof – rigorously defining givens (axioms), and methodically defining the logical steps in combining axioms to achieve new proofs. The second part he gives to Rene Descartes, the eponymous popularizer of our modern Cartesian coordinate system, and the merging of algebra and geometry that his coordinate system permitted. Part three gives us Gauss and the ideas of non-Euclidean geometry. Armed with algebra and calculus, Gauss could show that, for example, the three angles of a triangle need not add up to 180 degrees if the space in which the triangle occupies is not flat but curved. Part 4 brings us Albert Einstein, who was not a mathematician but who borrowed the ideas of Gauss and Georg Riemann in a revolutionary way that showed that non-Euclidean geometry was not a mathematical fiction but a genuine description of our universe. Part 5’s mathematician/physicist is Edward Witten, who placed string theory (specifically, M-theory) on firmer mathematical footing by describing the geometry of the branes (a more precise description of the strings of string theory).

Although there is a central character in each of the 5 portions of Mlodinow’s book, he leaves no glaring gaps in the narrative. Each part has its own back story, so that, along the way we learn about Thales, Pythagoras, Newton, Poincare, Heisenberg, and any number of other figures central to the development of mathematics and its influence on physics.

Mlodinow does two things particularly well in this book. One, he manages inject awe into the history of mathematics (a topic that probably sounds dry to most potential readers). For example, it is hard not to curse the short-sighted stupidity of townspeople and invaders who brought down the Great Library at Alexandria, and to wonder too how much farther mankind might be without the interregnum of the Dark Ages in Europe. As another example, it is hard not to marvel at the synchronicity of ideas percolating at just the right time on the eve of Einstein’s magnificent year (1905) and the 20-year period that gave birth to quantum mechanics. Mlodinow notes (on p. 142 of the paperback edition) that it was Gauss who forced Riemann to research the geometry of curved space when the latter was (at the time) more qualified to investigate other topics:

Riemann’s work on differential geometry became the cornerstone of Einstein’s general theory of relativity. Had Riemann not been so imprudent as to include geometry on his topic list, or Gauss not so bold as to choose it, the mathematical apparatus Einstein needed for his revolution in physics would not have existed.

Mlodinow highlights similar momentous conjunctions in history, such as young Pythagoras meeting and befriending an aging Thales, who would advise him to travel to Alexandria, Napoleon hearing of Gauss’s greatness, and sparing his city from destruction, and Einstein being on a well-timed trip abroad when Hitler’s men confiscated his possessions.

The second thing that Mlodinow does well, and this is critical in such a book, is that he has a gift for explaining difficult concepts while resorting to very few visuals and even fewer equations. This was present throughout the book but it really hit home for me in the Einstein chapter. I have read any number of popularizations about Einstein’s work, including from the great Asimov himself, but Mlodinow was the first to really get through to me why there is no such thing as simultaneity in the universe. Mlodinow’s examples – particularly his examples of he and his wife photographing his two children on a moving train (attempting to do so at the same *time*), and also his example of Einstein’s son riding a merry go round and letting go of a ball – made clear to me for the first time the physical necessity of different clocks and what exactly was an “inertial frame of reference”. Moldinow was equally clear in describing non-Euclidean geometry and the Poincare lines of hyperbolic space. As to string theory, Mlodinow at least made the concepts less objectionable on the face of it.

The level of the book is perfectly pitched to what I imagine is the intended audience – the educated layperson not intimidated by math, but also one who is unwilling to work especially hard while reading the book. The book is not intended for mathematicians, other than to perhaps give them a sense for the historical progression of ideas in their own field. The non-mathematician can get a lot out of the book without working very hard, and can get even more out of the book with some focused attention on the particularly difficult concepts. If you’ve ever been intrigued by general relativity, or curved space, string theory, or what the Greeks could do without numerals, this is a book worth reading.

Four and a half stars out of five.