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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)
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 =
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 Scholium and Remarks.
1. The value — — 1, = do, is called the compression of
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
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 =
tions, compared with Bouguer's, give
compared with the degree of Delambre and Mechain
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
number of observations differ but little from which the
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′′.
Ex. 2. Going along a straight and horizontal road which passed by a tower, I wished to find its height, and for this purpose measured two equal distances each of 84 feet, and at the extremities of those distances took three angles of eleva tion of the top of the tower, viz, 36° 50′, 21°24′ and 14°. What is the height of the tower? Ans. 53.96 feet.
Ex. 3. Investigate General Roy's rule for the spherical excess, given in the scholium to prob. 8.
Er. 4. The three sides of a triangle measured on the earth's surface (and reduced to the level of the sea) are 17, 18, and 10 miles: what is the spherical excess?
Ex. 5. The base and perpendicular of another triangle are 24 and 15 miles. Required the spherical excess.
Ex. 6. In a triangle two sides are 18 and 23 miles, and they include an angle of 58° 24′36′′. What is the spherical excess?
Ex. 7. The length of a base measured at an elevation of 38 feet above the level of the sea is 34286 feet: required the length when reduced to that level.
Er. 8. Given the latitude of a place 48°51′N, the sun's declination 18° 30′N, and the sun's altitude at 10h 11m 265 AM, 52°35'; to find the angle that the vertical on which the sun is, makes with the meridian.
Ex. 9. When the sun's longitude is 29° 13′ 43′′, what is his right ascension? The obliquity of the ecliptic being 23° 27' 40'.
Ex. 10. Required the longitude of the sun, when his right ascension and declination are 32° 46′ 52′′, and 13° 13′27′′N respectively. See the theorems in the scholium to prob. 12.
Ex. 11. The right ascension of the star a Ursæ majoris is 162° 50′ 34′′, and the declination 62° 50′ N: what are the longitude and latitude? The obliquity of the ecliptic being as above.
Ex. 12. Given the measure of a degree on the meridian in N. lat. 49°3', 60833 fathoms, and of another in N. lat. 12°32', 60494 fathoms to find the ratio of the earth's axes.
Ex. 13. Demonstrate that, if the earth's figure be that of an oblate spheroid, a degree of the earth's equator is the first of two mean proportionals between the last and first degrees of latitude.
Ex. 14. Demonstrate that the degrees of the terrestrial meridian, in receding from the equator towards the poles, are
increased very nearly in the duplicate ratio of the sine of the latitude.
Ex. 15. If p be the measure of a degree of a great circle perpendicular to a meridian at a certain point, m that of the corresponding degree on the meridian itself, and d the length of a degree on an oblique arc, that arc making an angle a with the meridian, then is d = Required a
demonstration of this theorem.
PRINCIPLES OF POLYGONOMETRY.
THE theorems and problems in Polygonometry bear an intimate connection and close analogy to those in plane trigonometry; and are in great measure deducible from the same common principles. Each comprises three general cases.
1. A triangle is determined by means of two sides and an angle; or, which amounts to the same, by its sides except one, and its angles except two. In like manner, any rectilinear polygon is determinable when all its sides except one, and all its angles except two, are known.
2. A triangle is determined by one side and two angles; that is, by its sides except two, and all its angles. So likewise, any rectilinear figure is determinable when all its sides. except two, and all its angles, are known.
3. A triangle is determinable by its three sides; that is, when all its sides are known, and all its angles, but three. In like manner, any rectilinear figure is determinable by means of all its sides, and all its angles except three.
In each of these cases, the three unknown quantities may be determined by means of three independent equations; the manner of deducing which may be easily explained, after the following theorems are duly understood.
In Any Polygon, any One Side is Equal to the Sum of all the Rectangles of Each of the Other Sides drawn into the Cosine of the Angle made by that Side and the Proposed Side*.
This theorem and the following one, were announced by Mr. Lexel of Petersburg, in Phil. Trans. vol. 65, p. 282: but they were first demonstrated by Dr. Hutton, in Phil. Trans. vol. 66, pa. 600.