eBwovered that the jjiameter at the equator 
is to the diameter at the poles, or the axis, as 
»',(). to 24 ). But this computation supposes 
rhe earth to be every where of an uniform 
density ; whereas if the earth is more dense 
near the centre, then bodies at the poles will 
be more attracted by this additional matter 
being nearer : and therefore the excess of the 
sefni-diumeter of the equator above the semi- 
axis will be different. According to this 
proportion between the two diameters, New- 
ton* farther computes, from the different 
measures of a degree, dial the equatorial 
diameter will exceed the polar, by 34 miles 
and 4-- ' , 
Nevertheless, Messrs. Cassini, both father 
and son, the cue in 1701, and the other in 
1713, attempted to prove, in the Memoirs of 
the Royal Academy of Sciences, that the 
earth was an oblong spheroid ; and in 1718, 
M, Cassini again undertook, from observa- 
tions, to shew that, on the contrary, the 
longest diameter passes through the poles ; 
which gave occasion for Mr. John Ber.nouhli, 
in his Essai d’une Nouvelie Physique Celeste, 
printed at Paris in 1735, to triumph over the 
British philosopher, apprehending that these 
observations would invalidate what Newton 
had demonstrated. And in 1720, M. De 
Mairan advanced arguments, supposed to be 
strengthened by geometrical demonstrations, 
farth r to contirm the assertions ol Cassini. 
But in 1735 , two companies of mathematicians 
were employed, one for a northern, and ano- 
ther for a southern expedition, the result ot 
whose observations and measurement plainly 
proved that the earth was Halted at the 
poles. 
The proportion of the equatorial diameter 
to the polar, as stated by the gentlemen em- 
ployed on the northern expedition for measur- 
ing a degree of the meridian, is as 1 to 0'9891 
by the Spanish mathematicians as 266 to 265, 
or as 1 to 0 99624; by M. Bouguer as 179 
to 178, or as 1 to Q‘99441. 
As to all conclusions deduced from the 
length of pendulums in different places, it 
is to be observed, that they proceed upon 
the supposition of the uniform density of the 
earth, which is a very improbable circum- 
stance ; as justly observed by Dr. Horsley, in 
his letter to captain Phipps. “ You finish 
your article, he concludes, relating to the 
pendulum, with saying, ‘ that these observa- 
tions gave a figure of the earth nearer to sir 
Isaac Newton’s computation, than any others 
that have hitherto been made and then you 
state the several figures given, as you imagine, 
by former observations, and by your own. 
Now it is very true, that ij the meridians be 
ellipses, or if the figure of the earth be that 
of a spheroid generated by the revolution of 
an ellipsis, turning on its ^shorter axis, the 
particular figure, or the ellipticity of the 
generating ellipsis, which your observations 
give, is nearer to what sir Isaac Newton saith 
it should be, if the globe were homogeneous, 
than any that can be derived from former 
observations. But yet it is not what you 
imagine. Taking the gain of the pendulum 
in latitude 79° 50', exactly as you fate it, the 
difference between tire equatorial and the 
polar diameter, is about as much less than the 
Newtonian computation makes it, mid the 
hypothesis of homogeneity would require, as 
you reckon it to be greater. I he proportion 
of 212 to 21 1 should, indeed, according to 
Yon. I. 
EARTH, 
vom* observations, be the proportion of the 
force that acts upon the pendulum at the 
pole's, to the force acting upon it at the equa- 
tor. But this is by no means t he same with 
the proportion of the equatorial diameter to 
the polar. If the globe were homogeneous, 
the equatorial diameter would exceed the 
polar by -Mj-g- of the length ot the latter: and 
the polar force would also exceed the equa- 
torial by the like part. But if the difference 
between the polar and equatorial force be 
greater than -A, (which may be the case in 
■=> - rrs - • J . 
an heterogeneous globe, and seems to be the 
case in ours), then the difference of the 
diameters should, according to theory, be 
less than and vice versa. 
‘ f I confess this is by no means obvious at 
first sight ; so far otherwise, that the mistake, 
which you have fallen into, was once very 
general. Many of the best mathematicians 
were misled by too implicit a reliance upon 
the authority of Newton, who had certainly 
confined his investigations to the homogene- 
ous spheroid, and had thought about the 
heterogeneous only in a loose and general 
way. "The late Mr. Clairau.lt was the first 
who set the matter right, in his elegant and 
subtle treatise ou the figure of the earth. 
