[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.
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 semi-side of a circumscribed regular 12-gon; then as AB:BO:OA::1: √3:2 he sought an approximation to √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 < 31⁄7:1. In a quite analogous manner he proved for the inscribed regular 96-gon that perimeter:diameter > 310⁄71:1. The conclusion from these therefore was that the ratio of circumference to diameter is < 31⁄7 and > 310⁄71. This is a most notable piece of work; the immature condition of arithmetic at the time was the only real obstacle preventing the evaluation of the ratio to any degree of accuracy whatever.[5]
No advance of any importance was made upon the achievement of Archimedes until after the revival of learning. His immediate successors may have used his method to attain a greater degree of accuracy, but there is very little evidence pointing in this direction. Ptolemy (fl. 127-151), in the Great Syntaxis, gives 3.141552 as the ratio[6]; and the Hindus (c. A.D. 500), who were very probably indebted to the Greeks, used 62832⁄20000, that is, the now familiar 3.1416.[7]
It was not until the 15th century that attention in Europe began to be once more directed to the subject, and after the resuscitation a considerable length of time elapsed before any progress was made. The first advance in accuracy was due to a certain Adrian, son of Anthony, a native of Metz (1527), and father of the better-known Adrian Metius of Alkmaar. In refutation of Duchesne(Van der Eycke), he showed that the ratio was < 317⁄120 and > 315⁄106, and thence made the exceedingly lucky step of taking a mean between the two by the quite unjustifiable process of halving the sum of the two numerators for a new numerator and halving the sum of the two denominators for a new denominator, thus arriving at the now well-known approximation 316⁄113 or 335⁄113, which, being equal to 3.1415929..., is correct to the sixth fractional place.[8]
The next to advance the calculation was Francisco Vieta. By finding the perimeter of the inscribed and that of the circumscribed regular polygon of 393216 (i.e. 6 × 216) sides, he proved that the ratio was > 3.1415926535 and < 3.1415926537, so that its value became known (in 1579) correctly to 10 fractional places. The theorem for angle-bisection which Vieta used was not that of Archimedes, but that which would now appear in the form 1 - cos θ = 2 sin² ½θ. With Vieta, by reason of the advance in arithmetic, the style of treatment becomes more strictly trigonometrical; indeed, the Universales Inspectiones, in which the calculation occurs, would now be called plane and spherical trigonometry, and the accompanying Canon mathematicus a table of sines, tangents and secants.[9] Further, in comparing the labours of Archimedes and Vieta, the effect of increased power of symbolical expression is very noticeable. Archimedes’s process of unending cycles of arithmetical operations could at best have been expressed in his time by a “rule” in words; in the 16th century it could be condensed into a “formula.” Accordingly, we find in Vieta a formula for the ratio of diameter to circumference, viz. the interminate product[10]—
½√½ · √½ + ½√½ · √½ + ½√(½ + ½√½) ...