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Euclid’s Elements Through the Ages

By Ernest Davis

Encounters with Euclid: How an Ancient Greek Geometry Text Shaped the World. By Benjamin Wardhaugh. Princeton University Press, Princeton, NJ, July 2021. 416 pages, $29.95.

Encounters with Euclid: How an Ancient Greek Geometry Text Shaped the World. By Benjamin Wardhaugh. Courtesy of Princeton University Press.
Euclid’s Elements—a mathematical treatise that comprises 13 separate books—is perhaps the most influential science or math text prior to Nicolaus Copernicus’ On the Revolutions of Heavenly Spheres. In the two millennia since its composition, Euclid’s tome has played a central role in countless elementary education curricula. Many students around the world—including numerous budding mathematicians and scientists—found it to be an enchanting introduction to the wonders of mathematics and rigorous proof; many others cursed it as horrible, meaningless drudgery. Euclid’s proofs are often seen as the paradigm for deductive argument. “As certain as a proposition in Euclid” is a byword for inarguable validity.

Yet despite his prominence, almost nothing is reliably known about Euclid’s life. He must have lived around 300 BCE. In addition to the Elements, three mostly intact books (Optics, Data, and Phaenomena) and four lost books (Conics, Pseudaria, Porisms, and Surface Loci) are attributed to him with varying degrees of confidence. Euclid is plausibly said to have been part of the court of Ptolemy I in Alexandria, but everything else about him is legend that dates from many centuries after his death.

Moreover, the original Greek text of the Elements is exceptionally hard to recover — much more so than many of the great ancient Greek works of philosophy, literature, and history. Since it is a textbook rather than a work of literature, teachers felt free to make additions and corrections (which were not always accurate) or omit material that they felt was superfluous. Furthermore, copyists who were mathematically uninformed and did not understand the diagrams introduced all kinds of errors into both the text and figures. More knowledgeable editors may have corrected these mistakes at a later date, but their corrections might have taken the text even further from the original.

As a particular example, Greek mathematician Theon of Alexandria produced an edition of the Elements in the fourth century CE. Benjamin Wardhaugh describes this version in Encounters with Euclid:

[Theon] cleaned up and smoothed out difficulties. He chose between variant versions of the texts, or combined the variants together ... He filled in gaps, real or imagined. While the Homeric editors saw themselves standing outside the epic tradition, mathematicians of later Alexandria tended to behave as though Euclid was a colleague, and they themselves members of a still-living tradition. Instead of textual purity and fidelity (“what did Euclid really write?”) they prized correctness, completeness, and usability.

In one specific instance, Theon notes that he inserted his own original proof. In other cases, however, it is essentially impossible to know which parts of the texts were Euclid’s and which were added or changed by Theon or additional editors, copyists, and printers.

Wardhaugh’s Encounters with Euclid is an account of how people have engaged with the Elements over millennia — what they thought of the work and how they studied, taught, edited, emulated, admired, or objected to it. Each chapter comprises roughly one episode in this long history.

Figure 1. A spread of pages from an edition of Euclid’s Elements that was printed in Venice in 1482, with extensive mathematical illustrations. Throughout the centuries, readers of the text added their own notes in the margins. This edition is on exhibit at the Folger Shakespeare Library in Washington, D.C., this summer. Figure courtesy of the Folger Shakespeare Library.
In my opinion, the most interesting and useful chapters are those that deal with people whose engagements with the Elements were particularly deep. For instance, the chapter on Theon of Alexandria is excellent, as is the chapter on Arethas of Patras—a 10th-century Byzantine bibliophile who commissioned and annotated what is now the oldest surviving complete copy of the Elements—and the chapter on Erhard Ratdolt, who produced a beautiful printed edition in 1482 (see Figure 1).

Christopher Clavius—a 16th-century Jesuit, scientist, Platonist philosopher, and one of the few contemporaries that Galileo admired—produced his own edition of the Elements. Under his influence, the text became part of the Jesuit educational curriculum. Wardhaugh writes:

Thus, the nearly quarter of a million children and adolescents in Jesuit care at any given moment would now be exposed to Euclid’s Elements in Clavius’ version; and as well as Clavius’ students at Rome and the long line of Jesuit mathematicians, the order would also go on to educate such non-Jesuits interested in mathematics as Descartes, Laplace, Diderot, and Voltaire. Clavius’ love for Euclid, and his distinctive vision of the Elements, would shape mathematical culture for two centuries.

