B, C. multicornis, natural size; p, polyp; gon, gonophores; rh, hydrorhiza.

Dz, by the hydranths, each with dactylozoid; gz, gastrozoid; b, mouth and tentacles; and, blastostyle; gon, gonophores; secondly, the " coenosarc," or rh, hydrorhiza.

In the north-west, by the Liberian frontier, the mountains in the Gon region rise over 6000 ft.

That " circles are to one another as the squares on their diameters " and that " in the greater of two concentric circles a regular 2n**-gon** can be inscribed which shall not meet the circumference of the less," however nearly equal the circles may be.

8), joined A and B with 0 the centre, bisected the angle AOB by OD, so that BD became the semi-side of a circumscribed regular 12**-gon**; then as AB :BO: OA :: I: d 3: 2 he sought an approximation to X 1 3 and found that AB: BO >153:265.

Next he applied his theorem 4 BO+OA: AB:: OB: BD to calculate BD; from this in turn he calculated the semi-sides of the circumscribed regular 24**-gon**, 48**-gon** and 96**-gon**, and so finally established for the circumscribed regular 96**-gon** that perimeter: diameter 3+V :I.

The proof of the correctness of the construction is seen to be involved in the following theorem, which serves likewise to throw new light on the subject: - AB being any straight line whatever, and the above construction being made, then AB is the diameter of the circle circumscribed by the square AB CD (self-evident), AB, is the diameter of the circle circumscribed by the regular 8**-gon** having the same perimeter as the square, AB, is the diameter of the circle circumscribed by the regular 16**-gon** having the same perimeter as the square, and so on.