5000 Years of Geometry

Ever since I began studying mathematics seriously as an undergraduate, I have strongly believed that knowing something about the history of the subject facilitates both understanding and appreciation of the substantive mathematics. This is especially true, I think, when it comes to the history of geometry. Of course, geometry occupied a particularly prominent role throughout much of the development of mathematics, and so the history of geometry, to some considerable extent, is the history of mathematics. The history of geometry is also replete with episodes that are not only important but also simply fascinating, including: the development of geometry as a deductive subject in ancient Greece; the revolution in mathematical thought that resulted in the acceptance and study of non-Euclidean geometries; the growth of analytic geometry and the resulting connections between geometry and algebra; and the relationship between the development of projective geometry and the artistic theory of perspective.

Given the importance and intrinsic interest of this subject, it is hardly surprising that, in addition to its discussion in mathematics history texts, there are several books discussing it in some detail, including Revolutions of Geometry by O’Leary, Geometry by its History by Ostermann and Wanner, Worlds out of Nothing by Jeremy Gray and Geometry: Our Cultural Heritage by Holme. And now we are fortunate to also have the book under review, which has actually been around for quite awhile, though not until now in English; this is a translation of the third edition of the text, the first edition of which was originally published in German in 2001. It is a very good book indeed, and an important and valuable addition to the literature on the subject.

The books mentioned in the preceding paragraph are primarily intended as textbooks for a course, complete (except for Holme’s book) with exercises for the reader (not many in the case of Worlds, but some). All four texts, particularly the first two mentioned above, are as much about substantive geometry as they are about history, and none are intended to be thorough histories of the subject; Worlds, for example, focuses on the development of geometry in the 19th century, and Geometry by its History has very little on the development of hyperbolic and elliptic geometry, a topic which surely ranks as one of the most important developments in the history of geometry.

The book under review, however, although having many of the features of a text (including exercises, many of them quite substantial) is also thorough enough to serve admirably as a reference. To denominate it as such, however, does not, by any stretch of the imagination, mean it is dry and uninvolving; on the contrary, the book is interesting and well-written, and filled with excellent figures (illustrations, maps, photos, etc.), quite a lot of which are in full color and visually striking. While this is not a book that most people would sit down and read from cover to cover, it amply rewards “drop-in” readers who want to learn about some particular aspect of geometric history.

And there are many such aspects to choose from here. The title is not misleading; the book really does cover thousands of years in the history of geometry, although the first half of this time period is covered rather quickly. After a 25-page chapter on the very early pre-history of the subject, discussing aspects of geometry in early Egypt and ancient Babylonia, the book moves forward about 2500 years in time and proceeds in chapter 2 to a fairly lengthy (100-page long) discussion of Greek mathematics. While most of this chapter is devoted to the mathematics of ancient Greece, ranging from Thales and Pythagoras to Apollonius and Archimedes, there is also a discussion of post-classical developments through the fall of the Roman empire and the resulting Byzantine empire.

Chapter 3, also about 100 pages in length, looks at developments in geometry in other cultures, both Asian (China, Japan, India and the Islamic countries) and the Americas (North and South). The authors then return to geometry in Europe, both during the Middle Ages (chapter 4) and the Renaissance (chapter 5). The latter chapter contains, among other things, a very interesting 20-page discussion of the role of geometry in Renaissance art.

One of the major such roles concerned the theory of perspective, which in turn provided an important impetus in the development of projective geometry. Mathematical advances in this area were made in the 17th century by people like Desargues (whose work, unfortunately, was not given the attention it deserved at the time, but which would be re-discovered in the 19th century). The work of Desargues is one of the topics covered in chapter 6 (“The Development of Geometry in the 17th and 18th Centuries”), along with other major geometric advances such as analytic geometry and a renewed interest (by people such as Saccheri, Lambert and others) in attempting to prove Euclid’s parallel postulate from the remaining axioms of Euclidean geometry, thereby “vindicating” Euclid. (For more details on this, see the informative review of Saccheri’s book Euclid Vindicated from Every Blemish.) Of course, all these attempts were in vain, and in the 19th century it was established that all such attempts had to be in vain, because models could be produced showing that hyperbolic geometry (a geometry we get when we replace the Euclidean parallel postulate by the contradictory assertion that through a point \(P\) not on line \(\ell\), there exist at least two lines parallel to \(\ell\)) was as consistent as Euclidean geometry.

These issues, and others, are, in turn, covered in chapter 7 (“New Paths of Geometry in the 19th Century”). The 19th century was a very eventful time for geometry, and, in addition to non-Euclidean geometry, this chapter discusses the various directions taken by the subject in this period, including: differential geometry, the use of vectors and the development of n-dimensional geometry, Klein’s “Erlangen Programme” and the theory of transformation groups, and the development of topology, both point-set and algebraic.

A final chapter (“Geometry in the 20th Century”) addresses modern issues, including foundational questions (there’s a good discussion of Hilbert’s Foundations of Geometry) as well as modern connections between geometry and science, art and computers. The discussion here goes a bit deeper than in a number of other books; for example, the authors describe as “generally known” the relationship between geometry and special and general relativity, and assert that one of the main goals of this section is to “make the reader look at completely different applications of geometry in physics, as well as chemistry, biology and geosciences.” However, recognizing that their assumption of general knowledge of the connections with relativity may be optimistic, the authors do survey this as well. (“Nevertheless, let us first look at what the reader expects.”)

The topic coverage throughout is excellent, and readers should learn a great deal from this book. For example, the discussion of the classical straightedge/compass problems of antiquity addresses not only the impossibility of trisecting an angle (which many students at least hear about as undergraduates, albeit in passing) but also the fact that the Greeks had developed tools other than compass and straightedge for this and other constructions (which, I think, many students do not know about). So, for example, there is mention of Archimedes’ trisection construction with a marked straightedge, Nicomedes’ construction using a conchoid, and Hippias’ quadratrix.

Because of the book’s chronological approach, a person interested in a particular topic may have to look through several chapters to get a full account of it. So, for example, somebody who is interested in the historical development of non-Euclidean geometry will find useful discussions in chapters 2 (talking about Euclid’s fifth postulate), 3 (early attempts by Islamic mathematicians to prove the fifth postulate), 6 (work of Saccheri and Lambert), 7 (Bolyai, Lobachevsky and the existence of models of hyperbolic geometry) and 8 (applications of non-Euclidean geometry to science and art). This is, of course, not a criticism; the subject of non-Euclidean geometry did not spring into existence all at once, and to the extent that the book conveys the slow process by which this revolution in mathematics took place, that is all to the good.

The translation from the German is occasionally a bit idiosyncratic (an exercise on page 559, for example, asks the student to “[l]ook for different positions of the Desarguesian figure in this model, so that the theorem is not fulfilled respectively”) but not such as to cause any real difficulties. The authors have, in fact, taken pains to make the exposition useful and reader-friendly. For example, in addition to the large number of attractive figures, each chapter is bookended with boxed summary pages: at the beginning of each chapter there is an historical timeline for the time period covered by that chapter, and at the end of almost every chapter there is a boxed chart entitled “Essential contents of geometry in …” with the appropriate time period filled in and the major developments of geometry in that time period summarized quickly. There are two very good indices (one devoted to names, the other subjects) and an extensive list of references. The authors’ attempt to relate geometry to culture should also have pedagogical benefits.

As I said before, this is an important and valuable book. If you like either geometry or the history of mathematics, take a look at it.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.