4/[pi] = 1 + 1/2 + 9/2 + 25/2 + 49/2 + .... (Brouncker, 1620-1684) [pi]/4 = 1 - 1/3 + 1/5 - 1/7 + .... (Gregory, 1638-1675) [pi]/6 = [sqrt](1/3) . (1 - 1/(3 . 3) + 1/(3^2 . 5) - 1/(3^3 . 7) + ...). [pi]/2 = (log _i_) / _i_. (Bernoulli) [pi]/(2[sqrt](3)) = 1 - 1/5 + 1/7 - 1/11 + 1/13 - 1/17 + 1/19..., thus connecting the primes. [pi]^2/16 = 1 - 1/2^2 - 1/3^2 + 1/4^2 - 1/5^2 + 1/6^2 - 1/7^2 - 1/8^2 + 1/9^2 + .... [pi]/2 = _x_/2 + sin _x_ + (sin^2 _x_) / 2 + (sin^3 _x_) / 3 + .... (0 < _x_ < 2[pi]) [pi]/4 = 3/4 + 1/(2 . 3 . 4) - 1/(4 . 5 . 6) + 1/(6 . 7 . 8) - .... 2[pi]^2/3 = 7 - (1/(1 . 3) + 1/(3 . 6) + 1/(6 . 10) + ...). [pi] = 2^_n_[sqrt](2 - [sqrt](2 + [sqrt](2 + [sqrt](2 + [sqrt](2...))))). Students of elementary geometry are not prepared to appreciate it, but teachers will be interested in the remarkable formula discovered by Euler (1707-1783), the great Swiss mathematician, namely, 1 + _e_^{_i_[pi]} = 0. In this relation are included the five most interesting quantities in mathematics,--zero, the unit, the base of the so-called Napierian logarithms, _i_ = [sqrt](-1), and [pi]. It was by means of this relation that the transcendence of _e_ was proved by the French mathematician Hermite, and the transcendence of [pi] by the German Lindemann. [Illustration] There should be introduced at this time, if it has not already been done, the proposition of the lunes of Hippocrates (_ca._ 470 B.C.), who proved a theorem that asserts, in somewhat more general form, that if three semicircles be described on the sides of a right triangle as diameters, as shown, the lunes _L_ + _L'_ are together equivalent to the triangle _T_. [Illustration] In the use of the circle in design one of the simplest forms suggested by Book V is the trefoil (three-leaf), as here shown, with the necessary construction lines. This is a very common ornament in architecture, both with rounded ends and with the ends slightly pointed. The trefoil is closely connected with hexagonal designs, since the regular hexagon is formed from the inscribed equilateral triangle by doubling the number of sides. The following are designs that are easily made: [Illustration] It is not very profitable, because it is manifestly unreal, to measure the parts of such figures, but it offers plenty of practice in numerical work. [Illustration: CHOIR OF LINCOLN CATHEDRAL] [Illustration: PORCH OF LINCOLN CATHEDRAL]