In order to render a station mark visible during the day, when its distance is so great as to prevent it from being otherwise seen, the reflexion of the sun's image from a plane mirror at the station has been employed. By night, the brilliant light produced by a stream of oxygen gas passing through burning alcohol and falling on a ball of lime has been used: this, from the name of the discoverer, is called the "Drummond light," and station points are said to have been distinguished by it at distances of nearly 70 miles.

Whatever be the direction of the series of triangles, the bearing, or angle which any station line makes with the meridian passing through one of its extremities is, when necessary, observed; and, a perpendicular being let fall upon the said meridian from the other extremity of the station line, the differences of the longitudes and latitudes between the stations are computed. These differences are then to be compared with the corresponding differences of longitude and latitude obtained by celestial observations at the stations themselves.

When the intention is to obtain the length of a degree of latitude, the distance between two points in the direction of the meridian must be computed, and compared with the difference between the observed latitudes of those points; then, by proportion, the length of the degree may be found. In like manner the computed length of a parallel of terrestrial latitude between two points is to be compared with the difference between the observed longitudes of the points in order to obtain the length of a degree of longitude on such parallel. And, from a comparison of the lengths of two degrees of latitude or two degrees of longitude (the determinations of the lengths of each pair being obtained from measurements made in regions distant from each other) the figure of the earth may be ascertained. This figure may also be determined by means of the measured or computed length of a degree of latitude, combined with the length of a degree of the terrestrial arc on a parallel of latitude, or with a degree of an arc formed by a section of the earth made by a plane passing through a normal to an elliptical meridian and perpendicularly to the plane of the latter.

In the English surveys the latitudes of the stations were obtained by observing the zenith distances of stars with what is called a zenith sector (arts. 107, 108.) whose radius, till lately, was 8 feet: such stars were chosen as pass the meridian near the zenith point in order to avoid the errors arising from any uncertainty in the value of the refraction. The differences of longitude were observed either by chronometers (art. 363.)

or by instantaneous signals (art. 367.) visible at the same time from two stations.

391. The operations which constitute an extensive trigonometrical survey demand the most minute attention, in order that they may possess the necessary accuracy; and, when performed, numerous corrections remain to be applied to the measured lines and angles on account of the imperfections of the instruments, and the various causes of error by which they are affected: corresponding attentions are also indispensable in making and reducing the astronomical observations with which the terrestrial operations are to be combined.

With respect to the base itself, the ground is never sufficiently level to allow the measurement to be made on its surface; and the only resource is to perform the operation on stages which are fixed for the purpose on supports whose heights are regulated so that the upper surfaces of the platforms may be parallel to the general surface of the ground in the direction of the intended base. The method of supporting the stages which was adopted by General Roy, in 1784, for the measurement of the base on Hounslow Heath, appears even now preferable to all others, and may safely be adopted for any similar undertaking. The rectilinear direction of the base having been determined by a line of pickets planted in the ground so as to appear to coincide with the crossing wires of a theodolite, or small transit instrument, which was fixed at one end of the base, with its telescope adjusted to move in a vertical plane passing through the assumed direction; a cord 200 yards long was then stretched on the ground in that direction, and small pickets were driven by the side of the cord at distances of 20 feet from each other. The principal tripods for supporting the stages were placed at 200 yards asunder, and these were invariably three feet high; but the intermediate tripods, which were placed at 20 feet asunder, were capable of having their upper extremities raised or lowered by means of screws, so as to allow them to be brought to a line passing through the air and connecting the tops of the former tripods. Each of the tripods rested on a platform of wood, which was laid on the heads of pickets driven into the ground till those heads were in one level surface; and the stages were trussed both horizontally and vertically, in order to prevent any lateral, as well as vertical deviation. On these the measuring rods were laid, and, while the measurement was being performed, they were clamped to the stages; the ends of every two rods were caused to abut against each other, or the end of one was allowed to fall beyond that of another on one side; and in the latter case, the measurement

