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 “π-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 |
| 100000000000000000000000000000000000 | |
| et minus | |
| quam | 314159265358979323846264338327950289 |
| 100000000000000000000000000000000000 ... | |
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]
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 part D′F outside the circle was equal to r; in other words, by a non-Euclidean construction he trisected the angle AOC, for it is readily seen that, since FD′ = FO = OC, the angle FOB = 1⁄3AOC.[14] This couplet of constructions is as important from the calculator’s point of view as it is interesting geometrically. To compare it on this score with the fundamental proposition of Archimedes, the latter must be put into a form similar to Snell’s. AMC being an arc of a circle (see fig. 11) whose centre is O, AC its chord, and HK the tangent drawn at the middle point of the arc and bounded by OA, OC produced, then, according to Archimedes, AMC < HK, but > AC. In modern trigonometrical notation the propositions to be compared stand as follows:—
2 tan ½θ > θ > 2 sin ½θ (Archimedes);
| tan 1⁄3θ + 2 sin 1⁄3θ > θ > | 3 sin θ | (Snell). |
| 2 + cos θ |