"Squaring the Circle": a History of the Problem/Chapter 3

by Ernest William Hobson

CHAPTER III

The Second Period

The new Analysis

The foundations of the new Analysis were laid in the second half of the seventeenth century when Newton (1642—1727) and Leibnitz (1646—1716) founded the Differential and Integral Calculus, the ground having been to some extent prepared by the labours of Huyghens, Fermat, Wallis, and others. By this great invention of Newton and Leibnitz, and with the help of the brothers James Bernouilli (1654—1705) and John Bernouilli (1667—1748), the ideas and methods of Mathematicians underwent a radical transformation which naturally had a profound effect upon our problem. The first effect of the new Analysis was to replace the old geometrical or semigeometrical methods of calculating $\pi$ by others in which analytical expressions formed according to definite laws were used, and which could be employed for the calculation of $\pi$ to any assigned degree of approximation.

The work of John Wallis

The first result of this kind was due to John Wallis (1616—1703), Undergraduate at Emmanuel College, Fellow of Queens' College, and afterwards Savilian Professor of Geometry at Oxford. He was the first to formulate the modern arithmetic theory of limits, the fundamental importance of which, however, has only during the last half century received its due recognition; it is now regarded as lying at the very foundation of Analysis. Wallis gave in his Arithmetica Infinitorum the expression

${\frac {\pi }{2}}={\frac {2}{1}}.{\frac {2}{3}}.{\frac {4}{3}}.{\frac {4}{5}}.{\frac {6}{5}}.{\frac {6}{7}}.{\frac {8}{7}}.{\frac {8}{9}}\ldots$

for $\pi$ as an infinite product, and he shewed that the approximation obtained by stopping at any fraction in the expression on the right is in defect or in excess of the value ${\frac {\pi }{2}}$ according as the fraction is proper or improper. This expression was obtained by an ingenious method depending upon the expression for ${\frac {\pi }{8}}$ the area of a semi-circle of diameter 1 as the definite integral $\int _{0}^{1}{\sqrt {x-x^{2}}}dx$. The expression has the advantage over that of Vieta that the operations required by it are all rational ones.

Lord Brouncker (1620—1684), the first President of the Royal Society, communicated without proof to Wallis the expression

${\frac {4}{\pi }}=1+{\frac {1}{2+}}{\frac {9}{2+}}{\frac {25}{2+}}{\frac {49}{2+}}$,

a proof of which was given by Wallis in his Arithmetica Infinitorum. It was afterwards shewn by Euler that Wallis' formula could be obtained from the development of the sine and cosine in infinite products, and that Brouncker's expression is a particular case of much more general theorems.

The calculation of π by series

The expression from which most of the practical methods of calculating $\pi$ have been obtained is the series which, as we now write it, is given by

$(-1\leq x\leq 1)$.

$\textstyle \tan ^{-1}{x}=x-{\frac {1}{3}}x^{3}+{\frac {1}{3}}x^{5}-\ldots$

This series was discovered by Gregory (1670) and afterwards independently by Leibnitz (1673). In Gregory's time the series was written as

$a=t-{\frac {t}{3r^{2}}}+{\frac {t}{5r^{4}}}-\ldots$

where $a$, $t$, $r$ denote the length of an arc, the length of a tangent at one extremity of the arc, and the radius of the circle; the definition of the tangent as a ratio had not yet been introduced. The particular case

${\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-\ldots$

is known as Leibnitz's series; he discovered it in 1674 and published it in 1682, with investigations relating to the representation of $\pi$, in his work "De vera proportione circuli ad quadratum circumscriptum in numeris rationalibus." The series was, however, known previously to Newton and Gregory.

By substituting the values ${\frac {\pi }{6}}$, ${\frac {\pi }{8}}$, ${\frac {\pi }{10}}$, ${\frac {\pi }{12}}$ in Gregory's series, the calculation of $\pi$ up to 72 places was carried out by Abraham Sharp under instructions from Halley (Sherwin's Mathematical Tables, 1705, 1706).

The more quickly convergent series

$\sin ^{-1}{x}=x+{\frac {1}{2}}{\frac {x^{3}}{3}}+{\frac {1{\,.\,}3}{2{\,.\,}4}}{\frac {x^{5}}{5}}+\ldots$,

discovered by Newton, is troublesome for purposes of calculation, owing to the form of the coefficients. By taking $\textstyle x={\frac {1}{2}}$, Newton himself calculated $\pi$ to 14 places of decimals.

