8
FACULTÉS
5. Dans l’ouvrage déjà cité (chapitre III, 39) j’ai prouvé que,
étant une fraction positive plus petite que
on a
![{\displaystyle \operatorname {Tang} .h={\frac {(+h)^{{\tfrac {1}{2}}|+1}}{(-h)^{{\tfrac {1}{2}}|-1}}}\;;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23a9772b3e3e54bb0f29fad58142549e05175925)
mais, suivant les réductions enseignées ci-dessus, on a
![{\displaystyle (+h)^{{\tfrac {1}{2}}|+1}={\frac {\left(h-{\tfrac {1}{2}}\right)!}{\left(h-1\right)!}}={\frac {2h}{1+2h}}\cdot {\frac {\left({\tfrac {1}{2}}+h\right)!}{h!}}\;;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd80bcf30db23c9f2a02efb475a683abf95bf88b)
![{\displaystyle (-h)^{{\tfrac {1}{2}}|-1}={\frac {(-h)!}{\left(-h-{\tfrac {1}{2}}\right)!}}={\frac {1-2h}{2-2h}}\cdot {\frac {(1-h)!}{\left({\tfrac {1}{2}}-h\right)!}}\;;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1370f4ac6da2ab35851846a8a4f8c0c321cd017f)
d’où résulte
![{\displaystyle \operatorname {Tang} .h\varpi ={\frac {h(1-h)}{\left({\tfrac {1}{2}}+h\right)\left({\tfrac {1}{2}}-h\right)}}\cdot {\frac {\left({\tfrac {1}{2}}+h\right)!\left({\tfrac {1}{2}}-h\right)!}{h!(1-h)!}}\;;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73c21756ebeeaa8759e75bcaf0cff084f5d1c33a)
Ainsi, si l’on demandait la tangente de
on aurait
d’où
![{\displaystyle \operatorname {Tang} .66^{\circ }.36'={\frac {0{,}37\times 0{,}63}{0{,}87\times 0{,}13}}\cdot {\frac {0{,}87!0{,}13!}{0{,}37!0{,}63!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2972d5962517263e63b3ccd269283a5b902fdc)
Voici le calcul :
![{\displaystyle {\begin{array}{rl}\operatorname {Log} .37&=1{,}56820\;17241,\\\operatorname {Log} .63&=1{,}79934\;05495,\\{\text{Comp. arith. }}\operatorname {Log} .87&=8{,}06048\;07474,\\{\text{Comp. arith. }}\operatorname {Log} .13&=8{,}88605\;66477,\\\operatorname {Log} .0{,}87!&=9{,}97856\;40362,\\\operatorname {Log} .0{,}13!&=9{,}97309\;61812,\\{\text{Comp. arith. }}\operatorname {Log} .0{,}37!&=0{,}05094\;51119,\\{\text{Comp. arith. }}\operatorname {Log} .0{,}62!&=0{,}04708\;93246,\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03a3c50faadb25526d2596c7a8b3ed77999f0bed)
![{\displaystyle \qquad \qquad \operatorname {Log} .\operatorname {tang} .66^{\circ }.36'=0{,}36377\;43220.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fed40aee7c06fbc68d2aeba7e426497157647622)