Any survey of a country, which is carried on from a single base by the computation of observed angular distances, may be properly called a trigonometrical survey; but the term is usually confined to measurements on a large scale, embracing a considerable extent of country, and requiring a combination of astronomical and geodetical operations.

There are two principal objects for which a trigonometrical survey may be undertaken; the first being to ascertain the exact situation of the different points of a country relatively to each other, and to the equator and meridians of the terrestrial spheroid, for the purpose of constructing an accurate map; and the second to determine the dimensions and form of the earth, by ascertaining the curvature of a given portion of its surface. Having already, in the article Figure of the Earth, given a general account of the principal operations, which, in different ages and countries, have been undertaken with a view to the second of these objects, and also explained the methods by which the elements of the earth's form and dimensions are computed from the geodetical measurements, and stated the results which are deducible from those measurements taken collectively, we shall not, in the present article, enter into the general subject of Geodesy, but shall confine ourselves to a more particular account than could have been properly given in the article cited, of the great survey which during the last half-century has been carrying on in our own country under the direction of the Board of Ordnance. We accordingly propose to give, in the first place, a brief abstract of the history of that operation, so far as it may be collected from the accounts published in the Transactions of the Royal Society, and in the three volumes of the work entitled "The Trigonometrical Survey of England and Wales," and, in the second place, to explain the methods by which the distances, latitudes, longitudes, bearings, and relative heights of the several positions are computed from the observations. The subject, though somewhat technical, is important both in a scientific and national point of view; and few persons, perhaps, have any just notions of the extremely refined nature of the operations to be executed, or of the difficulties to be surmounted, when the question is to determine terrestrial distances and positions with the extreme precision which alone can be tolerated in the present advanced state of mathematical and astronomical science.

From the account given by General Roy, it appears that the origin of the British trigonometrical survey goes back to the middle of the last century. The rise and progress of the rebellion which broke out in the Highlands of Scotland in the year 1745, convinced government of the importance of establishing military posts, and opening roads of communication in the remotest parts of the country; and a body of infantry having been encamped at Fort Augustus in 1747 with a view to these objects, Lieutenant-general Watson, who was then officially employed at that place as deputy quartermaster-general, conceived the idea of making a map of the Highlands. The proposal having met with the approval of the Duke of Cumberland, the survey necessary for the purpose was forthwith commenced under the direction of General Roy; and although it was originally intended to confine the operation to the Highlands, it was nevertheless extended to the Lowlands, and at length included the whole of the mainland of Scotland. The breaking out of the war in 1755 prevented the survey from being completed, and accordingly the projected map was never published. General Roy states, that although the work, which exists in manuscript, possesses considerable merit, and perfectly answered the purpose for which it was intended, yet, having been carried on with instruments of a common or even inferior kind, it is rather to be considered as a magnificent military sketch than a very accurate map of a country.

On the conclusion of the peace of 1763, the question of making a general survey of the whole island came for the first time under the consideration of the government; but although the utility of such a measure was acknowledged, no steps were taken to carry it into effect until after the termination of the American war in 1783. During this year memorial, drawn up by Cassini de Thury, was transmitted by the French ambassador to Mr Fox, then Secretary of State for the Foreign Department, setting forth the advantages that would accrue to astronomy by carrying a chain of triangles from the neighbourhood of London to Dover, to be connected with those of the French arc of meridian, which had now been extended from Collioure to Dunkirk, and thereby determining, by actual measurement, the relative positions of the Observatories of Greenwich and Paris. Cassini's proposal having been referred to the Royal Society, was warmly approved by that body; in consequence of which the government undertook to give the requisite assistance, and the execution of the operation was committed to General Roy, who was then employing himself in the measurement of a base for a projected survey of London, which he had undertaken with a view to connect the different private observatories in and about the capital with that of Greenwich, and, as he states, "that it might possibly serve as a hint to the public for the now almost forgotten scheme of 1763."

Although the operation which was thus resolved upon did not embrace a general survey of the kingdom, but was confined to the particular object of effecting a junction of the Greenwich Observatory with the French triangulation, it was still an object of great astronomical interest; and accordingly it was determined that the measurement should be conducted with the utmost possible care in all its details, and with the best instruments which could be provided by the celebrated Ramsden, at that time acknowledged to be the first artist in the world.

The first step in the operation was the accurate measurement of a base, from which the sides of a chain of triangles Hounslow might be successively computed. For this purpose General Roy selected a line on Hounslow Heath, a situation which presented the advantages of proximity to the capital and the Observatory of Greenwich; of great extent and levelness of surface; of being free from local obstructions, and commodiously situate for any future operations of a similar nature. The line extended from a place called King's Arbour, at the north-west extremity of the Heath, and terminated at Hampton poor-house, near Bushy Park, at the south-east extremity, the whole length being upwards of five miles.

The preliminary operations having been completed, and the terminal points having been marked by sinking wooden pipes into the ground, the measurement was commenced about the middle of July 1784. The measuring apparatus consisted of three deal rods, on which lengths of twenty feet were laid off by Ramsden, and a standard rod, with which the former were from time to time to be compared. The measuring rods were formed out of an old mast of Riga timber, their dimensions being 20 feet 3 inches in length (including the tippings, which were of bell-metal), about 2 inches deep, and

Trigonometrical Survey

1\(\frac{1}{2}\) inch broad, and trussed both laterally and vertically, so as to be rendered perfectly inflexible. They were constructed in such a manner that they might be used either by butting the end of one rod against the end of another, or by bringing fine transverse lines drawn upon them at the distance of an inch and half from each extremity into exact coincidence; but although the last method was considered by General Roy to deserve the preference, it was found on trial to be attended with so much inconvenience and loss of time that it was abandoned, and the measurement by contacts alone adhered to. The weather proved wet and unfavourable, and before the measurement was half completed, the deal rods were found, notwithstanding all the care which had been taken to prepare them of the best materials, so liable to sudden and irregular variations of length, from changes in the hygrometrical state of the atmosphere, as to leave no hopes of determining by their means the length of the base with the precision which was aimed at. As, however, so much of the work had been done, it was thought desirable to continue it until the whole base was measured, in order that the result might be compared with that which should afterwards be found by a different method.

General Roy cites some experiments to show how unfit deal rods are for such a purpose as the measurement of the base of a great trigonometrical operation. On one occasion the measuring rods, when compared with the standard, were found to exceed it, at a medium, one fifteenth part of an inch; so that if the whole base had been measured with the rods in that state, the difference would have amounted to more than \(7\frac{1}{3}\) feet, exclusive of any expansion or contraction of the standard, which was of the same material. Another experiment, made after the operation had been completed, was still more decisive. A line of 300 feet was accurately measured off in the garden of Sir Joseph Banks when the rods were in a dry state, the sun shining bright, and the temperature 68°. The rods were exposed to the dew during the night, and when lifted from the grass on the following morning were found to be quite wet excepting on the sides in contact with the ground. The line was then remeasured, and its length as given by the rods found to be less by 0.498 of an inch (or nearly half an inch) than on the preceding evening. Hence it appeared that the dew imbibed in one night, or a period not exceeding fourteen hours, caused such an expansion of the rods as in the whole base would have amounted to 45484 inches. (Trig. Survey, vol. i. p. 50.) These experiments were important, as deal rods had been employed in all the principal operations of the kind which had previously been undertaken for determining the magnitude and compression of the earth.

The measurement with the deal rods being completed, and the proper allowance (so far as it could be determined from the comparisons with the standard) made for the expansion, the distance between the centres of the pipes terminating the base, reduced to the level of the lower extremity at Hampton Court, and at the temperature of 63°, was found to be 27,406.26 feet of the standard scale from which the lengths of the measuring rods were laid off.

When it was discovered that the measurement by means of the deal rods would prove unsatisfactory, and General Roy was considering the different alternatives that might be adopted, it was suggested by Lieutenant-colonel Calderwood that glass rods should be substituted for those of deal. As it was found upon trial that there would be no difficulty in obtaining glass tubes of the desired length, and that they could be provided much sooner than rods of metal; and as it was obvious that the rate of expansion could be determined with equal certainty, it was resolved to adopt the suggestion. Accordingly, three hollow tubes, perfectly straight, upwards of twenty feet in length and about an inch in diameter, were selected, and converted by Ramsden into measuring rods. The tubes were placed in cases, to which they were made fast at the middle, and braced at several other points, so as to prevent them from bending or shaking, but not so closely as to prevent their free expansion or contraction. Both ends were ground perfectly smooth and at right angles to the axis of the bore; one end having a fixed apparatus or metal button attached to it for making the contacts, and the other end a moveable apparatus, or slider, which was pressed outwards by a slender spring, and against which the fixed extremity of the succeeding rod was pushed, until a fine line on the slider was brought into exact coincidence with another fine line on the glass rod, in which state the distance between the extremities was exactly twenty feet.

The ground on which the base was measured not being quite level, the whole distance was divided into hypotenuses or inclined lines in the same vertical plane, each containing thirty lengths of the measuring rods, or 600 feet; and the method pursued was to place the rods exactly in straight lines stretching from one extremity of a hypotenuse to another, and then to determine the relative heights of the two extremities of the hypotenuse by the spirit-level, for the purpose of reducing to the horizon. The cases containing the measuring rods were supported on trestles about two and a half feet above the surface of the ground.

The new measurement with the glass rods was begun on the 18th of August, and concluded on the 30th. On arriving at the south-east extremity, the end of the 137th rod was found to overshoot the centre of the pipe terminating the base, by 17.875 inches; and after the rate of expansion of the rods had been determined, and the proper equations applied, the difference between the present and the former measure with the deal rods was found to be 20.561 inches, of which the greater part is probably owing to the over-rated expansion of the deal rods, which, when brought into use, appear to have contracted sooner than was imagined, and thereby given a shorter distance than was assignable from the mean of any two comparisons with the standard. No use was made of the measurement with the deal rods in any of the subsequent operations.

After the measurement on the ground had been completed, it was necessary to determine by actual experiments the expansion of the glass rods, in order to obtain the exact length of the base. For this purpose an ingenious microscopic pyrometer, invented by Ramsden, was employed, by means of which the expansion of the brass scale, and of glass rods and various other substances, was ascertained for every degree of Fahrenheit from 32° to 212°. The glass measuring rods could not themselves be submitted to experiment on account of their great length; the rod on which the experiments were made was 5 feet in length, 0.93 inch in diameter, weighing 1 lb. 13\(\frac{1}{2}\) oz. and drawn from the same pot of metal as the measuring rods. These experiments were made in the winter of 1784 and spring of 1785.

In order to give a clearer idea of the number of minute circumstances to be attended to in an operation of this kind, we shall here state the final result in the words of General Roy. (Trig. Survey, vol. i. p. 84.)

| Hypotenusal length of the base as measured by 1869-925521 glass rods of 20 feet each | 27,402,749 feet | |-----------------------------------------------|----------------| | + 4.31 feet, being the distance between the last rod and the centre of the north-west pipe | 27,402,820 feet | | Reduction of the hypotenuses to be subtracted | 0.0714 feet | | Apparent length of base reduced to level of south-east extremity | 27,402,749 feet | | Add the difference between the expansion of the glass above, and contraction of it below, 63°, 4.1867 inches | 0.3489 feet |

Footnotes:

1. Trig. Survey, vol. i. p. 50. 2. Trig. Survey, vol. i. p. 84. In order that no precaution should be omitted to insure the accuracy of the operation, it was resolved to measure a base of verification towards the termination of the triangles. The ground selected for this purpose was Romney Marsh, Base of a tract which, on account of its levelness, was exceedingly well suited to the purpose. The marsh had been previously on Romney covered by the sea, and a considerable part of it, particularly Marsh, towards the bottom of the range of hills that separate it from the Wealds of Kent, is still lower than the sea at high water. A preliminary survey of the marsh having been made, a line was selected running from High Nook on the spire of Ruckinge church, and of nearly six miles in length. The terminal points were marked by sinking two wooden pipes into the ground. This base was not measured with the glass rods, but with a steel chain of 100 feet in length, made by Ramsden, and of which the accuracy had been tested by measuring with it a portion of the Hounslow Heath base simultaneously with the glass rods. The measurement was executed by Lieutenant Fiddes of the royal engineers. The apparent length of the base, or that given directly by the measurement, was 28,586 feet 8'385 inches; and after the proper reductions were made, the correct horizontal distance between the pipes, in feet of the standard brass scale, at temperature $62^\circ$, and at the level of the sea, was found to be 28,535 feet 8'128 inches.

On connecting the base with the series of triangles extending from Hounslow Heath, its length, as deduced from the former base, was computed to be 28,593-8 feet; so that the computed length fell short of the measured length about twenty-eight inches. This agreement is probably as near as was to be expected, and may be taken as conclusive proof of the general accuracy of the whole of the operations; nevertheless, as there were reasons for supposing that the accuracy attained in the base of verification was not equal to that of the original base, the whole of the triangles were computed from the latter. The measurement on Romney Marsh does not enter as a datum into any of the results of the survey.

By reason of the superior magnitude and excellence of Method of the instrument employed, the measurement of the angles observing was performed with a degree of accuracy which had probably never been equalled in any former survey. Although putting the reduction to the centre of the station requires a very simple calculation, it was thought desirable to avoid, as much as possible, reductions of every kind, and accordingly the centre of the great theodolite was adjusted by means of a plummet over the precise points marking the stations. The whole number of stations at which it was placed was twenty-three. In nine cases the instrument was elevated to the top of a tower, church steeple, or other building; and in the fourteen other cases, the station was marked by sinking a pipe into the ground, to indicate the precise spot over which the instrument had been placed, in order that the observations might be repeated, or the stations connected with others in any future operation. The sides of the triangles were computed by plane trigonometry; that is to say, the portion of the earth's surface over which the triangulation extended was regarded as a plane, and the measured bases as straight lines on that plane. This supposition, though in the small portion of surface in question it did not lead to errors of great magnitude, is inadmissible in an extensive survey. The spherical excess was indeed roughly computed for each triangle, but merely for the purpose of showing the amount of the errors of observation; and the observed angles of each triangle were adjusted so that the sum should equal $180^\circ$, by applying to each an arbitrary correction.

