The authors have developed an application which presents Euclid's *The Elements* visually in the form of a directed graph. Two nodes (propositions) are connected if one is used in the proof of the other. This article tells the story of the creation of the application, explains why we think it is a valuable addition to the literature, and describes briefly how to use it.

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.

On the inside of the front cover of *Euclid and his Modern Rivals* [1], Dodgson offered the visual display of the logical structure of the first book of *The Elements* shown in Figure 1.

**Figure 1.** Dodgson's partial graph of Euclid's Book I in his *Euclid and his Modern Rivals* (1879) [1]

It was this question of the logical structure of the propositions in *The Elements* that began our project. In the spring semester of 2010 the first author (hereafter to be referred to in the first person) was teaching a college course in geometry for the first time. While the class was discussing the Pythagorean Theorem (Book 1, Prop. 47), I mentioned to them that I had once been told that Book 1 is organized around the proof of the Pythagorean Theorem, but I did not know if this was true. This was just an off-the-cuff remark but I became so intrigued by the idea that during an idle moment I took out my copy of *The Elements* and began to sketch out which propositions the Pythagorean Theorem depends on. Had I been familiar with Dodgson's book at the time I would certainly have simply referred to it and this project would never have begun. Instead in about 15 minutes I had the sketch in Figure 2.

**Figure 2.** My first attempt at a partial graph of Euclid's Book I

This is obviously incomplete and almost certainly inaccurate. I was still intrigued but because I had to get to my next class I put my sketch aside.

I revisited this idea the following fall. As a course project, Brett Eyer, a student in my History of Mathematics course that semester, extended my diagram by creating a complete dependency graph of all of the propositions in the first book of Euclid. He presented a short talk with the chart in Figure 3 at the Fall 2010 meeting of the Eastern Pennsylvania and Delaware (EPaDel) section of the MAA.

**Figure 3.** Eyer's graph of Euclid’s Book I

Eyer's chart, more complete than either mine or Dodgson's, clearly refutes the idea that Book 1 is organized around the Pythagorean Theorem. Only the purple-shaded propositions support the Pythagorean Theorem, either directly or indirectly. In particular, Propositions 12, 17, 21, 25, 28, 40 and 45 are proved but never used in Book 1. The Pythagorean Theorem is clearly *a* focal point of Book 1, but it is not *the* focal point.

This begs the question, “Are the propositions listed above used elsewhere in* The Elements*?” Also, “Are there propositions which are proved but never used anywhere in *The Elements*?”

At this point the nature of my interest in the project began to change. Finding those propositions which supported the Pythagorean Theorem gave way to the more general problem of finding the dependency structure among *all* of the propositions in all thirteen books of *The* *Elements.*

In my History of Mathematics course in Spring 2011, I proposed this as a project to Roberge, Brown, and Dahiya. We spent that semester and the following summer skimming, sometimes reading closely, through all thirteen books of *The Elements* [2] and gathering the information we would need to create a complete graph of the dependency structure of the propositions of Euclid. Milbrand learned of the project and joined the team that summer as well.

Very early on it became clear that a static graph would be neither interesting nor useful. Whatever we built would have to be interactive because, paradoxically, a complete, but static, graph of the dependencies in *The Elements* gives both too much and too little information. That is, it is so completely overwhelming as to be almost useless. The graph in Figure 4 is nearly complete. See for yourself.

**Figure 4.** A graph of all 13 books of Euclid’s *Elements*

Also, while the dependency structure is interesting, without the actual text of the propositions it is only a curiosity. We therefore set ourselves the task of implementing a dynamic, interactive visualization of the dependency structure of all axioms, definitions, postulates and propositions in all thirteen books of *The Elements.* In addition, the English text of each would also be available at the click of a mouse. We chose to use Fitzpatrick's [2] translation for this part because it is freely available.

When the code was nearly completed we presented it at the 2011 MathFest in Madison, Wisconsin, that August. To be sure, there was still some debugging to do but we were confident that it was nearly finished.

Alas, it was not so. Upon returning from Madison we tried eliminating the bugs in our code but each time one was tamped down (I wouldn't say “fixed”) another, sometimes several more, popped up. We were caught in a seemingly endless bout of “Whack-A-Bug.”

