This designation is usually applied to those surveys of a country which are effected by means of trigonometry. It is well known that when the base of a triangle has been measured or calculated, and the angles at the ends of the base have been measured, this science affords the means of ascertaining the length of the two remaining sides. (See Trigonometry). The distance, therefore, between two points on the surface of a country having been measured for a base, and the angles which it forms with some third point, taken as the vertex of the triangle, having been measured, the length of the other sides can be calculated. These sides will then serve as the bases of other triangles, the length of whose remaining sides may in the same way be ascertained; and by thus extending a series of triangles over a country, its dimensions may be obtained with the greatest accuracy.

Any survey carried on from a measured base, by means of angular measurements, may be called a trigonometrical survey; but the term is usually confined to such as extend over a considerable extent of country, and require a combination of astronomical and geodetical observations. When conducted on a large scale, and when extreme accuracy is required, the work, simple in itself, becomes one of immense labour and ingenuity. When it is necessary to have regard to the curvature of the earth's surface, the effects of temperature, refraction, altitude above the sea, &c., in order that minute accuracy may be attained, not only is the utmost care necessary in observation, with instruments of the very finest character, but long and intricate processes of calculation in the highest branches of mathematics are involved.

The fundamental operation, and the one demanding the greatest degree of accuracy, is the measurement of the base line, for an error committed here will affect all the distances deduced from it,—triangulation doing nothing more than determining how many times, or fractions of a time, the measured base is contained in the observed distances. To measure a line 5 or 6 miles in length, so that the greatest possible error cannot exceed a very few inches, is manifestly a work of extreme care, and requiring instruments of the very nicest description. In order to insure the greatest possible accuracy of observation, it has been usual, in the survey of this country, not only to observe each angle a considerable number of times, but also from each point to observe the angular distances of all the other principal stations which are visible from it. The discrepancies that then arise are reduced to the nearest probable mean by the theory of probabilities. The degree of accuracy that may be attained by such measurements is shown by the fact, that the length of the base on Salisbury Plain, as calculated by means of a series of triangles, extending from the Lough Foyle base, in the north of Ireland, differed only about 5 inches from its measured length.

The object of a trigonometrical survey may be either to ascertain the form and extent of a country, with the relative distances and bearings of its principal points, for the purpose of constructing an accurate map; or to determine the figure and magnitude of the earth by ascertaining the curvature of a given portion of its surface. The latter of these objects having been already sufficiently noticed under Figure of the Earth, we shall confine our attention in the present article to the former; and perhaps the subject cannot be treated better than by giving an account of that great trigonometrical survey that for the last three quarters of a century has been carried on in this country under the direction of the Board of Ordnance, and which is, undoubtedly, the most magnificent work of the kind that has hitherto been attempted in any country. We shall first, then, give a brief historical sketch of that great undertaking, and afterwards some account of the manner in which it is carried out.

The first Government trigonometrical survey in Great Britain History was commenced in the Highlands of Scotland in 1747. After the suppression of the rebellion of 1745, government directed its attention to the establishment of military posts, and the opening up of roads of communication in the remoter parts of the Highlands. Lieutenant-General Watson, who was stationed at Fort Augustus, with the view of carrying out these objects, conceived the idea of at the same time making a map of that part of the country; and the sanction of the Duke of Cumberland having been obtained, operations were forthwith commenced, under the direction of General Roy, the assistant quarter-master-general. Though the work was at first intended to comprise only the Highlands, it was subsequently extended to the Lowlands, and at length included the whole of the mainland of Scotland. The war of 1755, however, put a stop to the undertaking, and the map was never published. General Roy afterwards arranged himself as "a work of considerable merit," though never having been carried on with the 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 subject of a general survey of Great Britain engaged the attention of government, but the breaking out of the American war prevented any steps being taken to carry it into effect.

The present undertaking dates from the year 1783, when, on the conclusion of the peace in that year, a memorial drawn up by M. Cassini de Thury, the French astronomer, was presented to the English government, recommending a trigonometrical measurement of the distance between the observatories of Paris and Greenwich, with a view of determining the exact difference of longitude between them. The proposal having met with the warm approval of the Royal Society, to whom it was referred, government agreed to carry it out, and General Roy was appointed to superintend the operations.

The first step in the process was the accurate measurement of a Hounslow base; and, in order that this might form the basis of any future Heath base survey of the United Kingdom, it was resolved that it should be done with the utmost possible care, and with the best instruments that could be furnished by Ramsden, at that time acknowledged to be the first maker in the world. Hounslow Heath, which, besides its proximity to the capital and Greenwich Observatory, presented the advantages of great extent and levelness of surface, free from local obstructions, was selected as the site for a base, and a line upwards of 5 miles in length was marked off for measurement. Up to that time, deal rods had been generally employed in the measurement of those bases in other countries that had been effected with the greatest appearance of care and exactness; and three such rods, or the best seasoned timber, were carefully constructed, each 20 feet 3 inches long, tipped at each end with bell metal. The preliminary operations having been completed, the measurement was commenced about the middle of July 1784. The weather was extremely unfavourable, and, before the measurement was half completed, the deal rods were found, notwithstanding all the care that had been taken in their preparation, so liable to sudden and irregular variations of length from the state of the atmosphere, as to afford little assurance that that accuracy would be attained which was desirable. As, however, so much of the work had been done, it was resolved to complete it, so that the result obtained might afterwards be compared with that given by more accurate instruments. The measurement with the deal rods being completed, and allowance made for expansion, the distance between the centre 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 27,409-26 feet of the standard scale from which the lengths of the measuring rods were laid off.

At the suggestion of Lieut.-Col. Calderwood, hollow glass rods were next adopted. Three of these, about an inch in diameter and upwards of 20 feet in length, were selected and placed in cases to which they were made fast in the middle and braced at several other points, so as to prevent them from bending or shaking, but allowing them freely to expand or contract. At one end of each rod was a fixed apparatus or metal button, and at the other a move- able apparatus or slider, pressed outwards by a slender spring. In contact the fixed extremity of each rod was pressed against the moveable apparatus of the preceding, 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 20 feet. The new measurement was commenced on the 18th and concluded on the 30th of August 1784. After the rate of expansion of the rods had been determined and allowed for, the length of the base in temperature 62°, reduced to the level of the sea, was found to be 27,404-0137 feet, or 5-19 miles.

The triangulation was not proceeded with till the summer of 1787, when Ramsden had completed his theodolite for the measurement of the angles. This was the finest instrument of its kind that had yet been constructed, having its horizontal circle 3 feet in diameter, and was the first to show that, in consequence of the spherical form of the earth, the sum of the three angles of a triangle on its surface exceeds that of two right angles. The same year Mears Cassini, Mechain, and Legendre distinguished members of the French Academy of Sciences met General Roy at Dover, and the connection was established between the triangulation of the two countries. In order to ascertain the accuracy of the operation, a base of verification was measured on Romney Marsh, near the termination of the triangulation, and was found to be 28,535 feet ±12 inches, at temperature 62°, and at the level of the sea. The measurement was made by a steel chain 100 feet in length, constructed by Ramsden, and of which the accuracy had been previously tested by measuring with it a portion of the Hounslow Heath base simultaneously with the glass rods. The measured length was found to exceed the computed length by about 28 inches, which was probably as near a coincidence as was to be expected, and may be taken as a conclusive proof of the general accuracy of the whole of the operations.

The illness soon after this of General Roy, followed by his death in 1790, seem to have for a time put a stop to any attempt to extend the operations already begun to a survey of the whole island. Indeed, the renewal of operations is ascribed to the accidental circumstance of the Duke of Richmond, as Master-General of the Ordnance, having purchased a very fine theodolite by Ramsden, similar to that used by General Roy, but with some improvements; as also two new steel chains of 100 feet each, by the same great artist. This theodolite is said to have been ordered by the East India Company for the purpose of surveying their possessions in the East, but some misunderstanding having arisen about the price, the directors refused it, and it was thrown upon the hands of the artist.

In 1791 the Ordnance survey was commenced under Colonel Williams, Captain Mudge of the Royal Artillery, and Mr Dalby, by the remeasurement of the base on Hounslow Heath. Instead of measuring-rod, the two steel chains already mentioned were used. Each chain consisted of 40 links of 2½ feet each. At first both chains were used in measuring, but afterwards one only, the other being kept as a standard by which the measuring-chain was compared. In the act of measuring, the chain was laid out in a succession of deal coffers, supported by trestles and stretched by a weight of 28 lbs. After all reductions had been made, the length of the base by the new measurement was found to be 27,404-3155 feet, or about 2½ inches greater than by General Roy's measurement with the glass rods. The mean of the two results, or 27,404-2, was therefore assumed as the true length of the base in the future calculations.

In prosecuting the survey, it was resolved, in the first instance, to carry a series of triangles southwards from the base, in order to ascertain the position of some of the principal stations on the coast; and also to determine the length of a degree of longitude by measuring the distance between Beachy Head on the coast of Sussex, and Dunmore in the Isle of Wight,—two stations lying nearly east and west of each other, above 64 miles apart, and visible from each other by day and night. An opportunity would also thus be afforded of connecting this series with the triangles of General Roy, and of thereby testing the accuracy of both operations. The triangulation was commenced in 1792, and the principal work of the following year was determining the directions of the meridians of Dunmore and Beachy Head. In 1794, a base of verification was measured on Salisbury Plain by steel chains, in the same manner as on Hounslow Heath, the one being used for measuring, and the other kept as a standard. After making the different reductions, the correct length of the base was found to be 35,574-3 feet, and on computing the distance by means of different combinations of the triangles from the Hounslow Heath base, the greatest and least results were found to be 35,574-3 and 36,573-8 feet, the mean being 35,574-3 feet, or about one inch short of the measurement. The distance between Beachy Head and Dunmore, as deduced from a mean of four different series of triangles, was found to be 339,397-6 feet (±42-2 miles), and the greatest difference from the mean was less than four feet. On computing from this and the observed angles made by the straight line joining the two stations with the meridians at each, the length of a degree of the great circle, perpendicular to the meridian at latitude 50° 41' (nearly that of the middle point between Beachy Head and Dunmore), was found to be 61,182-3 fathoms = 357,033-8 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, 292,312 feet; Dunmore, 292,914 feet; the assumed latitudes being, respectively, 50° 44' 24" and 50° 37' 7" (see Figure of the Earth).

During the years 1793 and 1796, the triangulation was continued, 1795-96, along the coast of Dorsetshire, Devonshire, and Cornwall, to Land's End. 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 the operations from this point into the north of Devonshire until a new base had been measured, the triangles at present being dependent upon those made in Cornwall in the previous year. In 1798, a series of secondary triangles was observed for completing the survey of Kent and Essex; but the principal operation of the year was the measurement of a new base of verification on King's Sedgemoor, in Somerseshire. This measurement was conducted, like those on Hounslow Heath and Salisbury Plain, by steel chains—only, on account of the irregularities of the ground, it was thought expedient to have a new chain of 50 feet for those parts where the longer chain could not well be used. The reduced length of this base was found to be 27,680 feet, or nearly 51 miles, and it was supposed by Captain Mudge that the amount of error could not exceed nine inches.