The calculation of the triangles, and the determination Results of of the relative positions of the Paris and Greenwich Observatories, with the lengths of the degree of meridian and measurement perpendicular, are given in detail in the Phil. Trans. for 1790; but the results are not now of much importance, ins- much as the whole of the observations have been since repeated and recomputed by more scientific and accurate methods. General Roy's operations gave the difference of longitude between Greenwich and Paris = 2° 19' 51", or in time 9m. 19s.4. Legendre found, from the same operations, 9m. 21s (Mémoires de l'Acad. 1788). This last determination would appear to be very near the truth, the difference of the two meridians having been found by fire-signals, = 9m. 21s.46 (Henderson, Phil. Trans. 1827); by Captain Kater's remeasurement of Roy's triangles, = 9m. 21s.18 (Phil. Trans. 1828); and, lastly, by the transit of chronometers from Greenwich to Paris and back, = 9m. 21s.14 (Dent, Proceedings of the Astronomical Society, January 1838).

The account of the remeasurement to which we have just alluded is given by Captain Kater in the Phil. Trans. for 1828. In 1821 the Academy of Sciences of Paris communicated to the Royal Society their desire that the operations for connecting the meridians of Paris and Greenwich should be repeated jointly by both countries; and the proposal having been acceded to, Colonel Colby and Captain Kater were appointed by the Royal Society to co-operate with M. Arago and M. Mathieu, the commissioners chosen by the Academy of Sciences. The requisite assistance having been readily obtained from the Ordnance department, the operations were begun in the autumn of the same year.

The instrument employed on this occasion was the great theodolite belonging to the Royal Society, the same which had been used by General Roy. The signals for connecting the stations on the opposite coasts were lamps with compound lenses, constructed on the principle and under the direction of M. Fresnel; and Captain Kater remarks, that the light far exceeded that of any of our light-houses, appearing at the distance of forty-eight miles as a star of the first magnitude. Having selected convenient stations on Fairlight Down and near Folkestone turnpike, the party carried the instrument across the channel, and observed the angles made at the stations of Cape Blancnez and Montlambert with those on the English coast. They then recrossed the channel, and observed the angles subtended by the signals at Cape Blancnez and Montlambert from the stations on Fairlight Down and Folkestone. After some difficulty, General Roy's station was discovered on Fairlight Down, and the observations at both stations were satisfactorily completed on the 27th of October. These reciprocal observations sufficed to establish the connection between the two countries; and the object now proposed by Colonel Colby and Captain Kater was to connect the triangles with General Roy's base on Hounslow Heath. On examination, the guns marking the termination of the base were discovered; but in consequence of the erection of numerous buildings since 1783, one end of it could not be seen from the other; it was therefore necessary to adopt a side of one of General Roy's triangles as the measure of the linear distances, and that from Severndroog Castle, on Shooter's Hill, to Hanger Hill Tower, was selected, these being the nearest stations to General Roy's base which would be identified with sufficient precision. In the course of the two following summers the angles were observed at all the intermediate stations, and also the observations completed at Greenwich which were necessary for determining the azimuths or bearings of the sides of the triangles in respect of the meridian of the observatory. In the calculation of the triangles Captain Kater made use of the theorem of Legendre (which will be afterwards explained), whereas General Roy's calculations, as already stated, were made by considering the surface a plane. The latter method is obviously incorrect; yet, from the near agreement of the results, it is evident that no differences of any consequence arose from the different modes of computation adopted. The following comparative table of distances given by Captain Kater shows the agreement between the two independent operations. The measures are here given in imperial feet.

| Distance from | By Gen. Roy | By Capt. Kater | Difference | |---------------|-------------|---------------|------------| | Fairlight to Frant | 1138505-59 | 1138557-34 | 675 | | Fairlight to Tenterden | 71577-24 | 715809-75 | 331 | | Fairlight to Folkestone | 1548292-70 | 154867-99 | 430 | | Dover to Notre Dame | 137459-40 | 137471-99 | 1259 | | Calais | 45222-72 | 45221-91 | 171 |

Although the geodetical operations which had been carried on by General Roy, might be regarded as subservient to a general survey of the country, they did not form part of any systematic plan for accomplishing that object; and on the death of the General, which took place in 1790, some time elapsed before any measures were taken to prosecute them further. In the introduction to the account of the first part of the survey carried on under the direction of the Board of Ordnance, the renewal of the operations is ascribed to the accidental circumstance of the Duke of Richmond, then master-general, having had an opportunity of purchasing "a very fine instrument, the workmanship of Mr Ramsden, of similar construction with that used by General Roy, but with some improvements; as also two new steel chains of one hundred feet each, made by the same incomparable artist." The new instrument is said to have been ordered by the East India Company for the purpose of surveying their possessions in the East, and Ramsden had exerted all his ingenuity in endeavouring to render it perfect; but some misunderstanding having arisen about the price, the directors refused it, and it was thrown on the hands of the artist. The Duke of Richmond having been advised to purchase it for the Ordnance, the instrument thus became the property of the public, and has been employed as the principal instrument of the survey down to the present time. It has been already stated that the instrument used by General Roy was the property of the Royal Society.

In 1791 the Ordnance survey was begun, and its execution committed to Colonel Williams and Captain (afterwards General) Mudge of the Royal Artillery, and Mr Dalby. The first operation which they undertook was the remeasurement of the base on Hounslow Heath. It had been objected to the former measurement, by that some error might be supposed to arise from the end of the two consecutive rods being made to rest on the same trestle, because when the first rod was taken off the face of the trestle being pressed by one rod only would have a tendency to incline a little forward, the effect of which would be to shorten the apparent length of the base; 2d, it was supposed that some error might arise from the casual deviation of the rods from a straight line in the direction of the base; and 3d, it was supposed, that from the manner of supporting the rods on two trestles only, they would be liable to bend in the middle. For these reasons it was determined to remeasure the base by a totally different method. Instead of measuring rods, the two new steel chains above alluded to were used. These were 100 feet in length, and containing 40 links of 2½ feet each. The links were in form of a parallelogram of half an inch square, and their length was considered advantageous, as rendering them less liable to apply themselves to any irregularities of the coffers on which they were supported.

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1 See Professor Playfair's Review of Mudge's Account of the Survey, in the Edinburgh Review, vol. v.

In preparing the chains, five coffers were arranged in a straight line and supported by courses of bricks; the chain was then placed on the coffers, and stretched with a weight of fifty pounds. The method adopted for bringing the marks noting the two extremities of the measuring chain successively over the same point was this. In any position the chain was supported by a post at each end; that at the preceding end carried a pulley, over which passed the rope sustaining the weight which stretched the chain, while that at the following end supported a screw apparatus, by means of which the chain could be drawn back against the weight. Another post at each end, not connected with the former with the chain, supported a scale. Now, the chain being in one of its positions, the scale at the preceding end was moved by means of screws, until one of its divisions coincided exactly with the mark on the handle of the chain. His scale remaining in its place, the chain was then carried forward into its next position, and adjusted by means of its screw apparatus, until the mark on its following end coincided exactly with that division of the scale with which the mark on the preceding end had coincided. After forty-eight chains had been measured, one of the chains as laid aside, on account of one of the links appearing to be a little bent, and the remainder of the base measured with the other; the former, after the defect had been repaired, being kept as a standard. Experiments were carefully made for determining the comparative lengths of the two chains, and also the rate of expansion; and in the actual measurement five thermometers were laid close by the chain in the coffers, and suffered to remain till they all indicated nearly the same temperature. The time required for this was in general from seven to fifteen minutes. The two chains were again compared after the measurement, and it was found that the working chain had been lengthened, through the rubbing and wearing of the joints, to the extent of 97 divisions of the micrometer, corresponding to 0.0378 of an inch. The whole base was divided into thirty hypotheses, but of unequal lengths, and each was reduced to the horizon by calculation. After the measurement was completed, the lengths of the steel chains were ascertained by means of a comparison with a 40-inch brass standard scale. All reductions being made, the length of the base, by the new measurement, at the standard temperature of 62° (but not reduced to the level of the sea), was found to be 7,404.3155 feet, being about 2¾ inches greater than was found from General Roy's measurement with the glass rods. We shall afterwards see, however, that the glass rods and steel chains were not referred to the same standard, and that there is consequently reason to suspect a considerably greater discrepancy. The mean of the two results, or 7,404.2 feet, was assumed as the true length of the base for future calculations.

Of two operations which agree so closely in their results, it can hardly be said that the one is in point of fact better than respect than the other. The practical difficulty in measurements of this kind, is to form correctly the contacts or coincidences of the extremities of two contiguous rods or chains; and in this respect the chain has unquestionably an advantage, because, on account of its greater length, there are fewer coincidences to be made. Nevertheless, when it is considered that the chain is not uniformly supported at very point; and that notwithstanding the weight by which it is stretched, some doubt must remain whether all its points are in a straight line when brought into its position in the coffers, and also the liability to irregular wear from the stretching; we are inclined to think, that a measurement by means of rigid rods is to be preferred to one by a flexible chain.

After the measurement had been completed, the terminal points of the base were permanently marked, by removing the wooden pipes and sinking iron guns into the ground, in such a position that the axis of the cylinder was placed exactly in the same vertical line with the terminal points. The muzzles were left above ground, and iron caps were screwed over them to protect the cylinders from the rain.

In prosecuting the survey, it was resolved, in the first instance, to carry a series of triangles from the base southwards, for the purpose of determining some of the principal stations on the sea-coast, and also because this would afford an opportunity of connecting the series with the triangles of General Roy, and of thereby testing the accuracy of both operations. Another object of importance to general science, was to determine the length of a degree of longitude, by measuring the distance between Beachy Head on the coast of Sussex, and Dunmose in the Isle of Wight, two stations lying nearly east and west of each other, above sixty-four miles distant, and visible from each other in clear weather, so that they could be made the angular points of a large triangle. In the early part of the spring of 1792, the ground was examined, and the stations fixed upon; and the great theodolite having undergone some improvements by Mr. Ramsden, and a moveable observatory having been erected for its reception, the triangulation was begun in the summer of that year. Most of the angles were observed more than once. When the stations were not more than signals about fifteen miles distant, staffs were erected for signals, in which case the angles were repeated till their truth became certain. For the more distant stations lamps and white lights were employed. In the use of the latter, it was not always possible to repeat the measures; but every precaution was taken to place the lights in the proper positions, and the charge and firing of them were committed to soldiers selected for their steadiness. The angle was taken when the light was going out. In order to preserve the exact positions of the points over which the axis of the instrument had been placed, large stones from a foot and a half to two feet square were sunk in the ground, generally two feet under the surface, having a hole of an inch square made in each of them, the centre of which marked the precise point of the station.

In the course of the summer of 1792, the instrument was carried to twelve different stations, commencing with Hanover Hill, and ending with Chaconbury Ring, about six miles north-west of Shoreham. Early in the spring of the following year the operations were resumed; and the principal object of this year's business was to determine the directions of the meridians of Dunmose and Beachy Head, for the purpose of ascertaining the length of the degree of longitude. The method employed for obtaining the meridian, was that of observing with the theodolite the distance between a terrestrial object and the pole-star upon each side of the pole when the star is at its greatest elongation. This observation gives the double azimuth nearly, without any corrections for the star's apparent motion; and as the motion in azimuth when the star is near its greatest elongation is slow, the time was shown sufficiently near by a good pocket watch. In this manner the angle made by the arc of the great circle joining the two places with the direction of the meridian, was observed at both stations; and as the distance between the stations would become known from the triangulation, all the elements were obtained for determining the length of the degree of longitude on the terrestrial spheroid at that latitude. After the observations at those two places were concluded, the instrument was taken to a few stations in the neighbourhood of Salisbury, and the operations of the year terminated at Highclere.

The greater part of the summer of 1794 was consumed in the measurement of a base of verification on Salisbury Plain. The measurement was effected in the same manner as at Hounslow Heath, one of the steel chains being used for measuring the different hypotenuses, and the other kept as a standard. The working chain was compared with the standard previous to the operation, and again when the measurement was finished. The details of the operation are entirely similar to those respecting the Hounslow Heath base; and the apparatus was different in no material respect. The base contained ninety-two hypotenuses, and the apparent length was 366 chains, minus 9-839 feet. The absolute length of the standard chain was assumed to be the same as it was found by Ramsden in 1791, by comparison with the standard scale; and after all the different reductions had been applied, the correct length of the base, at the temperature of 62° of Fahrenheit, and at the same level as the base on Hounslow Heath, was found to be 36,574-4 feet (nearly seven miles).

On computing the distance between the terminal points of this base, deduced from the Hounslow Heath base by different combinations of the triangles, the greatest and least results were found to be 36,574-8 and 36,573-8; the mean being 36,574-3 feet, or about one inch short of the measurement. So near an agreement must doubtless be ascribed, in part at least, to casual compensation of error; but it nevertheless affords a very satisfactory proof of the great accuracy with which the different parts of the work had been conducted.

The distance between Beachy Head and Dunmose, on which the determination of the length of a degree of the great circle perpendicular to the meridian was to depend, being computed from four different combinations of triangles, the mean of the four results was found = 339,397-6 feet, and the greatest difference from the mean was less than four feet. On comparing from this, and the observed angles made by the straight line joining the two stations with the meridians at each, according to the method which will be subsequently explained, the length of a degree of the great circle perpendicular to the meridian at altitude 50° 44' (nearly that of the middle point between Beachy Head and Dunmose), was found to be 61,182-3 fathoms, = 367,093-8 feet; and hence the degree of longitude at the two stations (which is found by multiplying the degree of the perpendicular circle by the cosine of the latitude) was obtained as follows: Beachy Head 328,312 feet, Dunmose 292,914 feet, the assumed latitudes being respectively 50° 44' 24" and 50° 37' 7". These results are respectively greater by about 772 feet than the corresponding degrees on the spheroid, which best represents the whole of the measured arcs of meridian, and of which the elements are given in Figure of the Earth, p. 563.

In order to mark permanently the two important stations of Beachy Head and Dunmose, an iron gun was inserted in the ground at each of the places, having the diameter of the bore in the same vertical line with the point over which the axis of the instrument had been placed. Unfortunately this precaution did not prove sufficient; for when Captain Kater was about to remeasure the angles at Beachy Head in 1826, the gun was not to be found. "In consequence of some misapprehension, it had been removed along with some old guns which were formerly near that place, and thus one of the valuable parts of the survey of Great Britain was irrecoverably lost." (Phil. Trans. 1828, p. 154.)