Euler and others occupied themselves in deducing from Gregory's series formulae by which $\pi$ could be calculated by means of rapidly converging series.

Euler, in 1737, employed special cases of the formula

$\tan ^{-1}{\frac {1}{p}}=\tan ^{-1}{\frac {1}{p+q}}+\tan ^{-1}{\frac {q}{p^{2}+pq+1}}$,

and gave the general expression

$\tan ^{-1}{\frac {x}{y}}=\tan ^{-1}{\frac {ax-y}{ay+x}}+\tan ^{-1}{\frac {b-a}{ab+1}}+\tan ^{-1}{\frac {c-b}{cb+1}}+\ldots$,

from which more such formulae could be obtained. As an example, we have, if $a,b,c,\ldots$ are taken to be the uneven numbers, and ${\frac {x}{y}}=1$,

${\frac {\pi }{4}}=\tan ^{-1}{\frac {1}{2}}+\tan ^{-1}{\frac {1}{2{\,.\,}4}}+\tan ^{-1}{\frac {1}{2{\,.\,}9}}+\ldots$.

In the year 1706, Machin (1680—1752), Professor of Astronomy in London, employed the series

${\frac {\pi }{4}}=4\left({\frac {1}{5}}-{\frac {1}{3{\,.\,}5^{3}}}+{\frac {1}{5{\,.\,}5^{5}}}-{\frac {1}{7{\,.\,}5^{7}}}+\ldots \right)$${}-\left({\frac {1}{239}}-{\frac {1}{3{\,.\,}239^{3}}}+{\frac {1}{5{\,.\,}239^{5}}}-{\frac {1}{7{\,.\,}239^{7}}}+\ldots \right)$,

which follows from the relation

${\frac {\pi }{4}}=4\tan ^{-1}{\frac {1}{5}}-\tan ^{-1}{\frac {1}{239}}$,

to calculate $\pi$ to 100 places of decimals. This is a very convenient expression, because in the first series ${\frac {1}{5}},{\frac {1}{5^{3}}},\ldots$ can be replaced by ${\frac {4}{100}}$, ${\frac {64}{1000000}}$, &c., and the second series is very rapidly convergent.

In 1719, de Lagny (1660—1734), of Paris, determined in two different ways the value of $\pi$ up to 127 decimal places. Vega (1754—1802) calculated $\pi$ to 140 places, by means of the formulae

${\frac {\pi }{4}}=5\tan ^{-1}{\frac {1}{7}}+\tan ^{-1}{\frac {3}{79}}=2\tan ^{-1}{\frac {1}{3}}+\tan ^{-1}{\frac {1}{7}}$,

due to Euler, and shewed that de Lagny's determination was correct with the exception of the 113th place, which should be 8 instead of 7.

Clausen calculated in 1847, 248 places of decimals by the use of Machin's formula and the formula

${\frac {\pi }{4}}=2\tan ^{-1}{\frac {1}{3}}+\tan ^{-1}{\frac {1}{7}}$.

In 1841, 208 places, of which 152 are correct, were calculated by Rutherford by means of the formula

${\frac {\pi }{4}}=4\tan ^{-1}{\frac {1}{5}}-\tan ^{-1}{\frac {1}{70}}+\tan ^{-1}{\frac {1}{99}}$.

In 1844 an expert reckoner, Zacharias Dase, employed the formula

${\frac {\pi }{4}}=\tan ^{-1}{\frac {1}{2}}+\tan ^{-1}{\frac {1}{5}}+\tan ^{-1}{\frac {1}{8}}$,

supplied to him by Prof. Schultz, of Vienna, to calculate $\pi$ to 200 places of decimals, a feat which he performed in two months.

In 1853 Rutherford gave 440 places of decimals, and in the same year W. Shanks gave first 530 and then 607 places (Proc. R. S., 1853).

Richter, working independently, gave in 1853 and 1855, first 333, then 400 and finally 500 places.

Finally, W. Shanks, working with Machin's formula, gave (1873—74) 707 places of decimals.

