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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 1/4c, we can express BB₁, B₁B₂, . . . in terms of 1/4c, and the identity AB_∞ = AB + BB₁ + B₁B₂ + . . . 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), 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.^[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 − x²)^1/2 as a member of the series of curves y = (1 − x²)¹, y = (1 − x²)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 +  1²/2  
+  3²/2  
+  5²/2  
+  7²/2 . . .

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^[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

A_2n = ${\sqrt {A_{n}A'_{n}}}$ (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 − 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 − 1/3 · 3 + 1/5 · 3² − 1/7 · 3³ + . . . ),

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

16/5(1 – 1/3 · 5² + 1/5 · 5⁴ – 1/7 · 5⁶ + . . .) – 4/239(1 – 1/3 · 239² + 1/5 · 239⁴ – . . .),

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

a₁/b₁  /± a₂/b₂  /± a₃/b₃  /± . . .

be an interminate continued fraction, a₁, a₂, . . ., b₁, b₂ . . . be integers, a₁/b₁, a₂/b₂, . . . be proper fractions, and the value of every one of the interminate continued fractions

a₁/b₁  /±  . . .,a₂/b₂  /± . . ., . . . 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  /– m²/3n  /– m²/5n . . .

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 1/7 + 2 tan^–1 3/79 (Euler, 1779),

π/4 =   tan^–1 1/7 + 2 tan^–1 1/3 (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.

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