The account of the survey for the years 1791-4 inclusive, in addition to the particulars now stated, gives also the latitudes and longitudes of a great number of places determined by intersections made from the principal stations, and referred to the meridian of Greenwich, or, if towards the western extremity of the series, to that of Dunmose. At all the principal stations the angles of elevation and depression were observed; and these observations being reciprocal, gave not only the relative altitudes of the stations, but also the mean refractions, assuming the spherical surface of the earth. With a view to obtain the absolute altitudes, the height of the station at Dunmose above low water was ascertained by levelling down to the sea-shore near Shanklin, a distance of about a mile. The mean refractions were found to vary from 4th to 13th of the contained arc. It was also noticed, that the relative heights deduced from elevations and depressions cannot always be depended upon (on account of the variable state of the refraction) to less than about ten feet, even supposing them to be the mean of two or three independent results, except perhaps, reciprocal observations are made at the same instant of time. The observations from which the relative altitudes were deduced were made on cloudy days, or towards the evenings, when the tremulous motion of the air is commonly the least.

During the years 1795 and 1796, the triangulation was continued from the stations near the base on Salisbury Plain, along the coasts of Dorsetshire, Devonshire, and Cornwall, to the Land's End, as it was considered desirable to have an early determination of the latitudes and longitudes of the great headlands in the channel, and also of the Scilly Isles. The details of the operations, including the calculations of the side of the principal triangles, the heights of the stations, the mean refractions, the distances of a great number of intersected objects from the principal stations, are given in the Philosophical Transactions for 1797, and form the last part of the first volume of the Trigonometrical Survey. The same paper also contains the bearings and distances from the meridian and parallels, and also the latitudes and longitudes of a number of places observed in Kent in 1795, with a smaller theodolite (half the size of the principal one), for the purpose of completing the map of that county.

The next account of the operations connected with the Ordnance survey is contained in the Philosophical Transactions for 1800, and is given in the name of Captain William Mudge alone, Mr Dalby having now retired from the service. This account describes the operations in 1797, 1798, and 1799, and is divided into three sections, of which the first contains the calculations of the sides of the principal and secondary triangles extended over the country in those three years, together with an account of the measurement of a new base line on Sedgemoor. The second section contains the computed latitudes and longitudes of those places on the western coast which had been intersected in 1795 and 1796, and also of such other places as were found conveniently situated in respect of three new meridians, the determination of which forms part of the present account. The last section contains the triangles which were carried over Essex, the western parts of Kent, and portions of the counties adjoining Kent, Suffolk, and Hertfordshire.

With respect to the determination of meridians, just alluded to, it is to be observed, that by reason of the curvature of the earth, and the errors consequently arising from computing on the supposition that the earth is a plane, it becomes necessary that the direction of the meridian be determined anew where the operations are extended over distances of about sixty miles in an eastern or western direction. The distance from Dover to Land's End is somewhat about 300 miles, and between these places five intermediate meridians were observed, dividing the distance into six nearly equal parts. These were Beachy Head and Dunmose (already mentioned), Black Down in Dorsetshire, Butterton Hill in Devonshire, and St Agnes Beacon in Cornwall. In computing the longitudes and latitudes, the places were of course referred to the nearest of those meridians; and a place in the middle between two was referred to both, and the mean of the results taken.

The operations of 1797 commenced with the observation of the pole-star at Black Down, early in April; and in the course of the summer the great theodolite was taken to twenty-one other stations, at which the angles were determined, all included between the meridian of that station and St Agnes Beacon. It was judged inexpedient to carry In 1798 a series of secondary triangles was observed for completing the survey of Kent and Essex; but the principal operation of this year consisted in the measurement of a new base of verification on King's Sedgemoor, in Somersetshire. This new measurement was conducted in the same manner, generally, as those of the former bases; but on account of the irregularity of the ground, which was cut up in all directions by numerous ditches and large drains, it was thought expedient to have a new chain of fifty feet; and accordingly one was prepared by Ramsden, similar in construction to the two 100 feet chains used at Hounslow Heath and Salisbury Plain. This new chain was not used for the whole of the measurement, but only in a few cases when the handles of the 100 feet chain would have had their places over ditches, or in situations in which there would not have been the means of correctly placing the register heads under the handles. The measuring chains were compared with the standard, not before the measurement commenced, but after it had considerably advanced, and again at the end of the operation; and it was assumed that the length of the standard chain had remained unaltered since its last determination by Ramsden. The reduced length of the base was 27,580 feet, or nearly 5½ miles; and it was supposed by General Mudge that the error cannot exceed nine inches.

The next account which we have of the survey is contained in a paper which was read before the Royal Society in June 1804, entitled "An Account of the Measurement of an Arc of the Meridian, extending from Dunnose in the Isle of Wight, to Clifton in Yorkshire, by Major W. Mudge." This forms part of the second volume of the Survey. The measurement of an arc of the meridian was contemplated from the commencement of the survey, but had been delayed for some years on account of the zenith sector, with which the celestial arc was to be determined, not having been completed by Mr Ramsden, whose health was then declining, and who in fact died before the instrument was entirely finished.

For the purpose of determining the figure and dimensions of the earth by the measurement of meridional arcs, it is important that the arc be of considerable length, in order to diminish the influence of any error in the determination of the difference of the latitudes of its extreme points. On looking at the map of Great Britain, it will be seen that the longest meridional line contained in it, is one which passes from Lyme in Dorsetshire, northward into Scotland, and terminates at Aberdeen, comprehending an arc of nearly 4° 47′. This line, therefore, presented itself as the most eligible; but, on a closer examination, it was found that the arc would run through a country abounding in hills of considerable magnitude, and consequently that no advantage would probably be gained from observing the zenith distances of stars at any of the intermediate stations, on account of the irregular local attraction. General Mudge therefore selected the meridional line passing from Dunnose to the mouth of the Tees, as being the freest from apparent obstructions, and of sufficient length. The choice of Dunnose as the terminal station likewise presented a considerable advantage; for that station having been connected with the Royal Observatory by the previous operations, the astronomical observations made there would serve to correct the latitudes of the places formerly determined. The point selected for the northern extremity was Clifton, a small village in the vicinity of Doncaster, nearly on the meridian of Dunnose; and a level of sufficient extent for the measurement of a base of verification was found at Misterton Carr, in the northern part of Lincolnshire. The two extreme stations were connected by a chain of twenty-two triangles, lying nearly in the direction of the line to be measured. Of these triangles, eleven, extending from Dunnose to Arbury Hill, near the middle of the line, had been already observed, and their sides computed from the Hounslow Heath base. The angles of the remaining eleven were observed in the years 1801 and 1802, and the distances computed from the new base.

The base on Misterton Carr was measured in the summer of 1802, by the same apparatus, and exactly according to the same methods as had been employed in the three former bases, on Hounslow Heath, Salisbury Plain, and Sedgemoor. Previously to the commencement of the operations, the measuring chains (the hundred feet and fifty feet) were both compared with the standard chain; and a similar comparison was made after the work was completed. It was assumed that the length of the standard chain was the same as when it had been compared by Mr Ramsden; an assumption which was subsequently proved to be correct (within very small limits) by a new comparison of the standard chain with Ramsden's brass scale. The extremities of the base were marked by two large blocks of oak sunk into the ground, having each a square hole in its upper surface, into which lead was cast, and ground to a smooth plane; and the diagonals of the holes being drawn on the lead, the intersections of the diagonals formed the terminal points. After making the necessary reductions for temperature, the wear of the chains, &c., the true length of the base at temperature 62° was found = 26,342.7 feet. No reduction was made for height above the sea, the altitude of the ground on which the base was horizontally measured being only thirty-five feet above the surface of the sea in the mouth of the Humber at half tide. As the correct determination of this base was of great importance, every precaution was taken in the course of the operations, and General Mudge was of opinion that the error of the measurement in excess or defect could not exceed two inches.

On computing the first eleven triangles, beginning at Dunnose, the distance between the stations at Corley and Arbury Hill, near the middle of the arc, was found to be 117,463 feet; and this depended on the bases on Hounslow Heath and Salisbury Plain. On computing directly from the measured base at Misterton Carr, the remaining eleven triangles, the same distance between Corley and Arbury Hill was found to be 117,457.1 feet. The difference, therefore, falls short of six feet; but this cannot be regarded as great, when it is considered that the distance between the two stations is rather more than twenty-two miles, and that the whole line from Dunnose to Clifton is nearly 200 miles. Had the computation been carried on from Dunnose all the way to Clifton, the length of the base on Misterton Carr, deduced from those of Hounslow Heath and Salisbury Plain, would have been found to be about one foot greater than its measured length.

The whole terrestrial distance between the parallels of Method of the two stations was computed by the method of parallels computing and perpendiculars, with reference to the meridians of both the arc of the extreme stations. The bearings of certain sides of the triangles from the meridian of Dunnose were deduced from the observed angles; and the sides being respectively multiplied into the cosines of the bearings, gave the distances on the meridian; the sum of which distances, of course, was the length of the meridional arc, or rather the sum of the twenty-two chords deduced from the sides of the triangles. This method proceeds on the supposition of the earth's surface, to some distance on both sides of the line, being plane; but as the triangles ran nearly north and south, and most of them were intersected by the meridian, and, moreover, as the distance between the meridians of the two extreme stations was only a few feet, the supposition leads to no sensible error. General Mudge states that he took the trouble to calculate the length of the whole line by increasing each portion computed from the sides of the triangles by the difference between the chord and its arc, and that the result only exceeded the former computation by about two and a half feet.

The whole length of the arc, from Dunnose to the parallel of Clifton, was found to be 1,036,337 feet; from Dunnose to the point over which the sector was placed at Arbury Hill, 586,349-5 feet; and, consequently, from the parallel of Arbury Hill to that of Clifton, 450,017-5 feet.

The geodetical part of the operation being completed, it still remained to determine the amplitude of the celestial arc. The instrument employed for the purpose was a zenith sector, contrived and executed in greater part by Ramsden. A description of this superb instrument would here be out of place; but it may be proper to state the general principle of its construction. The object being to measure the zenith distances of stars which pass near the zenith (in order to diminish the effects of refraction), the sector is constructed by suspending a telescope nearly vertical, and in such a manner that its axis shall have a motion in a vertical plane, extending to only a few degrees on each side of the vertical. From the axis of motion, which is placed very near the top of the telescope, a plummet is suspended by a very fine wire; and on the lower end of the telescope a scale is placed in the plane of the motion, and directly before the plummet. Now, it will be obvious, that as the telescope carries the scale along with it when it is moved in the vertical plane, while the plummet remains at rest, the angular deviation of the axis of the telescope from the vertical will be measured on the scale. It is necessary that the instrument be capable of being turned half round in azimuth, for the sake of reversion. When in use, the plane of the sector is placed in the meridian at one extremity of the arc, and the zenith distances of some stars at their meridian passage observed. It is then transported to the other extremity, and the zenith distances of the same stars are there also observed; and the mean difference of these distances is the amplitude of the arc, which is by this means obtained immediately and independently of the latitudes of the stations or declinations of the stars. The telescope of the sector in question was nearly eight feet long, and had an object-glass of four inches in diameter.

The sector, after some preliminary examination at the Tower, was first set up at the Royal Observatory, for the purpose of observing the zenith distances of some stars to be afterwards observed at Dunnose, in order to determine the latitude of that station. It was then, in the spring of 1802, removed to Dunnose, where it was set up, at a distance of about six and a half feet from the station where the azimuth had been observed in 1791. The whole number of stars observed was twenty-seven; and a sufficient number of observations having been obtained by the end of June, the sector was then transported to Clifton, where seventeen of the same stars were observed. Subsequently, in the same year, the instrument was set up at the station at Arbury Hill, which is very nearly at the middle of the arc, and where twelve of the stars which had been seen at Clifton were observed. In October the party returned to London, and it was found, on examination, that the sector was in as perfect a state as when first sent into the field, so that no error was to be apprehended from derangement or injury during transportation to the different stations.

On computing the observations, the final results were found to be as follows (Measurement of an Arc of Meridian, Trigonometrical Survey, vol. ii. p. 107):

Mean amplitude of the celestial arc between Dunnose and Clifton ........................................... 2° 50' 23" 38 between Dunnose and Arbury Hill 1° 36' 19" 98 between Dunnose and Greenwich 0° 51' 31" 39

For the deductions from those amplitudes combined with the above geodetical measures relative to the degree of the meridian, we refer to the article Figure of the Earth, p. 562. It is however to be remarked, that since the publication of that article, a paper by the celebrated astronomer Bessel has appeared in the Astronomische Nachrichten (vol. xiv. No. 336), in which an error is pointed out in the determination of the celestial arc. It appears that one of the stars (α Aurigae), observed at all the stations, was incorrectly reduced to the beginning of 1802; and that the error (which probably arose from applying the correction for nutation with a wrong sign) amounted to 18". On correcting this error, and deducing the mean result by a more exact method of reduction, Bessel finds the amplitudes to be respectively as under:

Dunnose to Clifton ........................................... 2° 50' 23" 49" .............. to Arbury Hill ................................. 1° 36' 20" 98" .............. to Greenwich .................................. 0° 51' 31" 67"

Adopting the above amplitudes as corrected by Bessel, and assuming (according to the measurement) the distance between Dunnose and Clifton to be 1,036,337 feet, the length of a degree of the meridian at the latitude of the middle point (62° 2' 20") will be found = 364,925-25 feet, or 60,820-88 fathoms of the scale to which all the measures given in the survey are referred. General Mudge states this distance to be 60,820 fathoms. The length of the degree at the middle point of the arc between Dunnose and Arbury Hill being found in like manner, gave 60,844 fathoms, exceeding the above by forty-four fathoms. But as this latitude is farther to the south, the length of the degree ought, on the supposition of the earth's compression, to be less by about nine or ten fathoms. This anomaly, presented by the measurement, is usually ascribed to a deflection of the plumb-line of the sector, occasioned by local attraction.

The third volume of the Trigonometrical Survey appeared in 1811, under the joint names of Colonel Mudge and Captain Colby. It contains an account of the progress of the survey from 1800 to 1809 inclusive; and, when taken in connection with the accounts previously published, comprises the survey of almost all England, the south of Wales, and a part of Scotland. The volume embraces a great variety of interesting topographical matter. It gives the angles observed with the great theodolite at the principal stations; the calculation of the sides of the triangles, amounting in number to 281; the elevations and depressions of the stations, as seen from each other; a description of the situation of the stations; an account of the measurement of a new base on Rhuddlan Marsh, near St Asaph, in North Wales; and the prolongation of the arc of meridian from Clifton northward to Burleigh Moor in Yorkshire. It likewise contains the computed altitudes of the stations, and of many other remarkable hills, and also the latitudes and longitudes of all the principal places in the country.

