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

[theta] = tan - (1/3) tan^3 [theta] + (1/5) tan^5 [theta] - ...,

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 [theta] = (1/3)
sqrt(3), found the ratio equal to

__ / 1 1 1 \
\/12 ( 1 - ----- + ------ - ------ + ... ),
\ 3 . 3 5 . 3 squared 7 . 3 cubed /

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 \
-- ( ---- + ----- - ----- + ... ) -
5 \ 3.5 squared 5.5^4 7.5^6 /

4 / 1 1 \
--- ( 1 - ------ + ------- - ... ),
239 \ 3.239 squared 5.239^4 /

which long remained without explanation.[22] Fautet de Lagny, still
using tan 30 deg., advanced to the 127th place.[23]

Leonhard Euler took up the subject several times during his life,
effecting mainly improvements in the theory of