It is easy to understand how this happened. The code had grown organically as we learned more about the internal dependencies in *The Elements*, sprouting new features as they suggested themselves to us, rather than according to an overall design. We proceeded this way in the beginning because we didn't know exactly what we were creating. We were just having fun.

As a result our code had several features which were ultimately not very useful. Other, more useful features were buggy. That is, they didn't always work as intended and sometimes not at all.

However at this point we had gathered all of the dependency data from [2], *and* we had very precise ideas of what the code should and should not do. Clearly, it was time to scrap our first efforts and rebuild everything from the ground up, this time working from a clear set of design specifications. This was only possible because we had learned so much from our first efforts.

Unfortunately the students on the team were all near graduation and they were unaccountably unreceptive to the suggestion that they delay graduating in order to finish this project. Since I cannot program in Java (or any modern programming language) I had resolved to wait for another, equally motivated group of students with whom to complete the project.

Then the last member of our team, my daughter Mary Boman, joined the project.

She had been following our progress thoughout her senior year of high school, and was about to enter college as a Computer Science major. And, crucially, she could write Java code. When I expressed my disappointment that finishing the project would be delayed, she volunteered to take a stab at the re-write. I explained the design specifications to her and she completed the new version within a few months.

**Figure 5.** Euclid's first use of his Parallel Postulate

It is well known that Euclid delayed using his fifth postulate, the Parallel Postulate,

And that if a straight-line falling across two (other) straight-lines makes internal angles on the same side (of itself whose sum is) less than two right-angles, then the two (other) straight-lines, being produced to infinity, meet on that side (of the original straight-line) that the (sum of the internal angles) is less than two right-angles (and do not meet on the other side),

as long as he could, but few can name the proposition where it is first used. In a traditional book it is very difficult to find the first use of Postulate 5. In ours it is a triviality. Start the code* and click on “Axioms.” Then click on “Axiom 5.” You will be presented with the graph in Figure 5 from which it is clear that the first use of Postulate 5 is in Book 1, Proposition 29. If you want to learn what Proposition 29 says, merely double-click on the node marked “1.29.” The statement,

A straight-line falling across parallel straight-lines makes the alternate angles equal to one another, the external (angle) equal to the internal and opposite (angle), and the (sum of the) internal (angles) on the same side equal to two right-angles,

and its proof will open in a separate window.

If you would like to see how Book 1, Proposition 29 is proved from the axioms simply right-click on it and select “Display This Node” in the drop-down menu. You will be presented with the graph in Figure 6.

**Figure 6.** The proof of Book I, Proposition 29. Notice that each proposition appears as a yellow node, labelled with the book and proposition number.

The propositions supporting Proposition 29 appear as nodes above it. Those it supports appear below. All can be opened and read by double-clicking. Right-clicking on any node opens the drop-down menu from which you can extend the graph by adding or deleting nodes associated with other propositions. Left-click and drag a node to move it. Continue in this fashion until you have followed the proof back to the Axioms, Definitions, and Common Notions or until it intersects with your personal knowledge.

A table of contents is not needed. Or rather, our book is its own table of contents. All you need to know before opening it is that it is Euclid's *The Elements*. Upon opening, the organizational structure of our book is clear because it mirrors the organizational structure of the topic.

* If you have not already done so, please 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.

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

We have learned from Colin McKinney [5] that in fact Euclid never gave explicit references in his proofs, although he did sometimes allude to previous propositions. Apparently, he assumed the reader would master everything that came before each proposition.

Making explicit reference to prior propositions appears to be an innovation of Euclid's modern editors, but they do not agree. More precisely, different editors chose different dependencies to make explicit. Since our book uses Fitzpatrick's [2] translation, which is based on Heiberg's edition of the Greek text, the dependency structure we display is necessarily incomplete. For example, in Heiberg's translation Axiom 4 is *never* explicitly referenced, whereas in Heath's translation [3] it is.

It would be interesting to gather the dependency data from another authoritative translation, Heath's for example, for comparison. If they are combined with the data we already have, an even more complete dependency graph would be possible. That project is ongoing.