The measurement of an arc of the meridian had been contemplated from the commencement of the survey, but was 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 altogether finished. The meridional line passing from Dunmore to the mouth of the Tees was selected, as being the freest from obstruction and of sufficient length. The points selected, viz., the northern extremity of Clifton, a small village in the vicinity of Doncaster, nearly on the meridian of Dunmore; and a plain of sufficient extent for the measurement of a base of verification was found at Misterton Carr, in the northern part of Misterton Lincolnshire. The two extreme stations were connected by a chain Carr base, of twenty-two triangles, lying nearly in the direction of the line to be measured. Of these, eleven, extending from Dunmore to Arbury Hill, near the middle of the line, had already been observed, and their sides computed from the Hounslow Heath base; the angles of the remaining eleven were measured in 1801 and 1802, and the distances computed from the new base. The base on Misterton Carr was measured in the same manner as Sedgemoor, and after making the necessary reductions, the length was found to be 26,342-7 feet. As the correct determination of this base was of great importance, every precaution was taken to insure accuracy, and General Mudge was of opinion that the error in excess or defect could not exceed two inches. On computing the distance between Arbury Hill and Corley, near the middle of the arc, by the first eleven triangles, it was found to be 117,463, and by the other eleven triangles, from Misterton Carr, 117,457-1 feet, a difference of less than six feet, upon an extent of somewhat more than 22 miles; a discrepancy which cannot be considered great, seeing that the length of the whole line from Dunmore to Clifton is nearly 200 miles. Had the computation been carried on from Dunmore to Clifton, the length of the base on Misterton Carr, as deduced from Hounslow Heath, would have been found to be only about one foot greater than its measured length. The whole length of the arc from Dunmore to the parallel of Clifton was found to be 1,036,337 feet, or 196-29 miles.

In 1806, another base was measured on Rhuddlan Marsh, near St Rhuddlan Asaph, North Wales, in the same manner as the former. After Marsh base necessary reductions were effected, its length was found to be 24,514-6 feet. As evidence of the agreement between that and former measurements, it was found that the distance between Castle King and Weaver Hill (two stations in Shropshire), as computed from the base on Misterton Carr, was 111,144-1 feet, and 111,148-4 feet, as computed from the new base on Rhuddlan Marsh; the difference in a line upwards of 21 miles in length being only 4-3 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 towards the east, and united with those proceeding westward from the base on Misterton Carr; and a fourth series was carried through Lancashire, Westmoreland, and Cumberland, into Scotland, and connected with another series extending from Misterton Carr base, through Yorkshire and Northumberland, and the east part of Scotland, as far as the north side of the Firth of Forth. Thus, down to 1809, the survey of nearly all England, the south of Wales, and a part of Scotland, had been made. The arc of meridian between Dunmose and Clifton was also extended northward to Burleigh Moor, about 3 miles north of Gibborough in Yorkshire; and another meridional length was determined between Dunmose and Delamere Forest in Cheshire, about 5 miles north of Tarporley. The triangulation of Scotland was continued first along the east coast to the borders of Ross-shire, and was subsequently extended to the Shetland Islands. A series was also carried from the Cumberland triangles along the west coast, through Dumfriesshire, and to the summit of Ben Lomond, connecting the remaining triangles of Perthshire. From 1811 to 1816 the survey was all but stopped on account of the war. In 1817, a new base of verification was measured with the steel chains on Belhelvie Links, near Aberdeen, the length of which, after making the various reductions, was found to be 26,515-5509 feet.

In 1818 and 1819, the principal triangulation of Scotland was proceeded with, but was suspended in 1820. In 1821, it was recommenced in the Shetland, Orkney, and Western Islands of Scotland, and was carried on in these districts in 1822. In 1823, the use of the large theodolites being required in order to proceed with the triangulation in South Britain, the principal triangulation in Scotland was suspended. In 1824, the scene of operations was transferred to Ireland, an accurate survey of that country being considered of more urgent importance.

When the Ordnance survey was first undertaken, it was intended to produce only a military sketch map of the south of England; but as the work advanced it came into favour as a road map, a travelling map, and a general geographical map. When the survey of Ireland was undertaken, it was thought advisable to secure some farther social advantages, especially to make it subservient to the proper assessment and collection of the grand jury cess and other local taxes. Accordingly a committee of the House of Commons, of which Lord Montesquie was chairman, having examined the question, recommended that a scale of 6 inches to the mile should be adopted for Ireland, the 1-inch scale on which the English apparatus was constructed was not sufficiently large to admit of the insertion of the townland boundaries and other minute territorial divisions, which were required for the purpose of making a valuation. (A townland is the smallest of the territorial divisions, and is almost always co-extensive with the private estates. There are 60,760 townlands in Ireland, and 2,460 parishes.) The original intention was, that the survey should be confined to an exact description of the contents and boundaries of the townlands; but as it advanced, it was found desirable to include the subdivisions of fields and homesteads; and that was accordingly done in the centre and southern parts of the island. The advantages of including the subdivisions of fields were found to be so great that they were afterwards extended to the northern parts.

In 1827, a new base was measured on the east shore of Lough Foyle, in the north of Ireland. Doubts having arisen respecting the accuracy attainable by means of steel chains, Colonel Colby adopted an ingenious compensating apparatus, consisting of two bars, one of iron and the other of brass, 10 feet in length, connected together at their centres, and having at each end an ingenious apparatus for measuring the degree of expansion or contraction of each. (An account of this apparatus will be found in the article FIGURE OF THE EARTH.) The whole line measured was nearly 8 miles in length (exactly 41,640-8873 feet); and it was calculated that the greatest possible error could not exceed two inches. It was also prolonged northward by triangulation to Mount Sandy, whereby two additional miles were given to it. From this base a series of triangles were extended all over the island, and were connected with those formerly observed in Wales, the Isle of Man, Scotland, and the Hebrides.

In 1838, the principal triangulation of Ireland having been completed, that of Great Britain was again taken up. All England, with the exception of the six northern counties, had been surveyed on the scale of 2 inches, and engraved and published on the scale of 1 inch to the mile; and a small portion of Scotland had been surveyed on a similar scale. It was found, however, that the thickly peopled manufacturing, mining, and colliery districts of the northern counties could not be properly represented on the 1-inch scale, and representations were made to the Treasury by various scientific societies and others to that effect. The matter being referred to the Duke of Wellington, the Treasury, in accordance with his advice, issued a minute, of date the 6th October 1840, stating that "My lords are satisfied, from the consideration they have given to the subject, that the scale on which the English survey has hitherto been conducted falls much short of what is required by the existing circumstances of the country. That scale was originally fixed principally with a view to military considerations, while the demand of the present day is for such a national survey as shall be permanently useful in aiding the improvement of the country, by serving as a basis and guide in the formation of railroads, canals, and other public works, besides assisting in the development of the geological structure of the country, and promoting in various other ways the progress of science and statistical knowledge. . . . My lords have no hesitation in giving their consent to the remainder of England and the whole of Scotland being surveyed on the Irish scale of 6 inches to the mile." Many my lords think themselves fortunate in having their opinion on the important subject confirmed by the concurrence of so high an authority as that of his Grace the Duke of Wellington."

At this time the county of Wigton, and one half of the counties of Ayr and Kirkcudbright, had already been drawn on the 1-inch scale in Scotland. In 1849, therefore, the survey of Scotland was commenced anew for the 6-inch scale; but the work for some years proceeded very slowly. The old base on Salisbury Plain not being found to agree very accurately with that on Plain base Lough Foyle, it was resolved to re-measure the former with the remeasured compensation bars employed in the measurement of the latter. The re-measurement was effected in 1849, and either from the guns which marked the extremities of the base having shifted their position, or from the original measurement having been defective, the result obtained was about a foot in excess of the previous measurement. This difference in a length of nearly 7 miles may seem small, but it sufficed to remove the greater part of the discrepancy which had been detected between the English and Irish bases.

The general work of the survey was proceeding so slowly, that in 1851 only Lancashire and Yorkshire in England, and Wigtonshire, Kirkcudbrightshire, Edinburghshire, and the Isle of Lewis in Scotland, had been surveyed. In that year, therefore, a committee of the House of Commons, Lord Elcho being chairman, was appointed to inquire into the state of the survey, and in particular to determine whether a 1-inch or a 6-inch scale should be adopted. The committee, in their report, came to the unanimous conclusion, that the scale of 6 inches, and the system of contouring, ought to be abandoned, on the ground that the 6-inch map alone was not of sufficient public utility to justify the large expenditure of public money, and that the 1-inch map was better adapted to geographical purposes. They appear also to have been influenced by the belief that the latter would be much more speedily executed. Orders were accordingly issued by the Treasury and Ordnance, in conformity with this report, that the remaining four counties of England, and the Isle of Scotland, should be done on the 1-inch scale. These instructions produced great satisfaction, both in Scotland and the unsurveyed parts of England, and numerous representations were made by various public bodies and others, in 1852 and 1853, to induce the Treasury to rescind the orders in favour of the 1-inch scale. Some of these memorials were referred by the Treasury to the Board of Ordnance in June 1852; and a determination was meanwhile come to by these departments, that the survey of Fife, which was then going on, should be conducted on the 6-inch scale, and that the question as to the scale of the rest of the country should be reserved for future consideration. By this time the desirability of even a larger than the 6-inch scale was agitated in influential circles, and in February 1853 Lord Elcho, then a lord of the Treasury, drew up a very able memorandum on the survey, and suggested a scale of 2½ inches as that best adapted for correct plans of estates, and for other civil purposes. This, with numerous letters on the subject, was transmitted, with a Treasury circular letter, dated 16th April 1853, to a large number of scientific societies and experienced persons in the kingdom, requesting them to state their opinion on the comparative merits of a 6-inch, as contrasted with any larger scale, for the purposes of a national survey. At the same time, an order was issued to the survey department to carry on the survey of Ayrshire and Dumfriesshire "with that degree of accuracy which would admit of the plans of the cultivated districts being hereafter drawn on the scale of 2¼ inches to a mile if desired." Of the replies to the circular letter, 20 were in favour of a 6-inch scale, and 120 in favour of a larger scale, among whom were the Registrar-general, the President of the Geographical Society, Sir Henry de la Beche, Mr A. C. Ramsay director of the Geological Society, the President of the Geological Society, the Society for the Amendment of the Law, the Metropolitan Commissioners of Sewers, the Ecclesiastical Society, the Poor-Law Board, the Board of Supervision for Relief of the Poor, the Commissioners of Works, the General Board of Health, the Statistical Society, the Statistical Conference, &c. The weight of authority being thus decidedly in favour of a larger scale, and generally for a scale, in the manuscript plans, of 2¼ inches to a mile. The progress which has been made in the survey, and the scales upon which the maps and plans have hitherto been drawn and published.

