**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