Another series which has also been employed for the calculation of $\pi$ is the series

$\tan ^{-1}{t}={\frac {t}{1+t^{2}}}\left\{1+{\frac {2}{3}}{\frac {t^{2}}{1+t^{2}}}+{\frac {2{\,.\,}4}{3{\,.\,}5}}\left({\frac {t^{2}}{1+t^{2}}}\right)^{2}+{\frac {2{\,.\,}4{\,.\,}6}{3{\,.\,}5{\,.\,}7}}\left({\frac {t^{2}}{1+t^{2}}}\right)^{3}+\dots \right\}$.

This was given in the year 1755 by Euler, who, applying it in the formula

$\textstyle \pi =20\tan ^{-1}{\frac {1}{7}}+8\tan ^{-1}{\frac {3}{79}}$,

calculated $\pi$ to 20 places, in one hour as he states. The same series was also discovered independently by Ch. Hutton (Phil. Trans., 1776). It was later rediscovered by J. Thomson and by De Morgan.

An expression for $\pi$ given by Euler may here be noticed; taking the identity

$\tan ^{-1}{\frac {x}{2-x}}=2\int _{0}^{x}{\frac {dx}{4+x^{4}}}+2\int _{0}^{x}{\frac {xdx}{4+x^{4}}}+\int _{0}^{x}{\frac {x^{2}dx}{4+x^{4}}}$,

he developed the integrals in series, then put $\textstyle x={\frac {1}{2}}$, $\textstyle x={\frac {1}{4}}$, obtaining series for $\textstyle \tan ^{-1}{\frac {1}{3}}$, $\textstyle \tan ^{-1}{\frac {1}{7}}$, which he substituted in the formula

${\frac {\pi }{4}}=2\tan ^{-1}{\frac {1}{3}}+\tan ^{-1}{\frac {1}{7}}$.

In China a work was published by Imperial order in 1713 which contained a chapter on the quadrature of the circle where the first 19 figures in the value of $\pi$ are given.

At the beginning of the eighteenth century, analytical methods were introduced into China by Tu Tê-mei (Pierre Jartoux) a French missionary; it is, however, not known how much of his work is original, or whether he borrowed the formulae he gave directly from European Mathematicians.

One of his series

$\pi =3\left(1+{\frac {1^{2}}{4{\,.\,}6}}+{\frac {1^{2}{\,.\,}3^{2}}{4{\,.\,}6{\,.\,}8{\,.\,}10}}+{\frac {1^{2}{\,.\,}3^{2}{\,.\,}5^{2}}{4{\,.\,}6{\,.\,}8{\,.\,}10{\,.\,}12{\,.\,}14}}+\ldots \right)$

was employed at the beginning of the nineteenth century by ChuHung for the calculation of $\pi$. By this means 25 correct figures were obtained.

Tsêng Chi-hung, who died in 1877, published values of $\pi$ and $1/\pi$ to 100 places. He is said to have obtained his value of $\pi$ in a month, by means of the formula

${\frac {\pi }{4}}=\tan ^{-1}{\frac {1}{2}}+\tan ^{-1}{\frac {1}{3}}$

and Gregory's series.

In Japan, where a considerable school of Mathematics was developed in the eighteenth century, $\pi$ was calculated by Takebe in 1722 to 41 places, by employment of the regular 1024agon. It was calculated by Matsunaga in 1739 to 50 places by means of the same series as had been employed by Chu-Hung.

The rational values $\textstyle \pi ={\frac {5419351}{1725033}}$, $\textstyle \pi ={\frac {428224593349304}{136308121570117}}$, correct to 12 and 30 decimal places respectively, were given by Arima in 1766.

Kurushima Yoshita (died 1757) gave for $\pi ^{2}$ the approximate values ⁠227/13⁠, ⁠10748/1089⁠, ⁠10975/1112⁠, ⁠98548/9885⁠.

Tanyem Shǒkei published in 1728 the series

$\pi ^{2}=8\left(1+{\frac {1}{6}}+{\frac {1{\,.\,}4}{6{\,.\,}15}}+{\frac {1{\,.\,}4{\,.\,}9}{6{\,.\,}15{\,.\,}28}}+\ldots \right)$,
$\pi ^{2}=4\left(1+{\frac {2}{6}}+{\frac {2{\,.\,}8}{6{\,.\,}15}}+{\frac {2{\,.\,}8{\,.\,}18}{6{\,.\,}15{\,.\,}28}}+\ldots \right)$

due to Takebe, and ultimately to Jartoux.