3. The change or changes in those scales, or in any of the details of the survey, which, according to your judgment, should be made.

4. The estimated cost of completing the surveys of England and Wales, Scotland, and Ireland respectively, on the scales and in the manner which you recommend to be adopted.

The commissioners found that the purposes which a national survey should subserve may be divided into two classes. "The first includes the wants of the State either for military purposes, for levying taxes or rates on real property, or in carrying into effect any legislative measure relating to land over which the State may exercise a direct superintendence, as, e.g., measures for the registration of the titles to lands, measures giving facilities to the transfer of landed property, and so forth." "The second class includes the wants of the public as individuals, where such wants are not confined to any particular section of the community, and where they cannot be satisfied by private enterprise." The principal among these is the necessity for a general geographical map of the country; next, the supply of commercial data, or definitions of the positions of well marked points by which private surveys may be facilitated, and surveys of limited districts may be connected as parts of one great plan; third, Such surveys of special districts as are likely to produce their contingent applications important public benefits, whether as aiding in other surveys, in tracing the course of railways or canals, in fixing materials for plans in reference to water supply to towns, drainage, and so forth, or in promoting the sciences of geology, gentry, and others. When the Tithe Commutation Act passed in 1836, the Ordnance Survey maps were found to be too small for the purpose contemplated in the Act, and hence a demand arose for first class plans of a scale as large as 20 to 25 inches to the mile. About two million pounds are said to have been expended in procuring these plans, which were for the most part hastily and carelessly got up, so that many of them are imperfect and inaccurate, and only about one-sixth of them are described as being first class plans. They had, moreover, been constructed on no uniform principle, and could never be juxtaposed so as to form a national cadastre. The commissioners found further, that owing to the frequent alterations in the instructions issued to the superintendent, the survey was in a very unsatisfactory state as regards uniformity. Thus, all England, with the exception of the six northern counties, had been surveyed for and mapped on the scale of one inch to the mile only. Of the six northern counties, the whole of Yorkshire and Lancashire had been plotted and engraved and published on the 6-inch scale, while Durham, the greater part of Westmoreland, and the southern part of Northumberland, had been plotted on the 25-inch scale. In Scotland, the counties of Edinburgh, Haddington, Fife, Kinross, Wigton, Kirkcudbright, and the Island of Lewis, had been plotted for, engraved, and published on the 6-inch scale; the county of Linlithgow had been published both on the 6 and 25 inch scales; and the counties of Ayr, Dumfries, Renfrew, Peebles, Berwick, Lanark, Forfar, Roxburgh, and Selkirk, had all been plotted for the 25-inch scale, and some of them had already been in course of publication on the 25 and 6 inch scales. In Ireland, the whole country had been plotted, engraved, and published on the 6-inch scale, and seven of the counties in which accurate details had not been taken in the first instance, have been revised and made complete for valuation and assessment purposes.

Armagh alone remained for revision, but also, it has been necessary to enlarge the plans of about 700 towns from the 6-inch scale, on account of its insufficiency for valuation purposes. In the case of transfers of land also, under the Encumbered Estates Act, the 6-inch plans of upwards of 2,000 acres had to be enlarged for the purposes of sales. Several towns throughout the United Kingdom had been surveyed, plotted, and published on various scales, from about 60 to 125-72 (1/2) inches to the mile, and some of these, and indeed of other surveys performed by the Ordnance Corps had been paid for by the parties requiring them. They recommended, 1st, That the 1-inch map of the United Kingdom be forthwith completed, engraved, and published, as beyond question the most important object in a national point of view. 2d., That the survey of the northern counties of England and the counties of Scotland proceed contemporaneously, and be completed and published.—the cultivated districts on the 25-inch scale, and the whole on the 6 and 1 inch scales, except the Highlands of Scotland, to be surveyed on the 1-inch scale only. 3d., That the revision of the 6-inch plans of Ireland be completed. 4th., That the final determination of the question as to the expediency of extending the survey on the 1/2-inch scale to Great Britain or the whole of the United Kingdom, be left to the decision of the legislature, when the contemplated measures with which it is more immediately connected may have been adopted. The expense required for the 25-inch scale exceeded by so small proportion that required for the 6-inch,

as to leave no doubt that when either is adopted, the preference should be given to the former, which will also furnish materials for the 6-inch scale. The estimated expense of carrying out the various plans was—

1. The completion of the United Kingdom on the 1-inch scale only ........................................... L279,972 2. The completion of those portions of the United Kingdom not already surveyed on the following scales, viz.—Cultivated parts 25-inch, moorland parts 1-inch, .................................................. 533,068 3. The completion of the United Kingdom on the 25-inch scale (towns on the ½-inch scale,) except those parts which have already been completed on the 6-inch scale, .................................................. 2,285,129 4. The completion of Great Britain on the 25-inch scale (towns on the ½-inch scale,) and of Ireland with revision on the 6-inch scale, .................................................. 2,430,764 5. The completion of the survey of the United Kingdom on the 25-inch scale, with towns on the ½-inch scale, .................................................. 2,686,764

The sums expended on account of the survey up to 31st March 1858, were—

England and Wales .................................................. L1,051,678. Scotland .................................................. 374,746. Ireland .................................................. 979,165.

On the receipt of the report of the Royal Commission, the Treasury, at the suggestion of the Secretary-of-State for War, directed the readoption of the 25-inch scale for the cultivated districts in the meantime, till the matter could be brought before parliament. Some doubts having been expressed in parliament as to the accuracy of the method of reducing the plans by means of photography, a committee of scientific gentlemen was appointed to inquire into and report upon the subject. They stated in their report that, in their opinion, the system of making reductions by photography is more accurate than by the pentagraph or any other known means of reducing plans; that, as regards time, it is for plans of rural districts only one fourth, and for plans of towns only one ninth, of the time required to make reductions with the pentagraph; that the saving effected by the introduction of photography has been at the rate of L1,615 per annum, and that the saving which will be effected by its introduction during the progress of the survey will amount to at least L31,952.

The present state of the survey will be best shown by giving a few extracts from the last published report of its progress, which comes down to 31st December 1858:—"The surveys of the six northern counties of England and Scotland are proceeding according to the recommendation of the royal commissioners on the survey, and the orders contained in the Treasury minute of the 11th September 1858. The detail survey of the six northern counties of England will be finished in the present year, and with the exception of the hill sketching of a small portion of country, we shall this year complete the 1-inch map of England. We shall also complete this year the outline 1-inch map of Ireland; 108 sheets are already published, and a great number of others are in progress. By the 31st March next we shall have finished the detail survey of the counties of Dumfries, Stirling, and Clackmannan. All the southern counties of Scotland are now closed, and we are now engaged in Perthshire and Forfarshire. With the exception of Aberdeen, all the great towns of Scotland have been surveyed, and as the greater part of the remainder of Scotland consists of mountainous country and very open work, I have no doubt but that we shall complete the survey of Scotland within the time, and at cost within the estimates which I gave the royal commissioners on the survey." He strongly recommends that the subject of proceeding with the survey of England on the 25-inch scale should be taken into consideration by Government and Parliament this year, that he may be enabled to make the preliminary arrangements such as ascertaining the boundaries of the townships, parishes, &c., for proceeding without any costly delays with the survey of the remaining three-fourths of England and Wales, the estimated cost of completing which, on the scales now adopted, being estimated at L1,450,000.

"17," he adds, "the remaining three-fourths are not finished in the same way, it will be the only portion of the United Kingdom without a cadastral survey, and we shall be almost the only state in Europe without one." As compared with 1857, the year ending 30th September 1858 exhibited a reduction on the actual cost of the plans of from 11½d. to 9½d. per acre, or about one-sixth on the ½-inch scale; and from 6½d. to 5½d. per acre, or about one-seventh on the 6-inch scale. As regards Scotland, the estimates by these two items alone will be reduced by L70,000. The saving effected by means of photography had also been increased, and was estimated to amount to L35,000 on the cost of the survey.

The engraved plans of Ireland on the 6-inch scale have been found of great value for many important purposes connected with the valuation of property, &c.; but for the transfer of land under Encumbered Estates Court, they were found to be utterly insufficient. The judges of the court therefore applied to the Lords Commissioners of the Treasury to have the plans required for the transfer of property under the authority of the court made as part of the ordnance survey, and on the ½-inch scale; and by Treasury letter of 11th September 1858, their lordships approved of this arrangement, provided that effectual care be taken that the whole of the expense incurred in preparing the plans be repaid by the Encumbered Estates Court. The plans are first drawn on the ½-inch scale by means of the notes of the original 6-inch survey, and are then subjected to a rigid examination on the ground in order to correct them to the present date, and also to perfect the details in a manner suitable to the increased scale.

The sums stated as requiring to be provided for in the estimates for 1859-60, were as follows:—

For the prosecution of the survey in England .................................................. L1,29,000 Scotland .................................................. 30,500 Ireland .................................................. 13,500

Towards engraving a geographical map of Great Britain, 4 miles to an inch .................................................. 1,000 Engraving geological survey, &c. .................................................. 600 Publication of the maps and levelling .................................................. 5,600 Survey of military stations .................................................. 4,000 Annual repairs to survey offices at Southampton and Dublin .................................................. 500 Topographical department—Salaries, extra pay of officers, and contingencies, including the purchase of maps and books .................................................. 5,400

L90,000

The number of royal engineers, civil assistants, and labourers employed on the surveys in each of the three kingdoms were—

| England | Scotland | Ireland | Total | |---------|----------|---------|-------| | Royal Engineers | 184 | 114 | 49 | 347 | | Civil Assistants | 425 | 192 | 136 | 753 | | Labourers | 253 | 209 | 55 | 517 | | Total | 872 | 515 | 249 | 1637 |

The principal triangulation of the United Kingdom, Principal which was commenced in 1783 under General Roy, has re-triangulatedly been completed, and an account of all the operations connected therewith is given in a volume entitled, Account of the Observations and Calculations of the principal Triangulation; and of the Figure, Dimensions, and Mean Specific Gravity of the Earth, as derived therefrom. Drawn up by Captain Alexander Ross Clarke, R.E., under the direction of Lieut.-col. H. James, R.E., F.R.S., &c. London, 1858, 4to, pp. 782. It consists of a series of great triangles, extending over the whole country, connecting and showing the distances and bearings of all the principal points. There are in all 218 points in the principal triangulation, and the observed bearings amount to 1554.

