hich serves likewise to
throw new light on the subject:--AB being any straight line whatever, and
the above construction being made, then AB is the diameter of the circle
circumscribed by the square ABCD (self-evident), AB1 is the diameter of
the circle circumscribed by the regular 8-gon having the same perimeter
as the square, AB2 is the diameter of the circle circumscribed by the
regular 16-gon having the same perimeter as the square, and so on.
Essentially, therefore, Descartes's process is that known later as the
process of _isoperimeters_, and often attributed wholly to Schwab.[18]

In 1655 appeared the _Arithmetica Infinitorum_ of John Wallis, where
numerous problems of quadrature are dealt with, the curves being now
represented in Cartesian co-ordinates, and algebra playing an important
part. In a very curious manner, by viewing the circle y = (1 - x squared)^1/2
as a member of the series of curves y = (1 - x squared)1, y = (1 - x
squared) squared, &c., he was led to the proposition that four times the
reciprocal of the ratio of the circumference to the diameter, i.e. 4/[pi],
is equal to the infinite product

3 . 3 . 5 . 5 . 7 . 7 . 9 ...
-----------------------------;
2 . 4 . 4 . 6 . 6 . 8 . 8 ...

and, the result having been communicated to Lord Brounker, the latter
discovered the equally curious equivalent continued fraction

1 squared 3 squared 5 squared 7 squared
1 + --- --- --- --- ...
2 + 2 + 2 + 2

The work of Wallis had evidently an important influence on the next
notable personality in the history of the subject, James Gregory, who
lived during the period when the higher algebraic analysis was coming
into power, and whose genius helped materially to develop it. He had,
however, in a certain sense one eye fixed on the past and the other
towards the future. His first contribution[19] was a variation of the
method of Archimedes. The latter, as we know, calculated the perimeters
of successive polygons, passing from one polygon to another of double
the number of sides; in a similar manner Gregory calculated the areas.
The general theorems which enabled him to do this, after a start had
been made, are

_____
A2n = \/AnA'n (Snell's _Cyclom._),

2 An A'n 2 A'n A2n
A'2n = ---------- or ----------- (Gregory),
An + A'2n A'n + A2n

where An, A'n are the areas of the inscribed and the circumscribed
regular n-gons