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 deg. and the vertical angle 36 deg., 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 2_s_ sides, and then of 4_s_ sides, and then of 8_s_ sides, and in general of 2^{_n_}_s_ 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_. We therefore have the following regular polygons: From the equilateral triangle, regular polygons of 2^_n_ . 3 sides; From the square, regular polygons of 2^_n_ sides; From the regular pentagon, regular polygons of 2^_n_ . 5 sides; From the regular pentedecagon, regular polygons of 2^_n_ . 15 sides. This gives us, for successive values of _n_, the following regular polygons of less than 100 sides: From 2^_n_ . 3, 3, 6, 12, 24, 48, 96; From 2^_n_, 2, 4, 8, 16, 32, 64; From 2^_n_ . 5, 5, 10, 20, 40, 80; From 2^_n_ . 15, 15, 30, 60. [Illustration: ROMAN MOSAIC FOUND AT POMPEII]