This method is obviously independent of the declination of the star: it is therefore most commonly adopted in trigonometrical surveys, in which the telescopes employed are

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 de tailed 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

arc.

Scholium.

=

=

Though it is not consistent with the purpose of this chapter 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, altitude, λ= 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 ecliptic, then the following equations, most of which are new, obtain generally, for all the stars and heavenly bodies.

=

1. tan a = tan λ. cos i — tan l. sec a. sin i.

2. sin d = sin λ. cos l. sin i + sin l. cos i.

3. tan

sin i

tan d. sec a + tan a. cos i. sin d. COS i sin a. cos d. sin i.

4. sin 5. cotan p = cos d. sec a. cot i 6. cotan pcos . sec A. cot i 7. cos a COS d = cos l. cos λ. 8. sin p. cos d = sin i. cos λ. 9. sin p. cos λ= sini. cos a. 10. tan tan λ. cos i. Į when 11. cos λ = cos a cos d.

sin d. tan a.

=

sin . tan λ.

70, as is always the case with the sun. The

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

PROBLEM XIII.

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.

E

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 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 r respectively; MP, mp, ordinates perpendicular to AB: suppose further CD=C; CB = d; d2. c2 = e2; CP = x; cpu; the radius or sine total = 1; the known angle вSM, or the latitude of the middle point M, L; the known angle Bsm, or the latitude of the point m, = 7; the measured lengths of the arcs м and m being denoted by those letters respectively. Now the similar sectors whose arcs are м, m, and radii of curvature R, r, give R:r:: M: m; and consequently Rm= The central equation to the ellipse investigated at p. 29 of this volume gives PM = ¿√(d2 — x2); pm = ́√ (d2 — u2);

ΤΜ.

c2x

c'u

C

also SP sp(by th. 17 Ellipse). And the method

=

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

On the other hand,

c4d

the triangle SPM gives SP: PM :: cos L: sin L;

· : √√ (d2 — x2): : cos L : sin L; whence a2 =

And from a like process there results, u2 =

that is,

d4 cos2 L

d-e sin L'

d4 cos2 l d2-e2 sin2 l*

Substituting in the equation Rmrм, for R, and r their

values,

values, for 2 and u2 their values just found, and observing that sin' L + cos2 L = 1, and sin2 + cos2 = 1, we shall find

=

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

=

(m3 sin L + m3 sin ?). (m3sin L– m‡ sin 1)

(m3

cos 1+ M3 cos L).(m3 cos 1 – หรื M3 COS L) Whence, by extracting the root, there results finally ́ ( m3 sin L + m3 sin 1) . (m3 sin L –

(m3 cos l + M3

M sin l)

cos L). (m3 cos / - MCOS 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 circles of curvature 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 counterbalance the small one to which we have just adverted.

Cor. Knowing the number of degrees, or the parts of degrees, in the measured arcs M, m, and their lengths, which are here regarded as the lengths of arcs to the circles which have R, 7, 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 exterminating e and x2, there results R =

d5

.There

c(d2-(d2-c2) 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.

General

1

General Scholium and Remarks.

d

d-c

1. The value — — 1, = do, is called the compression of

C

c

C

the terrestrial spheroid, and it manifestly becomes known when the ratio is determined. But the measurements of philosophers, 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 and Delambre, gives for the compression also d3271208 toises, c 3261443 toises, d-c = 9765 toises. General Roy's sixth spheroid, from the degrees at the equa

1

$34'

tor and in latitude 45°, gives 309-3′ 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 compression =

1 309.4

tions, compared with Bouguer's, give

Swanberg's observa

compared with the degree of Delambre and Mechain

1

1

1

1

307.4

Compared with Major Lambton's degree 307-17 A minimum of errors in Lapland, France, and Peru gives 323.4. Laplace, from the lunar motions, finds compression = From the theory of gravity as applied to the latest observations of Burg, Maskelyne, &c, From the variation of the pendulum

give results varying from to : but far the greater

178.4

577

1

number of observations differ but little from which the

304

computation from 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 VOL. III.

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 am 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 axes approaching nearly to either the ratio of 1 to 680 or of 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 arc of the parallel of latitude that crosses it. The theorems connected with these and some other methods are investigated by Professor Playfair in the Edinburgh Transactions, vol. v, to which, together with the books mentioned at the end of the 1st section of this chapter, the reader is referred for much useful information on this highly interesting subject.

Having thus solved the chief problems connected with Trigonometrical Surveying, the student is now presented with the following examples by way of exercise.

Ex. 1. The angle subtended by two distant objects at a third object is 66°30′39′′; one of those objects appeared under an elevation of 25'47", the other under a depression of 1". Required the reduced horizontal angle.. Ans. 66°30′37′′.