390 ASTRO 
of this correction is technically called the fecular equation. 
Kepler firft obferved this circumflance, from examining 
the obfervations of Regiomontanus and Waltherus; for, 
he conftantly found Jupiter forwarder, and Saturn back- 
warder, than they ought to have been from the mean mo¬ 
tions determined from the obfervations of Ptolemy and 
Tycho. He faid the fame of Mars ; but M. de la Lande 
©bferves, that he cannot find there is any fecular equation 
wanted for that planet. Flamftead alio obferved, that in 
all the belt Aflronomical Tables the mean motions of Sa¬ 
turn were too fwift, and of Jupiter too flow; whence it 
came to pals, that the computations gave thofe conjunc¬ 
tions which happened when the planets were direCt fome 
days fooner, and, when retrograde, fome days later, than 
the time from obfervation. Phil. Tranf. No.cxlix. He- 
velius alfo obferved the fame. M. Maraldi perceived al- 
fo, that the mean motions of Saturn, if we fuppofe them 
uniform, would agree both with the obfervations of Ty¬ 
cho and thofe of this age. Dr. Halley, in his Aftronomi- 
cal Tables, applied a fecular equation of 9^-° for 2000 
years to Saturn, and 3°4o' to Jupiter, but he does not 
give the obfervations from which lie deduced thefe con- 
clufions. M. de la Lande, from comparing the oppoli- 
tions in the years 1594, 1595, 1596, and 1597, with thofe 
in 17x3, 1714, 1715, 1716, and 1717, found the mean mo¬ 
tion of Saturn to be 12 0 13' 19" 14"', which is 16" in a 
year lefs than that given by Caflini; and the duration of 
the revolution greater by near four days. He chofe thofe 
oppofltions which happened near the mean diflance (as 
Caflini did alfo), becaufe the ti ne and mean motions being 
then equal, the conclufions would be more accurate. He alfo 
chofe other oppofltions at the diflance of about 120 years, and 
when Jupiter and Saturn were in fimilar fituations, fo that 
no error was to be apprehended from their mutual attrac¬ 
tion, this being the fame in each cafe. Now, if with the 
mean motion found in 120 years, the place of Saturn, from 
where it is now found to be, be computed for the time of 
the obfervation before-mentioned in the year 228 before 
Chrift, the longitude will be found to be too great by 7 0 ; 
this, therefore, is the fecular equation for 2000 years, ac¬ 
cording to this mean motion. But, from other obf^rva- 
tions, he concluded the mean motion to be 12 0 13' 26- 558". 
With this mean motion he finds the fecular equation to be 
47", in the firft century from which this motion was dedu¬ 
ced ; for, with this mean motion and fecular equation, the 
calculations beft agree with the ancient obfervations. From 
the theory of attraction it appears, that, fuppofing the 
aphelion of Saturn and Jupiter to be fixed, the fecular 
equation varies as the fquare of the time, which M. de la 
Lande thinks may be deduced from this confideration, that 
the velocity loft by Saturn, in confequence of the caufe 
which produces the equation, being fo very fmall, may be 
confidered equal in equal times; whence, from the prin¬ 
ciple of the law of falling bodies, the fpace loft nnift va¬ 
ry as the fquare of the time. Now, from from five ob¬ 
fervations by Ptolemy, he found the fecular equation for 
the firft 100 years to be 47" ; hence 100 2 :/ 2 :: 47“ : the fe¬ 
cular equation for t years. But the logarithm of 47 minus 
the logarthm of 100 2 is 7-6720979; hence, if to this con- 
llant logarithm we add twice the logarithm of t, we fhall 
have the logarithm of the fecular equation for t years from 
the commencement of the 100 years, to be iubtraCted from 
the mean longitude. 
But, befides the fecular equation, the mean motion of 
Saturn is alfo fubjeCt to other irregularities which are 
found to follow from the attraction of Jupiter. Dr. Hal¬ 
ley, in his AftrqhOmical Tables, obferves, that Jupiter, 
from his oppofitibn in 1^77 to that in 1689, was found, 
from indubitable obfervations, to be 12' flower than in the 
preceding or fubfequent revolutions. Alfo the periodic 
time of Saturn, between the years 1668 and 1698, was 
nearly a week fhorter than its mean revolution, and the 
periodic time between the year 1689 and that of 1719 was 
nearly as much greater; fo that between the two revolu¬ 
tions there was a ditference of more than thirteen days. 
N O M Y. 
