Henry Billingsley’s First English Translation of Euclid’s Elements

For a class project, this week I had the opportunity to view an original copy of Henry Billingsley’s first English translation (1570) of Euclid’s Elements (~300 BCE). This edition was printed very finely, and it even included foldable pop-up geometric diagrams! I wish all geometry textbooks could be up to this standard.

Billingsley Elements

Foldable pop-ups in the eleventh book.

Euclid’s Elements has a pretty interesting history of transmission. The original Greek manuscript of Elements was lost to Western Europe in the Middle Ages. In 8th century, Elements was translated into Arabic and became known to Byzantine scholars. From the Arabic version, English monk Adelard of Bath produced the first Latin translation in 12th century. The Latin translation of Elements was first set in type in Venice in 1482 under the title Elementa Geometriae. Later in 1533, a Greek edition by Theon of Alexandria was fortunately recovered, and Billingsley’s first English edition was translated from the Greek edition in 1570. It’s really fascinating to think that we are still able to read something from more than two thousand years ago.

Billingsley Elements (1)

The handsome woodcut title page shows Billingsley’s ideal of the beauty of mathematics.

Billingsley Elements (2)

An example of complicated geometric figures in the book.

Patching a Hole in Escher’s Artwork with Conformal Geometry

Last December, I happened to be in Milan just in time for a major Escher exhibition at Palazzo Reale. The Metamorphosis series was my all-time favorite, and I really liked his non-mathematical early works of Italian scenery, but I’d like to share something else in this post.

In “Print Gallery” (1956), Escher left a hole in the center of the lithograph because he did not know how to complete the picture consistently. This is a beautifully done paradox of looking at pictures in a print gallery while being in a picture.


In 2002, number theorist Hendrik Lenstra from Universiteit Leiden found a way to describe the geometry in “Print Gallery” by a complex exponential function. Using conformal mappings, the Leiden group generated a new rendition of “Print Gallery” with the hole filled in.

You can play around with the computer program and zoom in at the project website: