On vérifie cette correction en posant
d’où
il vient ainsi
![{\displaystyle {\frac {a}{1}}-{\frac {a^{2}}{2}}+{\frac {a^{3}}{3}}-{\frac {a^{4}}{4}}+\ldots =\operatorname {Log} .a(1+a),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e844fab9146668e84111f2fe7f10aaae60c4f29a)
comme cela doit être.
J’allais fermer ma lettre lorsque la remarque suivante m’a frappé. On a, comme l’on sait.
![{\displaystyle \operatorname {Cos} .2p=2\operatorname {Cos} .^{2}p-1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e50a600651e4f5fe40f2a63c518851beed72e113)
d’où il résulte
![{\displaystyle \operatorname {Arc} .(\operatorname {Cos} .=p')={\frac {1}{2}}\operatorname {Arc} .\left(\operatorname {Cos} .=2p'^{2}-1\right):}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e74f5648c2388b2dd59c404e6eebaaa2c18304d4)
puis donc qu’on a
![{\displaystyle {\frac {\operatorname {Cos} .x}{1}}+{\frac {1}{2}}{\frac {\operatorname {Cos} .3x}{3}}+{\frac {1.3}{2.4}}{\frac {\operatorname {Cos} .5x}{5}}+\ldots ={\frac {1}{2}}\operatorname {Arc} .(\operatorname {Cos} .=2\operatorname {Sin} .x-1),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb804c24dc4e0883f21e830da08a85a7f903ee80)
on aura aussi
![{\displaystyle {\frac {\operatorname {Cos} .x}{1}}+{\frac {1}{2}}{\frac {\operatorname {Cos} .3x}{3}}+{\frac {1.3}{2.4}}{\frac {\operatorname {Cos} .5x}{5}}+\ldots =\operatorname {Arc} .(\operatorname {Cos} .={\sqrt {\operatorname {Sin} .x}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ce6c45021cbab1fb403769925cac4c04574378d)
résultat extrêmement simple.
Voici une singulière conséquence de ce résultat. On a
![{\displaystyle \operatorname {Arc} .(\operatorname {Cos} .=p)={\frac {\pi }{2}}-\operatorname {Arc} .(\operatorname {Sin} .=p)={\frac {\pi }{2}}-\left({\frac {p}{1}}+{\frac {1}{2}}{\frac {p^{3}}{3}}+{\frac {1.3}{2.4}}{\frac {p^{5}}{5}}+\ldots \right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c567d6e8975be724958cd1c2120fb24ced49377)
en faisant donc
on aura
![{\displaystyle {\frac {\operatorname {Cos} .x}{1}}+{\frac {1}{2}}{\frac {\operatorname {Cos} .3x}{3}}+{\frac {1.3}{2.4}}{\frac {\operatorname {Cos} .5x}{5}}+\ldots =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f94368b01a899bc64e3cd1484703b34c68282d4)
![{\displaystyle {\frac {\pi }{2}}-\left({\frac {\operatorname {Sin} .^{\frac {1}{2}}x}{1}}+{\frac {1}{2}}{\frac {\operatorname {Sin} .^{\frac {3}{2}}x}{3}}+{\frac {1.3}{2.4}}{\frac {\operatorname {Sin} .^{\frac {5}{2}}x}{5}}+\ldots \right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fadcab44d564fd4c43a26332b5e77101347d3d2)
équation d’où on tire cette valeur remarquable du quart de la circonférence