630 
NAVIGATION. 
any two given infants of time will he in like proportion. For, 
if an obferver knew that, at the fame inftant that it was 
two o’clock in the afternoon under the meridian where he 
was, it was only mid-day at another place, it would be 
clear he w'as 30 0 to the eaftward of the given place : fince 
24I1 : ah :: 360° : 30 0 , and the longitude is eaft, fince the 
time at the place of obfervation is lateft. 
To afcertain the difference of longitude between the 
firft meridian and a given place, the angular diftance of 
the moon from the fun or a fixed ftar is to be obferved. 
For, as the diftance of the moon from the fun and feveral 
fixed liars eaft and weft of her is given in the Nautical 
Almanac for every three hours, calculated for the meri¬ 
dian of the Royal Obfervatory at Greenwich, it is clear 
that the diftance between the fame objeCts being obferved 
at any other place, the time at Greenwich may be deduced 
therefrom, which, compared with the apparent time, 
points out the difference of time, and, confequently, the 
difference of longitude between the two places. 
Of the Methods of clearing the Apparent Diftance between 
the Moon and the Sim, or a fixed Star, from the EjfeEls 
of lief ruction and Parallax. 
Since the obferved altitude of a celeftial objeCt is affeCt - 
ed by two phyfical caufes, the refraction and parallax, 
whole effeCts are produced in a vertical direction, it is 
therefore obvious that the obferved diftance between any 
two objeCts will be alfo affeCted by thefe caufes. Indeed, 
with regard to the fixed liars, the parallax vaniihes ; and, 
therefore, thefe objeCts are affeCted by refraCtion only. 
But, in obfervations of the moon particularly, the effeCt of 
parallax is very fenfible, upon account of its proximity 
to the earth. Therefore, by reafon of the above caufes, 
the true diftance between the moon and any celeftial ob- 
jeCt is, for the molt part, confiderably different from that 
obferved. 
Let Z, fig. 11, reprefent the zenith, s the apparent place 
of the fun, and m that of the moon ; the arch sm will, 
therefore, be the apparent diftance between thefe objeCts. 
Alfo, ZS,ZM, being vertical circles, paffmg through the 
centres of the fun and moon, the true and apparent places 
of thefe objeCts will be found therein. 
Now, fince the refraCtion is ever greater than the paral¬ 
lax of the fun at the fame altitude, the true place of the 
fun will therefore be lower than the apparent place, which 
let be S ; and, becaufe the moon’s parallax, at any given 
altitude, is greater than the refraCtion at that altitude, 
its true place will, therefore, be higher than the appa¬ 
rent place, which let be M; hence SM will be the true 
diftance. 
The method of reducing the apparent to the true dif¬ 
tance, or, in other words, that of clearing the apparent 
diftance from the effeCts of refraCtion and parallax, being 
the mo.lt tedious part of the calculus for afcertaining the 
longitude, when the calculation is performed by the 
common fpherical analogies, many eminent aftronomers 
and mathematicians have, therefore, given compendiums 
to facilitate the folution of this problem ; among which 
are tliofe by the chevalier de Borda, the abbe de la Caille, 
Meffrs. Delambre, Dunthorne, Elliot, Emerlon, Jeaurat, 
Krafft, De la Lande, Legendre, Lyons, Maikeiyne, Ro- 
bertfon, Romme, Witchel, Vince, See. but the largeft 
and molt elaborate work that has hitherto appeared for 
the purpofe of correcting the apparent diftance is the 
Cambridge Tables. Thefe tables were calculated by 
Meffrs. Lyons, Parkinfon, and Williams, under the in- 
fpeCtion of Dr. Shepherd, Plumian Profeffor of Aftro- 
nomy at Cambridge, by the rule formerly given by Mr. 
Lyons in the firft edition of the Requifite Tables. 
All the methods that have hitherto been given, for the 
purpofe of reducing the apparent to the true diftance, de¬ 
pend on one or other of the two following principles ; of 
which the firft (by Dr. Mackay) appears to be the molt 
fimple and accurate. 
i. With the apparent zenith diftances Z m, Z s, fig. 11. 
and the apparent diftance between the objeCts ms, com¬ 
pute the vertical angle mZs-. with which, and the true 
zenith-diftances, ZS, ZM, the true diftance Zmmavbe 
found. 
