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 semi-side of a circumscribed regular 12-gon; then as AB:BO:OA::1: sqrt(3):2 he sought an approximation to sqrt(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 < (3-1/7):1. In a quite analogous manner he proved for the inscribed regular 96-gon that perimeter:diameter > 3-(10/71):1. The conclusion from these therefore was that the ratio of circumference to diameter is < 3-1/7 and > 3-(10/71). 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 < 3-(17/120) and > 3-(15/106), 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 3-(16/113) or 355/113, 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 X 2^16) 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 [theta] = 2 sin² ½[theta]. 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 ...

From this point onwards, therefore, no knowledge whatever of geometry was necessary in any one who aspired to determine the ratio to any required degree of accuracy; the problem being reduced to an arithmetical computation. Thus in connexion with the subject a genus of workers became possible who may be styled "[pi]-computers or circle-squarers"--a name which, if it connotes anything uncomplimentary, does so because of the almost entirely fruitless character of their labours. Passing over Adriaan van Roomen (Adrianus Romanus) of Louvain, who published the value of the ratio correct to 15 places in his _Idea mathematica_ (1593),[11] we come to the notable computer Ludolph van Ceulen (d. 1610), a native of Germany, long resident in Holland. His book, _Van den Circkel_ (Delft, 1596), gave the ratio correct to 20 places, but he continued his calculations as long as he lived, and his best result was published on his tombstone in St Peter's church, Leiden. The inscription, which is not known to be now in existence,[12] is in part as follows:--

... Qui in vita sua multo labore circumferentiae circuli proximam rationem ad diametrum invenit sequentem--

quando diameter est 1 tum circuli circumferentia plus est

quam 314159265358979323846264338327950288 1OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

et minus quam 314159265358979323846264338327950289 1OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO ...

This gives the ratio correct to 35 places. Van Ceulen's process was essentially identical with that of Vieta. Its numerous root extractions amply justify a stronger expression than "multo labore," especially in an epitaph. In Germany the "Ludolphische Zahl" (Ludolph's number) is still a common name for the ratio.[13]

[Illustration: FIG. 9.]

[Illustration: FIG. 10.]

Up to this point the credit of most that had been done may be set down to Archimedes. A new departure, however, was made by Willebrord Snell of Leiden in his _Cyclometria_, published in 1621. His achievement was a closely approximate geometrical solution of the problem of rectification (see fig. 9): ACB being a semicircle whose centre is O, and AC the arc to be rectified, he produced AB to D, making BD equal to the radius, joined DC, and produced it to meet the tangent at A in E; and then his assertion (not established by him) was that AE was nearly equal to the arc AC, the error being in defect. For the purposes of the calculator a solution erring in excess was also required, and this Snell gave by slightly varying the former construction. Instead of producing AB (see fig. 10) so that BD was equal to r, he produced it only so far that, when the extremity D' was joined with C, the