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!! 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:
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