Customer Reviews

Geometry Civilized: History, Culture, and Technique

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All books are unique, as George Orwell might have said, but some are more unique than others. And Heilbron's "Geometry Civilized" may be the most unique of all. It is, on the one hand, a coffee table book, in size and presentation, with beautiful illustations. On the other hand, it is a serious geometry text with full proofs of many theorems in Euclidean geometry, and plenty of interesting exercises for the reader. But perhaps most of all, it is a fascinating ramble through a wide range of topics, written by a leading historian of science with a strong esthetic sense and equally strong views on math and science education. He is, in the words of W.S. Gilbert, "Teeming with a lot o' news", including "Many cheerful facts about the square of the hypotenuse" -- the title of his chapter on the Pythagorean Theorem. Another chapter, "From Polygons to Pi," includes the exact geometry of a Gothic arch and much of the accompanying ornamentation, as well as other topics ranging from Stonehenge to the Pentagon building, and from the idea behind burning mirrors attributed to Archimedes and actually constructed by Lavoisier and others, to the octagonal room designed by Thomas Jefferson. Anybody who enjoyed geometry in high school should love this book, and many people who feared or hated high school geometry may discover what they missed by not having a John Heilbron to show them the wonderful richness and flavor of what, presented badly, can appear a dry and useless subject.

The rise of geometry was simultaneous with the rise of civilization. When people are aggregated into permanent population centers, it becomes necessary to precisely measure areas so that the proper taxes can be collected and also measure in the third dimension so that magnificent buildings can be constructed. While both require knowledge of geometry, they are experiential in nature.
The true rise of Western civilization as we know it took place in Greece, and the most permanent feature of that culture was the development of abstract mathematics. From this point one, mathematics was an endeavor that involved objects whose true nature was only in the mind. Any geometric diagram could only at best be a crude approximation of the true situation. It is hard to underestimate how much of a change this was from earlier forms of reasoning. I for one, firmly believe that all of the other ideas of democratic government, ethics, logic and philosophy that arose at the same time and place were a consequence of the new, more abstract and theoretical thinking that was taking place in geometry.
This book is a combination of history and geometry, showing how intertwined the two are. It is also one of the more extensive collections of solved geometry problems that exists today. Heilbron poses many problems, solving nearly all of them immediately after they are posed. As you step through the solutions, it is with a sense of wonderment as the steps are so direct, sequential and easy to understand, which is the hallmark of good geometric proofs.
Packed with figures, this book is suitable as a text for courses in geometry that the students will love, and I will point it out to anyone who claims that geometry is dull. With supplements, it could also be used as a text for courses in the history of mathematics.

The rise of geometry was simultaneous with the rise of civilization. When people are aggregated into permanent population centers, it becomes necessary to precisely measure areas so that the proper taxes can be collected and also measure in the third dimension so that magnificent buildings can be constructed. While both require knowledge of geometry, they are experiential in nature.
The true rise of Western civilization as we know it took place in Greece, and the most permanent feature of that culture was the development of abstract mathematics. From this point one, mathematics was an endeavor that involved objects whose true nature was only in the mind. Any geometric diagram could only at best be a crude approximation of the true situation. It is hard to underestimate how much of a change this was from earlier forms of reasoning. I for one, firmly believe that all of the other ideas of democratic government, ethics, logic and philosophy that arose at the same time and place were a consequence of the new, more abstract and theoretical thinking that was taking place in geometry.
This book is a combination of history and geometry, showing how intertwined the two are. It is also one of the more extensive collections of solved geometry problems that exists today. Heilbron poses many problems, solving nearly all of them immediately after they are posed. As you step through the solutions, it is with a sense of wonderment as the steps are so direct, sequential and easy to understand, which is the hallmark of good geometric proofs.
Packed with figures, this book is suitable as a text for courses in geometry that the students will love, and I will point it out to anyone who claims that geometry is dull. With supplements, it could also be used as a text for courses in the history of mathematics.

I'll be brief since others have written a bit more. The book is beautifully illustrated and argued. Perhaps 25% of the book is about applications of geometry and 75%, especially the problems, is solving "famous" or intriguing geometry problems. It does have a significant of Chinese geometry, but the focus is Euclidian and its descendants, though it does have lot of intriguing methods used by medieval masons/architects. If you are interested in an analogous book, consider Squaring the Circle, but its focus is less on classic geometry and more on finding applications of geometry in art, so you get far more on the golden ratio, etc. I thought Geometry Civilized could add a lot of intriguing applications to a sophisticated geometry class.

Heilbron's greatest accomplishment in this work is the very thorough cutting and pasting that brings us many pretty pictures, especially from the worlds of art and architecture and old textbooks. Other than that there is little of value. The bulk of the book is the same old terse Euclidean geometry that you can find in just about any geometry book. You might as well read Euclid because Heilbron adds basically nothing in terms of insight and readability when it comes to the geometry itself. In fact, he repeatedly manages to create technical obstacles even in clear terrain; see for example what must surely be the most incomprehensible introduction ever of radian angle measure on page 278. Still, the book also discusses many diverse applications which perhaps makes it worthwhile? Unfortunately, no. First of all there are some horrendously formulated statements, such as the claim that pi "cannot be expressed as a number, even an irrational one" (p. 241) and the implicit claim that three points need not lie in a plane: "Assuming, what is more or less true, that Rhodes and Alexandria lie on the same noon circle or meridian (that is, that Rhodes, Alexandria, and the centre of the earth lie in the same plane), ..." (p. 66). One wonders how such things survived into the "corrected" paperback edition. More seriously, Heilbron frequently breaks the rule that in science and mathematics everything should be explained and nothing should be pulled out of a hat. He is forced to do so because he doesn't have very many interesting applications of Euclidean geometry to offer and so has to discuss applications that are thoroughly incompatible with the mathematics covered. This is completely unnecessary since Euclidean geometry has many wonderful applications, but Heilbron simply ignores them: remarkably, conic sections, for example, are never mentioned even though there is a section on burning mirrors (!?), where we are told in a parenthesis that "a slightly different surface, whose intersection with the plane of the paper makes a parabola, gives a more intense focus" than a spherical mirror (p. 282, this is the only occurrence of the word parabola in the book). Instead, for example, we learn that "a Dutch geometer named Willebrord Snel" simply "proposed" the law of refraction (p. 107), apparently on a whim, and there is no indication of why nature would choose to obey this curious law. Later this law is fundamental when we study the rainbow, yet another topic that our methods are completely incapable of handling. Lacking calculus, we resort to the use of a table of values and read off that "evidently" the properties of refraction in raindrops are such-and-such (p. 197). If this is a valid method then why did we bother toiling with proofs of the Pythagorean theorem, for example? We could have just thrown up a bunch of numerical calculations and said that "evidently" the theorem is true. Geometry is not "civilized" by betraying the very soul of rational inquiry.