It’s a bit late to be reviewing Euclid’s *Elements, *something over two millennia late. Still, I cannot resist calling attention to the publication of one of the most spectacularly beautiful books to appear in the last decade. Its full title is: *The First Six Books of the Elements of Euclid in which Coloured Diagrams and Symbols Are Used instead of Letters for the Greater Ease of Learners.*

It was in 1847 that Oliver Byrne, identified on the title page as “Surveyor of Her Majesty’s Settlements in the Falkland Islands and Author of Numerous Mathematical Works,” published Euclid’s first six books in a quarto edition, with woodblock illustrations of the theorems in brilliant solid regions of red, blue, yellow and black. It was published by a well-known London firm, William Pickering, and printed by an equally well-known printer, C. Whittingham at the Chiswick Press. A high point of Victorian printing, it must have cost a fortune to produce.

A triumph of book production of its time, the book was to become a tragic confirmation of the poor quality of paper used in the mid-19^{th} century, resulting in foxing, a brownish discoloration common to books of the period. Most copies of the 1847 edition suffer from this defect to varying degrees.

There are no such problems with this new facsimile just published by Taschen in a sumptuous edition where the original pages, before being photographed, were “washed” to remove evidence of foxing. The new paper is slightly off-white, which approximates the paper one might expect in a high-quality art book.

The facsimile is bound in elegant black cloth (unlike the original that was in fragile and thin cloth, either red or blue, that showed wear early on so that many extant copies have had to be rebound). The new edition is housed in a similarly designed black clamshell box which also contains a 96-page paperbound booklet by Werner Oechslin, in three languages (English, French, and German), telling the story of editions of Euclid and about the Byrne edition in particular.

The first copy of the original I ever saw was in a small bookshop in Cecil Court, just off Charing Cross Road, in London in 1960. I wandered into this particularly cluttered and dusty shop (Seligmann), where the proprietor looked more antiquarian than the books. I inquired, not hoping for much, whether he had any early mathematics. He demurred, but after a minute or so he said I should wait; he thought he might have something in the basement. So after some noisy rummaging about down there he emerged from what looked like a hole in the floor, holding a somewhat scuffed and worn copy of the Byrne *Euclid*. I quickly bought it for a little over $12, four pounds, ten shillings to be exact.

These days one occasionally finds a copy at an international book fair; recent copies are priced at $20,000 to $25,000. It has become a highly desirable and valuable book in its original form. With a listed price of $60 (and with discounts sometimes available) the new Taschen facsimile now makes it possible for just about anyone to own one.

It was Byrne’s original idea to create something that would make learning geometry easier. So he recast in brilliant colors not only the illustrations of the geometrical figures he was about to prove, but the details of the proofs as well. They consist of lines that read like “If [blue line segment] = [red line segment]” or “If [yellow sector of a circle] = [black sector of a circle],” and so on. It’s not clear to me that his system works all that well pedagogically, but it was a clever idea. Given pages with so much color, however, and the deft layout, with ornamental engravings signaling the beginning of every theorem, it is an aesthetic experience well worth the price.

A familiar illustration for the proof of the Pythagorean theorem adorns the title page of the original and the current publishers have promoted it to the front cover of the volume itself, again in brilliant primary colors, this time dramatically contrasting with the black background.

Euclid has always inspired printers to produce great editions of the *Elements*, beginning with the classic first printed edition by Erhard Ratdolt in 1482. It is strikingly beautiful with left or right margins reserved for illustrative geometric figures, and lavishly engraved borders and initial letters of paragraphs. It is interesting to note that the initial letters in the Byrne volume resemble very closely those in the edition of 1482.

All versions prior to Ratdolt appeared in manuscript form, of course. Following Ratdolt there was one additional incunable edition, in 1491, followed by 60 additional versions prior to 1600. One of the most interesting is the first edition in English, printed in 1570 by John Day in London, a particularly elaborate edition with fold-up “volvelles” so that three-dimensional figures could be pulled up off a page to illustrate theorems in the text. It also has an especially elaborate engraved title page that shows not only characters from the Bible but also from Greek mythology (Mercury), scientists and scholars (Ptolemy, Aratus, Hipparchus, Marinus, Strabo, and Polibius), along with allegorical figures representing the quadrivium of the seven liberal arts (Geometria, Arithmetica, Musica, and Astronomia).

A German edition of 1562, by Jacob Kundig in Basel, used red generously on the title page, but most of these early editions used color, if at all, only for text, where they often took the form of rubricated initials. Almost a century after Byrne, however, in 1944, the American book and font designer Bruce Rogers produced a very beautiful edition of the first six books of Euclid for Random House, which had a most interesting introduction by the French poet Paul Valéry and displayed beautiful geometric illustrations in exquisite muted colors. All in very good taste, but by comparison to Byrne it lacks exuberance.

