That the side of a regular inscribed hexagon equals the radius must have been recognized very early. The common divisions of the circle in ancient art are into four, six, and eight equal parts. No draftsman could have worked with a pair of compasses without quickly learning how to effect these divisions, and that compasses were early used is attested by the specimens of these instruments often seen in museums. There is a tradition that the ancient Babylonians considered the circle of the year as made up of 360 days, whence they took the circle as composed of 360 steps or grades (degrees). This tradition is without historic foundation, however, there being no authority in the inscriptions for this assumption of the 360-division by the Babylonians, who seem rather to have preferred 8, 12, 120, 240, and 480 as their division numbers. The story of 360° in the Babylonian circle seems to start with Achilles Tatius, an Alexandrian grammarian of the second or third century A.D. It is possible, however, that the Babylonians got their favorite number 60 (as in 60 seconds make a minute, 60 minutes make an hour or degree) from the hexagon in a circle (1/6 of 360° = 60°), although the probabilities seem to be that there is no such connection.[83]
The applications of this problem to mensuration are numerous. The fact that we may use for tiles on a floor three regular polygons—the triangle, square, and hexagon—is noteworthy, a fact that Proclus tells us was recognized by Pythagoras. The measurement of
the regular hexagon, given one side, may be used in computing sections of hexagonal columns, in finding areas of flower beds, and in other similar cases.
This review of the names of the polygons offers an opportunity to impress their etymology again on the mind. In this case, for example, we have "hexagon" from the Greek words for "six" and "angle."
Problem. To inscribe a regular decagon in a given circle.
Euclid states the problem thus: To construct an isosceles triangle having each of the angles at the base double of the remaining one. This makes each base angle 72° and the vertical angle 36°, the latter being the central angle of a regular decagon,—essentially our present method.
This proposition seems undoubtedly due to the Pythagoreans, as tradition has always asserted. Proclus tells us that Pythagoras discovered "the construction of the cosmic figures," or the five regular polyhedrons, and one of these (the dodecahedron) involves the construction of the regular pentagon.
Iamblichus (ca. 325 A.D.) tells us that Hippasus, a Pythagorean, was said to have been drowned for daring to claim credit for the construction of the regular dodecahedron, when by the rules of the brotherhood all credit should have been assigned to Pythagoras.
If a regular polygon of s sides can be inscribed, we may bisect the central angles, and therefore inscribe one of 2s sides, and then of 4s sides, and then of 8s sides, and in general of 2ns sides. This includes the case of s = 2 and n = 0, for we can inscribe a regular polygon of two sides, the angles being, by the usual formula, 2(2-2)/2 = 0, although, of course, we never think of two equal and coincident lines as forming what we might call a digon.