The base on Rhuddlan Marsh was measured in October 1806, with the apparatus so often referred to in the preceding statement. After the necessary reductions were applied, its true length at temperature 62° was found to be 24,514-26 feet. The ground was a flat about four miles north-west of St Asaph, and its mean height above low-water mark being only twenty-five feet, no reduction was requisite for height above the sea. In proof of the agreement between the present and the former measurements, the following instances are given. Computing from the base on Misterton Carr, the distance between Castle Ring and Weaver Hill (two stations in Staffordshire) was found to be 111,144-1 feet; the same distance deduced from the triangulation proceeding from the new base in Rhuddlan Marsh was 111,148-4; whence the difference between the two results in a line exceeding twenty-one miles is only 4-3 feet. Again, the distance from May Hill to the station on Malvern Hills, depending on the Hounslow Heath base, was found = 82,290 The same distance, found by computing from the Rhuddlan Marsh base, was 82,251 feet; the difference being thus 89 feet. From this base several series of triangles were carried in different directions. One series extended to Anglesea, and thence by Snowdon, down the western coast of Wales, joining, near Aberystwyth, a series proceeding from the triangles formerly observed in Gloucestershire. A second series proceeded southward from the base, and joined the southern triangles in Glamorganshire. A third series branched out towards the east, and united with those proceeding westward from the base on Misterton Carr; and a fourth series was carried from the Rhuddlan Marsh base, through Lancashire, Westmoreland, and Cumberland, into Scotland, and connected with another series extending from the Misterton Carr base, through Yorkshire and Northumberland, and the east part of Scotland, as far as the north side of the Frith of Forth.

In the different series of triangles now mentioned, the third angle of the triangle was not observed in many instances, the peculiar character of the country rendering it impracticable, or at least very difficult, to carry the large theodolite to the station. It is stated in the preface to the volume, that "with an instrument so excellent as that used in the survey, except for the purpose of intersecting surrounding objects, it is perhaps not necessary to adhere tenaciously to the practice of observing the third angle of every triangle." Notwithstanding this opinion, we think it is much to be regretted that the important verification obtained from the observation of the third angle should have been dispensed with on any account whatever; and in fact the observation of many of those very angles, and a consequent recomputation of the sides, have since been found necessary, in order to reconcile the results.

One of the most interesting portions of the volume is the account of the extension of the arc of meridian northward from Clifton to Burleigh Moor, a place situated about three miles north from the town of Gisborough in Yorkshire. In July 1806, the sector was erected at this station, and the zenith distances of several stars observed. The observations were reduced to the first of January 1806, and those formerly made at Dunnose having been reduced to the same epoch, the mean of the whole gave the difference of latitude between the parallels of Dunnose and Burleigh Moor = 3° 57' 13" 6. The terrestrial distance between the parallels of the two places was calculated both on the meridian of the station at Burleigh Moor and on that of Dunnose, and the mean of the two results was found = 1,442,852 feet. This determination agrees very closely with the former determination of the arc between Dunnose and Clifton; for it has been stated that, in respect of that arc, the terrestrial distance was 1,096,837 feet, and the amplitude of the corresponding arc in the heavens = 2° 50' 23" 38'; and assuming the length of the arc to be proportional to its amplitude (which is not sensibly erroneous for an arc of three or four degrees), the difference of the latitudes of Dunnose and Burleigh Moor will be found by a simple proportion = 3° 57' 13" 6. But the difference of the latitudes, as determined by the sector observations, was 3° 57' 13" 1, and therefore disagreeing only to the extent of half a second, or about fifty feet on the ground. The resulting length of the meridional degree at the mean latitude is 364,943 feet.

Another determination of the length of a meridional degree was obtained by erecting the sector at the station of Delamere Forest, in Cheshire, about five miles north of Tarporley. Eight of the stars which had been observed at Dunnose were observed at this station; and the mean of the observations gave 2° 36' 12" 2 as the difference in latitude between the stations on Dunnose and Delamere Forest. The direction of the meridian was likewise observed at this station, and the bearing of one side of a triangle thereby determined; which being carried through the series of triangles connecting Delamere Forest with Black Down in Dorsetshire, the distance between the parallels of Dunnose and Delamere Forest was found = 925,189 feet. By carrying on the calculation through the same series of triangles, but with an azimuth determined at a different station, the distance between the parallels was obtained = 925,180 feet. The mean is 925,184 feet. Now if we seek an arc having the same proportion to the arc between Dunnose and Burleigh Moor which the distance between those stations has to this distance, we shall find 2° 36' 13" 2 for the amplitude. But the amplitude derived from the sector observations is 2° 36' 12" 2. Hence the difference between the observed and computed amplitude is only 1", which corresponds to an error of 0" 4 in one degree. The agreement may be considered as satisfactory; but by reason of the distance of Dunnose from the meridian of Delamere, this determination of the length of the meridional arc cannot be admitted as of equal certainty with that of the meridian of Burleigh Moor, in any deductions relative to the figure of the earth. The result, however, gives 60,883 fathoms, or 364,938 feet, for the length of a meridional degree in latitude 53° 34', or nearly the centre of England.

It would far exceed our limits to give a detailed description of the immense mass of observations and results contained in this volume, and it is beside our purpose to enter into any critical examination of its contents. In a work of such magnitude, and abounding with such a multiplicity of minute details, it will not be expected that errors can be entirely avoided; and a considerable number, it must be admitted, have found their way into the volume. In some respects, also, the deductions have not been made so as to furnish results possessing all the accuracy possible to be attained by means of the improved science of the present day; but it is to be observed, that many of the more refined methods of calculation have been introduced into the practice of geodesy since the volume was published. The method of assigning weights to the several results, of estimating the probable errors, and applying corrections according to a uniform fixed rule derived from the theory of chances, and thereby avoiding arbitrary adjustments, was then scarcely known. It is due, however, to the present able and enlightened superintendent of the survey to state, that all these less perfect methods of computation and reduction have long been abandoned, and that every advantage which can be derived from the refined theories of Gauss and Bessel has been brought to bear on the work; and we have little doubt, that when the next account of it shall appear, the scientific skill displayed in the reductions will be found to be quite on a par with the instrumental precision.

The preceding abstract brings down the history of the survey to the end of the year 1809; and we regret that our account must here be brought to a conclusion; for although the work has continued to be prosecuted with more or less intermission since that time, no further detailed account has been given to the public. It may not be without interest, however, to state generally a few particulars respecting its subsequent progress.

Since 1809, the prosecution of the survey has been continued under the able direction of Colonel Colby of the Royal Engineers. The triangulation which had been carried ve to the south-east part of Scotland, as far as the Frith of Forth, was first continued along the east coast to the borders of Ross-shire, and subsequently extended to the Shetland Islands. A series was also carried from the Cumberland triangles, along the western coast, through Dumfriesshire, and to the summit of Ben Lomond, connecting all the remarkable points in Perthshire. In 1817 a base of verification was measured with the steel chains by Colonel Colby, on Belhelvie Links, near Aberdeen. Shortly after this, the The Ordnance survey of Ireland is perhaps the most complete operation of the kind that has ever been executed in any country. Hitherto the different bases of verification had all been measured with the same apparatus, and according to the same methods, as were used on Hounslow Heath; but doubts having arisen respecting the accuracy attainable by means of the steel chains, Colonel Colby, on entering upon this new field, resolved to have recourse to an entirely different method of proceeding, and, instead of chains, to adopt the ingenious compensating apparatus which has been described in Figure of the Earth, p. 553. By this method, metallic bars of ten feet in length, and defined by points whose distance remains invariable in all temperatures, are placed accurately along the line to be measured; and the distance between the terminal points of the preceding and following bars is measured by a micrometer microscope, with a certainty altogether unattainable by measuring rods abutting against each other, or by observing the coincidence of straight lines on the handles of the steel chains. The ground selected for the base lies on the east shore of Lough Foyle, on the north coast of Ireland, and the line itself forms part of a straight line drawn from Shaw Mountain to Mount Sandy, nearly north and south. The whole line measured was nearly eight miles in length. For a verification, it was divided into two parts, one about half the length of the other, and the length of the one deduced from the other in various ways by triangulation. By this means it was estimated that the greatest possible error could not exceed two inches. It was also prolonged by triangulation from the north end to Mount Sandy, whereby two additional miles were given to it, so that it may be considered as a base of ten miles, the probable accuracy of the last part being quite as great as that of the part determined by the actual application of the measuring bars. From this base a series of triangles commenced, which proceeded all over the island, and were connected with those formerly observed in Wales, the Isle of Man, the west coast of Scotland, and the Hebrides. Some of these triangles were of enormous extent, the sides exceeding a hundred miles.

The survey of Ireland having been completed, at least so far as regards the primary triangles, that of Scotland was recommenced in 1838, and has already been carried from the eastern counties westward to the island of Lewis in the north. At the present time (1840), surveying parties are also employed at different places in Wales and the north of England, in observing the angles which had formerly been omitted, and verifying others which had not been satisfactorily observed.

We cannot conclude this brief and imperfect sketch without alluding to the admirable maps which have been published from time to time, and are now in progress. These are all designed from the materials collected in the field, and are not only drawn, but also engraved and printed, at the Ordnance Map Offices in the Tower, and at Dublin. Of the maps of England, thirty-eight counties, embracing all the southern part of the island, have now been published, engraved on a scale of one inch to the mile. The maps of Ireland, of which eighteen counties are published, are given on the magnificent scale of six inches to the mile, and not only exhibit, with the utmost distinctness, the natural features of the country, but even the minutest topographical details. This series of maps forms a splendid national work; and as no part of the map of Scotland has yet been engraved, we trust that the whole of that country will be given on the same scale.

Of the Standards of Length used in the Trigonometrical Survey.

In the British survey, all the linear measures are expressed in feet measured from a certain scale; and in order that the results relative to the dimensions and figure of the earth may be comparable with those of other similar operations, it is necessary to determine the relation of this unit to other known measures.

General Roy's measurement of the Hounslow Heath base was performed with glass rods, on which lengths of twenty feet had been set off by Ramsden, from a standard scale, which is described "as a finely divided brass scale" of the length of forty-two inches, with a Vernier's division of 100 at one end and one of fifty at the other, whereby the thousandth part of an inch is perceptible. It was originally the property of Mr Graham, the celebrated watchmaker; has the name of Jonathan Sisson engraved on it, but was known to be divided by Bird." From this scale the measure was transferred to the glass rods as follows. A deal plank upwards of thirty feet long, nine or ten inches broad, and about three inches thick, was set up edgewise on stands, and planed perfectly straight and smooth. A silver wire was then stretched along the middle of the edge from one end to the other, and six distances of forty inches each were marked off by the side of the wire, at which points brass pins were driven into the wood, and their tops polished. A fine dot being then made on one of the extreme pins, and the silver wire being stretched over the dot and the middle of the other pins, the extent of forty inches was, with the utmost care, taken from the scale, by means of a pair of beam-compasses, whose micrometer screw showed very perceptibly a motion of the 5000th part of an inch, and transferred to the following brass pin on the plank. In this manner all the six lengths were laid off. Two brass rectangular cheeks were then placed on the plank so as to bisect the two extreme dots, and present relatively to each other surfaces perfectly parallel. The glass rods were then placed between the cheeks, and the bell-metal buttons which formed their extremities ground down until the length (determined by the coincidence of a fine line on the glass rod with another on the moveable apparatus on the extremity of the rod) was accurately fitted between the cheeks, and was consequently twenty feet of the brass scale.

The scale which it has been usual of late years to adopt in works of science as the standard of English measure goes by the name of Sir George Shuckburgh's scale, and is described at length in the Phil. Trans. for 1798. The relation between this scale and several other standards was investigated by Captain Kater, whose experiments and results are given in the Phil. Trans. for 1821. From these experiments it was found that thirty-six inches of General Roy's scale are equal to 36'000930 inches of Sir George Shuckburgh's; whence, in order to reduce distances expressed in terms of General Roy's scale to Sir George Shuckburgh's, we must multiply by 9999/942.

Now it has been stated that the length of the Hounslow Heath base, as measured by the glass rods, was found to be 27404-08 feet. Using the above multiplier, the same distance expressed in terms of Shuckburgh's scale is 27403-38 feet.

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1 From the "Estimates of the Office of Ordnance" for the year 1840-41, printed by order of the House of Commons, it appears that the sums of money already granted for the surveys of England, Wales, and Scotland, amount to L320,163, and that the sum proposed to be taken in the present year is L18,400. On the survey of Ireland there has been expended the large sum of L619,520, including L60,000 for the services of the present year. The number of persons employed on the Irish survey, at the date of the estimates, is stated to be 2037; and the number employed on the British survey 82. In the Ordnance survey, all the bases, excepting the recent one in Ireland, were measured with a steel chain, which was compared from time to time, generally before and after the operation, with a similar steel chain of 100 feet, which was not otherwise used in the measurement. The primary measure in this case was a brass standard forty-inch scale, belonging to Mr Ramsden, and not General Roy's. From this standard, six lengths of forty inches were transferred, by means of the beam-compass, to prismatic bar of cast iron, into which brass points were inserted at the proper distances, so that the bar formed a measure of twenty feet. In order to transfer this to the steel chain, a portion of the chain was placed on rollers exactly parallel to the iron bar, and stretched, as in the measurement of the several bases, with a weight of fifty-six pounds. From the extreme points on the edge of the bar, marking the distance of twenty feet, fine wire plummetts were suspended, so as nearly to touch the chain. One end of the chain, determined by a fine line drawn on the brass handle, being then brought under the wire by means of an adjusting screw, a point coinciding with the other wire was made in the chain. This part of the chain was then shifted, and another twenty feet measured in the same manner; thus the whole chain was determined at five successive operations.

The brass scale which was used in this operation is supposed to be lost, but the prismatic bar is still in existence, and forms, therefore, the only authentic standard of the survey, the standard steel chain being probably liable to some deterioration from rust or wear at the joints. This bar was examined by Captain Kater, who found the mean yard to be equal to 36.00254 inches of Shuckburgh's scale. Assuming this determination to be correct, it follows that every distance given in the Ordnance survey, in order to be expressed in terms of Shuckburgh's scale, must be multiplied by 9999694. Now the Hounslow Heath base was found by the measurement with the steel chains to be 7,404.3165 feet of Ramsden's standard scale. The length, therefore, in terms of Sir George Shuckburgh's scale, is 7,402.88 feet.

