8979323846264338327950288
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 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 = (1/3)AOC.[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 1/2[theta] > [theta] > 2sin 1/2 [theta] (Archimedes);

3 sin [theta]
tan (1/3)[theta] + 2sin (1/3)[theta] > [theta] > --------------- (Snell).