Encyclopaedia Britannica > Volume 7, ETM-GOA

Part II. G E O M
Lines and likewife given, being the dlftance of the ftation C and
^ Angles- D ; therefore (by the firft cafe of oblique-angled tri-
W--.,—... angjes ;n trigonometry) the fide AC will be found.
Wherefore, in the right-angled triangle ABC, all the
angles and the hypothenufe AC are given ; confe-
quently, by the fourth cafe of trigonometry, the height
fought AB will be found; as alfo (if you pleafe) the
diltance of the ftation C, from AB the perpendicular
within the hill or inacceifible height.
PROPOSITION XV.
Fig. i 9. From the top of a given height, to meafure the
di/lance BC.—Let the angle BAC be obferved by the
12th prop, of.this; wherefore in the triangle ABC,
right-angled at B, there is given by obfervation tire
angle at A ; whence (by the 59th art. of Part. I.) there
will alfo be given the angle BCA : moreover the fide
AB (being the height of the tower) is fiippofed to be
given. Wherefore, by the 3d cafe of trigonometry,
BC, the diftance fought, will be found.
PROPOSITION XVI.
Fig. 20. To meafure the dijlance of two places A and
B, of which one is acceffible, ly the graphomder.— Let
there be erefted at two points A and C, fufficiently
diftant, two vifible figns.; then (by the 12th prop, of
this Part) let the two angles BAC, BCA, be taken by
the graphometer. Let the diftance of the ftations A
and C be meafured with a chain. Then the third angle
B being known, and the fide AC being likewife
known ; therefore, by the firft cafe of trigonometry,
the diftance required, AB, will be found.
PROPOSITION XVII.
Fig. 2 1. To meafure hy the graphomder the dijlance
«f two places, neither of which is accefftble.— Let two
ftations C and D be chofen, from each of which the
places may be feen whofe diftance is fought; let the
angles ACD, ACB, BCD, and likewife the angles
BDC, BDA, CD A, be meafured by the graphometer;
let the diftance of the ftations C and D be meafured
by a chain, or (if it be neceffary) by the preceding
practice. Now, in the triangle ACD, there are given
two angles ACD and ADC; therefore, the third CAD
is likewufe given ; moreover, the fide CD is given ;
therefore, by the firft cafe of trigonometry, the fide
A will be found. After the fame manner, in the tri¬
angle BCD, from all the angles and one fide CD given,
the fide BD is found. Wherefore, in the triangle
ADB, from the given fides D A and DB, and the angle
ADB contained by them, the fide AB (the dillance
fought) is found by the 4th cafe of trigonometry of
oblique-angled triangles.
PROPOSITION XVIII.
Fig. 22. It is required by the graphomder and qua¬
drant lo meafure an accejfible height AB, placed fo on
a feep, that one can neither go near it in an horizontal
plane, nor recede from it, as we fuppofed in the folution of
the icphprop.—Let there be chofen any fituation, as
C, and another D ; where let fame mark be erefted :
let the angles ACD and ADC be found by the grapho¬
meter ; then the third angle DAC will be known.
Let the fide CD, the diftance of the ftations, be mea¬
fured with a chain, and thence (by trigon.) the fide
AC will be found. Again, in the triangle ACB, right-
angled at B,, having found, by the quadrant the angle
ACB, the other angle CAB is known likewife: but
the fide AC in the triangle ADC is already known ;
E t a Y . 6; :*
therefore the height required AB will be found by the Lines-and
4th cafe of right-angled triangles. If the height of AnS es f
the tower is wanted, the angle BCF will be found by
the quadrant: which being taken from the angle ACB
already known, the angle ACF will remain : but the
angle FAC wTas known before ; therefore the remain¬
ing angle AFC will be known. But the fide AC was
alfo known before ; therefore, in the triangle AFC,
all the angles and one of the fides AC being known*
A F, the height of the tower above the hill, will be
found by trigonometry.
SCHOLIUM.
It were eafy to add many other methods of meafu-
ring heights and diftances ; but if what is above be
underftood, it will be eafy (efpecially for one that is
verfed in the elements) to contrive methods for this
purpofe, according to the occafion : fo that there is
no need of adding any more of this fort. We ftiall
fubjoin here a method by which the diameter of the
earth may be found out.
PROPOSITION XIX.
Fig. 1. To find the diameter of the earth from one ol-
fervation.—Let there be chofen a high hill AB, near ^
the fea-lhore, and let the obfervator on the top of it,
with an exaft quadrant divided into minutes and fe-
conds by tranfverfe divifions, and fitted with a tele-
fcope in place of the common fights, meafure the angle
ABE contained under the right line AB, which goes to
the centre, and the right line BE drawm to the fea, a '
tangent to the globe at E ; let there be drawn from
A perpendicular to BD, the line AF meeting BE in F.
Now in the right-angled triangle BAF all the angles
are given, alfo the fide AB, the height of the hill;
which is to be found by fome of the foregoing me¬
thods as exaftly as pofiible ; and (by trigonometry)
the fides BF and AP are found. But by cor. 36th
3. Eucl. AF is equal to FE ; therefore BE will be
known. Moreover, by 36th 3. Eucl. the re£fangle
under BA and BD is equal to the fquare of BE. And
thence by 17th 6. Eucl. as AB : BE : : BE : BD.
Therefore, fince AB and BE are already given, BD
w’ill be found by 1 ith 6. Eucl. or by the rule of three;
and fubtra&ing BA, there will remain AD the diame¬
ter of the earth fought.
SCHOLIUM.
Many other methods might be propofed for mea-
furing the diameter of the earth. The moft exaft is
that propofed by Mr Picart of the academy of fciences -
at Paris.
“ According to Mr Picart, a degree of the meridian
at the latitude of 490 21' was 57,060 French toifes,
each of which contains fix feet of the fame meafure :
from which it follows, that if the earth be an exaft
fphere, the circumference of a great circle of it will be
123,249,600 Paris feet, and the femidiameter of the
earth i9,6i5,8oofeet: but the French mathematicians,
who of late have examined Mr Picart’s operations, af-
fure us, that the degree in that latitude is 57,183
toifes. They meafured a degree in Lapland, in the
latitude of 66° 20', and found it of 57,438 toifes. By
comparing thefe degrees, as well as by the obfervations
on pendulums, and the theory of gravity, it appears
that the earth is an oblate fpheroid;. and (fuppofing
thofe degrees to be accurately meafured) the axis or
diameter that pafies through the poles will be to the
4 Q^_2 diameter