It thus appears that the two measurements of this important base, when reduced to the same standard, give the following results:

with the glass rods..................27,403.88 feet, with the steel chain..................27,402.88 feet.

The difference is one foot, which is larger than can well be considered probable, supposing the steel measuring chain had been exactly five times the length of each of the glass rods. In the account of the survey, the difference is stated to be 2½ inches; but this was on the supposition that the two standards were precisely of the same length.

Prior to Captain Kater's experiments, it had been assumed, on some comparisons with an intermediate scale belonging to the Royal Society, that Ramsden's scale exactly agreed with General Roy's. The latter states (Trigonometrical Survey, vol. i. p. 16), that he placed the two scales (his own and the Society's) together on a table with micrometers alongside, and that after they had remained in this position two days, Mr Ramsden carefully took a length of three feet from the Royal Society's standard with his beam-compasses, and applied it to the other scale, and it was found to reach exactly thirty-six inches, the temperature being 65°. In like manner, when the length of the steel chains was determined, Mr Ramsden compared his own brass standard with that of the Royal Society; "and after the two standards had been allowed to remain together about twenty-four hours, they were found to be precisely of the same length." The temperature is not stated. Now this perfect agreement leads to the inference that the comparisons were not made with the requisite care, or that the means employed were insufficient.

In fact, it was found by Captain Kater, that the difference between General Roy's scale and that of the Royal Society amounted to 0.0047 of an inch; a quantity, however, within the limits of the distance measurable with the beam-compasses. The principal cause of the difference between the two determinations of the base may probably be ascribed to errors committed in transferring the length of the scale to the twenty-feet iron prism; and perhaps also in part to an erroneous comparison of this rod with Shuckburgh's scale; for it is said that the points on the brass pins inserted in the rod are so worn and enlarged by the application of the beam-compass as to render any exact comparison impossible.

The standard measure of the Irish base is a distance of ten feet, defined by two fine points on a bar of cast iron. We are not acquainted with the length of the mean foot in terms of Shuckburgh's scale; but as none of the results yet published are given in terms of this base, its accurate length, in relation to other standards, is at present unimportant.

Of the Selection of the Stations and the Signals.

In conducting a survey over an extensive country, the choice of stations must in a great measure be determined by the nature of the ground; for it is obvious that signals can only be erected at places mutually visible from each other. But although the natural irregularities of the surface of the ground render an entirely arbitrary disposition of the signals impracticable, there will frequently be room for a choice between two or more points; and it is therefore important to determine the conditions which must be fulfilled, in order that the inevitable errors of observation may have the least effect on the measured distances.

Let A, B, C be the three angles of a triangle, and \(a, b, c\) the sides respectively opposite. In each case the data are the three angles, and one of the sides, as \(a\); and the elements to be calculated are \(b\) and \(c\), the remaining sides. The question therefore is to determine the species of triangle, in order that the computed lengths of \(b\) and \(c\) may be the least affected by small indeterminate errors in the measurement of the angles. Suppose each of the angles to have been observed, and let the errors of observation be respectively \(\alpha, \beta, \gamma\). When the sum of the observed angles differs from 180°, or rather from 180° + the spherical excess, each angle is corrected by applying to it one third of the excess or defect. After the angles have thus been corrected, they still remain affected with the errors of observation; but as the sum is now correct, it is plain that the sum of the errors is nothing, or that \(\alpha + \beta + \gamma = 0\), and consequently \(\alpha = - (\beta + \gamma)\). Now, putting the given side \(a = 1\), the side \(c\) is found in parts of \(a\) from the formula \(c = \frac{\sin(C + \gamma)}{\sin(A + \alpha)} = \frac{\sin C \cos \gamma + \cos C \sin \gamma}{\sin A \cos \alpha + \cos A \sin \alpha}\); and

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1 On this subject, see an excellent paper by Mr Bailly in the ninth volume of the Memoirs of the Royal Astronomical Society. 2 In the article Figure of the Earth, the lengths of the English arcs of meridian are copied from the Trig. Survey, and are consequently expressed in feet of Ramsden's standard scale, while all the foreign measurements, as well as those of the Indian arcs, are in feet of Shuckburgh's scale reduced according to the comparative values of the different standards given by Captain Kater. This inadvertence does not affect the values there given of the earth's axes, and the lengths of the equatorial and meridional degrees, all of which are understood to be expressed in feet of Shuckburgh's scale. since \(a\) and \(\gamma\) are very small angles, we may assume \(\cos a = 1\), \(\sin a = a\), \(\cos \gamma = 1\), \(\sin \gamma = \gamma\), by which the formula becomes, on substituting \((3 + \gamma)\) for \(a\),

\[ c = \frac{\sin C}{\sin A} - (3 + \gamma) \cos A \]

Dividing the numerator of this expression by the denominator, and rejecting the terms in the quotient which contain the squares and higher powers of \(\beta\) and \(\gamma\), we get

\[ c = \frac{\sin C}{\sin A} + \frac{\beta \cos A \sin C}{\sin^2 A} + \frac{\gamma \sin B}{\sin^2 A} \]

so that the error of the side \(c\) is

\[ \frac{\beta \cos A \sin C}{\sin^2 A} + \frac{\gamma \sin B}{\sin^2 A} \]

Similarly, the error of the side \(b\) is found to be

\[ \frac{\gamma \cos A \sin B}{\sin^2 A} + \frac{\beta \sin C}{\sin^2 A} \]

It now remains to determine the conditions under which these expressions have the smallest values. Since \(a + \beta + \gamma = 0\), in every case two of these quantities must have the same sign, and the third the opposite. If therefore \(a\) and \(\beta\) are both positive, \(\gamma\) negative; and if they are both negative, \(\gamma\) positive; so that as \(\gamma\) must of necessity be either positive or negative, there are only two chances for \(a\) and \(\beta\) having the same sign. In like manner, there are also two chances only for \(a\) and \(\gamma\) having the same sign and \(\beta\) the opposite, and for \(\beta\) and \(\gamma\) having both the same sign and \(a\) the opposite. It thus appears, that for any two specified errors, there are only two chances for both having the same sign, and four chances for their having opposite signs; so that in the above expressions the probability that \(\beta\) and \(\gamma\) are of opposite signs, is twice as great as the probability of both having the same sign. Now if they have opposite signs, both expressions will be diminished if \(\cos A\) be positive, that is, if the third angle \(A\) be less than 90°, or less than a right angle. With respect to the relation between \(B\) and \(C\), which gives the smallest chances of error, it may be assumed as probable that the two errors \(\beta\) and \(\gamma\), though they have opposite signs, will not differ greatly in magnitude. But if they are nearly equal, the two expressions for the errors of \(b\) and \(c\) will have the smallest values when \(B\) and \(C\) are nearly equal. Hence the conditions which afford the greatest probability of the smallest errors are these:

1. That the angle opposite to the measured side be less than a right angle; and, 2. that the angles adjacent to that side be nearly equal. These conditions will be best fulfilled on the average of a series, by making the triangles as nearly as possible equilateral.

If only two angles \(B\) and \(C\) have been observed, then there is the same probability of \(\beta\) and \(\gamma\) having the same sign or opposite signs. In this case the chance of smallest error is obtained by making \(\cos A = 0\), that is, when \(A\) is a right angle.

In the earlier part of the British survey, the signals made use of were of various kinds, but principally flag-staffs carrying reverberatory lamps, and furnished with concave copper reflectors about nine or ten inches in diameter, well polished and silvered, and enclosed in tin cases, having plates of ground glass in front, to prevent the bad effects of unequal and unsteady light from exposure to the wind. Such signals can be seen at the distance of twenty or twenty-four miles. For more distant stations, Bengal or white lights were fixed in small sockets, supported on a tripod of about five or six feet in height. By means of a plummet, these could be readily placed precisely over the point marking the station. The now well-known Drummond light (for the description of which see Mr Drummond's paper in the Phil. Trans. for 1826) was practically applied as a night-signal at some of the stations in Ireland, and the west of Scotland. But the difficulties and inconveniences of night-signals are so great, and the observations attended with so much uncertainty from the unsteadiness and scintillation of the light, that the practice of observing by night has of late years been abandoned. At present the signals are usually formed by building a conical pile of stones, in some instances exceeding twenty feet in diameter at the base, over the point which marks the centre of the station. Such signals are of course subject to the great inconvenience, that they can only be seen in favourable states of the atmosphere; but in all other respects they are found preferable to any others. In some of the late operations where the distances were not very great, a plate of metal was used, having a narrow vertical slit cut in it; and in this case the signal consists of the line of light passing through the slit, and which can be rendered more brilliant by means of reflectors suitably disposed. Colonel Colby has also on some occasions successfully employed the heliotrope; a method recommended by Gauss, and which consists in throwing a beam of solar light upon a distant station by reflection from a small plane mirror, as that of a sextant or reflecting circle. A description of this ingenious but not very convenient method of forming signals for geometrical purposes is given in Zach's Correspondance Astronomique, vol. vi. p. 374.

Of the Calculation of the Sides of the Triangles.

In measuring angular distances with the theodolite, the plane of the instrument is carefully adjusted to the horizon, so that the angle observed is the horizontal angle at the station; and consequently, no previous reduction is necessary, on account of the three summits of the triangle being computed being unequally distant from the earth's centre, as is the case when the angles are measured with a sextant or repeating circle. The practice which was always followed in the Ordnance survey, of placing the centre of the theodolite directly under the centre of the signal, also obviated the necessity of a calculation for reducing the observed angle to the centre of the station. Formulas for both these reductions will however be found in Figure of the Earth, p. 555.

The three angles of any spherical triangle, as is well known, exceed 180° by a certain quantity, called the spherical excess, which is proportional to the area of the triangle. Let \(A\), \(B\), \(C\) denote the angles of a spherical triangle, \(r\) the radius of the sphere expressed in feet, \(\sigma = 3:14159\) the ratio of the circumference to the diameter, and \(S\) the number of square feet in the surface or area of the triangle, then, by trigonometry,

\[ S = \frac{A + B + C - 180^\circ}{180^\circ} r^2 \sigma \]

Let \(E\) denote the spherical excess \(= A + B + C - 180^\circ\); then \(E = \frac{S \times 180^\circ}{r^2 \sigma}\) in degrees, or \(E = \frac{S \times 648000}{r^2 \sigma}\) expressed in seconds. In any triangle which can be measured on the surface of the earth, \(S\) is very small in comparison of \(r\), and therefore \(E\) is a very small quantity. (In practice it seldom exceeds four or five seconds, though in some of the large triangles observed in the west of Scotland, whose sides exceed 100 miles, it amounted to thirty or forty seconds.) Hence an approximate value of \(S\) will enable us to compute \(E\) with sufficient precision. For this purpose, therefore, the triangle may be regarded as a plane one; and on denoting by \(a\), \(b\), \(c\) the number of feet in the sides respectively opposite to \(A\), \(B\), \(C\), we shall have for the area, \(S = \frac{ab \sin C}{2}\). Substituting this in the formula for the spherical excess, we get, in seconds,

\[ E = \frac{ab \sin C \times 648000}{2 \sigma} \] This formula is deduced on the hypothesis that the triangle is on a spherical surface, but it applies also to triangles on the surface of a spheroid; for it may be demonstrated that the spherical excess is the same for triangles on a spheroid and on a sphere, when the latitudes of the stations and their differences of longitudes are the same.

In order to compute the spherical excess of any triangle, it is necessary to know the value of \( r \), the radius of curvature of the spherical surface. Now the curvature of the arc joining any two stations on a spheroid varies with the latitude of the stations, and also with the direction of the line in question with respect to the meridian; but for the present purpose it will, in general, be sufficient to assume the value of \( r \), which corresponds to the curvature of the meridian at the mean latitude of the stations, and even to suppose it constant for a whole series of triangles contained between two parallels of latitude not distant more than a few degrees. If, however, the triangles are very large, it may be necessary to compute more accurately; and in such cases the nearest approximation to the true spherical excess will be found by computing, for the mean latitude of the three stations, the curvature of the meridian, and of the circle perpendicular to the meridian, and taking the mean of the two for the value of \( r \); or, which is nearly the same thing, by computing the radius of the vertical circle which is the meridian at an angle of 45° at that mean latitude.

The determination of the curvature of the meridian, the circle perpendicular to the meridian, and the geodetic line making any oblique angle with the meridian, are required for various other purposes connected with the survey as well as the present; we shall here subjoin the formulae by which they are severally computed from assumed dimensions and ellipticity of the earth.

Let \( ALP \) be the arc of the meridian passing through the station \( L \), \( AC \) the semidiameter of the equator, \( CP \) the semi-axis, \( M \) the normal at \( L \), meeting \( PC \) produced in \( N \). Assume \( a = C P \), \( b = AC \), and \( e = \text{the ellipticity, or } e = a (1 + e^2) \), and let \( \lambda \) be the latitude of \( L \), and \( R \) the radius of curvature of the meridian at \( L \); then it is shown in Figure of the Earth, p. 559, that

\[ R = a (1 - e + 3e \sin^2 \lambda) \ldots \ldots \ldots (2). \]

Let \( R' \) be the radius of curvature of the arc perpendicular to the meridian at \( L \); and it is shown in the same figure, p. 558, that \( R' = LN \), the normal extended to its intersection with the polar axis. Now let \( n = LM \), the normal at \( L \); then, by conic sections, \( R' = n = b^2 / a^2 \); whence \( R' = (1 + e)^2 \). But \( n = a (1 - e \cos^2 \lambda) \) (Figure of the Earth, p. 559); therefore, rejecting terms containing the square of \( e \), as insignificant, we find

\[ R' = a (1 + e + e \sin^2 \lambda) \ldots \ldots \ldots (3). \]

To find the curvature of the oblique circle, let \( r \) be the radius of curvature at the point \( L \) of a section of the spheroid containing \( LN \), and making with the meridian an angle \( \delta \); we have the following expression found by Euler (Lacroix, Calcul Différentiel et Integral, vol. i. p. 578).

\[ r = \frac{R'R'}{R \sin^2 \delta + R' \cos^2 \delta} \ldots \ldots \ldots (4). \]

This last expression may be put under a form more convenient for calculation. Dividing both terms by \( R' \), substituting \( 1 - \sin^2 \delta \) for \( \cos^2 \delta \), and converting the result into series, all the terms of which after the second may be neglected, we get

\[ r = R \left( 1 + \frac{R'R}{R'} \sin^2 \delta \right) \ldots \ldots \ldots (4). \]

Since in any circle the length of a degree is proportional to the radius (it is found by dividing the radius by the constant number 57.29578), if we make \( M = \) the length in feet of a degree of the meridian at \( L \), \( P = \) the length of a degree of the perpendicular arc, and \( D = \) the degree of an arc which makes with the meridian an angle \( \delta \), we shall have also

\[ D = M \left( 1 + \frac{P - M}{P} \sin^2 \delta \right) \ldots \ldots \ldots (5), \]

which is the expression usually given, and by means of which the length of the oblique degree is found in terms of the degrees of the meridian and perpendicular.

