Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/45

where $p_{n}$ is the perimeter of a regular inscribed polygon of $n$ sides, and $C$ is the circumference of the circle.

The work of Gregory

The last Mathematician to be mentioned in connection with the development of the method of Archimedes is James Gregory (1638—1675), Professor in the Universities of St Andrews and Edinburgh, whose important work in connection with the development of the new Analysis we shall have to refer to later. Instead of employing the perimeters of successive polygons, he calculated their areas, using the formulae

$A_{2n}'={\frac {2A_{n}A_{n}'}{A_{n}+A_{2n}}}={\frac {2A_{n}'A_{2n}}{A_{n}'+A_{2n}}}$;

where $A_{n}$, $A_{n}'$ denote the areas of in- and circum-scribed regular $n$-agons; he also employed the formula $A_{2n}={\sqrt {A_{n}A_{n}'}}$ which had been obtained by Snellius. In his work Exercitationes geometricae published in 1668, he gave a whole series of formulae for approximations on the lines of Archimedes. But the most interesting step which Gregory took in connection with the problem was his attempt to prove, by means of the Archimedean algorithm, that the quadrature of the circle is impossible. This is contained in his work Vera circuli et hyperbolae quadratura which is reprinted in the works of Huyghens (Opera varia I, pp. 315—328) who gave a refutation of Gregory's proof. Huyghens expressed his own conviction of the impossibility of the quadrature, and in his controversy with Wallis remarked that it was not even decided whether the area of the circle and the square of the diameter are commensurable or not. In default of a theory of the distinction between algebraic and transcendental numbers, the failure of Gregory's proof was inevitable. Other such attempts were made by Lagny (Paris Mém. 1727, p. 124), Saurin (Paris Mém. 1720), Newton (Principia I, 6, Lemma 28), and Waring (Proprietates algebraicarum curvarum) who maintained that no algebraical oval is quadrable. Euler also made some attempts in the same direction (Considerationes cydometricae, Novi Comm. Acad. Petrop. xvi, 1771); he observed that the irrationality of $\pi$ must first be established, but that this would not of itself be sufficient to prove the impossibility of the quadrature. Even as early as 1544, Michael Stifel, in his Arithmetic integra, expressed the opinion that the construction is impossible. He emphasized the distinction between a theoretical and a practical construction.