observers can take angles to within 1" or 2" of the truth. But this kind of instrument, however great its ingenuity in theory, has the accuracy of its observations necessarily limited by the imperfections of the small telescope which must be attached to it. General Roy and Colonel Mudge made use of a very excellent theodolite constructed by Ramsden, which, having both an altitude and an azimuth circle, combines the powers of a theodolite, a quadrant, and a transit instrument, and is capable of measuring horizontal angles to fractions of a second. This instrument, besides, has a telescope of a much higher magnifying power than had ever before been applied to observations purely terrestrial; and this is one of the superiorities in its construction, to which is to be ascribed the extreme accuracy in the results of this trigonometrical survey.

Another circumstance which has augmented the accuracy of the English measures, arises from the mode of fixing and using this theodolite. In the method pursued by the Continental mathematicians, a reduction is necessary to the plane of the horizon, and another to bring the observed angles to the true angles at the centres of the signals: these reductions, of course, require formulæ of computation, the actual employment of which may lead to error. But, in the trigonometrical survey of England, great care has always been taken to place the centre of the theodolite exactly in the vertical line, previously or subsequently occupied by the centre of the signal: the theodolite is also placed in a perfectly horizontal position. Indeed, as has been observed by a competent judge, " In no other survey has the work in the field been conducted so much with a view to save that in the closet, and at the same time to avoid all those causes of error, however minute, that are not essentially involved in the nature of the problem. The French mathematicians trust to the correction of those errors; the English endeavour to cut them off entirely; and it can hardly be doubted that the latter, though perhaps the slower and more expensive, is by far the safest proceeding."

14. In proof of the great correctness of the English survey, we shall state a very few particulars, besides what is already mentioned in art. 12.

General Roy, who first measured the base on HounslowHeath, measured another on the flat ground of RomneyMarsh in Kent, near the southern extremity of the first series of triangles, and at the distance of more than 60 miles from the first base. The length of this base of verification, as actually measured, compared with that resulting from the computation through the whole series of triangles, differed only by 28 inches.

Colonel

Colonel Mudge measured another base of verification on Salisbury plain. Its length was 36574.4 feet, or more than 7 miles; the measurement did not differ more than one inch from the computation carried through the series of triangles from Hounslow Heath to Salisbury Plain. A most remarkable proof of the accuracy with which all the angles, as well as the two bases, were measured!

The distance between Beachy-Head in Sussex, and Dunnose in the Isle of Wight, as deduced from a mean of four series of triangles, is 339397 feet, or more than 644 miles. The extremes of the four determinations do not differ more than 7 feet, which is less than 14 inches in a mile. Instances of this kind frequently occur in the English survey*. But we have not room to specify more. We must now proceed to discuss the most important problems connected with this subject; and refer those who are desirous to consider it more minutely, to Colonel Mudge's "Account of the Trigonometrical Survey;" Mechain and Delambre, "Base du Systéme Métrique Décimal;" Swanberg, "Exposition des Opérations faites en Lapponie;" and Puissant's works entitled " Geodesie" and "Traite de Topographie, d'Arpentage, &c."

SECTION II.

Problems connected with the detail of Operations in Extensive Trigonometrical Surveys.

PROBLEM I.

It is required to determine the Most Advantageous
Conditions of Triangles.

C

1. In any rectilinear triangle ABC, it is, from the proportionality of sides to the sines of their opposite angles, AB BC:: sin c: sin A, and consequently AB. Sin A = BC. sin c. Let AB be the base, which is supposed to be measured without perceptible error, and which therefore is assumed as constant; then finding the extremely A

B

small variation or fluxion of the equation on this hypothesis, it is AB cos A. A = sin c. BC + BC.cos c. c. Here, since we are ignorant of the magnitude of the errors or variations expressed by a and c, suppose them to be equal (a probable supposition, as they are both taken by the same instrument), and each denoted by v: then will

or, finally, BC = v. BC (cot A - cot c).

This equation (in the use of which it must be recollected that v taken in seconds should be divided by R", that is, by the length of the radius expressed in seconds) gives the error BC in the estimation of BC occasioned by the errors in the angles A and c. Hence, that these errors, supposing them to be equal, may have no influence on the determination of BC, we must have A = c, for in that case the second member of the equation will vanish.

