A2n = √AnA′n (Snell’s Cyclom.),
| A′2n = | 2An A′n | or | 2A′n A2n | (Gregory), |
| An + A′2n | A′n + A2n |
where An, A′n are the areas of the inscribed and the circumscribed regular n-gons respectively. He also gave approximate rectifications of circular arcs after the manner of Huygens; and, what is very notable, he made an ingenious and, according to J.E. Montucla, successful attempt to show that quadrature of the circle by a Euclidean construction was impossible.[20] Besides all this, however, and far beyond it in importance, was his use of infinite series. This merit he shares with his contemporaries N. Mercator, Sir I. Newton and G.W. Leibnitz, and the exact dates of discovery are a little uncertain. As far as the circle-squaring functions are concerned, it would seem that Gregory was the first (in 1670) to make known the series for the arc in terms of the tangent, the series for the tangent in terms of the arc, and the secant in terms of the arc; and in 1669 Newton showed to Isaac Barrow a little treatise in manuscript containing the series for the arc in terms of the sine, for the sine in terms of the arc, and for the cosine in terms of the arc. These discoveries formed an epoch in the history of mathematics generally, and had, of course, a marked influence on after investigations regarding circle-quadrature. Even among the mere computers the series
θ = tan - 1⁄3 tan3 θ + 1⁄5 tan5 θ - ...,
specially known as Gregory’s series, has ever since been a necessity of their calling.
The calculator’s work having now become easier and more mechanical, calculation went on apace. In 1699 Abraham Sharp, on the suggestion of Edmund Halley, took Gregory’s series, and, putting tan θ = 1⁄3√3, found the ratio equal to
| √12 ( 1 - | 1 | + | 1 | - | 1 | + ... ), |
| 3 · 3 | 5 · 3² | 7 · 3³ |
from which he calculated it correct to 71 fractional places.[21] About the same time John Machin calculated it correct to 100 places, and, what was of more importance, gave for the ratio the rapidly converging expression
| 16 | ( 1 - | 1 | + | 1 | - | 1 | + ... ) - | 4 | ( 1 - | 1 | + | 1 | - ... ), |
| 5 | 3 · 5² | 5 · 54 | 7 · 56 | 239 | 3 · 239² | 5 · 2394 |
which long remained without explanation.[22] Fautet de Lagny, still using tan 30°, advanced to the 127th place.[23]