364 ASTRO 
are not ellipfes, tire difference of the diameters may indeed, 
or it may not, be proportional to the difference between 
the polar and the equatorial force; but it is quite an un¬ 
certainty, what relation fubfifts between the one quantity 
and the other; our whole theory, except fo far as it relates 
to the homogeneous fpheroid, is built upon falfe affump- 
tions, and there is no faying what figure of the Earth any 
obfervations of the pendulum give.” 
He then lays down the following Table, which fhews 
the different refults of obfervations made in different lati¬ 
tudes ; in which the firfl three columns contain the names 
of the feveral obfervers, the places of obfervation, and 
the latitude of each; the fourth column fliews the quan¬ 
tity of P—IT in fuch parts as II is 100,000, as deduced from 
comparing the length of the pendulum at each place of 
obfervation, with the length of the equatorial pendulum 
as determined by M. Bouguer, upon the fuppofition that 
the increments and decrements of force, as the latitude is 
increafed or lowered, obferve the proportion which theory 
afiigns. Only the fecond and the laft value of P—n are 
concluded from comparifons with the pendulum at Green¬ 
wich and at London, not at the equator. The fifth co¬ 
lumn fliews the value of ^ correfponding to every value of 
P—II, according to Clairault’s theorem: 
Obfcrvtrs. 
Placts. 
Latitudes. 
P—II 
£ 
Bouguer 
Bouguer 
Green 
Bouguer 
Abbe de la 1 
Caille / 
The Acade-1 
micians / 
Capt. Phipns 
Equator 
Porto Bello 
Otaheite 
St. Domingo 
Cape of Good "1 
Hope J 
Paris 
Pello 
o° o' 
9 3 + 
17 29 
18 27 
33 S 3 
48 50 
66 48 
79 .SO 
741-8 
563-2 
5 9 x-° 
731-5 
585-1 
565-9 
471-2 
rhr 
T26 - 
TbT 
ih: 
1_ 
32 a 
1 
2? 1 
“ Bv this Tab'e it appears, that the obfervations in the 
middle parts of the globe, letting afide the Angle one at 
the Cape, are as confident as could reafonably be expect¬ 
ed , and ihey repreient the ellipticity of the Earth as about 
yt-g. But, when we come within ten degrees of the equa¬ 
tor, it fhould feem that the force of gravity fuddenly be¬ 
comes much lets, and within the like diftance of the poles 
much greater than it could be in fuch a fpheroid.” 
The following Problem, communicated by Dr. Leather- 
land to Dr. Pemberton, and publiflied by Mr. Robertfon, 
ferves for finding tire proportion between the axis and the 
equatorial diameter, from meafures taken of a degree of 
tire meridian in two different latitudes, fuppofing the Earth 
an oblate fpheroid. 
Let AP ap be an ellipfe reprefenting a feCtion of the 
Earth through the axis P p ; the equatorial diameter, or 
the greater axis of the ellipfe, being A a ; let E and F be 
two places where the meafure of a degree has been taken; 
tliefe meafures are proportional to the radii of curvature 
in the ellipfe at thofc places; and if CQ^ CR, be conju. 
N O M Y. 
gates to the diameters whofe vertices ara E and F, CQ will 
be to CR in the fubtriplicate ratio of the radius of curva¬ 
ture at E to that at F, by Cor.j, prop. 4, part 6, of 
Milnes’s Conic Sections, and therefore in a given ratio to 
one another; alfo the angles QCP, RCP, are the latitudes 
of E and F; fo that, drawing QV parallel to P p, and 
QJCYW to A a, thefe angles being given, as well as the 
ratio of CQjo CR, the rectilinear figure CVQXRY is 
given in fpecies; and the ratio of VC*—ZC‘ (=OXx 
XW) to RZ’—QV (—RXxXS) is given, which is the 
ratio of CA“ to CP*; therefore the ratio of CA to CP is 
given. Hence, if the fine and coline of the greater lati¬ 
tude be each augmented in the fubtriplicate ratio of the 
meafure of the degree in the greater latitude to that in the 
leffer, then the difference of the fquares of the augmented 
fine, and the fine of the leffer latitude, will be to the dif¬ 
ference of the fqu-ares of the cofine of the leffer latitude 
and the augmented cofine, in the duplicate ratio of the 
equatorial to the polar diameter. For, C q being taken in 
CQ^equal to CR, and qv drawn parallel to QV, Cv and 
vq, CZ and ZR will be the fines and cofines of the re- 
fpedlive latitudes to the fame radius; and CV, VQ^, will 
be the augmentations of Cv and C q in the ratio named. 
Hence, to find the ratio between the two axes of the 
Earth, let E denote the greater, and F the lefler, of the 
two latitudes, M and N, the refpective meafures taken in 
each; and let P denote 3 ^~i then ^_ cof -' F ~ p ’Xcof.*E 
leffer axis 
is=-■. 
greater axis 
It alfo appears by the above problem, that, when one of 
the degrees meafured is at the equator, the cofine of the 
latitude of the other being augmented in the fubtriplicate 
ratio of the degrees, the tangent of the latitude will be to 
the tangent anfwering to the augmented cofine, in the ra¬ 
tio of the greater axis to the lef's. For fuppofing E the 
place out of the equator ; then, if the femicircle Plmnpbe 
deferibed, and 1 C joined, and mo drawn parallel to aC .- Co 
is the cofine of the latitude to the radius CP, and CY that 
cofine augmented in the ratio before named ; YQJpeingto 
Y/, that is, Ca to Cn or CP, as the tangent of the angle 
YCQ^ the latitude of the point E, to the tangent of the 
angle YC/, belonging to the augmented cofine Thus, if 
M repreient the meafure in a latitude denoted by E, and 
N the meafure at the equator, let A denote an angle whofe 
r . - IM tan. A . leffer axis 
meafure is cof. Ex ! . —. Then-is—- 
\ N 
P*Xfin.*E—lin.* F 
tan. E 
greater axis 
But M, or the length of a degree, obtained by aftual 
menfuration in different latitudes, is known from the fol¬ 
lowing Table: 
Names. 
Latitudes.} Value of M. 
Maupertuis and Affoc. 
CafTini and J 
La Caille \ 
Bofcovich 
De la Caille 
Juan and Uiloa 
Bouguer 
Condamine 
0 1 
66 20 
49 22 
45 00 
43 00 
33 18 
OO OO 
00 00 
OO OO 
toifes 
^=57438 
M =57°74 
M =5 7050 
M—56972 
M—57037 
M=56 7 6S'| Att , 
M —56753 fequator 
M— 56749 J e( l uat01 - 
Now, by comparing the firfl: with each of the following 
ones; the fecond with each of the following ; and in like 
manner the third, fourth, and fifth, with each of the fol¬ 
lowing; there will be obtained twenty-five refults, each 
(hewing the relation of the axes or diameters ; the arith¬ 
metical means of all of which will give that ratio as 1 to 
•9951989. If the meafures of the latitudes of 49 0 22', 
and of 45 0 , which fall within the meridian line drawn thrd* 
France, and which have been re-examined and corrected 
fmee the northern and feuthern expedition, be compared 
with 
