Euclid21: Euclid's Elements for the 21st Century - What We Have Wrought

When this project began the authors (hereafter referred to as “we”) were playing. That is, it seemed like a fun project and we needed no other justification. However, as the project drew to a close, the question “What have we created?” pressed itself more and more heavily on us. Was this “thing” we'd created just a toy or was it something more?

It would be easiest at this point if the reader would download our code from: Euclid21 executable code only

This is a compressed file. Extract it and double-click the file “euclid21.jar” inside the folder “Euclid21-executable.” This will start the code and it should be clear how to interact with it from there. Play with it for a little while, and then come back.

If you would like to see – and perhaps modify – our source code, please download it from: Euclid21 source code

Our code certainly seems to be more than a toy – after all, all thirteen books of The Elements are available through this application – but exactly what it is was very hard to pin down at first.

In Euclid and his Modern Rivals, Charles Dodgson (Lewis Carroll) said [1]:

In one respect this book is experimental …. I mean that I have not thought it necessary to maintain throughout the gravity of style which scientific writers usually afffect …. I could never quite see the reasonableness of this …. Subjects there are, no doubt, which are in their essence too serious to admit of any lightness of treatement – but I cannot recognize Geometry as one of them.

We agree. One reason, by no means the only one, that The Elements is but little used in schools and universities these days is that it is so very august and imposing. Modern editions tend to be densely written and heavily footnoted. They're just no fun to read. This needn't be so but it frequently is.

To be sure there have always been attempts to redress this. Dodgson [1] made heroic efforts in that direction. More recently, David Joyce of Clark University created an online version of The Elements [4] that includes a set of Java applets. We know that for the Greeks a diagram was an integral part of a proof, not the secondary visual aid it has become in modern times (see [6]). Joyce's applets animate Euclid's original diagrams, making them interactively manipulable, which allows the reader to see how the proposition remains true however the figure is drawn. In a sense this restores some of the primacy of Euclid's diagrams. More importantly, they are fun to interact with. Joyce's webpage is in a sense, a book. All that it lacks is paper, which is, after all, the technology of the last millennium.

Books, for all of their profound importance to our history and culture, have begun to undergo a rapid and radical change. Indeed, it will soon be very difficult to define exactly what a “book” is, or whether there is any need for such a definition. In [8] and again in [9] the noted journalist Adam Penenberg said:

Coming soon ... the end of the book as we know it, and you'll be just fine. But it won't be replaced by the e-book, which is, at best, a stopgap measure. ...
Take note: The first battlefield tanks looked like heavily armored tractors equipped with cannons; early automobiles were called “horseless carriages” for a reason; the first motorcycles were based on bicycles; the first satellite phones were as clunky as your household telephone. A decade ago, when newspapers began serving up stories over the Web, the content mirrored what was offered in the print edition. What the tank, car and newspaper have in common is they blossomed into something far beyond their initial prototypes. In the same way that an engineer wouldn't dream of starting with the raw materials for a carriage to design a rad new sports car today, newspapers won't use paper or ink anymore. Neither will books. But mere text on a screen, the stuff that e-books are made of, won't be enough. ...
Like early filmmakers, some of us will seek new ways to express ourselves through multimedia. Instead of stagnant words on a page we will layer video throughout the text, add photos, hyperlink material, engage social networks of readers who will add their own videos, photos, and wikified information so that these multimedia books become living, breathing, works of art.

The electronic mathematics books published heretofore fulfill all of the requirements of a book published, like Dodgson's, in 1879. Or one written in 1679. There is a table of contents, there are page numbers, and, if it is a scholarly book, there is a bibliography and probably an index, all arranged in linear order. We suggest that this is not necessarily the best way to present mathematics in the 21st century and we offer this version of The Elements as an example of how mathematics might be better presented to an inquisitive mind, be it the mind of a student, a teacher, or a researcher.

The structure of The Elements has always been a directed graph. Each proposition depends on those which come before, and most support those that come after. It makes sense to let the form follow the function. This is what we have done.