Archimedes, we may direct our attention either to the infinite
series of geometrical operations or to the corresponding infinite
series of arithmetical operations. Denoting the number of units
in AB by ¼*c*, we can express BB_{1}, B_{1}B_{2}, . . . in terms of ¼*c*, and
the identity AB_{∞} = AB + BB_{1} + B_{1}B_{2} + . . . gives us at once
an expression for the diameter in terms of the circumference by
means of an infinite series.^{[1]} The proof of the correctness of the
construction is seen to be involved in the following theorem,
which 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), AB_{1} is the diameter of the
circle circumscribed by the regular 8-gon having the same
perimeter as the square, AB_{2} 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.^{[2]}

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*²)^{½} as a member of the series of
curves *y* = (1 − *x*²)¹, *y* = (1 − *x*²)², &c., he was led to the proposition
that four times the reciprocal of the ratio of the circumference
to the diameter, *i.e.* 4/π, 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 + 1²2

+ 3²2

+ 5²2

+ 7²2 . . .

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^{[3]} 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

_{2n}= (Snell’s

*Cyclom.*),

*A*′_{2n} = 2A_{n} A′_{n}A_{n} + A′_{2n} or 2A′_{n} A_{2n}A′_{n} + A_{2n} (Gregory),

where A_{n}, 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.^{[4]} 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 − 13 tan^{3} θ + 15 tan^{5} 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 = 13√3, found the ratio equal to

from which he calculated it correct to 71 fractional places.^{[5]}
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

^{4}– 17 · 5

^{6}+ . . .) – 4239(1 – 13 · 239² + 15 · 239

^{4}– . . .),

which long remained without explanation.^{[6]} Fautet de Lagny, still using tan 30°, advanced to the 127th place.^{[7]}

Leonhard Euler took up the subject several times during his
life, effecting mainly improvements in the theory of the various
series.^{[8]} With him, apparently, began the usage of denoting
by π the ratio of the circumference to the diameter.^{[9]}

The most important publication, however, on the subject
in the 18th century was a paper by J. H. Lambert,^{[10]} 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

*a*_{1}*b*_{1} ± *a*_{2}*b*_{2} ± *a*_{3}*b*_{3} ± . . .

be an interminate continued fraction, *a*_{1}, *a*_{2}, . . ., *b*_{1}, *b*_{2} . . .
be integers, *a*_{1}/*b*_{1}, *a*_{2}/*b*_{2}, . . . be proper fractions, and the value
of every one of the interminate continued fractions

*a*_{1}*b*_{1} ± . . .,*a*_{2}*b*_{2} ± . . ., . . . 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**n* = *m**n* – *m*²3*n* – *m*²5*n* . . .

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.^{[11]}

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} 17 + 2 tan^{–1} 379 (Euler, 1779),

π/4 = tan^{–1} 17 + 2 tan^{–1} 13 (Hutton, 1776),

neither of which was nearly so advantageous as several found by Charles Hutton, calculated π correct to 136 places.^{[12]} This achievement was anticipated or outdone by an unknown calculator, whose manuscript was seen in the Radcliffe library, Oxford, by Baron von Zach towards the end of the century, and contained the ratio correct to 152 places. More astonishing still have been the deeds of the π-computers of the 19th century.

- ↑ See Euler, “Annotationes in locum quendam Cartesii,” in
*Nov. Comm. Acad. Petrop.*viii. - ↑ Gergonne,
*Annales de math.*vi. - ↑ See
*Vera Circuli et Hyperbolae Quadratura*(Padua, 1667); and the*Appendicula*to the same in his*Exercitationes geometricae*(London, 1668). - ↑
*Penny Cyclop.*xix. 187. - ↑ See Sherwin’s
*Math. Tables*(London, 1705), p. 59. - ↑ See W. Jones,
*Synopsis Palmariorum Matheseos*(London, 1706); Maseres,*Scriptores Logarithmici*(London, 1791–1796), iii. 159 seq.; Hutton,*Tracts*, i. 266. - ↑ See
*Hist. de l’Acad.*(Paris, 1719); 7 appears instead of 8 in the 113th place. - ↑
*Comment. Acad. Petrop.*ix., xi.;*Nov. Comm. Ac. Pet.*xvi.;*Nova Acta Acad. Pet.*xi. - ↑
*Introd. in Analysin Infin.*(Lausanne, 1748), chap. viii. - ↑
*Mém. sur quelques propriétés remarquables des quantités transcendantes, circulaires, et logarithmiques.* - ↑ See Legendre,
*Eléments de géométrie*(Paris, 1794), note iv.; Schlömilch,*Handbuch d. algeb. Analysis*(Jena, 1851), chap. xiii. - ↑
*Nova Acta Petrop.*ix. 41;*Thesaurus Logarithm. Completus*, 633.