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Commentary on Proposition IV.16¶

Here, as in III. 10, we have the term “circle” used by Euclid in its exceptional sense of the circumference of a circle, instead of the “plane figure contained by one line” of I. Def. 15. Cf. the note on that definition (Vol. I. pp. 184—5).

Proclus (p. 269) refers to this proposition in illustration of his statement that Euclid gave proofs of a number of propositions with an eye to their use in astronomy. “With regard to the last proposition in the fourth Book in which he inscribes the side of the fifteen-angled figure in a circle, for what object does anyone assert that he propounds it except for the reference of this problem to astronomy? For, when we have inscribed the fifteen-angled figure in the circle through the poles, we have the distance from the poles both of the equator and the zodiac, since they are distant from one another by the side of the fifteen-angled figure.” This agrees with what we know from other sources, namely that up to the time of Eratosthenes (circa 284—204 B.C.) $24^\circ$ was generally accepted as the correct measurement of the obliquity of the ecliptic. This measurement, and the construction of the fifteen-angled figure, were probably due to the Pythagoreans, though it would appear that the former was not known to Oenopides of Chios (fl. circa 460 B.C.), as we learn from Theon of Smyrna (pp. 198—9, ed. Hiller), who gives Dercyllides as his authority, that Eudemus (fl. circa 320 B.C.) stated in his ἀστρολογίαι that, while Oenopides discovered certain things, and Thales, Anaximander and Anaximenes others, it was the rest (οἱ λοιποί) who added other discoveries to these and, among them, that “the axes of the fixed stars and of the planets respectively are distant from one another by the side of a fifteen-angled figure.” Eratosthenes evaluated the angle to $\frac{11}{83}$rds of $180^\circ$, i.e. about $23^\circ\ 51'\ 20''$, which measurement was apparently not improved upon in antiquity (cf. Ptolemy, Syntaxis, ed. Heiberg, p. 68).

Euclid has now shown how to describe regular polygons with $3, 4, 5, 6$ and $15$ sides. Now, when any regular polygon is given, we can construct a regular polygon with twice the number of sides by first describing a circle about the given polygon and then bisecting all the smaller arcs subtended by the sides. Applying this process any number of times, we see that we can by Euclid’s methods construct regular polygons with $3 \cdot 2^n, 4 \cdot 2^n, 5 \cdot 2^n, 15 \cdot 2^n$ sides, where $n$ is zero or any positive integer.