Euclid's Fourteenth Book

When a famous mathematician has something new to say, the whole world pays attention.

Euclid's Elements, which presented the state of the art in geometry around 300 B.C., has been extraordinarily influential. This massive, 13-volume compendium set the standard for mathematical exposition and precise discourse for many centuries. More than 2,000 editions have been published, and new, interactive versions now appear on the World Wide Web.

What's more, there is compelling evidence that Euclid of Alexandria (c. 365–275 B.C.) had a fourteenth book in mind. Officials of the Avril Foundation for Old Occidental Languages have announced that, after a year devoted to authentication and analysis, they are prepared to release the text of a manuscript that appears to be a Latin translation of research notes jotted down by Euclid in preparation for writing a fourteenth volume of Elements.

Ivars Peterson goes on:

It's clear that Euclid was uncomfortable with the fifth and most complicated of the five postulates that begin Elements. This postulate states, "If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles."

Euclid's newly discovered notes propose an alternative way of expressing this notion: "Through any given point can be drawn exactly one straight line parallel to a given straight line." Euclid went on to consider two other cases: In one case, no parallel line can be drawn through the point, and in the other case, more than one parallel line can be drawn. In these two situations, he says, the sum of the interior angles of a triangle is no longer exactly equal to two right angles.

On the surface of a sphere, for example, the sum of the angles of any triangle is greater than two right angles, Euclid notes. Such a geometry has no parallel lines, yet it obeys his first four postulates.

The manuscript hints that Euclid also explored the curious geometry that arises from the existence of multiple parallels. "I have discovered things so wonderful that I was astounded," he wrote at one point. "Out of nothing I have created a strange, new world." Unfortunately, much of this section of the manuscript has deteriorated beyond repair, so only a few tantalizing fragments remain.

Read the rest and then check the date.