Simanaitis Says

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ONE OF the best examples of the elegance of mathematics, The Elements of Euclid, was composed more than 2300 years ago. The Greek mathematician Euclid lived in Alexandria around 300 B.C. when he put together 13 books on geometry, number theory and what is now called geometric algebra. Euclid didn’t discover all this mathematics on his own. Rather, he collected, codified and, in some cases, corrected earlier works.

And note this term “discover”—as opposed to “invent” or “concoct.” Mathematicians like to say something is discovered, with the charming conceit that it already existed, waiting to be found.

The everlasting claim to fame of The Elements is its being the oldest extant treatment of mathematics that’s axiomatic and deductive. That is, rather than being simply a hodge-podge of facts, techniques or formulas, The Elements set the standard for logical development of a topic from carefully crafted assumptions (axioms), through statements of fact (theorems) with deductive verifications of these statements (proofs).

This approach is crucial to the development of modern sciences. Said another way, a “science” that fails to be axiomatic and deductive isn’t much of a science.

The approach is elegant because it derives so much from so little. It’s said “Beauty is the quotient of Order/Complexity” (see If so, imagine the wealth of order derived from simple Euclidian assumptions such as “A point is that which has no parts” and “A line is length without breadth.”


The Billingsley edition of The Elements of Geometrie of the Most Auncient Philosopher Euclid of Megara, the first in the English language, 1570

The Elements is also considered the most successful and influential textbook ever written. It was one of the first mathematical treatises set in movable type—in 1482, less than 40 years after the Gutenberg Bible. What’s more, it’s said to be second only to the Bible in the number of editions published.

My own Elements, a special gift from wife Dottie, is a facsimile of the 1847 Oliver Byrne edition. As the complete (and very Victorian) title describes, this idiosyncratic rendering replaces a lot of the verbiage with color-coded symbols and diagrams. When it was published, the book was called “one of the oddest and most beautiful books of the century.” I think of it as a combination geometry text and Mondrian exhibit.


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, by Oliver Byrne, William Pickering, 1847; a facsimile with added commentary by Werner Oechslin, Taschen, 2010. It’s listed at both and The link: Oliver Byrne: Six Books of Euclid

Byrne’s Elements and its Taschen facsimile have not been without controversy. It’s debatable whether Byrne’s graphical proofs are easier or more difficult to follow than traditional ones.


The Pythagorean Theorem, proved in Byrne’s graphical manner.

I enjoy them because of their pictorial elegance, just as the originals have their logical elegance. Some bibliophiles have questioned Taschen choices taken in the facsimile. To my eye, these matters tend to be arcane and hardly detract from the book’s beauty.

There’s also an online version of Byrne’s Elements ( It’s fun to view what has become a collectible book worth thousands of dollars. On the other hand, turning the real pages of the facsimile is a real pleasure. ds

© Dennis Simanaitis,, 2013

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This entry was posted on January 8, 2013 by in Sci-Tech and tagged , , .
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