Who was Oliver Byrne? The publisher of the new edition says that little is known of his life, but that he was an author and engineer. I have a mid-19^{th} century lithograph showing him, probably in his late 30s, seated at a table, with a quill pen and looking heavenward for inspiration, perhaps more in the pose of a poet than an engineer. But this is no picture of a poet with Byrne’s name added at the bottom: the manuscript he’s working on has a geometrical illustration. And at the base of the picture he is identified as “Late Professor of Mathematics, College for Civil Engineering [at Putney].” Beyond that it tells us that he is “Author of ‘The New and Improved System of Logarithms,’ ‘The Doctrine of Proportion,’ ‘The Practical, Complete and Correct Gager,’ ‘The Elements of Euclid by Colours,’ ‘A Practical Treatise on Spherical Trigonometry,’ ‘How to Measure the Length of a Degree on the Earth’s Surface by the assistance of Railroads,’ &c., &c, &c.” And, as if that were not enough, we are told that he is the “Inventor of ‘The Patent Calculating Instruments,’ ‘The System of Facilitating the Acquirement of Geometry, & of other Linear Arts and Sciences, by Colours, &c.,’ ‘Proposer of the New Theory of the Earth, which accounts for many Astronomical, Geographical, and Geological Phenomena, hitherto unaccounted for.’ ”

He obviously kept busy; with only his surveying duties in the Falkland Islands, he probably had time on his hands. What is tantalizing here is the second entry in the list of inventions — his system for “Facilitating the Acquirement” of these subjects “by Colours.” Does something of this “invention” survive? Some answers appear in Oechslin’s tri-lingual and richly illustrated essay that accompanies the facsimile.

Oechslin is a professor at the Swiss Federal Institute of Technology in Zürich, specializing in architectural theory. In his essay he addresses the question, “Who was Euclid?,” the answer to which is not as obvious as it might seem. First, he is not Euclid of Megara, but the Euclid of Alexandria (who came along a century later). There was confusion over this that lasted for many years, with early editions of the *Elements* attributed to “Euclides Megarensis.” Some have claimed that Euclid as a person never existed; the work is that of a group of scholars.

Oechslin looks at Byrne’s book as it relates to philosophy, the history of 19^{th} century mathematics, the teaching of geometry, and the relationship between the illustrations in this book and the work of artists of the 1920s: Piet Mondrian, Le Corbusier, and Theo van Doesburg. There are also plates to illustrate the similarity of Byrne’s figures to the colored illustrations in books on chemistry in the 19^{th} century and, much later, the work of Gyorgy Kepes’ *Language of Vision*. I suspect that Byrne himself would be surprised to read of some of these connections since his concern was to make the teaching of geometry easier. In his time, however, he was pretty much ignored by his contemporaries in mathematics. Augustus de Morgan was particularly dismissive, probably encouraging Florian Cajori to call Byrne “a curiosity.” There appears to be no mention of Byrne in Cajori’s history of mathematics teaching in that period in America or in his classic, *History of Mathematics*, and only brief mention in his book on mathematical notation.

Strangely enough, Oechslin provides no illustration from the 1482 *editio princeps*, but instead includes a page from the first Euclid in Greek, dated 1533, and a full-page portrait of the wrong Euclid, claimed to have been derived from pictures of Euclid of Megara on old coins! I assume that the later Euclid of Alexandria did not make it onto coins.

Not surprising is the fact that Oechslin does refer to books by Byrne that did not accompany his portrait; they were published late in life. One that I have, the *Dual* *Arithmetic*, of 1863, is among them, along with his *The Young Geometrician* of 1865 and his *Geometry of Compasses* of 1877, the latter two showing pages with color. Apparently the cost of the *Euclid* did not deter later publishers from putting out money for color printing.

It is disappointing that Oechslin, someone interested in art and architecture, did not comment on the superb quality of the printing of the Euclid by the Chiswick Press. The great problem in color printing is registration and, though there are the occasional “misses,” the hits far outnumber them, making the experience of seeing these pages all the more enjoyable.

Taschen deserves much credit for bringing out this edition, in cooperation with John Windle, a prominent San Francisco bookseller, who has long been interested in this work of Byrne. It’s a book for mathematicians or anyone interested in geometry, pedagogy, the history of printing, art, or the ancient world. I’m using it for Christmas presents this year.

**Reference:**

Charles Thomas-Stanford. *Early Editions of **Euclid**’s* Elements. Alan Wofsy Fine Arts, San Francisco, 1977.

Gerald L. Alexanderson is the Michael and Elizabeth Valeriote Professor in Science at Santa Clara University. He has served the MAA as President, Secretary, and editor of *Mathematics Magazine.*