2. But, as the two errors, denoted by A, and c, which we have supposed to be of the same kind, or in the same direc tion, may be committed in different directions, when the equation will be BC = ± BC (cot A + cot c); we must enquire what magnitude the angles A and c ought to have, so that the sum of their cotangents shall have the least value possible; for in this state it is manifest that Be will have its least value. But, by the formulæ in chap. 3, we have

And hence, whatever be the magnitude of the angle B, the error in the value of BC will be the least when cos (ASC) is the greatest possible, which is, when a = c.

We may therefore infer, for a general rule, that the most advantageous state of a triangle, when we would determine one side only, is when the base is equal to the side sought.

3. Since, by this rule, the base should be equal to the side sought, it is evident that when we would determine two sides, the most advantageous condition of a triangle is that it be equilateral.

4. It rarely happens, however, that a base can be commodiously measured which is as long as the sides sought. Supposing, therefore, that the length of the base is limited, but that its direction at least may be chosen at pleasure, we proceed to enquire what that direction should be, in the case where one only of the other two sides of the triangles is to be determined.

1

Let it be imagined, as before, that AB is the base of the triangle ABC, and BC the side required. It is proposed to find the least value of cot A + cot c, when we cannot have a = c. Now, in the case where the negative sign obtains, we have

cot Acot c =

AB-BC. COS B

BC. sin B

BC-AB. COS B

AB.SMB

AB2-BC2 AB BC. sin B

This equation again manifestly indicates the equality of AB and BC, in circumstances where it is possible: but if AB and BC are constant, it is evident, from the form of the denominator of the last fraction, that the fraction itself will be the least, or cot A - cot c the least, when sin B is a maximum, that is, when B = 90°.

5. When the positive sign obtains, we have cot A + cot c =

cot A +

AB SIN A

BC2
A B2 sin? A

BC2-AB2 Sin2 A) = cot A + √( 1). Here, the least value of the expression under the radical sign, ⚫ is obviously when A = 90°. And in that case the first term, cot A, would disappear. Therefore the least value of cot A + cot c, obtains when A = 90°; conformably to the rule giyen by M. Bouguer (Fig. de la Terre, pa. 88). But we have already seen that in the case of cot A - cot c, we must have в = 90. Whence we conclude, since the conditions A = 90°, B = 90°, cannot obtain simultaneously, that a medium result would give A = B.

If we apply to the side AC the same reasoning as to BC, similar results will be obtained: therefore in general, when the base cannot be equal to one or to both the sides required, the most advantageous condition of the triangle is, that the base be the longest possible, and that the two angles at the base be equal, These equal angles however, should never, if possible, be less than 23 degrees.

PROBLEM 11.

To deduce, from Angles measured Out of one of the stations, but Near it, the True Angles at the station.

When the centre of the instrument cannot be placed in the vertical line occupied by the axis of a signal, the angles observed must undergo a reduction, according to circumstances. 1. Let c be the centre of the station, P the place of the centre of the instrument, or the summit of the observed angle APB: it is required to find c, the measure of ACB, supposing there to be known APB = P, BPC = P, CP = d,

BCL, AC = R.

Since the exterior angle of a triangle is equal to the sum of the two interior opposite angles (th. 16 Geom.), we have, with respect to the triangle FAP, AIB = P + IAP; and with regard to the triangle BIC, AIB = C + CBP. Making these two values of AIB.equal, and transposing IAP, there results

C = P + IAP

But the triangles CAP, CBP, give

CP

sin CAP sin IAP = sin APC =

AC

d.sin (P+p)

;

R

And, as the angles car, CBP, are, by the hypothesis of the problem, always very small, their sines may be substituted

for their arcs or measures: therefore

The use of this formula cannot in any case be embarrassing, provided the signs of sin p, and sin (p + p) be attended to. Thus, the first term of the correction will be positive, if the angle (p + p) is comprised between 0 and 180°; and it will become negative, if that angle surpass 180°. The contrary will obtain in the same circumstances with regard to the second term, which answers to the angle of direction p. The letter R denotes the distance of the object a to the right, L the distance of the object B situated to the left, and p the angle at the place of observation, between the centre of the station and the object to the left.

2. An approximate reduction to the centre may indeed be obtained by a single term; but it is not quite so correct as the form above. For, by reducing the two fractions in the second member of the last equation but one to a common denominator, the correction becomes