[ 9ii ] 
pofe, of which a conftruftion upon the principles of 
the flereographic projection of the fphere is exhibited 
by Mr. Collins, in his Mariner’s Plain Scale new 
planed (£). And as the direct method of folving both 
thefe problems by numbers requires a diverfity of ti 1- 
gonometrical operations, a fet of tables has lately been 
publifhed for a more compendious way of computa.- 
tion in the problem, where the interval 01 time is 
given, whereby the {hip’s true latitude may be very 
expeditioufly derived from the (hip’s dead reckoning, 
provided the obfervations are made within certain li- 
mits of time. > 
But however worthy of notice this method may 
be, new tables for the purpofe are altogether unne- 
cessary. It confifts of two parts : the firft computes, 
from the latitude exhibited by the dead reckoning of 
the {hip, the diftance from noon of the middle time 
between the obfervations, and thence the time of 
either: the fecond operation computes, from one of 
thefe obfervations, what fhould be the fun s meri- 
dian altitude, had the (hip’s reckoning given the true 
latitude ; but if the latitude affumed from that reckon- 
ing is erroneous, the altitude thus computed will not 
be conformable to it ; however, if the times for the 
obfervations are properly chofen, it will much better 
agree to the true latitude, and thereby the affumeds 
latitude may be more or leis coriedted. 
But both thefe operations are an immediate con- 
fequence from the propofition in fpherical trigono- 
metry, ufually delivered under the name of the fourth, 
axiom, which is this, That the fquare of the radius 
is to the reft angle under the fines 01 the hdes con- 
(b) Book iii. p. 35. 
taining' 
