 Euclid's elements, Book 2, Propositions 12 and 13, are a little complicated, but if we take them apart, we see that these are actually equivalent to what we think about as the law of cosides. And so we might say that Euclid's element incorporates some trigonometry. However, the real development of plane trigonometry occurred later. Much of this later development of trigonometry was due to Ptolemy, who wrote the oldest surviving comprehensive work on what we might call trigonometry. Ptolemy lived in the 2nd century AD in Alexandria, Egypt. Here's an image of him, as he would have appeared if he lived in 15th century Europe. Ptolemy called his work on mathematics the great mathematical treatise. Now it's worth pointing out that Ptolemy was not being arrogant here. Great in this context means big, and by implication, comprehensive. Now Ptolemy actually wrote in Greek, so the title was actually Megales in Taxis. And because the Greeks weren't fond of long titles any more than anybody else's, the title was often shortened to Megiste, which we might read as the big book. Now while Ptolemy wrote the work in Greek, it turns out that the most influential versions of the book came from Islamic translators. And Arabic translators use the same title, Megiste, but they prefaced it with the definite article, Al. And so it became known as Al-Megiste. And so generally we refer to this now as the Almagest. Now the Almagest was actually written as an astronomical treatise, but part of astronomy was computing the positions of the planets, and for that Ptolemy has to consider the lengths of chords in a circle. Now in general we can describe a chord by the central angle subtending the chord. That's the angle whose sides mark off the endpoints of the chord. Alternatively any angle inscribed in a circle actually defines three chords, the two legs of the angle inside the circle, and then the chords subtended by the inscribed angle. Euclid and his predecessors had found several theorems regarding chords in a circle, and a couple of the following are useful. First there's Staley's theorem, almost one of the very first theorems in geometry, which is that an angle inscribed in a semicircle is a right angle. And then it's also useful to keep in mind that inscribed angles subtending equal arcs are equal. In other words, as long as the arcs are equal it really doesn't matter where we place the angle. Ptolemy considered the problem of finding the lengths of chords in a circle. Now in order to do this he begins with the construction of a regular pentagon inscribed in a circle. And Ptolemy's construction is the following. Let A C be the diameter with center D. We'll construct the perpendicular D B. We'll bisect D C and E and join B E. We'll find F so that E F is equal to E B. Then B F will be the side of a regular pentagon, and F D the side of a regular decagon. This last observation comes from one of the propositions in Euclid's elements. Next, Ptolemy found some chord lengths. Now since the length of the chord in a circle depends on the length of the diameter, Ptolemy chose to make the diameter 120 P parts. And this allowed him to state the following. The chord on the arc of 60 degrees, in other words the chord joining two points on the arc of the circle, where the central angle is 60 degrees, has a length of 60 parts. Now from the construction of the pentagon, Ptolemy can also compute the chord of the arc of 72 degrees, has a length of 70, 32, 3, expressed in hexagazimals. He also found the chord of 90 degrees, has a length of 84, 51, 10, and the chord of 120 degrees, has a length of 103, 55, 23. Now except for the chord of 72 degrees, Ptolemy doesn't give a lot of details on how he can find these chord lengths, but most of them emerge from simple geometry. For example, the length of the chord on the arc of 90 degrees when the circle has a diameter of 120 parts. So let's draw our circle with the chord whose central angle is 90 degrees. So notice that this 90 degrees, well that's the measure of the central angle, and so this chord is one side of a right triangle where the two legs have a length equal to the radius of the circle. And that means we can use the Pythagorean theorem where A and B are the radii of the circle with diameter 120. And so we can find C, and in modern notation we can find that as a decimal. Now Ptolemy probably found this hexagazimal expansion directly using some variation of Theon's method of finding square roots, but to see where these come from we'll go ahead and do the modern arithmetic to get there. So we'll convert this to a hexagazimal, and we want the fractional part to be expressed in 60ths. And if we want to go to two hexagazimal places, that's something 60th plus something 60 squares. And to find D1, we can multiply everything by 60. And since D1 has to be a whole number, we can let D1 equal 51, the whole number portion. And that means this second hexagazimal is the decimal portion, so once again we can multiply by 60. And again, since D2 should also be a whole number, we'll let D2 equal 10. And that gives us our hexagazimal 84, 51, 10.