Having computed the spherical excess \( E \) from approximate values of the lengths of the sides (obtained by supposing the triangle a plane one), the sum of the three observed angles should be \( 180^\circ + E \). But as every observation is attended with some degree of uncertainty, the probability is infinitely small that the sum will be precisely equal to this quantity in any case. The difference (which in general will amount to some seconds) is the error of the observed angles; and the next question to be considered is, how should the error be apportioned among the three angles, so that the probability of the result being true may be greater than if any other mode of distribution were adopted? If no reason exists for supposing that one angle has been determined more accurately than another, the error should of course be equally divided among the three angles; but in practice this is seldom the case, for it will usually happen that one or other of the angles has been determined by a greater number of observations, or by observations made under more favourable circumstances than the others, and consequently the three determinations are not affected with the same probable errors. In the earlier period of the Ordnance survey, and indeed so far as the published account extends, the apportionment of the error appears to have been made in a manner entirely arbitrary, or at least according to the observer's judgment of the relative goodness of the observations; but this objectionable practice is now abandoned, and a uniform method, founded on the theory of chances, adopted. Suppose several observations to have been made of the same angle, and that the seconds of reading are \( l, l', l'' \), &c., and let \( m \) be the average or arithmetical mean of the whole; then \( m - l, m - l', m - l'' \), &c., are the errors of the individual observations, and the weight of the determination, or of the average \( m \), is equal to the square of the number of observations divided by twice the sum of the squares of the errors. See Probability, vol. xviii. p. 635, No. 145. In this manner the weight is found for each angle, and the error of the triangle, that is, the difference between the sum of the three angles (each being the average of the observed values) and \( 180^\circ + E \), is divided into three parts respectively proportional to the reciprocal of the weights, which parts form the corrections to be added to or subtracted from the angles to which they respectively correspond. We have then three corrected spherical angles, the sum of which is exactly \( 180^\circ + E \).

The three spherical angles of the triangle being thus determined, the next step in the operation is to compute the lengths calculating of the two remaining sides; one side being always known, the sides either by the measurement of a base, or by the previous computation of another triangle. Three different methods of computation have been practised. That which first suggests itself is to transform the side whose length is already known in feet, into an arc of a circle (which is done by comparing it with the radius of the earth), and solving the triangle by the usual formulae of spherical trigonometry. This was the method followed by Boscovich in his measurement of the Italian arc of meridian; and it was also practised in some instances by Delambre; but as it involves a somewhat tedious process of calculation, it is not that which is generally adopted. As the distance of any two stations mutually visible from each other is very small in comparison of the whole circumference of the earth, the chord of the intercepted arc will differ from the arc itself by a quantity which may be computed from the known ratio of the chord to the radius of the earth, but which in general is so small as to be insensible. If, therefore, from the observed spherical angles we deduce in each case the corresponding angles formed by the chords, and with these compute the sides by plane trigonometry, we shall obtain the chords of the arcs intercepted between the stations, and thence the arcs themselves, or the true geodetical lines on the spheroid. On this principle all the triangles of the Ordnance survey have been computed. A third method, which has been followed in all the recent continental surveys, and which was also adopted by Captain Kater in the verification of General Roy's triangles, depends on a theorem first demonstrated by Legendre; namely, that a triangle on a sphere or spheroid, which is small in comparison of the whole spherical surface, differs insensibly from a plane triangle of which the sides are respectively equal in length to the sides of the triangle on the sphere, and whose angles are respectively equal to those of the spherical triangle, each diminished by one third of the spherical excess. We shall here give formulae for the two last methods, with an example of their application.

To reduce an observed horizontal angle to the angle formed by the chords of the containing sides, let \( r \) be the radius of the sphere on which the triangle is described; \( a, b, c \), the sides (the radius and sides being both expressed in feet); and, for the sake of abridging, let

\[ \frac{a}{r} = \alpha, \quad \frac{b}{r} = \beta, \quad \frac{c}{r} = \gamma; \]

so that \( a, \beta, \gamma \) represent the sides of a similar triangle on a sphere whose radius \( = 1 \). Let \( A \) be the horizontal angle opposite the side \( a \), and \( A - x \) the corresponding angle formed by the chords; the object is to find the correction \( x \) to be applied to the observed angle \( A \). From a well-known formula of plane trigonometry, we have

\[ \cos(A - x) = \frac{\text{chord}^2 \beta + \text{chord}^2 \gamma - \text{chord}^2 a}{2 \text{chord} \beta \times \text{chord} \gamma}; \]

or, since chord \( a = 2 \sin \frac{1}{2} a \),

\[ \cos(A - x) = \frac{\sin^2 \frac{1}{2} \beta + \sin^2 \frac{1}{2} \gamma - \sin^2 \frac{1}{2} a}{2 \sin \frac{1}{2} \beta \times \sin \frac{1}{2} \gamma}. \]

Now \( \cos(A - x) = \cos A \cos x + \sin A \sin x \); and since \( x \) is small, \( \cos x = 1 \), and \( \sin x = x \) (very nearly); therefore

\[ \cos(A - x) = \cos A + x \sin A, \]

and consequently

\[ \cos A + x \sin A = \frac{\sin^2 \frac{1}{2} \beta + \sin^2 \frac{1}{2} \gamma - \sin^2 \frac{1}{2} a}{2 \sin \frac{1}{2} \beta \times \sin \frac{1}{2} \gamma}...(A). \]

Now in any spherical triangle

\[ \cos A = \frac{\cos \alpha - \cos \beta \cos \gamma}{\sin \beta \sin \gamma} \]

\[ = \frac{1 - 2 \sin^2 \frac{1}{2} \alpha - (1 - 2 \sin^2 \frac{1}{2} \beta)(1 - 2 \sin^2 \frac{1}{2} \gamma)}{4 \sin \frac{1}{2} \beta \cos \frac{1}{2} \beta \times \sin \frac{1}{2} \gamma \cos \frac{1}{2} \gamma} \]

\[ = \frac{\sin^2 \frac{1}{2} \beta + \sin^2 \frac{1}{2} \gamma - \sin^2 \frac{1}{2} a}{2 \sin \frac{1}{2} \beta \cos \frac{1}{2} \beta \times \sin \frac{1}{2} \gamma \cos \frac{1}{2} \gamma}, \]

therefore, by reason of the equation (A),

\[ \cos A = \frac{\cos A + x \sin A}{\cos \frac{1}{2} \beta \cos \frac{1}{2} \gamma} = \frac{\sin \frac{1}{2} \beta \sin \frac{1}{2} \gamma}{\cos \frac{1}{2} \beta \cos \frac{1}{2} \gamma}; \]

whence we find

\[ x \sin A = \sin \frac{1}{2} \beta \sin \frac{1}{2} \gamma - (1 - \cos \frac{1}{2} \beta \cos \frac{1}{2} \gamma) \cos A. \]

Now the arcs \( \beta \) and \( \gamma \) being small, we may assume, without sensible error, \( \sin \frac{1}{2} \beta = \frac{1}{2} \beta, \cos \frac{1}{2} \beta = 1 - \frac{1}{8} \beta^2 \);

whence (making also the like assumption for \( \gamma \)), we get

\[ x \sin A = \frac{\beta \gamma}{4} - \frac{\beta^2 + \gamma^2}{8} \cos A; \]

therefore, since \( \frac{1}{8} (\beta + \gamma)^2 = \frac{1}{16} (\beta + \gamma)^2 + \frac{1}{16} (\beta - \gamma)^2, x \sin A = \frac{1}{16} (\beta + \gamma)^2 - \frac{1}{16} (\beta - \gamma)^2 \cot \frac{1}{2} A \)

and consequently, dividing both sides by \( \sin A \),

\[ x = \frac{1}{16} (\beta + \gamma)^2 \tan \frac{1}{2} A - \frac{1}{16} (\beta - \gamma)^2 \cot \frac{1}{2} A. \]

In this formula \( x \) is expressed in parts of the radius \( = 1 \); but in the applications \( x \) must be expressed in seconds of arc. Now if \( R^o \) denote the number of seconds in the radius \( = 203,2645^o \), then \( 1 : x :: R^o : \text{seconds in arc } x \); where \( x R^o = \text{seconds in arc } x \). Multiplying therefore the above equation by \( R^o \), and substituting for \( \alpha, \beta, \gamma \), their values, we have ultimately

\[ x = \frac{1}{16} \left( \frac{b + c}{r} \right)^2 \tan \frac{1}{2} A \cdot R^o - \frac{1}{16} \left( \frac{b - c}{r} \right) \cot \frac{1}{2} A \cdot R^o \quad (6), \]

from which formula \( x \) is found in seconds.

In like manner, the corrections are found for the two other horizontal angles \( B \) and \( C \), each being expressed in terms of the angle itself and its containing sides. Let these corrections be respectively \( x' \) and \( x'' \), then it is evident that \( x + x' + x'' = E \), the spherical excess, and consequently, that when the true horizontal angles are respectively diminished by \( x, x', \) and \( x'' \), the sum of the three angles thus reduced will be exactly \( 180^o \). These reduced angles are the angles for calculation; and on computing from them the sides, we obtain the chords of the spherical arcs intercepted between the stations, whence the arcs themselves are easily obtained by means of the following formula:

Let \( \varphi \) be any small arc in parts of the radius \( = 1 \); then chord \( 2\varphi = 2 \sin \varphi \); but we have also \( \varphi = \varphi - \frac{1}{2} \varphi^2 \); therefore chord \( 2\varphi = 2\varphi - \frac{1}{2} \varphi^2 \). Now, if we assume \( \varphi = \frac{a}{r} \), or \( \varphi = \frac{a}{2r} \); then chord \( \frac{a}{r} = \frac{a^2}{24r^3} \) and consequently

\[ a = \text{chord } a + \frac{a^3}{24r^3} \quad \ldots \quad (7). \]

In the last term of this expression the chord may be taken for the arc, and we have therefore this rule for finding the arc in terms of the chord. Divide the cube of the chord expressed in feet by twenty-four times the square of the radius (in feet), and add the result to the computed length of the chord; the sum is the corresponding length of the arc.

We shall now give an example of the application of the preceding formulae by computing the sides of one of the large triangles connecting the west of Scotland with Ireland, with the requisite data for which we have been obligingly furnished by Colonel Colby. The three stations or angular points are Benlomond in Shetlandshire, Cairnmuir-on-Dengh in Kirkcudbright, and Knocklayd in the county of Antrim; the longest side, from Benlomond to Knocklayd, exceeding ninety-five miles. The data are as follows.

At Benlomond the angle subtended by the stations on Knocklayd and Cairnmuir-on-Dengh was determined from three observations, giving respectively (we repeat only the seconds) \( 56^\circ 43' 29'' 97, 27' 04', 26'' 72 \). The mean is \( 56^\circ 43' 28'' 58 \); whence the errors are respectively \( +139, -156, +014 \); the squares of the errors \( 14931, 23716, 0196 \); the sum of the squares \( 43233 \); the weight (the square of the rule of observations divided by twice the sum of the squares of the errors) \( = 9 \div 84466 = 0411 \); and the reciprocal of the weight \( = 961 \).

At Cairnmuir-on-Dengh, one observation gave the angle \( 79^\circ 42' 28'' 69 \). The weight is assumed \( = 1 \), whence the reciprocal of the weight is 10.

At Knocklayd, two observations gave the angles \( 43^\circ 34' 35'' 36, 35'' 43 \). The mean is \( 43^\circ 34' 36'' 69 \); the errors respectively \( +147 \) and \( -146 \); the weight found as above \( = 4660 \); and the reciprocal of the weight \( = 2146 \).

We have therefore the following data for the computation of the triangle: Hence \( x = 11^\circ 583 - 1^\circ 231 = 10^\circ 352 \). As the corrections for the other angles B and C are found precisely in the same manner, it is unnecessary to give the calculation. The results are respectively \( x' = 14^\circ 684 \), \( x'' = 9^\circ 724 \).

From this we have the spherical excess \( E = (x + x' + x'') = 34^\circ 760 \). But the excess of the sum of the three observed angles above 180° was 34° 16', consequently the error of the observations is — 0° 60'. Now, on dividing this error into three parts proportionally to the three numbers 961, 10, 2146 (the reciprocals of the weights), we find the corrections for the observed angles to be respectively \( + 0^\circ 04 \), \( + 0^\circ 46 \), and \( + 0^\circ 10 \). On applying these corrections, we obtain the true spherical angles; and on diminishing each of these last by the corresponding corrections for reduction to the chords, viz. 10° 35', 14° 68', 9° 73', we obtain the chord angles, or angles for calculation. The results are as follows:

| Observed Angles | Spherical Angles | Chord Angles | |-----------------|------------------|-------------| | A | 56° 43' 28" 58 | 56° 43' 18" 27 | | B | 79° 42' 28" 69 | 79° 42' 14" 47 | | C | 43° 34' 36" 89 | 43° 34' 27" 26 |

With the angles in the last column of this table, we are enabled to calculate the lengths of the chords opposite A and B. It is now necessary to use logarithms to seven decimals at least. The calculation is as follows.

Log. chord \( c = 35203348 = 54563840 \) Log. sin. A \( (56° 43' 18" 27) = 99222144 \) Log. sin. B \( (79° 42' 14" 47) = 99929499 \) Co.log.sin.C \( (43° 34' 27" 26) = 01615956 \)

Log. chord \( a = 56303940 \) Log. chord \( b = 57011295 \)

and the results are, chord \( a = 426,966-69 \) feet, chord \( b = 502,492-46 \) feet, chord \( c = 352,033-48 \) feet.

