asiatique_, June
1841), and probably also the Greeks. Among the ancient Egyptians, as
would appear from a calculation in the Rhind papyrus, the number
(4/3)^4, i.e. 3.1605, was at one time in use.[1] The first attempts to
solve the purely geometrical problem appear to have been made by the
Greeks (Anaxagoras, &c.)[2], one of whom, Hippocrates, doubtless raised
hopes of a solution by his quadrature of the so-called _meniscoi_ or
_lune_.[3]

[Illustration: Fig. 6.]

[Illustration: Fig. 7.]

[The Greeks were in possession of several relations pertaining to the
quadrature of the lune. The following are among the more interesting. In
fig. 6, ABC is an isosceles triangle right angled at C, ADB is the
semicircle described on AB as diameter, AEB the circular arc described
with centre C and radius CA = CB. It is easily shown that the areas of
the lune ADBEA and the triangle ABC are equal. In fig. 7, ABC is any
triangle right angled at C, semicircles are described on the three
sides, thus forming two lunes AFCDA and CGBEC. The sum of the areas of
these lunes equals the area of the triangle ABC.]

As for Euclid, it is sufficient to recall the facts that the original
author of prop. 8 of book iv. had strict proof of the ratio being <4,
and the author of prop. 15 of the ratio being >3, and to direct
attention to the importance of book x. on incommensurables and props. 2
and 16 of book xii., viz. 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.

[Illustration: FIG. 8.]

With Archimedes (287-212 B.C.) a notable advance was made. Taking the
circumference as intermediate between the perimeters of the inscribed
and the circumscribed regular n-gons, he showed that, the radius of the
circle being given and the perimeter of some particular circumscribed
regular polygon obtainable, the perimeter of the circumscribed regular
polygon of double the number of sides could be calculated; that the like
was true of the inscribed polygons; and that consequently a means was
thus afforded of approximating to the circumference of the circle. As a
matter of fact, he started with a semi-side AB of a circumscribed
regular hexagon meeting the circle in B (see fig. 8), joined A and B
with O the centre, bisected the angle AOB by OD, so that BD became the
s