The following series published in 1739 by Matsunaga may be mentioned:

$\pi =9\left(1+{\frac {1^{2}}{3{\,.\,}4}}+{\frac {1^{2}{\,.\,}2^{2}}{3{\,.\,}4{\,.\,}5{\,.\,}6}}+{\frac {1^{2}{\,.\,}2^{2}{\,.\,}3^{2}}{3{\,.\,}4{\,.\,}5{\,.\,}6{\,.\,}7{\,.\,}8}}+\ldots \right)$.

The work of Euler

Developments of the most far-reaching importance in connection with our subject were made by Leonhard Euler, one of the greatest Analysts of all time, who was born at Basel in 1707 and died at St Petersburg in 1783. With his vast influence on the development of Mathematical Analysis in general it is impossible here to deal, but some account must be given of those of his discoveries which come into relation with our problem.

The very form of modern Trigonometry is due to Euler. He introduced the practice of denoting each of the sides and angles of a triangle by a single letter, and he introduced the short designation of the trigonometrical ratios by $\sin \alpha$, $\cos \alpha$, $\tan \alpha$, &c. Before Euler's time there was great prolixity in the statement of propositions, owing to the custom of denoting these expressions by words, or by letters specially introduced in the statement. The habit of denoting the ratio of the circumference to the diameter of a circle by the letter $\pi$, and the base of the natural system of logarithms by $e$, is due to the influence of the works of Euler, although the notation $\pi$ appears as early as 1706, when it was used by William Jones in the Synopsis palmariorum Matheseos. In Euler's earlier work he frequently used $p$ instead of $\pi$, but by about 1740 the letter $\pi$ was used not only by Euler but by other Mathematicians with whom he was in correspondence.

A most important improvement which had a great effect not only upon the form of Trigonometry but also on Analysis in general was the introduction by Euler of the definition of the trigonometrical ratios in order to replace the old sine, cosine, tangent, &c., which were the lengths of straight lines connected with the circular arc. Thus these trigonometrical ratios became functions of an angular magnitude, and therefore numbers, instead of lengths of lines related by equations with the radius of the circle. This very important improvement was not generally introduced into our text books until the latter half of the nineteenth century.

This mode of regarding the trigonometrical ratios as analytic functions led Euler to one of his greatest discoveries, the connection of these functions with the exponential function. On the basis of the definition of $e$ by means of the series

$1+z+{\frac {z^{2}}{2!}}+{\frac {z^{3}}{3!}}+\ldots$,

he set up the relations

$\cos {x}={\frac {e^{ix}+e^{-ix}}{2}}$, $\sin {x}={\frac {e^{ix}-e^{-ix}}{2}}$,

which can also be written

$e^{ix}=\cos {x}+i\sin {x}$, $e^{-ix}=\cos {x}-i\sin {x}$.

The relation $e^{i\pi }=-1$, which Euler obtained by putting $x=\pi$, is the fundamental relation between the two numbers $\pi$ and $e$ which was indispensable later on in making out the true nature of the number $\pi$.

In his very numerous memoirs, and especially in his great work, Introductio in analysin infinitorum (1748), Euler displayed the most wonderful skill in obtaining a rich harvest of results of great interest, largely dependent on his theory of the exponential function. Hardly any other work in the history of Mathematical Science gives to the reader so strong an impression of the genius of the author as the Introductio. Many of the results given in that work are obtained by bold generalizations, in default of proofs which would now be regarded as completely rigorous; but this it has in common with a large part of all Mathematical discoveries, which are often due to a species of divining intuition, the rigorous demonstrations and the necessary restrictions coming later. In particular there may be mentioned the expressions for the sine and cosine functions as infinite products, and a great number of series and products deduced from these expressions; also a number of expressions relating the number $e$ with continued fractions which were afterwards used in connection with the investigation of the nature of that number.

Great as the progress thus made was, regarded as preparatory to a solution of our problem, nothing definite as to the true nature of the number $\pi$ was as yet established, although Mathematicians were convinced that $e$ and $\pi$ are not roots of algebraic equations. Euler himself gave expression to the conviction that this is the case. Somewhat later, Legendre gave even more distinct expression to this view in his Éléments de Géométrie (1794), where he writes: " It is probable that the number $\pi$ is not even contained among the algebraical irrationalities, i.e. that it cannot be a root of an algebraical equation with a finite number of terms, whose coefficients are rational. But it seems to be very difficult to prove this strictly."