ASTRONOMY. 
add it to the Moon’s place at the firft obfervation, and the 
difference between that fum and the Moon’s place at the 
fecond obfervation fhevvs the effect of the equation of the 
orbit between thefe two fituations of the Moon. Repeat 
this for a great many intervals, and mark thole where the 
difference between the fum before-mentioned and the 
Moon’s true place is greatefl both in excels and defeat. 
If the greatefl excefs and defedt be equal, it is a proof 
that at the time of the firft obfervation, the Moon was in 
its apogee or perigee, and that its true and mean places 
were the fame. In this cafe each of thefe differences is 
the greatefl equation of the Moon’s orbit. If the greateft 
excefs and defedt be not equal, half the fum will meafure 
the greatefl equation ; and, if front the greateft equation 
we fubtracl the lead of the differences, we (hall have the 
equation of the Moon at the time of the firft obfervation. 
M. Cafilni ufes the place of the Moon as determined from 
its eclipfes, feledting thofe which were proper for this pur- 
pofe; and, although the apogee has moved in the interval, 
yet, as the true and mean place of the Moon always coin¬ 
cide at the apogee, it will not affedt the conclufion. Flem. 
d'AJlron. p. 297. 
Hence to find the place of the apogee, let AM PV be 
the orbit of the Moon, A the apogee, P the perigee, C 
the centre of the orbit, T the Earth in the focus, F the 
other focus, M the place of the Moon 
at the time of the firft obfervation ; 
produce TM to R, take MRrzMF, 
and join R F. From the greateft e- 
quation find the ratio of AC to CT; 
tiiis being known, we have, As TF 
is to T R lo is fin. T R F to fin. T F R, 
or AFR; now FRT=|FMT the 
equation of the Moon at the firft ob¬ 
fervation, upon the fimple elliptic hy- 
pothefis ; hence we know AFR, from v\ hich fub tract 
FRT, and we get ATM the Moon’s diflance from its 
apogee. 
Let the firft eclipfe, with which the others are to be 
compared, be a total one, the middle of which happened 
at Paris on December 10, 1685, at job. 38'. 10". mean 
time. The true place of the Sun at that time, by calcu¬ 
lation, was 8f. 19 0 . 40'. and confequently the Moon’s 
place was 2L 19 0 . 40'. Let the next eclipfe be the total 
one on May 16, 1696, the middle of which was 12I1. 7'. 
5-6". mean time at Paris, and the Moon’s place was 7f. 
a< 5 °. 53'. 35". Now in this interval of ten years (of which 
three were biffextiles) 157b. ih. 29'. 46". the mean mo¬ 
tion of the Moon, omitting the complete revolutions, was 
5L 12 0 . 53'. 10". this added to 2 f. 19 0 . 40'. the place at 
the firft eclipfe, gives Sf. 2 0 . 33'. 10". for the mean place 
at the fecond eclipfe, the difference between which and 
the true place 7C 26°. 53'. 3 5". is 5 0 39' 35". The next 
eclipfe compared with the firfi was that on March 15, 1699, 
the middle of which was at 7b. 14'. mean time at Paris, 
at which time the Moon’s true place was 5f, 25 0 . 28'. 41". 
Now in this interval of thirteen years (of which three were 
biffextiles) 94d. 20I1. 35'. 50’'. the mean motion of the 
Moon, omitting the revolutions, was 3L i°. 24'. 47". this 
■added to 2L 19'. 40'. the place at the firft eclipfe, gives 
5L 21 0 . 4'. 47". for the mean place at this third eclipfe, 
the difference between which and 5L 25 0 . 28'. 41". the 
true place is 4 0 . 23'. 54". Now in the former cafe, the 
true place was lefs than the mean place by 5L 39 0 . 35'. and 
in the latter cafe, the mean place is the leaft by 4 0 23 54". 
Thefe are the greateft differences of all the eclipfes be¬ 
tween 1685 and'1720. Now the fum of thefe differences 
is io° 3' 29”, and the half fum is 5 0 1' 44-5" the greateft 
equation of the Moon’s orbit deduced from theft?obfterva- 
tions. And if from 5 0 1' 44'5" we take 4 0 23' 54'', the 
leaft difference, we have 37' 50-5" for the equation of the 
Moon at the time of the firff eclipfe ; and this taken from 
af. 19 0 . 40'. the true place of the Moon at that time, gives 
af. 19 0 . 2'. 10". for the mean place of the Moon on Dec. 