There are other documents and books, especially in mathematics, which are organized in a similar fashion. Our code could be modified to handle any such document, however we are not currently investigating this. Instead our code is freely available, licensed under the Gnu Open Software License so that others can make any such changes should they choose to. That is, anyone is free to use and/or modify the code for any non-commercial purpose.

Finally, while *The Elements* may or may not be, as Dodgson believed, the perfect text for teaching plane geometry, it is unique in the history of our species. It may well be the most read, most published book in history, outstripping even the various religious documents ([7], p. 55). For that reason alone, quite apart from its intrinsic value to mathematics, we should continue to find ways to pass it on to each successive generation. This is our contribution to that cause.

[1] Charles A. Dodgson. *Euclid and his Modern Rivals.* Macmillan and Co., 1879.

[2] Richard Fitzpatrick, editor. *Euclid’s Elements of Geometry.* Lulu.com or http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf, 2007.

[3] Robert Maynard Hutchins, editor. *The Thirteen Books of Euclid’s Elements. Great Books of the Western World.* Encyclopedia Britannica, Inc., 1952. Translated by Sir Thomas L. Heath.

[4] David E. Joyce. *Euclid’s Elements.* Interactive webpage, 1998: http://aleph0.clarku.edu/~djoyce/java/elements/.

[5] Colin McKinney. Personal communication, March 2014. Conversation at the 2014 HPM Americas meeting.

[6] Reviel Netz and William Noel. *The Archimedes Codex.* Da Capo Press, 2007. ISBN: 978-0-786-74538-8.

[7] Donal O’Shea. *The Poincar**è** Conjecture.* Walker Publishing Company, Inc., New York, 2007.

[8] Adam L. Penenberg. “Forget e-books: The future of the book is far more interesting.” Blog entry, December 2009: http://www.fastcompany.com/1493951/forget-e-books-future-book-far-more-interesting.

[9] Adam L. Penenberg. *Viral Loop: From Facebook to Twitter, How Today’s Smartest Businesses Grow Themselves.* Hyperion Books, 2009. ISBN: 978- 1-4013-2349-3.

**Eugene Boman** earned his BA from Reed College in 1984 and his Ph.D. from the University of Connecticut in 1993. He has been teaching and researching mathematics at the Pennsylvania State University since 1995 and feels very strongly that the best and most interesting work he has done has always been his work with undergraduate students. He is a recipient of the 2008 MAA Allendoerfer Award (with student Derek Seiple and colleague Richard Brazier) for the paper "Mom! There’s an Astroid in my Closet!"

**Siddharth Dahiya** is originally from the town of Rohtak near the Indian capital city of New Delhi. He earned BAs in Computer Science and Mathematics from Penn State Harrisburg. He is currently completing his MA in Computer Science, also from Penn State Harrisburg. After graduating, he will be joining Microsoft Corporation in Redmond, Washington, as a Software Engineer.

**Tyler Brown** is originally from Elizabethtown, Pennsylvania. He studied Mathematics and Computer Science at Penn State Harrisburg earning a BS in both fields in May 2014. Currently he is enrolled as a PhD student in Iowa State University's Pure Mathematics Program. After graduating, he hopes to find a place in academia as a lecturer or postdoctoral fellow following his passion for teaching and research.

**Alexandra Milbrand** grew up in the suburbs of Harrisburg, Pennsylvania. Staying close to home, she graduated with her BS in Mathematics from Penn State's Harrisburg campus. She now studies at Florida Atlantic University, working towards her Ph.D. in Mathematics. Once completed, she wishes to teach mathematics at the collegiate level.

**Joseph L Roberge, Jr.** is originally from Rome, New York. He graduated Valedictorian from Dauphin County Technical School, majoring in Electronics Technology. In 2012, he graduated from Penn State's Harrisburg Campus with a BS in Computer Science. He is presently completing his MS in Computer Science, also from Penn State, Harrisburg. He currently works in Information Technology for the Pennsylvania State Police. After graduation, he hopes to continue his pursuit of lifelong learning and fully invest himself in the world of computer technology and security.

**Mary Boman** is originally from the town of DuBois, Pennsylvania. She graduated from Hershey High School in 2013 and expects to receive her BA in Computer Science from Bryn Mawr College in 2017. In the meantime she has applied for summer internships for the upcoming summer of 2015.