We shall have the same result as that which we have already obtained by resolving the Example according to the method employed in Case 3, page 166; but that method is rather long, and, on that account, it is superseded by the present, which, in cases like the preceding, gives results sufficiently exact. It must not, however, be forgotten, that, in fact, the preceding method is a method of approximation.

One or two cases may be brought under the conditions of the foregoing. If two sides of a spherical triangle are each nearly 180°, and the third side is very small, then, if we produce the two longer sides till they meet, a small spherical triangle will be formed, which may be resolved by the preceding method; and, from the sides and angles of this second triangle those of the first may be immediately deduced. Again, if two angles of a spherical triangle are very acute, the polar or supplemental triangle will have two sides nearly = 180°, and the third a very small side: this case, therefore, is under the conditions of the preceding one.

Triangles solved by the foregoing methods will be spherical triangles, their sides arcs of great circles, and the computed arc of the meridian also an arc of a great circle. But, the foregoing methods have not always been employed, and they are not indispensably necessary. Delambre, in France, and Colonel Mudge, in England, do not consider the line of the meridian as a curve, butras formed of the chords of curves; and they have resolved their triangles not as spherical triangles, either by the direct, or by Legendre's approximate process, but as rectilinear triangles, the sides of which are the chords of arcs. This method then requires Theorems and formulæ different from those that have been already explained. It, however, to a certain point, goes along with the former; for it supposes the oblique angles to be reduced, either instrumentally * or by calculation, (page 186) to horizontal

* Ramsden's Theodolite, by means of an azimuth circle, gives at once the horizontal angle when the oblique angle is observed; this instrument first, we believe, shewed the spherical excess, which rarely amounts to 3". Delambre's and Mechain's Theodolites were Borda's repeating circles of small dimensions; with these the oblique angle was observed, and then reduced by calculation (see page 186.) to the horizontal angle.

BB

angles; and, the horizontal angles being, as it has been already explained, spherical angles, must be reduced to the angles contained by the chords of the spherical sides of the triangle. To effect this reduction, we must resolve this Problem.

Given the angle contained by the spherical arcs a n O, bmo, (Fig. 15), the angle a Ob contained by the chords a O, bo is required.

Let the spherical sides ab, bm O, an O, be represented by

then cos. c = sin. a. sin.b.cos. C + cos. a. cos. b (page 139.).

From this expression, e may be found by the method given in Case 3, page 171; thus, taking the instance given in page 187, the distance of Calais from Watten and Fiennes being in minutes and seconds of the Earth's meridian respectively 15' 59", 7' 26", we have = 7' 59".5,

b

= 3′ 43′′.

... 90°

a+b

4

= 89° 54′ 8.75;

+M=180°-(33° 21′ 10′′.25); ... sin. = 9.7401996

M =

33 9 27.75.....sin. 9.7379440

... log. sin.

2)19.4781436 = 9.7390718

and 0 = 66° 30′ 36′′.68.

For reasons, however, similar to those which we used when speaking of the reduction of oblique angles to horizontal angles, the direct and ordinary rules of Spherical Trigonometry are too tedious for the occasion: @ differs very little from C, and a small correction only is required. To enable the computist to obtain such correction and with little trouble, M. Delambre, as in the former case of reduction, page 187, has investigated a formula, and constructed Tables resembling the former Tables.

In this method, then, M. Legendre's Theorem is not employed, nor is the Theorem relative to the spherical excess requisite ; for there is this criterion of the accuracy of the observationsthe sum of the three angles contained by the chords deduced from the spherical angles, ought, if the observed angles were truly observed, to equal 180°. The remainder of the calculation in this method is by the common rules of Plane Trigonometry; and the arc of the meridian, if that is proposed to be determined, is composed of the sides of an irregular polygon inscribed in a circle.*

We may then, in a Trigonometrical survey, calculate the arcs and angles according to the exact rules of Spherical Trigonometry, as Boscovich has done; or, we may reduce the observed angles to angles of the chords, and calculate, by Plane Trigonometry, such reduced angles and the chords, as Delambre and Mudge have done; or we may, by a slight transformation, give to spherical triangles the properties of plane triangles, and resolve them as such, according to Legendre's method.

Delambre, indeed, indefatigable in quest of accuracy, informs us, Mem. Inst. page 512, 1806, that he computed by the three different methods, the whole series of triangles that extend from Dunkirk to Barcelona.

* The whole of the process of calculation in a Trigonometrical survey has not been described. Since the Earth is not a perfect sphere, some small correction is due to its spheroidical form; the investigation of these corrections is not of great difficulty, but is omitted here, since it would be foreign to the plan and matter of the present Treatise.

1

CHAP. XIII.

On the Relations between the corresponding Variations of the Angles and Sides and Triangles; and, on the Means of selecting, in the application of Trigonometrical Formulæ, the Conditions that are most favourable to accuracy of result.

THE preceding Chapter contains some illustration of the use of Trigonometrical formule. These formulæ are applied to certain data or conditions furnished by observation. Now, the Mathematical process is sure and infallible; but all instrumental observation, in a greater or less degree, is liable to error. The practical result then cannot be perfectly exact: but it will not, necessarily, be inexact to the full extent of the error of the observation. That error, according to the conditions of the case, will be variously modified by the Mathematical process. If it changes its magnitude by changing the conditions, it will be least when the conditions are of certain values. Hence, if it should happen, that we could vary the conditions, it would undoubtedly be expedient to assign them such a magnitude, that the errors of observations should least vitiate the results: that, in the words of a Mathematical statement, the error of the result should be least with a given error of observation.

:

These remarks stand, perhaps, in need of some illustration. The height of a tower may be determined by observing the angle which its summit makes with the horizon, and by measuring the horizontal distance between its base and the station of the observer. Now, in observing the angle, a certain error may be committed: but the error of the result (that which is Mathematically obtained) will vary as the distance between the tower and the observer is varied. If, therefore, we have it in our power, to regulate that condition, that is, if we can observe the height of the tower at what distance we please from its base, we plainly ought to select that which renders least inaccurate the result.