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For, if P be the elevated pole, st
the circle described by the star, PR
Ez the latitude: then since PS =
Pt, PR must be
1 (Rt+RS).
This method is obviously indepen- H
dent of the declination of the star :
it is therefore most commonly adopt-
ed in trigonometrical surveys, in
which the telescopes employed are

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of such power as to enable the observer to see stars in the
day-time the pole-star being here also made use of.


Numerous other methods of solving this problem likewise are given in books of Astronomy; but they need not be detailed here.

Corol. If the mean altitude of a circumpolar star be thus measured, at the two extremities of any arc of a meridian, the difference of the altitudes will be the measure of that arc : and if it be a small arc, one for example not exceeding a degree of the terrestrial meridian, since such small arcs differ extremely little from arcs of the circle of curvature at their middle points, we may, by a simple proportion, infer the length of a degree whose middle point is the middle of that




Though it is not consistent with the purpose of this chap-
ter to enter largely into the doctrine of astronomical spherical
problems; yet it may be here added, for the sake of the young
student that if a = right ascension, d declination, l
titude, a longitude, p =
angle of position (or, the angle at
a heavenly body formed by two great circles, one passing
through the pole of the equator and the other through the
pole of the ecliptic), i inclination or obliquity of the eclip-
tic, then the following equations, most of which are new, ob-
tain generally, for all the stars and heavenly bodies.

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8. sin p. cos d

cos l. cos λ.

sin i. cos a.

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9. sin p. cos a = sin i . cos a.

10. tan a tan λ. cos i.
cos ¿. ? when l=0, as is always the case

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The investigation of these equations, which is omitted for the sake of brevity, depends on the resolution of the spherical triangle whose angles are at the poles of the ecliptic and equator, and the given star, or luminary.


To determine the Ratio of the Earth's Axes, and their Actual Magnitude, from the Measure of a Degree or Smaller Portion of a Meridian in Two Given Latitudes; the earth being supposed a spheroid generated by the rotation of an ellipse upon its minor axis.

Let ADBE represent a meridian of the earth, DE its minor axis, AB a diameter of the equator, m, m, arcs of the same number of degrees, or the same parts of a degree, of which the lengths are measured, and which are so small, compared with the magnitude of



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the earth, that they may be considered as coinciding with arcs of the osculatory circles at their respective middle points; let mo, mo, the radii of curvature of those middle points, be=R and respectively; мp, mp, ordinates perpendicular to AB : suppose further CDC, CBd; d2-c2=e2 cr=x; cpu ; the radius or sine total = 1; the known angle вSм, or the latitude of the middle point м, L; the known angle вsm, or the latitude of the point m, =l; the measured lengths of the arcs м and m being denoted by those letters respectively.

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Now the similar sectors whose arcs are м, m, and radii of curvature R, r, give R : r :: м: m; and consequently Rm=rм. The central equation to the ellipse investigated at p. 533 of volume first gives PM —

c2 u

also sp =

c2 x d2

; sp



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ã√ (d2 = x2); pm= √ (d3 — u2) ;

(by th. 17 Ellipse). And the method

of finding the radius of curvature (Flux. art. 74, 75), applied to the central equations above, gives

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R = (d1 — e2x2) #


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d4 c

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d3 av (d2 — x2):: cos L: sin L; whence x2 =

the triangle SPM gives SP: PM: COS L : sin L; that is,

And from a like process there results, u2 =

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Substituting in the equation Emry, for R, and r their va-
lues, for xa and a2 their values just found, and observing that
šin2 L + cos2 £= 1, and sin2 7+ cos2 l = 1, we shall find

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or m3 (d2-e2 sin2 7)

(d2 —e2 sin2 L).

