the various series.[24]
With him, apparently, began the usage of denoting by [pi] the ratio of
the circumference to the diameter.[25]

The most important publication, however, on the subject in the 18th
century was a paper by J.H. Lambert,[26] read before the Berlin Academy
in 1761, in which he demonstrated the irrationality of [pi]. The general
test of irrationality which he established is that, if

a1 a2 a2
-- -- -- ...
b1 +- b2 +- b3 +-

be an interminate continued fraction, a1, a2, ..., b1, b2 ... be
integers, a1/b1, a2/b2, ... be proper fractions, and the value of every
one of the interminate continued fractions

a1 a2
-- , -- , ... be < 1,
b1 +- ... b2 +- ...

then the given continued fraction represents an irrational quantity. If
this be applied to the right-hand side of the identity

m m m squared m squared
tan --- = --- ---- ---- ...
n n - 3n - 5n

it follows that the tangent of every arc commensurable with the radius
is irrational, so that, as a particular case, an arc of 45 deg., having
its tangent rational, must be incommensurable with the radius; that is to
say, [pi]/4 is an incommensurable number.[27]

This incontestable result had no effect, apparently, in repressing the
[pi]-computers. G. von Vega in 1789, using series like Machin's, viz.
Gregory's series and the identities

[pi]/4 = 5tan^{-1} (1/7) + 2tan^{-1} (3/79) (Euler, 1779),
[pi]/4 = tan^{-1} (1/7) + 2tan^{-1} ( 1/3) (Hutton, 1776),

neither of which was nearly so advantageous as several found by Charles
Hutton, calculated [pi] correct to 136 places.[28] This achievement was
anticipated or outdone by an unknown calculator, whose manuscript was
seen in the Radcliffe library, Oxford, by Baron von Zach towards the end
of the century, and contained the ratio correct to 152 places. More
astonishing still have been the deeds of the [pi]-computers of the 19th
century. A condensed record compiled by J.W.L. Glaisher (_Messenger of
Math._ ii. 122) is as follows:--

+-----+------------+-----------------+--------------------------------------------+
| | |No. of fr. digits| |
|Date.| Computer. +--------+--------+ Place of Publication. |
| | | calcd. |correct.| |
+-----+------------+--------+--------+-------