Leonhard Euler took up the subject several times during his life, effecting mainly improvements in the theory of the various series.[24] With him, apparently, began the usage of denoting by π 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 π. The general test of irrationality which he established is that, if
| a1 | a2 | a3 | ... | |||
| 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 | ... | ||
| b1 | ± ..., | b2 | ± ..., |
be < 1, then the given continued fraction represents an irrational quantity. If this be applied to the right-hand side of the identity
| tan | m | = | m | m² | m² | ... | ||
| 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°, having its tangent rational, must be incommensurable with the radius; that is to say, π⁄4 is an incommensurable number.[27]
This incontestable result had no effect, apparently, in repressing the π-computers. G. von Vega in 1789, using series like Machin’s, viz. Gregory’s series and the identities
π⁄4 = 5 tan-1 1⁄7 + 2 tan-1 3⁄79 (Euler, 1779),