of a degree has been taken : these measures 
are proportional to the radii of curvature in 
the ellipse at those places ; and ifCQ, CR.be 
conjugates to the diameters whose vertices 
are. E and F, CQ will be to CR in the sub- 
tnpacate ratio of the radius of curvature at 
F, to that at F, bv cor. 1, prop. 4, part 6, of 
Jvl lines s Conic Sections, „ 
EARTH. 
w,V , “ V u , v h prop. 4, part o, ot 
- ii.nes s Conic Sections, and therefore in a 
flpp nr° D t0 on ? another: a >so, the angles 
* i * \ are fh e latitudes of E and F ; so 
that, drawing QV parallel to Vp, and QXYVV 
to Aa, these angles being given, as well as the 
t0 CR ’ the rect il inear figure 
( \7%, K 4' s S lven 'n species ; and the ratio 
?-RxT Z ?^ (=QX X W? t0 
yr'^f X AS) is given, which is the ratio of 
. . to Ci ; therefore the ratio of CA to CP 
is giveiv 
Hence, if the sine and cosine of the greater 
latitude be each augmented in the subtripli- 
cate ratio of the measure of the degree in the 
greater latitude to that in the lesser, then the 
difference of the squares of the augmented 
sine, and the sine of the lesser latitude, will 
lie to the difference of the squares of the 
cosine ot the lesser latitude and the augmented 
cosine, in the duplicate ratio of the equatorial 
to the polar diameter. For, C q being taken 
™ “ 9, C( l ua * CR, and qv drawn parallel to 
<4 V , Cv, and rg, CZ and ZR will be the signs 
and cosines of tlie respective latitudes to the 
same radius; and CV, VQ, will be the aug- 
mentations of Ct> 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 lesser of the two latitudes, M and N 
the respective measures taken in each, and 
letP denote v / ^ : then 
V N 
/ cos - 2 T — P~ X cos. 2 E* . lesser axis 
V P 2 x sin. 2 E — sin. 2 F S greater axis 
It also appears by the above problem that 
when one ot the degrees measured is at the 
equator, the cosine of the latitude of the other 
being augmented in the subtriplicate ratio of 
the degrees, the tangent of the latitude will 
be to tiie tangent answering to the augmented 
cosine, in the ratio of the greater axis to the 
less. For supposing E the place out of the 
equator ; then if the semicircle P hnnp be 
described, and 1C joined, and m > drawn 
parallel to aC : Co is the cosine of the lati- 
tude to the radius CP, and CY that cosine 
augmented in the ratio before-named ; YQ 
being to \ /, that is Ca to C/i 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 cosine. Thus, 
it M represent the measure in a latitude de- 
noted by E, and N the measure at the equa- 
tor, let A denote an angle whose measure is 
cos Ex’ / tan - A lesser axis 
cos. l X -Then— — — - 
' iN tan. E greater axis 
But M, or the length of a degree, obtained 
by actual mensuration in different latitudes, 
is known from the following table •. 
Names. 
Maupertuis, &c. 
Cassini and > 
La Caille 
Boseovich 
Be la Caille 
Juan and Ulloa 
Bouguer 
Condamine 
5 
Latit. 
0 / 
66 20 
49 22 
45 00 
43 00 
33 18 
00 00 
00 00 
00 00 
Falue of M. 
toises. 
M= 57438 
M=57074 
M=57050 
M= 56972 
M— 57037 
M==56768 ) at the 
M— 56753 ,-equa- 
M— 56749 ) tor. 
Now, by comparing the first with each 
of the following ones ; the second with each 
the following ; an( ^ j n p^e manner> the 
third, fourth, and fifth, with each of the fol- 
lowing; there will be obtained 25 results, 
each shewing the relation of the axes or dia- 
meters ; the arithmetical means of all of 
which will give that ratio as 1 to 0-9951989. 
It the measures of the latitudes of 49° 22', 
and ot 45°, which fall within the meridian line 
drawn through France, and which have been 
re-examined and corrected since the northern 
and southern expedition, be compared with 
those of Maupertuis and his associates in the 
north, and that of Bouguer at the equator, 
there will result six different values of the 
ratio of the two axes; the arithmetical mean of 
all which is that of 1 to 0-9953467, which 
may be considered as the ratio of the greater 
axis to the less ; which is as 230 to 228-92974, 
or 215 to 214, or very near the ratio as- 
signed by Newton. 
Now, the magnitude as well as the figure 
of the earth, that is, the polar and equatorial 
diameters, may be deduced from the fore- 
going problem. For, as half the latus rectum 
of the greater axis A a is the radius of cur- 
vature at A, it is given in magnitude from the 
degree measured there, and thence the axes 
themselves are given. Thus, the circular arc 
whose length is equal to the radius being 
57-29578 degrees, if this number be multiplied 
by 56750 toises, the measure of a degree at 
the equator, as Bouguer has stated it, the 
product will be the radius of curvature there, 
or halt the latus rectum of the greater axis ; 
and this is to half the lesser axis in the ratio 
ol the less axis to the greater, that is, as 
0-9953467 to . 1 : whence the two axes are 
6533820 and 6564366 toises, or 7913 and 
7950 English miles ; and the difference be- 
tween the two axes about 37 mile's. 