The following table gives the lengths of the several bases Bases as measured and reduced to the same standard, and their lengths as shown in the triangulation, in accordance with the adopted scale of linear measure:—

| Date | Base | Length in terms of Ramsden's Scale | Ordnance Standard | Length in Triangulation | Difference | |------|------|----------------------------------|-------------------|------------------------|-----------| | 1791 | Hounslow Heath | 27404/24 | 27406/190 | 27406/190 | +0.173 | | 1794 | Salisbury Plain | 36574/23 | 36577/630 | 36577/630 | +0.283 | | 1801 | Musteron Carr. | 26342/10 | 26344/060 | 26344/060 | -0.191 | | 1806 | Rhuddlan Marsh. | 24514/25 | 24516/000 | 24517/770 | -1.596 | | 1817 | Belshavie | 25515/65 | 25517/530 | 25517/770 | -0.240 | | 1827 | Lough Foyle | 41640/887 | 41641/103 | 41641/103 | +0.216 | | 1843 | Salisbury Plain | 36577/638 | 36577/636 | 36577/636 | -0.02 |

1 About L3,000 of this sum is required to meet the expense of surveys made for the Landed Estates Court, Ireland, which will be repaid. 2 This amount will be covered by the sale of the maps. In the year ending 30th June 1859, the sum so received was L5,366, besides the value of maps supplied to public departments during the same period, amounting to L908, being in all L6,274. 3 The large number of assistants in England arises from the head-office being at Southampton. The base lines from which all the trigonometrical distances have been computed are those which were measured on Salisbury Plain, and on the shore of Lough Foyle in Ireland. They are respectively 693 and 789 miles long, and the difference between their measured lengths and their lengths as computed through the triangulation from each other is only about 5 inches. This difference was divided in proportion to the square roots of the lengths of the measured bases, whence was obtained the mean base which has been used in the triangulation, and there is therefore a difference of about 2½ inches between the measured and computed length of each of these bases. Of the four other base lines measured, the greatest difference between the measured lengths, and the lengths as computed from the mean base, does not amount to 3 inches in three of them. The fourth base, on Rhuddlan Marsh in North Wales, was measured in an unfavourable position, and the difference between its measured and computed length is 1,596 feet.

The mean length of each side of a triangle, in the principal triangulation is 354 miles. There are 37 sides, whose lengths are between 80 and 90 miles, 18 whose lengths are between 90 and 100 miles, and 11 exceeding 100 miles in length. The longest side in the triangulation is 111 miles. The sum of all the distances or sides in the principal triangulation is 206,710,000 feet, or, in round numbers, exactly ten times the radius of the earth. The horizontal angles, as well as the azimuthal bearings of the stations, have been principally obtained from observations with Ramsden's great three feet theodolites; and the perfection of these instruments may be judged from the fact, that the sum of the angles in the triangles rarely differed 3° 4' from the true sum. For an account of these, and the other instruments used in the survey, see the work already referred to. One of the operations frequently demanding great labour, was the setting up of the theodolite, which had sometimes to be raised by scaffolding to a great height; sometimes a solid foundation had to be made for it in a bog; and sometimes a site had to be prepared for it on the rocky summit of a mountain. In exposed situations on the tops of mountains, a stone wall had generally to be built close round the lower part of the observatory—a precaution which on one occasion saved the instrument from certain destruction. In placing the framework for receiving the feet of the instrument, every care had to be taken to have it as nearly concentric with the station as possible: the circular mahogany tray upon which the instrument rests was then placed on the table, and accurately brought over the centre mark by means of a plumb-line suspended from the orifice in its centre. The next step, after placing the instrument in position, was to select a spot for the referring object—an object which, from its position, should be visible under all circumstances, and to which the bearings of all other objects could be referred. It was so constructed as to present a fine vertical line of light of any required breadth, and it was usually placed at the distance of one or two miles. As the adjustments of the theodolite are liable to be deranged by travelling, the first operation at a station was to correct any deficiencies of this nature. The general adjustment consists in this, that the line of collimation of the telescope should be perpendicular to its axis of rotation; this axis perpendicular to the revolving axis of the instrument; and the latter perpendicular to the plane of the horizon. The runs of the micrometers should also agree with the divisions on the limb. The observations then proceed as follows:—

The instrument being perfectly levelled, and the lower part or body firmly clamped to the table, the observer directs his telescope upon the referring object, and, having carefully clamped the upper limb, brings the intersection of the cross-hairs, by the motion of the tangent-screw, to bisect the vertical line of light in the referring object. He then reads the degrees, minutes, and seconds given by the divided circle and each of the micrometer microscopes. The upper limb is then unclamped, and the telescope directed to the next object to be observed, which is bisected, and the readings recorded. Similarly all the other points to be observed, and then, finally, the referring object is bisected and read again. The agreement of this last reading with that at the commencement of the series is some test of the care of the observer and the steadiness of the instrument. Each such series is called an "arc," and they are numbered consecutively from the commencement to the close of the station. The first arc being completed, the telescope is reversed in its Y's, the horizontal circle turned through 180°, and the levels adjusted if necessary. The second arc is then proceeded with precisely in the same manner as the first, closing as well as commencing with the referring object. The second arc being completed, the lower limb of the instrument is moved through a small angle, 20° or 30°, and clamped, so as to get readings on another part of the circle; and in this position the third arc is taken. This being completed, the telescope and horizontal circle are reversed, as in the second arc, and so on. As the readings of the microscopes at each observation are read out by the observer, they are recorded—invariably in ink—by the booker in the observation-book. A copy of the day's work is made every evening, and, being carefully compared by the observer with the original, is transmitted to the head-office every two or three days. The original record of the observations is retained by the observer until the book is filled. From each point the angular distances of all the principal stations visible from it are observed, and in this way a number of values are obtained, which correct one another. Each of these values, again, is the result of numerous repetitions of the same observation, and thus the greatest possible degree of accuracy is secured. In order to obtain the mean of a number of observations, so as to give due weight to those of them which there is reason to believe to be most nearly correct, the "method of least squares," a result of the theory of probabilities, is adopted. See article PROBABILITY, § X.

Besides determining the directions and distances of the principal points of a country, it is necessary for the construction of a map to determine the latitudes and longitudes of the several points, and the inclinations of the sides of the triangles to the meridian. In order to effect this, the latitude and longitude of one station at least, and the inclination of a side of one of the triangles to the meridian, must be determined astronomically, and hence the geographical latitudes and longitudes of all the other stations, and the inclinations of their several sides to the meridians which pass through their extremities, may be computed. As, however, a small error in the computation would lead to errors of considerable magnitude in the latitudes and longitudes deduced from it, it is necessary to verify the computed results by a number of astronomical observations at different stations in course of the survey.

The latitudes of thirty-two of the stations were determined in course of the survey with Ramsden's zenith sector of 8 feet radius, which was destroyed in the fire at the Tower in 1841, and with Airy's zenith sector of 20-5 inches radius by Troughton and Simms. From the very close approximation of the observed zenith distances of the stars with each instrument, and the perfection of the instruments

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1 In Airy's Figure of the Earth the equatorial radius is 20,928,713 feet, and the polar radius 20,853,810 feet, giving an ellipticity of \( \frac{1}{298} \); but from the ordnance and other more recent surveys abroad the axes are found to be 20,926,500 feet and 20,855,400 feet respectively, and the ellipticity about \( \frac{1}{298} \). The mean density of the earth, as determined from observations at Arthur's Seat, is 5-316. The longitudes of the different stations were in the first instance calculated on Airy's figure of the earth; by using the observed latitude of Greenwich, and the observed azimuth of the meridian mark, and transferring this azimuth with the differences to each station in succession; the observed latitudes and the observed azimuths were then used to determine corrections to Airy's Figure, and the Greenwich latitude and azimuth; the longitudes and latitudes then received their necessary corrections.

The longitudes were also determined chronometrically by the transmission of chronometers, and by transit observations; the difference of longitude obtained in this way between the extreme points, viz., Greenwich and Peaghman in the island of Valentia, including an arc of about 10°, was found to correspond to a difference of 461 feet in the position of the station as determined geodetically.

The altitudes of many of the stations have been determined by actual levelling with the spirit-level, and are all referred to the mean tidal level at Liverpool; but the vertical angles or zenith distances of the stations were also reciprocally observed with the theodolites.

Each triangle of the primary triangulation is broken up into a series of smaller triangles, the sides of which are from 5 to 10 miles long, and this secondary triangulation is again broken up into smaller triangles, whose sides are about 1 mile long, and which form the tertiary or minor triangulation. The length of each side of the minor triangles are then actually measured on the ground, the men noting in their "field-books" every fence, stream, or other object they may cross. They then measure cross lines from one side of the triangle to another, and by taking offsets from the measured lines to every object on the face of the country, they obtain in their field-books the data for plotting accurate plans of the country upon any scale which may be required. The plotting is done in the office from the field-books. The survey is first laid down upon paper, and is then traced on thin tracing-paper. A man is then sent to walk over the ground to see that the surveyor has made no omission, and that everything which should be supplied is there; and also with the view of putting in the character of the trees, the woods, the gardens, the names, and so on; in fact to put in that detail which enables the draftsman in the office to make a perfect plan of the country. The computed trigonometrical length of each side of a triangle is thus checked by actual measurement; and the accuracy of the lines within each triangle is checked by the plotting, so that no errors which may be made by the surveyor can escape detection. Of course, the same extreme care is not necessary in dealing with the smaller triangles as is required in the principal triangulation, as errors here are more easy of detection, and cannot be propagated beyond very narrow limits; while the calculations are also comparatively simple. The difference, therefore, in the expense of surveying for a large and a small scale is comparatively slight, a fact which seems to have been lost sight of by those who raised a cry in parliament against the larger scales on the score of expense. It is of course equally laborious and costly to measure the distances of a given number of objects on a large or on a small scale; the additional expense arises only on the last operations, and between a 25-inch and a 6-inch scale, it is less than ten per cent.

But not only are all the surveying operations conducted under the Board of Ordnance, but also all the operations connected with the engraving and printing of the maps. The head-quarters of the survey of the United Kingdom is at Southampton, and from thence all orders connected with the administration and conduct of the survey are issued, and all the plans and maps of Great Britain are there engraved and printed; those of Ireland being engraved and printed at the survey-office in the Phoenix Park, Dublin. The superintendent resides at Southampton; and is assisted by an officer in charge of the general correspondence, and the publication of the maps of Great Britain at Southampton; an officer in charge of the trigonometrical branch of the work, and of the initial levelling; and an officer in charge of the examination and zincography of the manuscript plans, and of their reduction to the 6 and 1 inch scales. There is also an officer in charge of the office and publication of the maps in Dublin. Division officers are employed upon certain districts, and have under them numerous parties of surveyors, draftsmen, and computers, acting under the direction of non-commissioned officers; and by this arrangement the whole force is systematically and effectually regulated. The division officers are supplied from the head-office at Southampton, with the trigonometrical distances, and the positions of the trigonometrical stations on the plans; they personally examine the plans on the ground, and certify to their accuracy before they are forwarded for publication. Each officer makes up the accounts of his division, which, after being examined at the head-quarter office, and certified by the superintendent, are forwarded to the accountant-general.