Encounters with Euclid also includes excellent chapters on philosopher Baruch Spinoza and polymath Isaac Newton, both of whom modeled major texts—Ethics and Philosophiæ Naturalis Principia Mathematica, respectively—on Euclid. Another interesting chapter centers on the debate in 19th-century England about whether the Elements is a suitable geometry textbook. Mathematician Augustus De Morgan argued strongly in favor of Euclid, and author Charles Dodgson (better known by his pen name, Lewis Carroll) wrote a book called Euclid and His Modern Rivals that mocked competing geometry textbooks. However, they and other Euclid supporters lost the battle in the early 20th century.

Encounters with Euclid actually corrected a major misimpression of my own. I had thought that Euclid’s logical gaps—other than concerns about the parallel postulate—were not discovered until the drive toward rigor in the 19th century. On the contrary, medieval and early modern mathematicians were troubled by these issues and attempted to fix them. The complete axiomatization was finally realized in David Hilbert’s 1899 Grundlagen der Geometrie, which Wardhaugh astonishingly fails to note.

Regrettably, other chapters seem less successful. Quite a few of them are short biographies of otherwise interesting people whose interactions with Euclid were not especially noteworthy. Hrotsvitha of Gandersheim was an extraordinary medieval poet and dramatist in the 10th century, but her only connection to Euclid is a play that mentions perfect, deficient, and abundant numbers. George Eliot was a great novelist, but being reminded that the protagonists of The Mill on the Floss disliked their study of Euclid does not do much for my understanding of either the novelist or the novel. Max Ernst’s “Euclid” painting and Euclid’s remote relation to designer Elsa Schiaparelli’s couture likewise did not interest me. On the other hand, Wardhaugh’s short chapter on polymath Mary Fairfax Somerville led me to her fascinating memoirs [1], so I owe him a great debt just for that.

Incidentally, the flyleaf references Abraham Lincoln—who studied the Elements in his 40s and was proud to have mastered it—but the book neglects to mention him at all. In fact, Encounters with Euclid becomes increasingly focused on Great Britain in the 18th and 19th centuries. The U.S. is completely absent from the narrative until 1922, when Edna St. Vincent Millay wrote her poem “Euclid alone has looked on Beauty bare.”

Overall, Wardhaugh’s book is surprisingly disengaged from the actual mathematical content of the Elements; it does not quote or analyze any proofs and comments on very few propositions. Two of Euclid’s most famous results—his proof that there are infinitely many prime numbers, and the Euclidean algorithm for the greatest common divisor—are not mentioned at all. It would also have been interesting to read a direct comparison between the proof styles of Euclid and other classical mathematicians. Finally, the text contains 34 images of book pages and other artifacts, but the quality of the reproductions is not especially good.

Let me also frivolously suggest that the next edition include some lighthearted discussion of Euclid parodies. I can think of two candidates offhand: Lewis Carroll’s The Dynamics of a Particle, which focuses on party politics (“Let it be granted that a speaker may digress from any one point to any other point”); and Stephen Leacock’s Boarding-House Geometry (“Any two meals at a boarding-house are together less than two square meals”). I’m sure that other such examples exist.

Yet despite these minor shortcomings, Encounters with Euclid is enjoyable and informative. Wardhaugh’s prose is clear and elegant, and each chapter begins with a charming, evocative vignette that is written in the historical present with an impressionistic style. The book contains a lot of fascinating material with which I was not familiar, and the short sketches of the different intellectual worlds that Euclid touched—Alexandria in 300 BCE, Alexandria again in 400 CE, Baghdad in the 10th century, Beijing in the 17th century, and so forth—are vivid. If you are interested in the transmission of classic Greek mathematics throughout the centuries, it is certainly worth reading.

[1] Somerville, M. (2001). Queen of science: Personal reflections of Mary Somerville (D. McMillan, Ed.). Edinburgh, Scotland: Canongate Books.

Ernest Davis is a professor of computer science at New York University’s Courant Institute of Mathematical Sciences.

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