And very nearly the same ratio is deduced 
from the lengths of pendulums vibrating in 
the same time, in different latitudes; pro- 
vided it be again allowed that the meridians 
are real ellipses, or the earth a true spheriod, 
which, however, can only take place in the 
case of an uniform gravity in all parts of the 
earth. 
Thus, in the new Petersburgh acts, for the 
years 1788 and 1789, are accounts and cal- 
culations of experiments relative to this sub- 
ject, by M. Krafft. These experiments were 
made at different times, and in various parts 
of the Russian empire. This gentleman lias 
collected and compared them, and drawn the 
proper conclusions from them : thus he infers 
that the length .r of a pendulum that swings 
seconds in any given latitude x, and in a tem- 
perature of 10 degrees of Reaumur’s ther- 
mometer, may be determined by this equa- 
tion : 
x = 439-178 -f- 2-321 sine 2 x, lines of a 
French foot, 
or x = 39-0045-}- 0-206 sine % in English 
inches, 
in the temperature of 53 of Fahrenheit’s ther- 
mometer. 
This expression nearly agrees, not only 
with all the experiments made on the pendu- 
lum in Russia, but also with those of Mr. 
Graham in England, and those of Mr. Lyons 
in 79° 50’ north latitude, where he found its 
length to be 431-38 lines. It also shews the 
augmentation of gravity from the equator to j 
the parallel of a given latitude A: for, putting ; 
S for t,ie gravity under the equator, G for 
that under the pole, and y for that under the 
latitude x, M. Kralft finds 
V — (1 -4-0-0052848 sine 2 x)g; andtheref. G— • 
1 -0052848 g. ' 
From this proportion of gravity under dif- 
ferent latitudes, the same author infers that, 
in case the earth is a homogeneous ellipsoid 
its oblateness must be T l. T ; instead of 
which ought to be the result of this hypothe- 
sis : but 011 the supposition that the earth is a 
heterogeneous ellipsoid, he finds its oblate- 
ness, as deduced from these experiments, to be 
HT ' which agrees with that resulting from 
the measurement of some of the degrees of 
the meridian. This confirms an observation 
ot M. Be la Place, that if the hypothesis of 
the earth’s homogeneity is given up, then 
theory, the measurement of degrees of lati- 
tude, and experiments with the pendulum, all 
agree in their result with respect to the ob- 
lateness of the earth. 
In the Philos. Trans, for 1791, Mr. Balby 
has given some calculations on measured 5 
degrees of the meridian, whence he infers,, 
that those degrees measured in middle lati- 
tudes, wili answer nearly to an ellipsoid whose 
axes are in the ratio assigned by Newton 
viz. that of 230 to 229. And as to the devia- 
tions of some of the others, viz. towards the 
poles and equator, he thinks they are caused 
by the enors in the observed celestial arcs. 
Earth, Magnetism of. The notion of 
the magnetism of the earth was started by 
Gilbert; and Boyle supposes magnetic 
etlluvia moving from one pole to the other. 
See Magnetism. 
Earth, Magnitude of. This has beea- 
variously _ determined by different authors, 
both ancient and modern. The usual wav 
has been, to measure the length of one de- 
gree of the meridian, and multiply it by 360,. 
for the whole circumference. ' Biogenes 
Laertius informs us, that Anaximander, a 
sc holar of Thales, who lived about 550 years 
before the birth of Christ, was the first who 
gave an account of the circumference of the 
sea and land ; and it seems his measure was 
used by the succeeding mathematicians, till 
the time of Eratosthenes. Aristotle, at the 
end of Jib. 2 Be Ccelo, says, the mathemati- 
cians who have attempted to measure the 
circuit of the earth, make it 40000 stadia: 
which, it is thought, is the number determined 
by Anaximander. 
Snellius relates, from the Arabian geogra- 
pher Abulfedea, who lived about the 1300th. 
year of Christ, that about the 800th year of 
Christ, Almaimon, an Arabian king, having 
collected together some skilful mathemati- 
cians, commanded them to find out the cir- 
cumference of the earth. Accordingly these 
made choice of the fields of Mesopotamia, 
where they measured under the same meri- 
dian from north to south, till the pole was 
depressed one degree lower : which measure 
they found equal to 56 miles, or 56£: so that, 
according to them, the circuit of the earth is 
20160 or 20340 miles. 
It was a long time after this before any 
more attempts were made in this business. 
At length, however, Mr. Snell, professor of 
mathematics at Leyden, about the year 1620, 
with great skill and labour, by measuring 
large distances between two parallels, found 
one degree equal to 28500 perches, each of 