From this there arises e2 = d2-c2 (by hyp.) =

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d2 __ M3sin2 L—m3sin2l ___ (m3sinL+m3sin?) . (μ3sinL-m's sin l)





m3 cos3 l-μ3cos2 (m3 cos/+MCOSL). (m3 cos/-cos l Whence, by extracting the root, there results finally

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(m3 cos 1+ cos L). (m3 cos l-m3 cos L)

This expression, which is simple and symmetrical, has been
obtained without any developement into series, without any
omission of terms on the supposition that they are indefinitely
small, or any possible deviation from correctness, except what
may arise from the want of coincidence of the circle of cur-
vature at the middle points of the arcs measured, with the arcs
themselves; and this source of error may be diminished at
pleasure, by diminishing the magnitude of the arcs measured:
though it must be acknowledged that such a procedure may
give rise to errors in the practice, which may more than coun
terbalance the small one to which we have just adverted.

Cor. Knowing the number of degrees, or the parts of de-
grees, in the measured arcs M, m, and their lengths, which
are here regarded as the lengths of arcs to the circles which
have R, r, for radii, those radii evidently become known in
magnitude. At the same time there are given the algebraic
values of R and r; thus, taking R for example, and exterminat-
c2 d2
ing e2 and x2, there results R =



d2 — e2) sin2 L)
fore, by putting in this equation the known ratio of d to c;
there will remain only one unknown quantity d or c, which
may of course be easily determined by the reduction of the
last equation; and thus all the dimensions of the terrestrial,
spheroid will become known;

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terrestrial spheroid, and it manifestly becomes known when the ratio is determined. But the measurements of philoso



phers, however carefully conducted, furnish resulting compressions, in which the discrepancies are much greater than might be wished. General Roy has recorded several of these in the Phil. Trans. vol. 77, and later measurers have deduced others. Thus, the degree measured at the equator by Bouguer, compared with that of France measured by Mechain 1


and Delambre, gives for the compression , also d=3271208
toises, c=3261443 toises, d-c=9765 toises. General Roy's
sixth spheroid, from the degrees at the equator and in latitude
45°, gives
Mr. Dalby makes d = 3489932 fathoms,
c=3473656. Col. Mudge d=3491420, c=3468007, or 7935
and 7882 miles. The degree measured at Quito, compared
with that measured in Lapland by Swanberg, gives compres-
Swanberg's observations, compared with Bou-


1 309:4


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to the latest observation of Burg, Maskelyne, &c. 309.05 From the variation of the pendulum in different latitudes


*. Dr. Robinson, assuming the variation of gravity at 335-78

* This number 337 does not result from the variation of the pendulum in different latitudes, but is altogether erroneous in consequence of certain numerical mistakes in La Place's calculations. Ed.




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the phenomena of the precession of the equinoxes and the nutation of the earth's axis, gives for the maximum limit of the compression.

2. From the various results of careful admeasurements it happens, as Gen. Roy has remarked, "that philosophers are not yet agreed in opinion with regard to the exact figure of the earth; some contending that it has no regular figure, that is, not such as would be generated by the revolution of a curve around its axis. Others have supposed it to be an ellipsoid; regular, if both polar sides should have the same degree of flatness; but irregular if one should be flatter than the other. And lastly, some suppose it to be a spheroid differing from the ellipsoid, but yet such as would be formed by the revolution of a curve around its axis." According to the theory of gravity however, the earth must of necessity have its axis approaching nearly to either the ratio of 1 to 680 or 303 to 304; and as the former ratio obviously does not obtain, the figure of the earth must be such as to correspond nearly with the latter ratio.

3. Besides the method above described, others have been proposed for determining the figure of the earth by measurement. Thus that figure might be ascertained by the measurement of a degree in two parallels of latitude; but not so accurately as by meridional arcs. 1st. Because, when the distance of the two stations, in the same parallel is measured, the celestial arc is not that of a parallel circle, but is nearly the arc of a great circle, and always exceeds the arc that corresponds truly with the terrestrial arc. 2dly, The interval of the meridian's passing through the two stations must be determined by a time-keeper, a very small error in the going of which will produce a very considerable error in the computation. Other methods which have been proposed, are, by comparing a degree of the meridian in any latitude, with a degree of the curve perpendicular to the meridian in the same latitude; by comparing the measures of degrees of the curves perpendicular to the meridian in different latitudes; and by comparing an arc of a meridian with an are of the parallel of latitude that crosses it. The theorems connected with these and some other methods are investigated by Professor Play

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