Geometry Civilized: History, Culture, and Technique

Rating:
Summary: A truly unique book
Review: 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.

Rating:
Summary: A truly unique book
Review: 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.

Rating:
Summary: Geometry and history intertwined
Review: 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.