NAVIGATION. * 631 
Ex. 2. Let the apparent diftance between the centres of 
the fun and moon be 7a 0 ai' 40", the apparent altitude of 
the moon 19 0 19', that of the fun 25 0 16', and the moon’s 
horizontal parallax 56' 32". Required the true central 
diftance ? 
App. dift. =7a°ai' 4o"N.¥.8=696983 
Dif.app.alt. = j 57 o N.V.S.= 5387 
—-Log.dif.9'9978i4 
Cor. D’s alt. =— 50 39 Dif.=69ij96 - log. 5 83915a 
Q’s alt. =— 1 52 N.No. =688123 - 5‘S37666 
Dif. true alt. = 5 4 29 N.V.S.= 3920 
True dift. =72 3 50 N.V.S.=692043 
Method 2. To the fum of the apparent altitudes of the 
objefts add the correftion of the moon’s altitude, and fub- 
traft that of the fun or ftar, and half the aggregate will be 
half the fum of the true altitudes. 
To the apparent diftance add the apparent altitudes of 
the centres of the fun and moon; find the difference be¬ 
tween half the fum and the apparent diftance. 
Take the log. anfwering to the moon’s apparent alti¬ 
tude and horizontal parallax, to which add the log. co¬ 
fines of the above half-fum and difference, reject 10 from 
the fum of the index; thefe three logarithms and half 
the remainder will be the log. fine of an arch. 
Now add together the log. co-fines of the fum and dif¬ 
ference of this arch, and half the fum of the true alti¬ 
tudes; then will half the fum of thefe two logarithms be 
the log. fine of half the true diftance. 
Ex. Let the apparent diftance of the fun and moon’s 
centres be 38° 45' 40", the apparent altitude of the 
moon’s centre 29 0 31', that of the fun’s centre 35 0 43', 
and the moon’s horizontal parallax 57'43". Required 
the true diftance. 
Sum of app. alts. 65° 14' o" 
Corre£tion J) ’s alt. -j- 48 33 
Correction ©’salt.— 1 12 
App. diftance 
Aop. alt. ]) 
= 3 S° 45 ' 
= 29 31 
a r ,' 1 
40 
O 
Sum of true alt. 
66 1 21 
App. alt. © 
— 35 
43 
O 
Half fum true alt. 
O 
O 
CO 
Sum 
103 
59 
40 
Log. dif. 
- 
9-996578 
Half 
5 i 
59 
50 
- 
co-fine 
9-789369 
Difference 
13 
14 
10 
- 
co-line 
9-988307 
Half fum true alt. 3 3 
O 
4 °$ 
* 
. 
19774254 
Arch 
5 ° 
27 
192 
- 
fine 
9887127 
Sum 
83 
28 
O 
_ 
co-fine 
O'o<; 6 o 7 i 
Difference 
17 
26 
39 
- 
co-fine 
9979553 
19 
14 
I I 
- 
fine 
19-035624 
9-517812 
True diftance 
38 
28 
22 
Prob. II.. To find the Apparent Time at Greenwich, 
anfwering to a given Diftance between the Moon and 
the Sun, or one of the Stars ufed in the Nautical 
Almanac. 
Rale. If the given diftance is found in the Nautical 
Almanac in either of pages viii. ix. x. or xi. of the month, 
oppofite to the given day, or to that which immediately 
precedes or follows it, the time is found at the top of the 
page. But, if this diftance is not found exaftly in the 
ephemeris, then fubtraft the prop. log. of the difference 
between the two contiguous diftances, one of which, being 
greater and the other lefs than the given diftance, from 
the prop. log. of the difference between the given and 
preceding diftances, the remainder will be the prop. log. 
of the excefs of the time above that anfwering to the 
preceding diftance; lienee the apparent time is known. 
But, in ftriftnefs, this proportional part ought to be cor¬ 
rected by the equation of fecond difference. 
Ex. 9 December 7, 1804, the true diftance between 
the centres of the fun and moon was 59°43' 8". Required 
the apparent time at Greenwich. 
Given diftance =59°43' 8'' 
Dif.=o°28' 22"P.Log.=8o24 
Dift.atiii.hours=59 14 46 
Dif.= i 25 27 P.Log.= 3236 
Dift. at vi. hours=So 40 13 -- 
Excefs - - o 59 46 P.Log. 4788 
Preceding time - 300 - 
Apparent time at Greenwich 3 59 46 
Prob. III. The Latitude of a Place, and its Longitude by 
Account, being given, together with the Diftance be¬ 
tween, and the Altitudes of the Moon and the Sun, 
or one of the Stars in the Nautical Almanac ; to find the 
correCt Longitude of the Place of Obfervation. 
Rule. Reduce the eftimate time of obfervation to the 
meridian of Greenwich by the Nautical Almanac. To 
this time take from the Nautical Almanac, page vii. of 
the month, the moon’s horizontal parallax and femi-dia- 
meter. Increafe the femi-diameter by the augmentation 
anfwering to the moon’s altitude. 
Find the apparent and true altitudes of each object’s 
centre, and the apparent central diftance, with which let 
the true diftance be found by any of the methods given 
for that purpofe; and find the apparent time at Green¬ 
wich anfwering thereto by the lalt Problem. . 
If the fun or ftar be at a proper diftance from the meri¬ 
dian, when the diftance was obferved, compute the appa¬ 
rent time at the fliip. If not, the error of the watch may 
be found from obfervations of the altitudes of the fun or 
liars, taken either before or after that of the diftance. Or, 
the apparent time may be inferred from the moon’s alti¬ 
tude, taken at the fame time with the diftance. 
The difference between the apparent times of obferva¬ 
tion at the fhip and Greenwich will be the longitude of 
the Ihip in time, which is ealt or weft, according as the 
time at the Ihip is later or earlier than the Greenwich 
time. 
The only purpofe to which the longitude by account 
is applied is to reduce the time at the fhip to the meri¬ 
dian of Greenwich, in order to take the moon’s femi-dia¬ 
meter and horizontal parallax from the Nautical Alma¬ 
nac, agreeable to this time; and it is evident that, in 
moll cafes, an error of a few degrees in the intimated 
longitude will not produce any fenfible error in thefe 
quantities. But, in order to afeertain the apparent time 
at the (hip from the fun’s altitude, the declination of 
that objedl: ought to be taken from the Nautical Almanac, 
anfwering to the apparent time at Greenwich deduced 
from the diftance. 
Ex. % November 8, 1804, in latitude 34 0 53' N. and 
longitude 24 0 W. by account, about 3I1. 50m. P. M. the 
obferved diftance between the nearell limbs of the fun 
and moon was 67° 48'29"; the obferved altitude of the 
moon’s lower limb 31 0 10'; and that of the fun’s i4°46'; 
height of the eye 12 feet. Required the true longitude. 
Time at Ihip 3l>5o'P.M. Ob.dill. 0 & J ’s nearell 1 .67°48' 29" 
Lon. in time 1 36 Sun’s femi-diameter + 16 13 
- Moon’s femi-diameter +15 1 
Reduced time 5 26 P.M. Augmentation - + 7 
Apparent central diftance 68 19 50 
Alt. Sun’s lower limb 14 46 
Semi-diameter - + 16 
Dip - — 3 
Apparent alt. O’s centre 14 59 
Moon’s horizontal parallax 55' 6" 
App. 
Alt. ])’s 1 . limb 
31 
IO 
Semi-diam. 
+ 
IS 
Dip 
— 
3 
Ap.alt, 5 ’s cen. 
3 i 
22 
Ap.alt.O’s cen. 
14 
59 
Diff. app. alt. 
1(5 
2.3 