That work Hath now been many years in the 
hands of mathematicians, among whom 1 
imagine there are none, who have considered 
the subject attentively, that do not acquiesce 
in the author’s conclusions. 
“ In the second part of that treatise, it is 
proved, that putting P for the polar force, 
14 for the equatorial, 5 for the true ellipticity 
of the earth’s figure, and « for the ellipticity 
of the homogeneous spheriod, 
P— n ° , P-n 
— 2t — $ : therefore 5 — 2« 
n n 
and, therefore, according to your observation, 
^ , This is the just conclusion from 
your observations on the pendulum, taking it 
tor granted, that the meridians are ellipses: 
which is an hypothesis, upon which all the 
reasonings of theory have hitherto proceeded. 
But, plausible as it may seem, I must say, 
that there is much reason, from experiment, 
to call it in- question. If it were true, the 
increment of the force which actuates the 
pendulum, as we approach the poles, should 
be as the square of the sine of tiie latitude: 
or, which is the same thing, the decrement, 
as we approach the equator, should be as the 
square of the co-sine of the latitude. But 
whoever takes the pains to compare together 
such of the observations of the pendulum in 
different latitudes, as seem to have been 
made with the greatest care, will find that the 
increments and decrements do by no means 
follow these proportions ; and in those which 
l have examined, i find a regularity in the 
deviation which little resembles the mere 
error of observation. The unavoidable con- 
clusion is, that the true figure of the meridians 
is not elliptical. if the meridians are not 
ellipses, the difference of the diameters may, 
indeed, or it may not, be proportional to the 
difference between the polar and the equa- 
torial force ; but it is quite an uncertainty, 
! what relation subsists between the one quan- 
tity and the other ; our whole theory, except 
so’ far as it relates to the homogeneous 
spheroid, is built upon false assumptions, and 
there is no saying what figure of the earth any 
observations of the pendulum give.” 
4 D 
/ 
Ho then lays down the following table, 
which shews the different results ot observa- 
tions made in different latitudes; in this the 
first three columns contain the names of the 
several observers, the places ot observation, 
and the latitude of each ; the fourth column 
shews the quantity of P— n in such parts as 
I! is 1 00000, as deduced from comparing the 
length of the pendulum at each place of ob- 
servation, with the length of the equatorial 
pendulum, as determined by M. Bouguer, 
upon the supposition that the increments and 
decrements ot force, as the latitude is increased 
or lowered, observe the proportion which 
theory assigns ; only the second and the last 
value of P— n are concluded from com- 
parisons with the pendulum at Greenwich 
and at London, not at the equator. The 
fifth column shews the value of § correspond- 
ing to every value of P— -11, according t* 
Ciairaull’s theorem : 
JtjRjS, 
h 
1* 
b 
is 
H 
h 
t! 
GO CS Q 
O) 
CO 
•H M 
r— < 
LO 
1 
-f'O Cl 
co 
00 
to 
e- 
to io 
*>. 
*0 
2 
"b 
-H 
*n 
o 
GO 
o 
CO M M 
<o 
-r 
If) 
53 
o 
ClKOO 
CO 
oo 
to 
Ol 
CO 
sr 
to 
0s 
•a o ji c _ _ 
c*3 cx, O 35 O 
v y w i, „ 
I w 1 3 
O O O-H 
pg SC O P3 < 
u By this table it appears, that the ob- 
servations in the middle parts of the globe, 
setting aside the single one at the Cape, are- 
as consistent as could reasonably be expected ; 
and they represent the ellipticity ot the earth 
as about . T q~. But when we come, within 
10 degrees of the equator, it should seem 
that the force of gravity suddenly becomes 
much less, and within the like distance of the 
poles much greater, than it could be in such 
a spheroid.” 
The following problem, communicated by 
Dr. Leather-land to Dr. Pemberton, and pub- 
lished by Mr. Robertson, serves for finding 
the proportion between the axis and the 
•equatorial diameter, from measures taken of a 
degree of the meridian in two different 
latitudes, supposing the earth an oblate sphe- 
roid. 
Let AP ap (Plate Miscel. fig. 88.) be an 
ellipse representing a section ot the earth 
through the axis P p ; the equatorial diameter, 
or the greater axis of the ellipse, being Aa ; 
; let E and F be two places w here the measuee 