This Dr. Halley fuppofes to arife from the attraction of 
the greater bodies in the fyftem being different in different 
pofitions. For he obferves, that in 1683 there was a con¬ 
junction of Jupiter and Saturn, when, from the pofition 
of the apfldes, the planets approached neareft to each o- 
ther, and Saturn was moft urged towards the Sun, and 
Jupiter from it; fo that Jupiter’s velocity being increafed, 
and its force to the Sun diminiflied, its orbit was increafed, 
and confequently its periodic time; on the contrary, Sa¬ 
turn’s velocity being diminiflied, and its force to the Surx 
increafed, its orbit, and confequently its periodic time, 
was diminiflied. Now, fays be, if the fame thing fliould 
happen again, that is, if a conjunction fliould take place 
again in the fame point of the heavens, and the fame ef¬ 
fects fliould follow, we may hope that it can be accounted 
for from the laws of gravity ; but if, in like circumftances, 
the fame efteCts are not found to take place, other extra¬ 
neous caufes are to be fought for. But M. de la Place has 
difcovered, that thefe inequalities, as well as the fecular 
equations, may be reprefented by an equation, from Ju¬ 
piter’s attraction, of 48', which depends on five times the 
longitude of Saturn minus twice that of Jupiter, of which 
the period is 918 years. For this we nnift employ the mean 
annual motion of 12 0 13' 36-81". Thus all the irregula¬ 
rities of Saturn’s motion are confined to a certain period, 
after which they all return again. In the years 1701 and 
1760 the errors of Dr. Halley’s Tables were 8J'and 2i|', 
according to M. dela Lande, fo that the motion of Saturn 
was greater by 13', and its periodic time was fhorter 6£ 
days, than in its revolution between 1686 and 1745. Now, 
the mean anomaly in 1701 and 1760 was 3 0 1', and the an¬ 
gle at the Sun between Jupiter and Saturn was i9°in 1701, 
and 30 0 in 1760; fo that the error in the mean motion could 
not arife from any diflimilar fituations of Saturn in its orbit, 
by which the elements of the motions might err; nor from 
the different fituations of Jupiter, that difference not being 
liifficient to caufe fuch an error. 
The motion of Jupiter requires alfo a fecular equation, 
as Dr. Halley obferved, who made it 3 0 49' 24" for 2000 
years, or 34-4" for the firft century, fuppofing it to encreafe 
as the fquare of the time. M. Maraldi alfo obferved, that 
the modern obfervations gave the motion of Jupiter great¬ 
er than the ancient. M. de la Lande found, by comparing 
the obfervations made 240 years before Chrift, with thofe 
in the year 508, that Jupiter’s fecular motion in eighty- 
three years was 2' 04". And, comparing the obfervations 
in 508 with thofe in 1503 and 1504, we find nearly the 
fame refult. But, if we compare the conjunction of Ju¬ 
piter with Regulus on OCtober 12, 1623, with the like ob¬ 
fervation made in 1706, we find it 21' minutes for etghty- 
three years. Dr. Halley, in his Tables, fixed it at 12' 26" 
for eighty-three years, which makes the revolution eight 
hours fhorter than that deduced from the ancient obferva¬ 
tions. The oppofltions from 1689 to 1698 compared with 
thofe in 1749, give a mean motion equal to that in the Ta¬ 
bles of Caflini; which Tables give the place of Jupiter 
j' too much in 508. Thefe conclufions indicate a great ir¬ 
regularity in Jupiter’s motions; and this irregularity is far¬ 
ther confirmed, if we conlider that Wargentin makes the 
fecular equation for the firft 100 years to be 18"; M. Bad¬ 
ly makes it 12%" ; and M. de la I.aride fixes it at 30I" for 
the firft 100 years, or 3 0 23' 20'' for 2000 years, admitting 
it to increafe as the fquare of the time, which agrees near¬ 
ly with Dr. Halley’s determination. - M. dela Grange, 
from the theory of gravity, finds it to be 3' 18", which, 
as M. de la Lande obferves, agrees very well with the ob¬ 
fervations from 1590 to 1762, but not with the ancient ob¬ 
fervations. Euler determined it from theory to be 2' 23". 
M. de la Lande fays, that his own fecular equation, with 
the mean fecular motion of 5ft 6°. 27'. 30". agree as nearly 
as poflible to all the obfervations. M. de la Place found 
in 1786 an inequality of 20' from the attraction of Saturn, 
the period of which equation is 918 years, as in Saturn. 
Thus he made the fecular equation difappear, it being on¬ 
ly an irregularity whofe period is 918 years. This fuppofes 
