actional 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 sin squared 1/2[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]--

___________________
__________ / ___________
___ / ___ / / ___
1/2 \/ 1/2 . \/ 1/2 + 1/2\/ 1/2 . \/ 1/2 + 1/2 \/ 1/2 + 1/2 \/ 1/2 ...

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 31415926535