The plans of parishes on the 25-inch scale are published by lithography, zincography, or the anastatic process; the plans of the towns, counties, and the present map of the kingdom, are engraved on copper.

Zincography is now generally adopted in preference to zincolithography, on account of the facility of handling thin zinc plates rather than lithographic stones, which are necessarily very heavy, and are constantly liable to be broken. "We find," says Colonel James, "not only that the plates are cheaper than the stones, and not liable to break, but that the impressions are darker and clearer; and as we can at all times resort to new tracings, or to the anastatic process, for producing any additional copies that may be required, the same plates, after being cleared of the impressions, are again and again used for taking the impressions of other maps."

The peculiar advantage of the anastatic process consists in the means which it affords of reproducing from a drawing or print, however old, which has been made with a greasy ink, a new plate or any number of new plates of zinc, from which new impressions may be taken. It is first ascertained, by rubbing a piece of thin paper over some part of the drawing, whether the ink is so fixed as that no trace of it will come off by pressure; and if this is the case, the drawing is immersed for a few minutes in a hot solution of strontia (1 oz. of strontia to a quart of water), which has the effect of loosening the ink. It is then partially dried, and afterwards immersed in a solution of nitric acid (one to six of water). If the print be comparatively new, the strontia bath is not required, and it is only necessary to immerse it in the nitric acid. The drawing is then ready to be transferred to a zinc plate, previously polished as finely as possible with powdered emery, and etched by placing a sheet of paper over it damped with nitric acid, and passing it through the press. The transfer is effected by passing the plate through a copperplate printing-press, after which the drawing is removed, and the plate wiped over with gum-water. It is then charged with printing-ink, and subsequently etched with phosphoric acid, a few drops of which are mixed with gum-water, after which it is ready to be printed in the usual manner.

Plans of towns on the 1/8 inch scale, and of the cultivated dis-Photo-tricts on the 1/32 inch scale, are reduced to the 6-inch scale for photogravure by photography. "The advantages," says Colonel James, "derived from the introduction of this method will readily be understood by those who are at all conversant with the tedious methods of reducing plans of towns by the pentagraph or the eidograph, by proportional compasses or..." Trigonometrical Survey.

Squares, or any of the methods formerly employed; and it is no exaggeration to say, that the reduction of a plan of a large town on the \( \frac{1}{25} \) scale may be and is now made by photography, first to the \( \frac{1}{25} \) scale; and from this to the 6-inch scale, at one-hundredth of the cost at which we were previously able to do this work; and, what is scarcely less important, the reductions are made and copies printed so rapidly by this process, that no delay takes place in the publication of the same district on the different scales ordered—a most important consideration, when it is borne in mind that the survey is now proceeding at a rate of about 1,500,000 acres per annum, and that, in the conduct of so large a work, the rapid closing up and the final disposal of the work as it is produced are essentially necessary to success in the conduct of it. As a proof of the facilities which photography gives us in making the reductions, I may mention that during the last week one man, with the assistance of a printer and a labourer, reduced 32,000 acres from the 25-inch to the 6-inch scale; and that he produced 3 copies of 45 sheets, or 135 impressions, in six days, besides some other work. One hundred draftsman could not have produced so much work." "I have made experiments to ascertain the relative power of the different colours for producing photographs, and I found the following scale of colours produced a scale of shades from nearly perfect white to jet black: blue, purple, red, orange, yellow. I have, therefore, had all the houses on the MS. plans coloured yellow."—Report on the Ordnance Survey of the United Kingdom for 1855-6. The lens of the camera is a single achromatic meniscus \( \frac{3}{4} \) inches in diameter, with a principal focal length of 24 inches. The required scale of the reduction is obtained by tracing on the ground glass of the camera a rectangle corresponding on the reduced scale to the rectangle of the plan to be reduced. The curvature of the image and indistinctness of outline from spherical aberration are both remedied by reducing the diaphragm in front of the lens to a small aperture. The negative copy is taken upon glass by the collodion process, and is then placed in the printing-frame in contact with sensitive paper, and in this way as many positive prints are taken in succession as may be required. Recently, in place of nitrate of silver, a coating of bichromate of potash, gum, and lamp-black, or any other pigment, has been employed in preparing the printing-paper. The action of light on a coating of this composition has the peculiar effect of rendering it insoluble in water, and consequently, when a sheet of paper coated with it is placed in the printing-frame under the collodion negative, the outline of the plan is rendered insoluble in water, and remains on the paper when all the remainder of the composition is washed away; and thus a positive plan, in ink of any colour which may be required, is obtained. The prints, however, obtained by this process are less clear and sharp in their outline than those obtained from nitrate of silver; but, by the following mode of treatment, sharp, clear lines are produced:—The ink of the print, after being soaked in a saturated solution of caustic potash or soda, becomes, so to speak, disintegrated, and is then in a state in which the print may at once be rubbed down, and the outline transferred to the waxed surface of a copper plate for the engraver. This promises to be of great importance, as, after obtaining the photographed reductions of the maps, they have hitherto been obliged to make tracings from them in ink, for the purpose of transferring the plan to copper; the expense and delay of which will now be saved, whilst no risk is run of any error being made by the draftsman.

A new method of printing from a negative, extremely simple and inexpensive, has recently been tried, and promises to be of great use. By means of it the reduced print is in a state to be at once transferred to stone or zinc, from which any number of copies can be taken, or to the waxed surface of the copper plates. To effect this the paper, after being washed over with the solution of the bichromate of Trigonometrical Survey, being gum and dried, is placed in the printing-frame under the collodion negative, and after exposure to the light, the whole surface is coated over with lithographic ink, and a stream of hot water then poured over it. As the portion which is exposed to the light is insoluble, whilst the composition in all other parts, being soluble, is easily washed off, the outline of the map, in a state ready to be transferred either to stone, zinc, or the copper plate, is at once obtained, or the photograph can be taken on the zinc at once. "From the perfect manner," says Col. (now Sir) H. James, "in which we are able to transfer the impressions to zinc, we can, if required, print any number of faithful copies of the ancient records of the kingdom, such as Doomsday Book, the Pipe Rolls, &c., at a comparatively speaking, very trifling cost. I have called this new method photo-zincography, and anticipate that it will become very generally useful, not only to government, but to the public at large, for producing perfectly accurate copies of documents of any kind."—Report for 1859.

A considerable saving in the cost of engraving is effected by using steel punches to cut the woods, figures, rocks, &c., on the copperplates. A portion of the writing, also, is engraved by machine (Becker's patent); and the parks and sands are ruled by a machine with a steel dotting-wheel, the pressure of which, and the interval between the lines of dots, being regulated according to the tint required to be produced.

The process of electrotyping has been found to be eminently useful for the purposes of the survey. Instead of typing-printing from the copperplates, which speedily become obliterated in parts, and thus lead to a constant expense for repairing or re-engraving, an electrotype copy of the plate is taken, from which the impressions are printed. The original plate is thus preserved in its perfect state, from which, when required, another electrotype copy can be taken, or rather from the "matrix" of the original. Another great advantage is the facility it affords for making alterations, it being much easier to scrape off obsolete details, &c., from the electrotype cast of an engraved plate where they are in relief, than to cut them out from the original copperplate. Copies can also in this way be obtained from the plates in their different stages of progress, so that different classes of information can be engraved upon a map the same in all other respects, as for instance, one copy of a map may be obtained with contours, boundaries, &c., another with the hill features, a third with the geological lines, &c. This process has also been very usefully employed in joining two or more engraved plates together, so as to form a single copperplate for printing. In this way maps of several of the counties of Scotland, to serve as indexes to the 6-inch county maps, have been formed out of the copperplates of the 1-inch map. In one instance, no less than seven plates have been thus joined together.

The peculiar and practical advantage of the 25-inch scale of the parish plans over the smaller scales is, that it enables the acreage of every single enclosure and garden in the country to be given with a distinct reference to it. Besides, as every square inch represents an acre, any one may, in point of fact, ascertain exactly what each field contains. Tables of areas of each separate enclosure are also published with these plans. Each sheet contains 960 acres, being a mile wide and 1\( \frac{1}{2} \) mile long. The reference, therefore to any property is exceedingly simple, and may be described as consisting of the fields numbered so and so, represented on sheet No. —— of the Government Survey of the County of ———. Sixteen of the plans on the 25-inch scale go to form one sheet on the 6-inch scale, which therefore comprises 24 square miles, being 6 miles long by 4 wide. On the towns scale of \( \frac{1}{8} \), or 10 feet 6 inches to a mile, not only is every house laid down, but the actual divisions of each (on the ground floor) are also given. (D.K.) The word Trigonometry, from the Greek τριγωνομετρία, signifies the measurement of trigons or triangles. In general, a trigon is determined when three of its dimensions are known; the other dimensions may be computed from these data, and hence the complete measurement of a trigon includes the computation of those of its lines and angles which have not been ascertained by direct measurement. Now, this computation presents much greater difficulties than the operation of measuring does, and thus a treatise on trigonometry comes to be, virtually, a treatise on the art of computing the unknown parts of a trigon.

The ancient geometers sought to connect the numerical expression for an angle, with the rate of divergence of its sides, by cutting off from these two equal distances of a determinate length, and by ascertaining the value of the line joining their extremities. Since the extremities of these distances lie in the circumference of a circle, the connecting lines are the chords of the intercepted arcs; they were accordingly entered in tables as such.

The Arabian mathematicians, however, introduced tables of the half chords of the double arcs, to which they gave the name jeib, signifying in the bosom (whence the word jeb, a purse as carried in the bosom), which was translated into the Latin sinus, having the same signification. They also used tables of the lengths of the shadows of gnomons corresponding to various degrees of zenith distance, which are called by us tangents, but by them zybl, literally shadows. To these there have been added, in later times, tables of secants, or lines joining the vertex of the angle with the extremities of the tangents, in Arabic katy', that which cuts.

Almost all the operations of trigonometry are carried on by help of these tables of sines, tangents, secants, or, since the date of Napier's famous invention, by means of tables of their logarithms.

