Lecture 18 We finished looking at Book IV. We actually spent some time talking again about how to inscribe a regular pentagon in a given circle. The reason is that in order to construct the regular 15-gon inscribed in a given circle, it suffices to be able to construct the regular pentagon and the equilateral triangle. In fact, the construction of an equilateral triangle inscribed in a circle is equivalent to constructing an arc of length equal to 1/3rd of the given circumference. Similarly, the construction of a regular pentagon is equivalent to constructing an arc of length equal to 1/5th of the given circumference. Since 1/3-1/5 =2/15...we can construct an arc of length equal to 2/15th of the original circumference. If we now bisect that arc we get another arc of length equal to 1/15th of the original circumference. Thus, we can construct the side of the regular 15-gon. Take a look at the picture given in the following proposition!! Proposition 16. Since we can inscribe an equilateral triangle, a square, a pentagon, etc...using the bisecting procedure we can construct several regular polygons: the ones with 6, 12, 24, 48, etc ... sides (from the equilateral triangle); the ones with 4, 8, 16, 32, etc ... sides (from the square); the ones with 5, 10, 20, 40, etc ... sides (from the pentagon). People assumed that these were the only regular polygons that could be circumscribed in a circle. It came a big surprise when young Carl F. Gauss showed in 1796 how to inscribe into a given circle a regular 17-gon. We then mentioned Book V, where Euclid talks about the theory of proportions, introduced by Eudoxus of Cnidos to overcome the problem of the existence of incommensurable quantities discovered by the Pythagoreans. But we moved rapidly to Book VI, where Euclid applies the theory of proportions to geometric figures. We discussed Definition 1, namely: Similar rectilinear figures are such as have their angles severally equal and the sides about the equal angles proportional. Funny language!!, right? Anyway, observe that we need both requirements: a rectangle and a square have both the same angles...but they are certainly not similar. Similarly, a square and a rombus have sides with the same length...but they are certainly not similar! However....for triangles it is enough to have the same angles in order to be similar: Proposition 4 or it suffices to have the proportional sides Proposition 5. We then studied Proposition 8, which is the one we used to give an alternative proof of Pythagoras' Theorem: Proposition 8. With Book VI we finished the study of plane geometry. Now, Euclid undertakes (in the next 3 books) the `burden' of developing Number Theory. We started commenting a few definitions from Book VII: even and odd, composite, prime, perfect, ... numbers. Book VII has 22 Definitions and 39 Propositions. Then it starts right away with the Euclidean division algorithm and the method of how to compute the greatest common divisor of two given numbers. Click on the above links to go directly to the proofs and commentaries of the results.