2. Let Z, fig. 12, be the zenith ; Z m, Zs, the apparent 
zenith-diftances of the moon and ftar, and sm the appa¬ 
rent diftance. Let S,?, mn, be the refraCtions in altitude 
of thefe objeCts refpeCtively. Join Sn, and draw sa, mb, 
perpendicular thereto ; and Sa, n b, will be the effeCts of 
refraCtion ; which, therefore, being applied to the appa¬ 
rent diftance sm, will give Sn, the diftance corrected by 
refraCtion. 
Again, let n M be the parallax of the moon in altitude ; 
then S M, being joined, will be the true diftance. From M 
draw Me perpendicular to Sn, and cn will be the principal 
efteCt of parallax in diftance; which, being applied to the 
diftance corrected by refraCtion, will give the arch S c. 
Now', in the right-angled fpherical triangle SeM, Sc and 
Me being given, S M may be found, or rather the differ¬ 
ence between SM and Sc may be computed, which, being 
applied to Sc, will give the true diftance S M. 
If the objeCl with which the moon is compared be the 
fun, another correction depending on the parallax of 
that objeCt is neceffary. This correction may be com¬ 
puted on the fame principles as the efteCt of the moon’s 
parallax. 
The firft of the following methods is, perhaps, as eafy 
a folution of this problem as has hitherto appeared, efpe- 
cially when the table of natural verfed fines is ufed. The 
fecond is another folution, which will be found extremely 
eafy, by uling the table of log fines; and the third .and 
fourth methods are given, merely*becaufe they may be 
performed entirely by natural verfed fines. Thefe me¬ 
thods depend on the firft general principle. The other 
methods are deduced from the fecond general principle. 
Prob. I. The Apparent Diftance between the Moon and 
the‘Sun, or a fixed Star, together with the Altitude 
of each being given, to find the True Diftance. 
Method 1. Take the correction of the moon’s altitude 
from Mackay, Tab. IX. to which add the correction of 
the fun’s altitude, or the refraCtion of the ftar, from 
Mackay, Tabs. VI. VII. Now, this fum added to, or fub- 
traCled from, the difference of the apparent altitudes, ac¬ 
cording as the moon is higher or lower than the fun or 
ftar, will give the difference of the true altitudes. 
From the natural verfed fine of the obferved diftance, 
fubtraCl the natural verfed fine of the difference of the 
apparent altitudes, and to the log. of the remainder add 
the log. from Tab. XLII. anfwering to the moon’s appa¬ 
rent altitude and horizontal parallax, corrected by the 
number from Tab. XLIII. or XLIV. according as the 
diftance between the moon and the fun, or a fixed ftar, is 
obferved. Now, the natural number anfwering to the 
fum of thefe two logs, being added to the natural verfed 
fine of the difference of the two altitudes, will give the 
natural verfed fine of the true diftance. 
Mr. Keith, in his Trigonometry, page 297, fays, “ Dr. 
Mackay’s firft method, page 112 of his Treatife, (firft 
edition,) is the fimpleft I ever met with, where his Tables 
are ufed.” 
Ex. 1. Let the apparent diftance between the centres 
of the fun and moon be 8i° 23' 38", the apparent altitude 
of the fun 27° 43', the apparent altitude of the moon 
48° 22', and the moon’s horizontal parallax 58'45". Re¬ 
quired the true diftance ? 
App. dift. n:8i°23' 38"N.V.S.=2850359 
Dif.app. alt. 2=20 39 o N.V.S.= 64248 
-Log.dif. 9.994623 
Cor.))’s alt. =-(- 38 12 Dif. =786111 - log. 5.895484 
©’s alt. =-f- 1 40 N.No. =2776438 - 5-890107 
Dif. true alt. =21 18 52 N.V.S.= 68401 
True diftance, 81 4 26 N.V.S.=:844839 
Ex. 
