pôle du monde, il est clair qu’en menant par les points
P
{\displaystyle \mathrm {P} }
H
{\displaystyle \mathrm {H} }
P
H
,
{\displaystyle \mathrm {PH} ,}
P
H
{\displaystyle \mathrm {PH} }
H
,
{\displaystyle \mathrm {H} ,}
H
P
A
{\displaystyle \mathrm {HPA} }
Fig. 13.
Donc, si l’on nomme
l
a
l
a
t
i
t
u
d
e
d
u
p
o
i
n
t
H
{\displaystyle \mathrm {la\ latitude\ du\ point\ H} }
α
,
{\displaystyle \alpha ,\ \ }
l
a
l
o
n
g
i
t
u
d
e
d
u
m
e
^
m
e
p
o
i
n
t
{\displaystyle \mathrm {la\ longitude\ du\ m{\hat {e}}me\ point} }
β
,
{\displaystyle \beta ,\ \ }
l
′
a
r
c
H
M
,
c
′
e
s
t
−
a
`
−
d
i
r
e
l
′
o
u
v
e
r
t
u
r
e
d
u
c
e
r
c
l
e
M
N
{\displaystyle \mathrm {l'arc\ HM,\ c'est-{\grave {a}}-dire\ l'ouverture\ du\ cercle\ MN} }
ζ
,
{\displaystyle \zeta ,\ \ }
l
′
a
r
c
A
E
{\displaystyle \mathrm {l'arc\ AE} }
V
,
{\displaystyle \mathrm {V} ,\ }
l
′
a
r
c
H
A
{\displaystyle \mathrm {l'arc\ HA} }
W
,
{\displaystyle \mathrm {W} ,}
on aura d’abord, dans le triangle
H
A
P
{\displaystyle \mathrm {HAP} }
A
{\displaystyle \mathrm {A} }
fig . 13),
tang
A
P
=
tang
P
H
cos
H
A
P
et
sin
H
A
=
sin
P
H
sin
H
P
A
,
{\displaystyle \operatorname {tang} \mathrm {AP=\operatorname {tang} PH\cos HAP\quad {\text{et}}\quad \sin HA=\sin PH\sin HPA} ,}
c’est-à-dire
tang
V
=
tang
α
cos
β
et
sin
W
=
cos
α
sin
β
.
{\displaystyle \operatorname {tang} \mathrm {V} ={\frac {\operatorname {tang} \alpha }{\cos \beta }}\quad {\text{et}}\quad \sin \mathrm {W} =\cos \alpha \sin \beta .}
Ensuite on aura (fig . 12)
A
N
=
W
+
ζ
,
A
M
=
W
−
ζ
,
C
T
=
tang
V
O
N
=
tang
V
N
2
=
cot
W
+
ζ
2
,
{\displaystyle \mathrm {AN=W+\zeta ,\quad AM=W-\zeta ,\quad CT=\operatorname {tang} VON=\operatorname {tang} {\frac {VN}{2}}=\cot {\frac {W+\zeta }{2}}} ,}
et de même
C
S
=
cot
W
−
ζ
2
.
{\displaystyle \mathrm {CS=\cot {\frac {W-\zeta }{2}}} .}