The early trigonometers used a radius consisting of sixty parts, but we now regard the radius of the circle as unit, and divide it decimally—that is, into ten, one hundred, one thousand equal parts, and so on; the values of the sines, tangents, secants, being expressed in decimal fractions of the radius. Thus, (fig. 1), AOB being an angle at the centre of the circle described with the radius OA, the arc AB is homologous to it, so that if the whole circumference be divided into any number of degrees, say 360, the arc AB contains as many of those divisions as the angle AOB contains of 360th parts of the entire revolution. Having let fall from B the line BC perpendicular to OA, BC is the sine of the arc AB, or of the angle AOB; also, having drawn AD to touch the circle at A, and to meet the continuation of OB in D, AD is the tangent, OD the secant, and AC the versed sine of the same arc; these also, by an extension of the meaning of the words, are said to be the tangent, secant, and versed sine of the angle AOB to the radius OA; and if OA be taken as unit, we write, indifferently,

\[ \begin{align*} CB &= \sin AB; \\ AD &= \tan AB; \\ OD &= \sec AB; \\ AC &= \text{ver } AB; \end{align*} \]

\[ \begin{align*} CB &= \sin AOB; \\ AD &= \tan AOB; \\ OD &= \sec AOB; \\ AC &= \text{ver } AOB. \end{align*} \]

The defect of an acute angle from a right angle, or of its homologous arc from a quadrant, is called the complement of that angle or of that arc, and, for shortness sake, the sine, tangent, and secant of the complement are called the cosine, cotangent, cosecant of the angle or arc. If, then, OE be drawn perpendicular to OA, the angle BOE is the complement of AOB, the arc BE the complement of AB; while FB, EG, OG, EF, which are the sine, tangent, secant, versed sine of BE, are called the cosine, cotangent, cosecant, covered sine of AB; and, conversely, CB, AD, OD, AC are the cosine, cotangent, cosecant, covered sine EB.

Among these lines there are several very obvious relations; thus, putting \(a\) for the arc AB,

\[ \begin{align*} OC^2 + CB^2 &= OB^2 \text{ or } \cos^2 a + \sin^2 a = 1, \\ OD^2 - DA^2 &= OA^2 \text{ or } \sec^2 a - \tan^2 a = 1, \\ OG^2 - GE^2 &= OF^2 \text{ or } \cot^2 a - \tan^2 a = 1; \end{align*} \]

\[ \begin{align*} OC : OB :: OA : OD \text{ or } \cos a : \sec a = 1, \\ CB : BO :: OE : OG \text{ or } \sin a : \cot a = 1, \\ DA : AO :: OE : EG \text{ or } \tan a : \cot a = 1; \end{align*} \]

\[ \begin{align*} OC : CB :: OA : AD \text{ or } \cos a : \tan a = \sin a, \\ BC : CO :: OE : EG \text{ or } \sin a : \cot a = \cos a. \end{align*} \]

If one of the functions of an angle be given, we can, by help of these propositions, easily compute the values of the others. Thus, if \(a\) be an angle such that its sine is \(1/169\), we have \(\sin^2 a = 1/169\), whence \(\cos^2 a = 125/169\); \(\cos a = 12/13\); \(\tan a = \sin a / \cos a = 5/12\); \(\cot a = 12/5\); \(\sec a = 1/\cos a = 13/12\); \(\csc a = 1/\sin a = 13/5\); \(\text{ver } a = 1/13\); \(\text{cov } a = 8/13\).

But the real problems which lie at the root of all trigonometrical researches are these: having given the sine, or any other of these functions of an angle, to compute of how many degrees that angle consists; and conversely, having given, numerically, the value of an angle to find the values of its various functions. The practical solution of these problems is to construct a trigonometrical canon in which the values of the sines, tangents, and secants are set down for every minute, second, or other small portion of the circumference.

If the sines and cosines of two arcs be known, we can readily obtain those of their sum and of their difference. Thus, let AB, BC (fig. 2), be two arcs, of which the sines BD, CE, and consequently the cosines OD, OE, are known; we may obtain the sine CF of their sum AC, or CF' of their difference AC' by the following process.

Through E draw EG parallel to BD and HEH' perpendicular to it; then, since the trigons ODB, OGE, CHE are similar, we have

\[ \begin{align*} OB : OE :: BD : EG :: OD : OG, \text{ or, putting } a \text{ for the arc } AB, \\ \text{and } b \text{ for } BC \text{ or } BC'. \end{align*} \]

Again, \(1 : \cos b :: \sin a : EG :: \cos a : OG;\)

whence, \(EG = \sin a \cdot \cos b; OG = \cos a \cdot \cos b.\)

\[ \begin{align*} OB : EC :: OD : CH :: DB : EH; \end{align*} \]

or, \(1 : \sin b :: \cos a : CH :: \sin a : EH;\)

whence, \(CH = \cos a \cdot \sin b; EH = \sin a \cdot \sin b.\)

Now, \(FC = GE + CH; FC' = GE - CH; OF = OG - EH; OF'\)

\[ \begin{align*} = OG + EH; \text{ therefore, } \end{align*} \] If we can by any means obtain the sine of one degree, equations (1) and (3) enable us, by putting \( b = 1^\circ \), and \( a \) successively equal to 1, 2, 3, &c., degrees, to obtain the sines and cosines of every degree in the half quadrant; and so for any other division: the essential matter, then, is to obtain the sine of the smallest sub-division which we intend to use.

On putting \( b = a \), equations (1) and (3) become

\[ \sin 2a = 2 \sin a \cos a \\ \cos 2a = \cos^2 a - \sin^2 a \]

and, observing that \( \sin^2 a + \cos^2 a = 1 \), the latter of these may be put under the forms

\[ \cos 2a = 2 \cos^2 a - 1 \\ \cos 2a = 1 - 2 \sin^2 a \]

whence \( \cos a = \sqrt{\frac{1}{2} \left( 1 - \frac{1}{2} \cos 2a \right)} \)

\[ \sin a = \sqrt{\frac{1}{2} \left( 1 - \frac{1}{2} \cos 2a \right)} \]

by help of which we can compute the sine and cosine of an arc when the cosine of its double is known.

Again, on putting \( b = 2a \), equations (1) and (3) become, after transformations,

\[ \sin 3a = 3 \sin a - 4 \sin^3 a \\ \cos 3a = 4 \cos^3 a - 3 \cos a \]

by means of which the sine and cosine of the third part of an arc may be computed.

Similarly, on putting \( 3a \) for \( a \), \( 2a \) for \( b \), in the fundamental equations we obtain

\[ \sin 5a = 5 \sin a - 20 \sin^3 a + 16 \sin^5 a \\ \cos 5a = 16 \cos^3 a - 20 \cos^5 a + 5 \cos a \]

which enable us to compute the sine and cosine of the fifth part of an arc.

In the ancient system of graduation, still very generally adhered to, the quadrant is divided into 90 degrees, the degree into 60 minutes primi or minutes; the minute again into 60 seconds secundi or seconds; while, in the modern or centesimal system the quadrant is divided into 100 degrees, the degree into 100 minutes, the minute into 100 seconds. The former graduation is accomplished by means of the prime divisors 2, 3, 5, while the modern requires only the use of 2 and 5, so that the above equations contain all that is needed for the formation of the canon of sines according to either of the systems.

The tri-section of an angle implies the solution of an equation of the third degree irreducible by Cardan's rule. The quinsextion requires the solution of an equation of the fifth degree. As no practicable method for reducing such equations was then known, the computers of the actual tables of sines were forced to use only repeated bisections; these bisections they carried on until they arrived at arcs so small as to be proportional to their sines without sensible error, and thence, by a common proportion, they found the sine of one minute. This process, besides being indirect, is attended with a serious practical inconvenience when the arc to be bisected is small. The cosine of a small angle is nearly equal to unit, so that the quantity

\[ \frac{1}{2} \cos 2a \]

comes to have few effective figures, and can only give the sine of the angle \( a \) true to a small number of places. This circumstance compels us to carry the primary computations to a great number of decimal places. By employing the higher equations we avoid these inconveniences, as is seen from the following computation of the sine of one degree of the ancient division.

Since the chord of 60° is just equal to the radius, the sine of 30° must be \( \frac{1}{2} \), whence

\[ \cos 30° = \sqrt{\frac{3}{4}} = .866025403784439. \]

Observing that, according to equations (9) (10)

\[ \cos 15° = \sqrt{\frac{1}{2} + \frac{1}{2} \cos 30°} \\ \sin 15° = \sqrt{\frac{1}{2} - \frac{1}{2} \cos 30°} \]

we obtain

\[ \cos 15° = .965925826289068 \\ \sin 15° = .258819045102321. \]

By putting \( x \) for \( \sin 5° \) in equation (11), we have

\[ 4x^3 - 3x + \sin 15° = 0 \]

whence the following computation:

| \( 24 \) | \( 24y \) | \( 12y^2 - 3 \) | \( 4y^3 - 3y - \cos 15° \) | \( \cos 5° \) | |---|---|---|---|---| | \( 24 \) | \( 24y \) | \( 9 \) | \( 034074174 \) | \( 1 \) | | \( 24 \) | \( 24y \) | \( -072 \) | \( -072 \) | \( -072 \) | | \( 24 \) | \( 24y \) | \( 108 \) | \( 108 \) | \( 108 \) | | \( 24 \) | \( 24y \) | \( 9928108 \) | \( 007182066 \) | \( 997 \) | | \( 24 \) | \( 24y \) | \( -019320 \) | \( -019262040 \) | \( -7187127 \) | | \( 24 \) | \( 24y \) | \( 776 \) | \( 776 \) | \( 776 \) | | \( 24 \) | \( 24y \) | \( 9908680 \) | \( 9908853736 \) | \( 996194698 \) |

which gives \( \cos 5° = .996194698 \).

The sine and cosine of one degree are now to be found by help of equations (13) (14). Putting \( x \) for \( \sin 1° \), equation (13) becomes

\[ 16x^3 - 20x^2 + 5x - \sin 5° = 0 \]

whence the following very rapid computation:

| \( 1920 \) | \( 1920x \) | \( 960x^2 - 120 \) | \( 320x^3 - 120x \) | \( 80x^4 - 60x^2 + 5 \) | \( 16x^5 \) | \( \sin 1° \) | |---|---|---|---|---|---|---| | \( 1920 \) | \( 0 \) | \( -120 \) | \( 0 \) | \( +5 \) | \( -087155742 \) | \( -0 \) | | \( 1920 \) | \( +33408 \) | \( +29065 \) | \( +2688 \) | \( +1686 \) | \( +0181656 \) | \( -0174 \) | | \( 1920 \) | \( +33408 \) | \( +101 \) | \( +175 \) | \( -6273 \) | \( +49818417 \) | \( +0524 \) | | \( 1920 \) | \( +33408 \) | \( -1197076 \) | \( -2092587 \) | \( +49817322 \) | \( -000000031 \) | \( -0174524 \) | | \( 1920 \) | \( +33408 \) | \( +101 \) | \( +175 \) | \( -6273 \) | \( +49818417 \) | \( +0524 \) | | \( 1920 \) | \( +33408 \) | \( -1197076 \) | \( -2092587 \) | \( +49817322 \) | \( -000000031 \) | \( -0174524 \) | And again, putting \( y \) for \( \cos 1^\circ \) in equation (14)

\[ 16y^4 - 20y^3 + 5y - \cos 5^\circ = 0. \]

| 1920 | 1920y | 960y^2 - 120 | 320y^3 - 120y | 80y^4 - 60y^2 + 5 | 16y^6, &c. | \cos 1^\circ | |------|-------|--------------|----------------|------------------|----------|-------------| | 1920 | 1920-00 | 840-000 0 | 200-000 00 | 25-000 000 | -003 805 302 | 1- | | | - 29 | - 291 8 | - 127 68 | - 030 400 | - 3 800 | - 000 152 | | | | + 2 | + 9 | + 2 310 | | | | | | = 1 | | | | |

| 1920 | 1919-71 | 839-708 2 | 199-872 34 | 24-969 608 | -000 007 612 | - 999 848 | | | | | | | - 7 616 | - 305 | | | | | | | | | | | | | | | | - 999 847 695 |

So that \( \sin 1^\circ = 0.017 452 406 \) \( \cos 1^\circ = 0.999 847 695 \)

If we now, in order to compute the sine and cosine of one minute, proceed by bisection, we have

\[ \sin 30' = \sqrt{0.000 076 153}; \] \[ \cos 30' = \sqrt{0.999 923 847}. \]

The first of these equations is clearly unfit for giving a result with great precision, for which reason, if our intention had been to compute the sine of one minute, it would have been better to have gone on with the bisection of \( 15^\circ \); and then of \( 7^\circ 30' \), so as to obtain the sine and cosine of \( 3^\circ 45' \); thence by two trisections to reach \( 2^\circ \); and, lastly, by two quinquisections to have obtained the sine and cosine of one minute.

