*From `A Short Account of the History of Mathematics' (4th edition, 1908)
by W. W. Rouse Ball.*

*James Gregory*, born at Drumoak near Aberdeen in 1638, and
died at Edinburgh in October 1675, was successively professor at
St. Andrews and Edinburgh. In 1660 he published his Optica
Promota, in which the reflecting telescope known by his name
is described. In 1667 he issued his Vera Circuli et Hyperbolae
Quadratura, in which he shewed how the areas of the circle
and hyperbola could be obtained in the form of infinite convergent
series, and here (I believe for the first time) we find a distinction
drawn between convergent and divergent series. This work contains
a remarkable geometrical proposition to the effect that the ratio
of the area of any arbitrary sector of a circle to that of the
inscribed or circumscribed regular polygons is not expressible by
a finite number of terms. Hence he inferred that the quadrature
of the circle was impossible; this was accepted by Montucla, but
it is not conclusive, for it is conceivable that some particular
sector might be squared, and this particular sector might be the
whole circle. This book contains also the earliest enunciation of
the expansions in series of sin *x*, cos *x*,
*x* or arc sin *x*, and
*x* or arc cos *x*. It was reprinted in 1668 with an
appendix, Geometriae Pars, in which Gregory explained
how the volumes of solids of revolution could be determined. In
1671, or perhaps earlier, he established the theorem that

the result being true only if lie between - and . This is the theorem on which many of the subsequent calculations of approximations to the numerical value of have been based.

This page is included in a collection of mathematical biographies taken from A Short Account of the History of Mathematics by W. W. Rouse Ball (4th Edition, 1908).

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