10, 1685, at toh. 38'. 10". mean time at Paris. This 
439 
therefore may be conlidered as an epoch of the mean place 
of the Moon. This is the method ufed by M. Caflini. 
But the belt method is, to oblerve accurately the place of 
the Moon for a whole revolution as often as it can be 
done, and by comparing the true and mean motions, the 
greateft difference will be double the equation. If two 
obfervations be found, where the difference of the true 
and mean motions is nothing, the Moon muff; then have 
been in its apogee and perigee. Mayer makes the mean 
eccentricity o - 05503;68, and correfponding greateft equa¬ 
tion 6° 18' 31 -6". It is 6° 18' 32" in his laft tables pub- 
lilhedbyMr. Mafon, under the direction of Dr. Mafkelyne. 
To determine the place of the apogee, from M. Cafiini’s 
obfervations, we have the greateft equation =5° T 44*5", 
therefore, 57 0 17' 48• 8" : z° 30' :: AC— 100000 : 
C7'~4388 for the Moon’s eccentricity. Alfo TF~ 8776 : 
771=200000 :: fin. TRF— 18' S5' 2 5" '■ fin. 777 ?, or AFR, 
=7° 12' 20", from which take 7 " 7 ? 7 r =i 8 ' 55-25", and we 
have ATM=6° 53' 25'' the diflance of the Moon from its 
apogee; add this to zf. 19 0 . 40'. the true place of the 
Moon, and it gives 2 f. 26°. 33'. 25". for the place of the 
apogee on December 10, 1685, at ioh. 38'. 10". mean 
time at Paris. This therefore may be confidered as ail 
epoch of the place of the apogee. 
If we Compare the fame eclipfe in 1685 with two others, 
one of which happened on July 7, 1675, and the other on 
April 14, 1642, we fhall get the equation of the orbit 5 0 
2' 14", differing only thirty-feven feconds from the other 
determination. Alfo the place of the apogee at the eclipfe 
in 16S5, conies out 2L 25 0 . 57'. 58V._vvh.ich is 35' 27" lefs 
advanced than, by the former cafe. If the Moon’s place 
be determined by obfervation at any time when it is not 
eclipfed, the fame method may be applied, and the way 
to get at the greateft accuracy is to make a great number 
of inch comparifons, and take the mean. 
To determine the mean Motion of the Apogee. Find its place 
at different times, and compare the difference of the places 
with the interval of the time between. To do this, we 
muff firff compare obfervations at a fmall diflance from 
each other, left we fnould be deceived in a whole revo¬ 
lution ; and then we can compare thofe at a greater dif- 
tance. Now we may either compute the place of the apo¬ 
gee at feveral times, or we may find it from knowing the 
place once, according to the following method, given by 
M. Caffini in his Aftronomy, p.307. The place of the ' 
apfide has been determined for Dec. 10, 1685 ; and to find 
from thence its place at any other time,' obferve the true 
place of the Moon at that time ; and find the mean motion 
correfponding to that interval, and add it to, or fubtraft 
it from, the place of the apogee on Dec. 10, 16S5, accord¬ 
ing as the time was after or before that, and you have the 
mean place of the Moon at that time ; the difference be¬ 
tween which and the true place obferved, is the equation 1 
of the orbit at that time ; if the mean place be forwarder 
than the true, the Moon is in the firft fix figns ; if back- 
warder, in the laft fix. But the fame equation may an- 
fwer to two different mean anomalies ; this therefore leaves 
an uncertainty in refpect to the place of the apogee. Now 
from the mean place of the Moon fubtrafft each mean ano¬ 
maly, and it gives the place of the apogee correfponding 
to each ; confequently you get the motion of the apogee 
correfponding to each place thus found ; and to determine 
which is the true motion, repeat the operation foy fome 
other time compared with the place of the apogee on De¬ 
cember 10, 16S5, and you will get the motio.n correlpond- 
ing to two places again. Then compare thefe two motions 
with the other two, and thofe two which agree intill be 
the true motion. 
By thus comparing the place of the apogee on Dec. 21, 
1684, at ioh. 55'. 58". apparent time, with the place de¬ 
termined on Dec. 10, 1685, M. Caflini found the time of a 
revolution of the apfides to be either eight years and nearly 
nine months, or about tlu'ee years. And by comparing 
the place of the apogee on Nov. 29, 1686, at 1 ih. 7'. iS". 
apparent time, with the place on Dec. ip, 1685, he found 
that 