The construction of the canon of sines is greatly facilitated by employing the method of differences. Thus, if we take three equi-different arcs, \( a - b, a, a + b \), their sines with the first and second differences stand thus,

\[ \begin{array}{ll} \text{First Difference.} & \text{Second Difference.} \\ \sin(a-b) & \sin a - \sin(a-b) \\ \sin a & \sin(a+b) - \sin a \\ \sin(a+b) & \sin(a+b) - \sin a \end{array} \]

and this second difference may, by help of equations (1) and (2), be put under the form \( -2 \sin a (1 - \cos b) = -2 \sin a \cdot \text{ver } b \).

Now, since \( b \) is necessarily a small angle, its versed sine, that is, the defect of its cosine from radius, must also be very small, so that the multiplication by it is attended with little labour. Again, if \( a - 2b, a - b, a, a + b, a + 2b \), be five equidifferent arcs, the fourth difference of their sines is

\[ \sin(a+2b) - 4 \sin(a+b) + 6 \sin a - 4 \sin(a-b) + \sin(a-2b) \]

which may be put under the forms

\[ 2 \sin a [\cos 2b - 4 \cos b + 3] \quad \text{or}, \] \[ 4 \sin a [\cos b - 1]^2 = +4 \sin a \cdot \text{ver } b^2. \]

And, further, if we take seven equidifferent arcs from \( a - 3b \) to \( a + 3b \), the sixth difference of their sines is

\[ -8 \sin a \cdot \text{ver } b^3, \]

and this law of formation extends to all differences of an even order.

But in applying the method of successive differences, the minute errors unavoidable in the last placed figures would go on accumulating, wherefore we must provide periodical tests of the accuracy of the work. Now, we have already obtained the sines of \( 15^\circ, 30^\circ, 60^\circ, 75^\circ \), wherefore the sine of \( 45^\circ \) only is wanted to complete the series of arcs differing by \( 15^\circ \). But of \( 45^\circ \) the sine is, necessarily, equal to the cosine, and therefore

\[ \sin 45^\circ = \frac{\sqrt{2}}{2} = 0.707 106 781 \]

so that our first table of sines is this:

\[ \begin{array}{cccccc} \text{Arc} & \text{Sine} & \text{First Diff.} & \text{Second Diff.} & \text{Third Diff.} & \text{Fourth Diff.} \\ 0^\circ & 0.000 000 000 & 87 155 743 & - & - & + \\ 5^\circ & 0.087 155 743 & 85 492 435 & 663 308 & 658 260 & 10 059 \\ 10^\circ & 0.173 648 178 & 85 170 867 & 1 321 568 & 648 201 & 14 911 \\ 15^\circ & 0.258 819 045 & 83 201 098 & 1 969 679 & 633 210 & 19 808 \\ 20^\circ & 0.342 920 143 & 80 598 119 & 2 602 979 & 613 402 & 24 741 \\ 25^\circ & 0.426 118 262 & 77 381 738 & 3 215 881 & 588 921 & 29 691 \\ 30^\circ & 0.500 000 000 & 73 576 436 & 3 805 302 & 559 960 & 34 661 \\ 35^\circ & 0.573 576 436 & 69 211 174 & 4 389 002 & 529 741 & 39 651 \\ 40^\circ & 0.642 781 610 & 64 319 171 & 4 892 002 & 498 506 & 44 673 \\ 45^\circ & 0.707 106 781 & 58 937 662 & 5 381 509 & 448 552 & 49 724 \\ 50^\circ & 0.766 444 443 & 53 107 601 & 5 830 661 & 404 180 & 54 794 \\ 55^\circ & 0.818 152 044 & 46 873 360 & 6 234 241 & 358 736 & 59 884 \\ 60^\circ & 0.866 025 404 & 40 282 383 & 6 590 971 & 310 572 & 64 994 \\ 65^\circ & 0.903 629 149 & 33 384 834 & 6 847 542 & 262 408 & 69 114 \\ 70^\circ & 0.939 692 621 & 26 233 205 & 7 151 629 & 214 080 & 74 241 \\ 75^\circ & 0.965 925 826 & 18 881 927 & 7 351 982 & 165 704 & 79 374 \\ 80^\circ & 0.984 067 753 & 11 386 945 & 7 541 643 & 117 361 & 84 514 \\ 85^\circ & 0.996 194 698 & 3 805 302 & 7 731 643 & 68 661 & 89 663 \\ 90^\circ & 1.000 000 000 & 3 805 302 & 7 921 643 & 63 861 & 94 813 \end{array} \]

Having written the sines of \( 0^\circ \) and of \( 5^\circ \), we get the first differences; and the first of the second differences is obtained by taking the product of \( \sin 5^\circ \) by \( -2 \text{ ver } 5^\circ \), from which the second of the first differences and the sine of \( 10^\circ \) are had. The first of the third differences, viz., 658 260 is the product of the second of the first differences, viz., \( +86 492 435 \) by the same multiplier \( -2 \text{ ver. } 5^\circ \); from it we get the next second difference, the next first difference, and the sine of \( 15^\circ \), the coincidence of which with the previously determined value shows the work to have been accurately done. The first of the fourth differences, 10 059, is the product of \( -1321 568 \) by the same multiplier, or, as is preferable, of \( \sin 10^\circ \) by \( (2 \text{ ver. } 5^\circ)^2 \).

Each successive fourth difference is the product of the same multiplier \( (2 \text{ ver. } 5^\circ)^2 \) by the sine of the next arc; now this multiplier \( 000 057 921 \) has few effective figures, and thus the labour of the calculation is much reduced.

The same process may be extended to the sixth order of differences, which are obtained by multiplying the columns of sines by \( (-2 \text{ ver. } 5^\circ)^3 = 000 000 441 \), which has still fewer effective figures; but the advantage obtained by pushing the differences to a high order is counteracted by the circumstance, that the minute errors, unavoidable in the last place figures, accumulate to cause inconvenience.

When the subdivision is more minute, this method is still more rapid; thus, in constructing the table of the sines of arcs differing by one degree, we use the multipliers

\[2 \text{ ver } 1^\circ = 0.00304610\] \[(2 \text{ ver } 1^\circ)^2 = 0.00000093.\]

| Arc | Sine | 1st Diff. | 2nd Diff. | 3rd Diff. | 4th Diff. | |-----|------|-----------|-----------|-----------|-----------| | 0° | 0.00000000 | 17.452406 | - | - | + | | 1° | 0.017452406 | 17.447091 | 0.032 | 5.317 | | | 2° | 0.034899497 | 17.436459 | 15.941 | 5.309 | 8 | | 3° | 0.052335958 | 17.420518 | 21.249 | | | | 4° | 0.069758474 | 17.399260 | | | | | 5° | 0.087155743 | | | | |

It is always necessary to carry the computation of the differences to two or three places beyond what is intended to be used. This is well seen in the above example, in which the values of the sines are given true to the nearest figure in the ninth decimal place; the fourth differences found by help of these correct values, are 8, 1; whereas, by computation from the sines of 2° and 3°, they ought to have been 3.25 and 4.86, if the work had been carried two places farther.

This illustration is sufficient to exemplify the manner in which trigonometrical tables are constructed. Various contrivances are employed for diminishing the labour of making tables of secants, tangents, and logarithmic sines. A complete account of these would extend this article to an unreasonable length; we shall, therefore, at once address ourselves to the method of using those tables which have been already constructed.

Hitherto we have only considered the sines, tangents, and secants of arcs within the quadrant; and must now examine the cases of arcs extending to the whole circum- ference, or even beyond it.

Let a point start from A, and move round the circum- ference of the circle described from the centre O with the radius OA (fig. 3); when the moving point has passed over the arc AB, the sine has grown from zero to CB; as the arc continues to increase, the sine also in- creases, until the arc becomes a complete quadrant AD, at which time the sine is the radius OD. After the arc has passed into the second quadrant, the sine FE shortens to become zero, when the arc is the semicircumference AG; and it is obvious that the sine of the arc AE, which is greater than the quadrant, is also the sine of its supplement GE, which is as much less than the quadrant.

When the arc, as AH, exceeds the semicircumference, its sine IH appears on the other side of the diameter AOG, and this position is represented by the sign −. The sine of AK (three quadrants) is the radius OK, and is therefore represented by −1; and when the arc, as AL, exceeds three quadrants, its sine ML, actually decreasing, is, in algebraic language, said to be increasing, that is, becoming less sub- tractive. The sine of the whole circumference AA is again zero; and if the moving point be supposed to continue its progress, the same changes in the value of the sine are repeated in each succeeding revolution. The cosine of the arc undergoes corresponding changes; it is positive in the first quadrant, negative in the second and third, and again positive in the fourth quadrant.

Resolution of Right-angled Triangles.

In a right-angled triangle ABC, besides the right angle at C, we have one angle and three sides to consider: of these, two being given, the other two may be computed thus:

**Case 1.**—An angle and the hypotenuse being given.

If the hypotenuse AB were made the radius, BC would be the sine, and AC the cosine of the angle A; hence BC = AB · sin A, AC = AB · cos A (fig 4).

Let A be 27° 53′; AB = 57° 2′; we obtain BC and AC thus:

\[ \begin{align*} \log \sin 27° 53′ &= 0.4699462 \\ \log 57° 2′ &= 2.7580030 \\ \log \cos 27° 53′ &= 0.9484040 \\ BC &= 267.883 \\ AC &= 506.299 \end{align*} \]

**Case 2.**—An angle and the adjacent side being given.

If the given side, say AC, were made the radius, CB would be the tangent, and AB the secant of the adjacent angle A; hence CB = AC · tan A, AB = AC · sec A.