It now only remains to determine the spherical arcs corresponding to the above chords. Using the same value of the radius as above, namely, log. \( r = 732112 \), we find

\[ \frac{a^2}{24^2} = 739, \quad \frac{b^2}{24^2} = 1205, \quad \frac{c^2}{24^2} = 414, \]

and therefore, by equation (7), the distances on the surface of the spheroid are,

Cairnsmuir to Knocklayd \( a = 426,974-08 \) Benlomond to Cairnsmuir \( b = 502,504-51 \) Benlomond to Knocklayd \( c = 352,037-62 \)

The whole process is brought under one view, by arranging the calculations as in the subjoined table.

| Stations | Observed Angles | Observations | Chord Corrections | Appor. of Error | Angles for Calculation | Calculation | Chord Sides opposite each Angle | Lengths of Arcs | |----------------|-----------------|--------------|-------------------|----------------|------------------------|-------------|--------------------------------|---------------| | Benlomond | 56° 43' 28" 58 | 3 | 0-961 | 10-35 | 56° 43' 18" 27 | 54563840 | 426966-69 | 426974-08 | | Cairnsmuir | 79° 42' 28" 69 | 1 | 10-00 | 14-68 | 79° 42' 14" 47 | 99222144 | 502492-46 | 502504-51 | | Knocklayd | 43° 34' 36" 89 | 2 | 2-146 | 9-73 | 43° 34' 27" 26 | 01615956 | 352033-48 | 352037-62 | | | 180° 0 34' 16" | | | | | | | | | Error, — 0° 60'| | | | | | | | | The method now explained is that which has been followed in computing all the triangles of the Ordnance survey. The following, which is generally adopted in the continental measurements of geodetical arcs, is of somewhat easier application.

As before, let \(a \div r, b \div r, c \div r\), be represented respectively by \(a, b, c\). We have then

\[ \cos A = \frac{\cos a - \cos b \cdot \cos c}{\sin b \cdot \sin c} \]

and since the arcs \(a, b, c\) are small in comparison of \(r\), a sufficient approximation will be obtained, if, in the development of the sines and cosines in terms of the arc, we retain the terms only which involve powers not higher than the fourth, that is, if we make

\[ \cos a = 1 - \frac{a^2}{2} + \frac{a^4}{2 \cdot 3 \cdot 4}; \quad \sin b = b - \frac{b^3}{2 \cdot 3} \]

and also make a like substitution for \(\cos b, \cos c, \gamma\), and \(\sin \gamma\).

With these values, the above equation becomes

\[ \cos A = \frac{1}{2} (\beta^2 + \gamma^2 - a^2) + \frac{1}{2} (a^4 - b^4 - c^4) - \frac{3}{2} \beta^2 \gamma^2 \]

Restoring the values of \(a, b, c\), and denoting the numerators of the two terms by \(M\) and \(N\) respectively, we have

\[ \cos A = \frac{M}{2bc} + \frac{N}{24bc^2} \]

In a plane triangle whose sides are respectively equal in length to the three arcs \(a, b, c\), let \(A'\) be the angle opposite to the side \(a\); then

\[ \cos A' = \frac{b^2 + c^2 - a^2}{2bc} = \frac{M}{2bc} \]

On squaring both sides of this equation, and substituting \(1 - \sin^2 A'\) for \(\cos^2 A'\), we get

\[ -4b^2c^2 \sin^2 A' = a^4 + b^4 + c^4 - 2a^2b^2 - 2a^2c^2 - 2b^2c^2 = N \]

In consequence of these two values of \(M\) and \(N\), the equation (8) becomes

\[ \cos A = \cos A' - \frac{bc}{6r^2} \sin A' \]

If we now assume \(A = A' + x\), then \(x\) will obviously be a very small arc, and we may suppose \(\cos x = 1\), and \(\sin x = x\), without sensible error. Hence \(\cos A = \cos A' - x \sin A'\), and we have therefore \(x = \frac{bc}{6r^2} \sin A'\); whence

\[ A = A' + \frac{bc}{6r^2} \sin A' \]

Now, \(\frac{1}{2}\) be sin. \(A'\) is the area of the rectilineal triangle whose sides are \(a, b, c\), and which does not sensibly differ from the spherical triangle; therefore, denoting this area by \(S\), and observing that a similar result will be found for \(B\) and \(C\), we shall have \(A = A' + \frac{1}{2} (S \div r^2)\), \(B = B' + \frac{1}{2} (S \div r^2)\), \(C = C' + \frac{1}{2} (S \div r^2)\); therefore, since \(A' + B' + C' = 180^\circ\), \(A + B + C - (S \div r^2) = 180^\circ\), so that \(S \div r^2\) may be regarded as the excess of the three angles \(A, B, C\) of the spherical triangle above two right angles.

From this demonstration it follows, that if from each of the angles of any small triangle on the surface of a sphere or spheroid (for the proposition holds true for both surfaces) one third of the spherical excess be deducted, the sines of the angles thus diminished will be proportional to the lengths of the opposite sides, and consequently the sides may be computed as if the triangle were rectilineal. It is to be remarked, that the angles to be diminished by one third of the spherical excess, are not the observed angles, but the horizontal angles corrected for the errors of observation.

As an example of this method, we shall take the same triangle as before.

The three angles having been observed, the first step is to compute the spherical excess \(E\) from the formula (1), and for this purpose we take the same value of the radius \(r\) of the oblique circle as was determined above, namely, \(\log_2 r = 7.32112\). With this radius, the constant part of the spherical excess, namely, \(648000 \div 2\pi r\), becomes \(0.37116\), and hence the formula (1) becomes

\[ \log E = \log a + \log b + \log \sin C + 0.37116 \]

Approximate values of \(a\) and \(b\) (one of them may be the given side) must now be found. Now, we found above \(\log a = 5.63039\), \(\log b = 5.70111\); and we have also \(\log \sin C = 9.83842\). Adding these and the constant number \(0.37116\) into one sum, we get \(\log E = 1.54108\), and consequently \(E = 34^\circ 760\). This value of the spherical excess agrees exactly (as it ought) with the sum of the three chord corrections found above.

Having computed the spherical excess, the error of the three observed angles is found, and apportioned among the angles in the reciprocal proportion of their respective weights, and the correct spherical angles deduced in the same manner as already explained in describing the other method. Each of the spherical angles is then diminished by \(\frac{1}{3} E = 11^\circ 38\frac{1}{2}\); and the results are the mean angles from which the sides are to be calculated. They are as under:

| Stations | Corrected Spherical Angles | Mean Angles for Calculation | |----------|---------------------------|-----------------------------| | Benlomond, A... | 56° 43' 28"62 | 56° 43' 17"04 | | Cairnsmuir, B... | 79° 42' 29"15 | 79° 42' 17"56 | | Knocklayd, C... | 43° 34' 56"99 | 43° 34' 25"40 | | Sum........ | 180° 0' 34"76 | 180° 0' 0"00 |

With these mean angles, and the given side \(c = 352,037.62\) feet (in this case the arc is taken, and not the chord), we can now compute the other sides \(a, b\).

\[ \log c = 5.5203762 \]

\[ \log \sin A (56° 43' 17"04) = 9.8222127 \]

\[ \log \sin B (79° 42' 17"56) = 9.9929511 \]

\[ \log \sin C (43° 34' 25"40) = 0.1615997 \]

\[ \log a = 5.6304015 \]

\[ \log b = 5.7011399 \]

whence \(a = 426,974.06\) feet, and \(b = 502,504.42\) feet.

These results agree almost exactly with those obtained by the reduction to the chord angles, the greatest difference being less than a tenth of a foot in a distance exceeding ninety-five miles.

On comparing the two methods of calculation which have now been explained, it will be obvious that the latter, or that of Legendre, is the simpler of the two, inasmuch as it requires only the calculation of the spherical excess by a single operation, whereas the reduction to the chords requires an equivalent operation for each of the three angles.

The calculation of the chord angles is however, in practice, rendered easy by an auxiliary table, in which the correction is given for different values of the angle, and of the sum and difference of its containing sides. But the spherical excess may be entered in a table in the same manner. A sufficient approximation to this quantity, or rather to the area of the triangle on which the excess depends, may frequently be obtained by making a plan of the triangle, and measuring the length of the perpendicular upon the known side by means of a scale of equal parts. In fact, if the three angles have been equally well observed, so that the error is to be divided equally among them, the calculation of the spherical excess is not even necessary for computing the sides; as the same mean angles will be obtained by diminishing each of the observed angles by one third of the excess of the sum above \(180^\circ\). Tables for the reduction of the spherical angles to the plane of the chords, and Calculation of the Latitudes, Longitudes, and Azimuths.

The formulae which have now been given suffice for the computation of the sides of the principal triangles, or the distances of the stations from each other; but for the purpose of mapping the country, it is also necessary to determine the latitudes and longitudes of the several stations, as the inclinations of the sides of the triangles to the meridian. In order to effect this by means of the geodetical measurements, the latitude of one station at least, and the inclination of a side of one of the triangles to the meridian of that station, must be determined by astronomical means; as when this has been done, the geographical latitudes and longitudes of all the other stations in a chain of triangles, and the inclinations of their several sides to the meridians which pass through their extremities, may be computed. The computation, however, is of such a nature that a small error in determining the direction of the meridian leads to errors of considerable magnitude in the latitudes and longitudes deduced from it; and it accordingly becomes necessary, in the progress of a large survey, to verify the computed results by astronomical observations at various stations not very remote from each other in respect of longitude. It has been stated in the preceding historical sketch, that eight different meridians were determined by direct observation in the progress of the survey of England. The problem which usually occurs in practice may be enunciated as follows.

Given the latitude and longitude of a station A, and the distance of A from another station B, and also the direction of the meridian at A in respect of the straight line AB, to determine the latitude and longitude of B, and the direction of the meridian at B in respect of AB.

In the surface of a sphere whose centre is at the point where the normal at A intersects the polar axis of the spheroid, let PAX be the meridian of A, PBY the meridian of B, meeting the former in P; then in the spherical triangle PAB there are given PA the co-latitude of A, the side AB determined from the triangulation, and the observed angle PAB which (in the annexed figure) is the supplement of the azimuth at A, the azimuth being supposed to be reckoned from the south towards the west, all round the circle; and from these data we have to deduce PB the co-latitude of B, the angle APB the difference of the longitudes of A and B, and PBA, which, added to 180°, is the azimuth of A as seen from B.

Let \( l = 90^\circ - AP \) = latitude of A, \( l' = 90^\circ - BP \) = latitude of B, \( \lambda = l - l' \), \( A = 180^\circ - PAB \) = azimuth of B seen from A, \( d = \text{arc } AB \) in parts of the radius.

In the spherical triangle PAB we have

\[ \sin l = \cos l \sin d \cos A + \sin l \cos d, \]

whence

\[ \sin l - \sin l' = \cos l \sin d \cos A + \sin l (1 - \cos d). \]

But \( \sin l - \sin l' = 2 \sin \frac{1}{2} (\lambda - l') \cos \frac{1}{2} (\lambda + l') \)

\[ = 2 \sin \frac{1}{2} \lambda \cos \left( \frac{1}{2} \lambda \right) + \sin l \sin \frac{1}{2} \lambda, \]

therefore

\[ 2 \sin \frac{1}{2} \lambda \left( \cos l \cos \frac{1}{2} \lambda + \sin l \sin \frac{1}{2} \lambda \right) = \cos l \sin d \cos A + \sin l (1 - \cos d). \]

Now since \( \lambda \) the difference of the latitudes can only be a small arc, we may reject without sensible error all the terms in the development of the sine and cosine containing higher powers of \( \frac{1}{2} \lambda \) than the square; that is, we may assume \( 2 \sin \frac{1}{2} \lambda = \lambda \), \( \cos \frac{1}{2} \lambda = 1 - \frac{1}{2} (\frac{1}{2} \lambda)^2 \),

\[ \sin \frac{1}{2} \lambda = \frac{1}{2} \lambda. \]

In like manner, since \( d \) is small in comparison of the radius, we may put \( \sin d = d = \cos d = \frac{1}{2} d^2 \);

whence the above equation becomes, on dividing by \( \cos l \),

\[ \lambda = d \cos A + \frac{1}{2} (d^2 - \lambda^2) \tan l. \]

For a first approximation the second term may be neglected, and we have then \( \lambda = d \cos A \), which is equivalent to supposing the meridians at the two stations to be parallel. Substituting this value of \( \lambda \) in the second term, we get for a nearer approximation

\[ \lambda = d \cos A + \frac{1}{2} d^2 \sin^2 A \tan l. \]

This new value of \( \lambda \) may be again substituted in the equation (9), by which a still nearer value would be obtained; but it is unnecessary to carry the approximation farther.

The difference of latitudes \( \lambda \) is here expressed in parts of the radius 1. Let \( 70^\circ \) be the number of seconds in \( \lambda \); then \( \lambda = \lambda^\circ \sin 1^\circ = (l - l') \sin 1^\circ \) (being also expressed in seconds). The arc \( d \) likewise denotes the distance on the sphere from A to B in parts of the radius; therefore, if D be the number of feet in AB, and R' the number of feet in the radius of the circle perpendicular to the meridian at A, we shall have \( d = D \div R' \).

Making these substitutions, the last equation becomes

\[ l - l' = \frac{D \cos A}{R' \sin 1^\circ} - \frac{D^2 \sin^2 A \tan l}{2 R'^2 \sin 1^\circ}. \]

This formula gives the latitude of the station B, supposing the surface that of a sphere; but a small correction is required on account of the compression. To investigate this, let AM and BN be the normals to the spheroidal surface meeting the polar axis in M and N; join BM, and draw MX and NY parallel to the plane of the equator. The angle AMX is the astronomical latitude of the station A, or the latitude \( l \), and the angle BMX is the latitude of B, determined by the above formula.

But the true astronomical latitude of B is the angle BNY, which is less than BMX by the angle MBN; we have therefore to compute this angle, and add its value to \( \lambda \), the difference between \( l \) and \( l' \). Let MBN = \( \psi \). In the triangle BMN we have \( \sin \psi : \sin BNM :: MN : MB \).

Now \( \sin BNM = \cos BNY = \cos l' \); and if C be the centre of the spheroid, then \( MN = CM - CN \); and because AB is always a small arc on the spheroid, MB is sensibly equal to AM; therefore the above proportion gives \( AM \sin \psi = (CM - CN) \cos l' \).

