![{\displaystyle {\begin{array}{l}\mathrm {ou} \;a=+1.{\frac {3.3}{2.4}}\cdot {\frac {5.5}{4.6}}\cdot {\frac {7.7}{6.8}}\,\mathrm {etc.} \\b=-{\frac {1.1}{2.4}}\cdot {\frac {5.5}{2.8}}\cdot {\frac {7.7}{4.10}}\cdot {\frac {9.9}{6.12}}\,\mathrm {etc.} \\c=+{\frac {1.1}{4.6}}\cdot {\frac {3.3}{2.8}}\cdot {\frac {7.7}{2.12}}\cdot {\frac {9.9}{4.14}}\cdot {\frac {11.11}{6.16}}\,\mathrm {etc.} \\d=-{\frac {1.1}{6.8}}\cdot {\frac {3.3}{4.10}}\cdot {\frac {5.5}{2.12}}\cdot {\frac {9.9}{2.16}}\cdot {\frac {11.11}{4.18}}\cdot {\frac {13.13}{6.20}}\,\mathrm {etc.} \\e=+{\frac {1.1}{8.10}}\cdot {\frac {3.3}{6.12}}\cdot {\frac {5.5}{4.14}}\cdot {\frac {7.7}{2.16}}\cdot {\frac {11.11}{2.20}}\cdot {\frac {13.13}{4.22}}\cdot {\frac {15.15}{6.24}}\,\mathrm {etc.} \\f=-{\frac {1.1}{10.12}}\cdot {\frac {3.3}{8.14}}\cdot {\frac {5.5}{6.16}}\cdot {\frac {7.7}{4.18}}\cdot {\frac {9.9}{2.20}}\cdot {\frac {13.13}{2.24}}\cdot {\frac {15.15}{4.26}}\cdot {\frac {17.17}{6.28}}\,\mathrm {etc.} \\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a19410f69838762949ab9186f664e5ce03a106)
La quantité
ou le quart de la circonférence équivaut,
suivant le théorème de Wallis, à
![{\displaystyle {\tfrac {2.2}{1.3}}\cdot {\tfrac {4.4}{3.5}}\cdot {\tfrac {6.6}{5.7}}\cdot {\tfrac {8.8}{7.9}}\cdot {\tfrac {10.10}{9.11}}\cdot {\tfrac {12.12}{11.13}}\cdot {\tfrac {14.14}{13.15}}\;\mathrm {etc.} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/24d518d684f25a82b9a224cb47f5499b2a631d26)
Si l’on remarque maintenant quelles sont, dans les valeurs
de
etc., les facteurs que l’on doit écrire aux
numérateurs et aux dénominateurs, pour y compléter la
double série des nombres impairs et des nombres pairs, on
trouvera que les facteurs à suppléer sont :
![{\displaystyle {\begin{array}{lll}\left.{\begin{array}{lc}\mathrm {pour} \;b&\!{\dfrac {3.3}{6}}\\\mathrm {pour} \;c&\!{\dfrac {5.5}{10}}\\\mathrm {pour} \;d&\!{\dfrac {7.7}{14}}\\\mathrm {pour} \;e&\!{\dfrac {9.9}{18}}\\\mathrm {pour} \;f&\!{\dfrac {11.11}{22}}\\\end{array}}\right\}&\mathrm {et} \;\mathrm {l'on} \;\mathrm {en} \;\mathrm {conclut} &\left\{{\begin{array}{rcrc}a&\!\!=\!\!&2.\!\!&\!\!{\dfrac {2}{\pi }}\\b&\!\!=\!\!&\!-2.\!\!&\!\!{\dfrac {2}{3\pi }}\\c&\!\!=\!\!&2.\!\!&\!\!{\dfrac {2}{5\pi }}\\d&\!\!=\!\!&\!-2.\!\!&\!\!{\dfrac {2}{7\pi }}\\e&\!\!=\!\!&2.\!\!&\!\!{\dfrac {2}{9\pi }}\\f&\!\!=\!\!&\!-2.\!\!&\!\!{\dfrac {2}{11\pi }}\\\end{array}}\right.\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d474c6f0d3a3563f5c8c39146cb98621254a90eb)