was made by observing with a microscope the coincidence of two lines, one of which was drawn across the rod near each of its extremities. When chains were used, they were laid in troughs resting, like the stages, on the heads of the tripods, and were stretched in the direction of their lengths by means of weights suspended from their extremities, in order to ensure an equality of tension. The chains were 100 feet long; they were of steel, and were constructed similarly to that of a watch. The temperature of the air was always registered, and allowance was made for the variation produced in the lengths of the rods or chains by the variations of temperature, according to the known expansion of the material for an increment of temperature expressed by one degree of the thermometer. In the French surveys, the measurement was made with bars of platinum and brass; a bar formed of one of these metals being laid upon one formed of the other, and they were attached together at one extremity only. The difference in the expansions of the two metals was observed, and from this the expansion of each was determined. In the trigonometrical survey of Ireland, which is still being carried on, Colonel Colby employed for the measurement of his base a rod consisting of two bars, one of brass and the other of iron; and by an ingenious contrivance the necessity of making corrections on account of the expansion or contraction of the metal was obviated. This consisted in the rods being placed side by side with their greater breadths in vertical positions, and being riveted firmly together in the middle of their length, so that their expansions might take place from thence in opposite directions. A steel tongue was attached horizontally by pins to the upper edges of both bars near each of their extremities, so as to be perpendicular to the lengths of both at a certain temperature (60° Fahrenheit), and project beyond the compound bar on the side which was of iron. The amount of the expansion of each rod with equal increments of heat, was determined by experiment; and the expansion or contraction of brass being greater than that of iron, when the temperature was above or below that at which the tongue at each end was perpendicular to the bars, the tongue took an oblique position: a fine dot was made on the projecting part of each in such a situation that its distances from the two bars were proportional to the amount of their expansions; thus, whatever might be the temperature of the compound rod, the distance between the dots remained constantly the same. This distance was ten feet. As, when two such rods were laid on the stage, the dots could not be brought to coin

cidence, they were placed so that the dot at the extremity of one was at a constant distance from that which was at the extremity of the next; the positions being ascertained by means of microscopes fixed on a short compound rod, of a construction similar to that of the ten-feet rods.

392. The measured base consisting of lines neither coincident with the general surface of the terrestrial spheroid, nor having a uniform inclination in every part, it is necessary to consider the portion measured on each inclined plane as the hypotenuse of a small right-angled triangle, of which the difference of level between its extremities is the perpendicular, and a horizontal line passing through one extremity is the base. The length of each of these last lines is therefore to be computed from the given hypotenuse and perpendicular; and being further reduced to one particular level, as that of the nearest seas, the sum of all the reduced lines constitutes the required length of the base. Its extreme points should be rendered permanent by sinking in the ground, at each end, a large stone, or a gun with its axis in a vertical position; and each termination of the measurement is at a point on the first and last rod vertically above a dot made on a metal plate, which is let into the stone, or into a plug inserted in the bore of the gun.

A

h

B

The nature of the reduction of oblique to horizontal lines may be understood from the annexed figure, where ab, bc, cd, &c. represent several of the distances measured on inclining stages, all in a vertical plane passing through the extremities A and B of the base. Imagine the lines ba, cf, dg, &c. to be drawn from b, c, d, &c. perpendicularly to the radii or normals AC, bc, cc, &c. of the earth, and let A'B' be an

arc of a sphere whose surface would coincide with that of the neighbouring seas; then the relative heights of a, b, c, &c. with respect to each other (found by the spirit-level) will be the lengths of the perpendiculars Aa, bf, cg, &c.: hence the reduced lines ab, fc, gd, &c. may be computed; and the heights of A or B above the level of the sea (that is AA' or BB') being known by barometrical operations or otherwise (arts. 430. &c.), the heights A'a, b'f, c'g, &c. become known. Lastly, the radius CB' of the arc A'B' being given, we may by proportion obtain the reduced lines a'b', b'c', &c., and consequently the whole line or arc A'B', which is the required length of the base.

The formula for reducing any measured or computed arc to one, between the same normals, at the level of the sea, may be thus investigated, the earth being supposed to be a sphere. Let Ab be the measured or computed arc, and ab its value when reduced to the level of b; let CA' or CB', the radius of the earth at the level of the sea, be represented by r, and let a'a or b'b, the height of ab above that level, be equal to h, which is supposed to be known as above. Then r + hr :: ab: A'b'; and by division

r+h⋅ab).

But, since h is very small compared with r, we may without sensible error consider the difference between ab and A'b' as

h

equal to ab; whence A'b' may be found. In the like

r

manner all the lines or arcs b'c', c'd', &c. may be obtained.

393. It is necessary that all the angles of each triangle in the primary series of triangles should be actually observed; and in making a choice of stations, when all the angles are observed, it is to be remarked that the condition most favourable to the accuracy of computation is that in which each triangle is nearly equilateral. But when two angles only are observed, as is frequently the case in a secondary series of triangles, the unobserved angle ought to be nearly a right angle.

In taking the angles of the principal chain, the stations should be as near as possible to the permanent edifices of the country, as churches, mills, &c. These, however, cannot become the angular points of the triangles, since it is always impossible to bring the theodolite to the axis, or to any object which would indicate the centre of the station; therefore a portable scaffold, carrying a well-defined mark, is set up near each edifice, and the angles contained between every three such marks are observed. When, however, a trigonometrical survey of a country is to be executed, it is often convenient to observe the angle subtended by two permanent objects, such as the vanes of church-steeples, in which case these should become the vertices of the two remaining angles of the triangle; but, as the instrument cannot be placed vertically under them, it should be as nearly so as possible, and the angle thus observed must be reduced to that which would have been taken at the centre of the station.

394. The direct processes of trigonometry will always enable the geodetical surveyor to compute the values of the reduced angles; and it would appear, at first sight, so much