If we now assume \( a \) and \( b \) to denote respectively the polar and equatorial semi-diameters, we readily find, from the equation of the ellipse,

\[ CM = \frac{b^2 - a^2}{b^2} AM \sin l. \]

Substituting \( a (1 + e) \) for \( b \), developing, and rejecting terms multiplied by powers of \( e \), we have \( (b^2 - a^2) \div b^2 = 2e \); whence \( CM = 2e AM \sin l \). In like manner, \( CN = 2e AM \sin l \); consequently Now, \( \sin l - \sin l' = 2 \sin \frac{1}{2} (l - l') \cos \frac{1}{2} (l + l') \); and by reason of the smallness of \( e \) we may put \( \cos \frac{1}{2} (l + l') \approx \cos^2 \frac{1}{2} l \); therefore \( \sin \psi = 2e \sin \lambda \cos^2 l \), or \( \psi = 2e \lambda \cos^2 l \). This is the correction for the spheroidicity; the corrected difference of latitudes is consequently \( \lambda + \psi = \lambda (1 + 2e \cos^2 l) \). Hence on the spheroid the equation (11) becomes

\[ l - l' = \left( \frac{D \cos A}{R' \sin l'} + \frac{D^2 \sin^2 A \tan l}{2 R'^2 \sin^2 l'} \right) (1 + 2e \cos^2 l) \]

To find the difference of the longitudes of \( A \) and \( B \), or the angle at the pole, we have in the above triangle \( \sin P : \sin PAB : \sin AB : \sin PA \); that is, since \( P \) and \( PA \) are small, \( P : \sin A : d : \cos l' \); whence \( P = \frac{D \sin A}{R' \cos l' \sin l'} \).

Let \( B \) be the azimuth of \( AB \) on the horizon of the station \( B \). To find \( B \) we may employ the following property of spherical triangles, which also holds good for triangles on a spheroid, viz. \( \tan \frac{1}{2} P : \cot \frac{1}{2} (PAB + PBA) : \cos \frac{1}{2} (PB - PA) : \cos \frac{1}{2} (PB + PA) \); that is, since \( PAB = 180^\circ - A \), and \( PBA = B = 180^\circ \), and consequently \( PAB + PBA = B - A \),

\[ \tan \frac{1}{2} (B - A) = \tan \frac{1}{2} P \sin \frac{1}{2} (l + l') \cos \frac{1}{2} (l - l') \]

Now \( \cot \frac{1}{2} (B - A) = \tan \{ 90^\circ - \frac{1}{2} (B - A) \} \); and since the difference of the two azimuths, or, which is the same, the sum of the two angles \( PAB \) and \( PBA \), is always very nearly equal to \( 180^\circ \), the angle \( 90^\circ - \frac{1}{2} (B - A) \) is always very small, and we may therefore substitute the arc for the tangent without sensible error. For the same reason we may take \( \frac{1}{2} P \) for \( \tan \frac{1}{2} P \). Making these substitutions, doubling, and transposing, we get

\[ B = 180^\circ + A - P \sin \frac{1}{2} (l + l') \cos \frac{1}{2} (l - l') \]

The two formulae (13) and (14) require no correction on account of the ellipticity; for with respect to the first, the angle \( APB \) at the pole is the same on the spheroid as on the sphere; and with respect to the second, the correction is so small as to be altogether inconsiderable.

In deducing the above formulae, it has been supposed that the azimuth at the station \( A \) only has been observed. If the direction of the meridian at \( B \) has likewise been determined astronomically, then another value of \( P \) will be obtained by substituting \( B \) for \( A \) and \( l \) for \( l \) in the equation (13); and the mean of the results should of course be taken as the correct value.

We shall now apply the preceding formulae to the computation of the latitude, longitude, and azimuth of one of the principal stations of the survey; and we shall select that of Black Down in Dorsetshire, referring it to the station at Dunmose in the Isle of Wight. The following elements are given in the Survey, vol. ii. p. 88, et seq. Distance of Black Down from Dunmose \( = D = 314,367.5 \text{ feet} (= 59.528 \text{ miles}) \); latitude of Dunmose \( = l = 50^\circ 37' 7'' 30'' \); angle at Dunmose between Black Down and the direction of the meridian towards the north \( (= 180^\circ - A) = 84^\circ 54' 52'' 5 \).

From this last angle it is obvious that Black Down lies westward and a little to the north of Dunmose.

Assuming the same elements of the spheroid as before, we have \( e = 0.003824 \); hence we easily find \( 1 + 2e \cos^2 l = 1.00267 \), the logarithm of which is 0.00116. We have also from the formula (3) the radius of the circle perpendicular to the meridian \( = 20,963,000 \text{ feet}, \text{in round numbers}; \text{with this value of } R', \text{and the given value of } D, \text{the computation of the equation (12) is as follows:}

\[ \log D = 5.49735 \\ \log R' = 7.32145 \\ \log \sin^2 A = 9.99938 \\ \log (D - R') = 8.17590 \\ \log \cos A = 8.494763 \\ \log \sin l' = 5.31443 \\ \log (1 + 2e \cos^2 l) = 0.00116 \]

\[ -274.87 = 2.43912 \]

Hence \( l - l' = 274^\circ 87' + 28'' 10'' = 246^\circ 77'' \) (the first term being negative because \( \cos A \) is negative), and \( l' = l + 4^\circ 6' 77'' \). But \( l = 50^\circ 37' 7'' 30'' \); therefore \( l' = 50^\circ 41' 14'' 07'' \). In the Survey, vol. ii. p. 91, the latitude of Black Down is found by a different mode of computation \( = 50^\circ 41' 13'' 8'' \).

For the difference of longitude we have to compute the formula \( P = D \sin A / R' \cos l' \sin l' \) (13). Making use of the value of \( l' \) just found, we have

\[ \log (D - R') = 8.17590 \\ \log \sin A = 9.99829 \\ \log \cos l' = 0.19822 \\ \log \sin l' = 5.31443 \]

\( 4882.3 = 3.68684 \)

Hence \( P = 1^\circ 21' 28'' 3 \). In the Survey the value of \( P \) is found to be \( 1^\circ 20' 46'' 4 \). The difference, which amounts to \( 16'' \), arises from the different elements of the earth's figure assumed in the Survey, which give the length of a degree of the circle perpendicular to the meridian at the latitude of Dunmose \( = 367,038 \text{ feet}; \text{whereas the length of the degree corresponding to the more correct value of the radius assumed in the preceding calculation is only } 365,874 \text{ feet}. \)

It is now known that the differences of the longitudes of the stations on the southern coast of England are all given too small, in consequence of this erroneous assumption of the length of the perpendicular degree.

The longitude of Dunmose west of Greenwich having been found by a previous operation \( = 1^\circ 11' 36'' \), that of Black Down is consequently \( 2^\circ 32' 35'' 3 \).

It now remains to find the azimuth at Black Down, by computing the formula (14). Here we have

\[ \log \sin \frac{1}{2} (l + l') = 50^\circ 39' 10'' 68'' = 9.88836 \\ \log \cos \frac{1}{2} (l - l') = 0^\circ 22' 3'' 38'' = 0.00000 \\ \log P = 4882.3 = 3.68684 \]

\( 1^\circ 22' 40'' 12'' = 3760'' 12'' = 3.57520 \)

and consequently \( B = 180^\circ + 95^\circ 5' 7'' 50'' - 1^\circ 22' 40'' 12'' \), or \( B = 274^\circ 2' 27'' 38'' \). The observed angle \( PBA \) was \( 94^\circ 2' 22'' 75'' \). Hence the azimuth reckoned from the south towards the west was \( 274^\circ 2' 22'' 75'' \), which differs from the computed value by \( 4'' 63'' \). It may be remarked, that the method of determining the azimuths adopted in the Survey is attended with some degree of uncertainty. It consists, as has been already stated, in taking the mean of the two angles observed with the theodolite between a flag-staff and the pole-star at its greatest elongation west and east. Now, by reason of the altitude of the pole-star in our latitudes above the terrestrial signal, any error in the adjustment of the cross axis of the theodolite to horizontality produces a greater error in the resulting azimuth. Captain Kater (Phil. Trans. 1839, p. 188) considers the observations of the pole-star for determining the perpendicular degree in our latitudes, as wholly unworthy of credit.

Of the Heights of the Stations, and Refraction.

In order to complete the explanation of the methods of calculation employed in the Survey, it only remains to show how the relative altitudes of the principal stations have been Let A and B be two stations whose distance from each other is known, and let AC and BC be the directions of gravity, meeting in C, the earth being supposed spherical. Let AD and BD be the horizontal lines in the plane ACB, and suppose a and b to be the apparent planes of A and B respectively, as seen from each other. Now, if the rays of light proceeded in straight lines, the angle DAB would be the depression of B below the horizon of A, and DBA the depression of A below the horizon of B; and since AD and BD are perpendicular to AC and BC respectively, the sum of these two angles is equal to C, or to the arc of the great circle intercepted between A and B. But this arc is known from the measured distance of A from B, and the assumed dimensions of the earth; consequently the sum of DAB and DBA is given. If, therefore, from this sum we subtract the sum of the two observed depressions, namely, DAB and DBA, there will remain the sum of BAB and ABA, that is, the sum of the two refractions, the half of which is the mean refraction, and assumed to be the actual refraction at each station. Let us put

\[ d = DAB, \text{ the observed depression at } A, \] \[ d' = DBA, \text{ the observed depression at } B, \] \[ C = ACB, \text{ the measure of the contained arc,} \] \[ \varepsilon = \text{the mean refraction,} \]

and we have

\[ \varepsilon = \frac{1}{2} \left( C - (d + d') \right) \].................(15).

Now, let H be the point in CB which is on the same level surface with A, so that CH = CA; then HB is the altitude of B above A, which we have to determine. Join H, and the angle DAH is equal to half of C. But the angle BAH is obviously the difference between DAH and the sum of DAB and DBA. Hence if \( \phi \) denote the angle AH, we have

\[ \phi = \frac{1}{2} C - (d + \varepsilon) \]...................(16).

This gives the angular elevation of B above A; consequently if \( \phi \) is expressed in seconds, and AB in feet, then H will be obtained in feet from the formula

\[ BH = AB \times \phi \sin I^\circ \]...................(17).

It has been assumed that the objects observed at both stations are depressed. If one of them is elevated, then \( d \) and \( d' \) must be taken negatively. It is also to be remarked, that for the purpose of finding the refraction, each observation must be reduced, previously to the calculation, to the axis of the instrument; that is to say, allowance must be made for the difference of level between the axis of the instrument at either station, and the object (as the top of a flag-staff) at the same station observed from the other. This allowance is made by computing the angle subtended by that difference at the distance between the stations, and deducting it from the observed depression, or adding it to the observed elevation.

For the determination of the absolute altitudes, it is necessary that the heights of one or more of the stations be determined by actually levelling down to the surface of the sea. The heights of all the intermediate stations are then established in the manner now explained, by the reciprocal angles of elevation or depression, carried on from station to station; and it is obvious that a verification will be obtained for every three stations; for the difference of altitude between A and B, when found by direct observation, ought to be the same as when deduced from the difference of the heights of each of those stations, and a third station C.

For estimating the altitudes of places at which the theodolite was not placed, and where, consequently, reciprocal observations were not obtained, it was necessary to assume a mean value of the refraction \( \varepsilon \), in order to obtain the angle \( \phi \) in the equation (16). On account of the wide limits within which the horizontal refraction is found to vary, this is attended with some uncertainty, even when every attention is paid to the condition of the atmosphere. In general the effect of refraction was assumed to be from \( \frac{1}{4} \)th to \( \frac{3}{4} \)th of the whole difference of altitude.

As an example, we select the following case from the Survey. At the station of Black Comb in Cumberland, Scilly Bank appeared depressed 49° 14′; and at Scilly Bank, Black Comb was observed elevated 31° 31′; and the distance between the two stations is 121,028 feet. From these data we have to compute the mean refraction and the difference of altitude.

The height of the instrument at both stations was 5½ feet. Hence if \( R \) denote the number of seconds in the arc equal to the radius, and \( x \) the correction for the height of the instrument, we have this proportion, 121,028 : 5 : 5 :: \( R : x \); whence, since \( R = 206,264.8 \), \( x \) is found = 9″. Subtracting this from the seconds of observed depression, and adding it to the seconds of observed elevation, we find \( d = 2944″ \) and \( d' = 1900″ \); hence \( d + d' = 1044″ \). Now allowing the length of a second of arc at the mean latitude to be 101½ feet, the distance in seconds is 121,028 ÷ 101½ = 1192, or C = 1192″, and we have by formula (15) the mean refraction \( \varepsilon = \frac{1}{2} \left( C - (d + d') \right) = 74″ \), which is very nearly \( \frac{1}{4} \)th of the intercepted arc.

With this value of \( \varepsilon \) we obtain (16) \( \phi = -2422″ \); whence, since \( \sin 1° = 0.00000485 \), we have for the difference of altitudes (17)

\[ 121,028 \times 2422 \times 0.00000485 = -1422, \]

or the depression of Scilly Bank below Black Comb is 1422 feet. In the Survey, vol. iii. the altitudes of the two stations are stated to be respectively, Black Comb 1919 feet; Scilly Bank 500 feet; whence the difference is 1419 feet.

We have now explained, with as much detail as the limits Authors which must be prescribed to the present article would permit, the methods of calculation by which the mutual distances of the several stations of an extensive survey, and their geographical co-ordinates, that is to say, their latitudes, longitudes, and altitudes, are deduced from the observations. These form the principal and essential objects of a trigonometrical survey of a country. There are however various other points of great importance connected with geodetical measurements, into which, in the present outline, it is impossible for us to enter; and particularly the methods of estimating the probable errors of the observations, and of the results deduced from them, and of applying these results to the determination of the form of the earth's surface in the country over which the operation extends. For information on these points, and indeed on all other questions relating to the higher geodesy, we refer the reader to the following works: Delambre, Base du Système Métrique Décimal, Paris, 1806–10; Gauss, Supplementum Theoriae Combinationis Observationum Erroribus minimis obnoxiae, Göttingen, 1828; Struve, Breitengradmessung in den Ostseeprovinzen Russlands, Dorpat, 1831; Bessel, Gradmessung in Ostpreussen und ihre Verbindung mit Preussischen und Russischen Dreiecksnetzen, Berlin, 1838; Puissant, Traité de Géodesie, 2nd ed. Paris, 1822; and particularly the Nouvelle Description Géométrique de la France, Paris, 1832–40, by the last-named author,—a work which we could wish to see imitated in our own country.