**Example.**—Let A = 57° 41′, AC = 897.7

\[ \begin{align*} \log \tan 57° 41′ &= 0.9188839 \\ \log 897.7 &= 2.9531312 \\ \log \sec 57° 41′ &= 0.2719725 \\ CB &= 1419.107 \\ AB &= 1679.205 \end{align*} \]

**Case 3.**—An angle and the side opposite to it being given.

If BC were made the radius, CA would be the cotangent, AB the cosecant of the opposite angle A; whence CA = BC · cot A, AB = BC · csc A.

Thus, let BC = 419.72; A = 15° 31′; and the calculation may be arranged as under:

\[ \begin{align*} \log \cot 15° 31′ &= 0.6565214 \\ \log 419.72 &= 2.6229597 \\ \log \csc 15° 31′ &= 0.6726459 \\ AC &= 1511.754 \\ AB &= 1568.937 \end{align*} \]

In these three examples the logarithm of the given side is placed in the middle, so as to be conveniently added to the logarithm above for the one result, to the logarithm below for the other.

**Case 4.**—The hypotenuse and one side being given.

**Example.**—Let AB = 894.37; BC = 514.63; then when the angle as well as the other side is wanted, the computa- tion may be arranged thus:

\[ \begin{align*} \log \cos 35° 07′ 42″ &= 9.0126815 \\ \log 894.37 &= 2.9515172 \\ \log 514.63 &= 2.7114951 \\ \log \sin 35° 07′ 42″ &= 9.7599779 \\ \log AC &= \log 731.474 = 2.8541987 \end{align*} \]

Here, from the logarithm of BC we subtract that of AC, in order to obtain the logarithmic sine of the angle A, and thence the angle A itself. Then, taking the logarithmic cosine of that angle, we write it above or adjacent to the logarithm of AB, the space hav- ing been purposely left open for it, these added together give the logarithm of AC.

When the side AC alone is wanted, this process is somewhat long; it is preferable to proceed as under— This operation is founded on the fact, that the difference between the squares of two lines is equivalent to the rectangle under the sum and the difference of those lines.

**Case 5.**—When the two sides are given.

If AC = 980.91, CB = 762.43, the angle A and the hypotenuse AB may be found thus:

\[ \begin{align*} \text{Log sec } 37^\circ 51' 24'' 4 &= 0.1026219 \\ \text{Log } 980.91 &= 2.9916229 \\ \text{Log } 762.43 &= 2.8822000 \\ \text{Log tan } 37^\circ 51' 24'' 4 &= 9.8905708 \\ \text{Log } \log 1242.37 &= 3.0942511 \\ \end{align*} \]

**Resolution of Oblique-Angled Triangles.**

In general we have to consider two angles and the three sides of a triangle; of these, any three being given, the remaining two may be computed.

**Case 6.**—Two angles and a side being given.

The third angle is found by subtracting the sum of the two given ones from 180°.

Having described a circle round the triangle ABC (fig. 5), join the centre O with each of the corners, and let fall the perpendiculars OD, OE, OF; then the angles BOD, COE, AOF are, respectively, equal to BAC, CBA, ACB. Now, BD is the sine of the angle BOD to the radius OB; so is CE of COE, and AF of AOF, so that the three lines BD, CE, AF, or their doubles BC, CA, AB, are proportional to the sines of the opposite angles BAC, CBA, ACB.

When both of the other sides are wanted, the computation can be arranged neatly, so as to avoid unnecessary figuring, by observing that the cosecant of an angle ist he inverse of its sine.

**Example.**—Let AB = 1378.7; BAC = 47° 53'; ABC = 65° 19', and consequently ACB = 66° 44'.

\[ \begin{align*} \text{Log sin } 47^\circ 53' &= 9.8702756 \\ \text{Log csc } 66^\circ 48' &= 0.0366205 \\ \text{Log } 1378.7 &= 3.1394699 \\ \text{Log sin } 65^\circ 19' &= 9.9583869 \\ \text{BC} &= 1113.659 \\ \text{AC} &= 1362.941 \\ \end{align*} \]

The logarithms of the results are here obtained by adding together the three upper and the three under logarithms.

**Case 7.**—Two sides and an angle opposite to one of them being given.

As, in certain circumstances, this case admits of a double solution, it can only be safely used when there are the means of discriminating between the two results. Thus if, in the measurement of some triangle, we had found CAB = 37° 27', AB = 3881, BC = 2360, it would have been difficult to determine which of the two results should be taken. For in finding the angle C by means of the proportion BC : AB :: sin A : sin C, we obtain for the log sine of C the value 9.9999844, which is the log sine of 89° 30', and somewhere between 45° and 55°, and is also the log sine of the supplement, viz., 90° 30', and between 5° and 12°; the angle C may thus be either 89° 30' 50", or 90° 29' 10", and each of these values with an uncertainty of several seconds.

\[ \begin{align*} \text{Colog } 2360 &= 6.6270880 \\ \text{Log } 3881 &= 3.5889436 \\ \text{Log sin } 37^\circ 27' &= 9.7839528 \\ \text{Log } 9.9999844 &= 9.9999844 \\ \end{align*} \]

If we take the former value, we obtain 53° 02' 10" for the remaining angle B, and thence get the length of the third side AC, as follows:

\[ \begin{align*} \text{Log csc } 37^\circ 27' &= 0.2160472 \\ \text{Log sin } 53^\circ 02' 10" &= 9.9025547 \\ \text{Log } 2360 &= 3.3729120 \\ \text{AC} &= 3101.09 \\ \end{align*} \]

But if we take the latter value, we have C = 52° 03' 50", giving the following calculation for AC:

\[ \begin{align*} \text{Log csc } 37^\circ 27' &= 0.2160472 \\ \text{Log sin } 52^\circ 03' 50" &= 9.8969101 \\ \text{Log } 2360 &= 3.3729120 \\ \text{AC} &= 3051.04 \\ \end{align*} \]

Thus, it seems that this mode of measuring a triangle is to be avoided whenever the angle opposite to the other given side is nearly a right angle; both on account of the difficulty of discriminating between the two results, and of the inexactitude with which each of these is obtained.

When the side opposite the measured angle is the greater of the two, there can be no ambiguity.

**Case 8.**—Two sides and the angle contained by them being given.

When the sides BA, BC, and their contained angle ABC, are known, the other angles and the third side may be computed by drawing a perpendicular from B to AC, and by resolving the right-angled triangles thus formed. The following process is, however, generally preferred:

Having produced one side CB, lay off BD, BE, each equal to BA, join DA, EA, and through C draw CF parallel to EA. Then ABD, the supplement of ABC, is equal to the sum of the two angles BAC, BCA, therefore BHA is half the sum of A and C, while EAC or ACF is half their difference. Also, EAD is a right angle, so that FD is the tangent of DCF, FA the tangent of FCA to the radius OF; but FD : FA :: CD : CE; wherefore,

\[ \frac{BC + AB}{BC - AB} = \tan \frac{A + C}{2} : \tan \frac{A - C}{2} \]

By help of this proportion we can find the half difference of the two angles, and thence the angles A and B themselves. Afterwards the third side is usually computed by help of the law stated in Case 6; but the following process, which is believed to be new, enables us to avoid the seeking out of logarithms, and the opening of the trigonometrical canon at so many places.

In the triangle AEC we have sin CAE : sin AEC :: EC : CA; that is,

\[ \sin \frac{A - G}{2} : \sin \frac{A + G}{2} :: CB - BA : CA. \]

Or, in the triangle ADC, sin DAC : sin ADC :: DO : CA; that is,

\[ \cos \frac{A - G}{2} : \cos \frac{A + G}{2} :: CB + BA : CA. \]

By means of either of these proportions we can find the third side CA.

**Example.**—Let AB = 738.6; BC = 1079.3; ABC = 67° 42'; whence A + C = 112° 18'.

\[ \begin{align*} \text{AB} &= 738.6 \\ \text{BC} &= 1079.3 \\ \text{1817.9 colog } &= 6.7404300 \\ \text{349.7 log } &= 2.5323721 \\ \text{Log tan } 56^\circ 00' &= 9.1734677 \\ \text{Log tan } 15^\circ 36' 42.7'' &= 9.4462698 \\ \text{A} &= 71^\circ 45' 42.7'' \\ \text{B} &= 40^\circ 32' 17.3'' \\ \text{Log } 349.7 &= 2.5323721 \\ \text{Log sin } 56^\circ 00' 00'' &= 9.9193390 \\ \text{Log csc } 15^\circ 36' 42.7'' &= 0.3706553 \\ \text{AC} &= 1051.396 \\ \end{align*} \] When, as often happens, the third side only is wanted, the following process is perhaps the most expedient:

Through B draw BG parallel to CF, and, of course, bisecting DA.

Then \( DC^2 - CA^2 = DF^2 - FA^2 = 4 \cdot DG \cdot GF \), but \( DG = AB \cdot \cos \frac{B}{2} \).

\( GF = BC \cdot \cos \frac{R}{2} \); therefore \( AC^2 = DC^2 - 4 \cdot AB \cdot BC \left( \cos \frac{B}{2} \right)^2 \).

Hence, if we find M a mean proportional between DG and GF, we must have

\[ M = \cos \frac{B}{2} \cdot \sqrt{(AB \cdot BC)}, \text{ and } AC = \sqrt{(AB + BC + 2M)}. \]

\((AB + BC - 2M)\); whence the following computation:

\[ \begin{align*} \log & \quad 7386 = 1.4342045 \\ \log & \quad 10793 = 1.0165711 \\ \log \cos & \quad 33^\circ 51' = 9.9193390 \\ M & = 741.506 = 2.8701147 \\ AB + BC & = 1817.9 \\ 2M & = 1483.012 \\ AB + BC + 2M & = 3300.912 \log = 3.5186340 \\ AB + BC - 2M & = 3348.883 \log = 2.5248996 \\ & = 2|0435336 \\ AC & = 1051.397, 3.0217668 \end{align*} \]

**Case 9.**—The three sides being given.

Bisect the angles BAC, BCA, and BCR the supplement of BCA, by the lines AQQ, CO, and CQ (fig. 7); draw also the perpendiculars OP, QR; then it is easy to show that O is the centre and OP the radius of the inscribed circle, that Q is the centre and QR the radius of one of the circles of external contact; also, that AR is the semiperimeter of the triangle, and AP, PC, CR, the excesses of that semiperimeter above the sides CB, BA, AC respectively.

The triangle QCR is similar to COP, wherefore \( OP : PC :: CR : RQ \); and \( OP : RQ = PC : CR \). Again, \( AR : AP :: RQ : OP \); \( OP : RQ : OP^2 \), wherefore \( AR : AP :: PC : CR : OP^2 \), or \( OP = \sqrt{\left( \frac{AP \cdot PC \cdot CR}{AR} \right)} \). But \( AP : PO :: R : \tan \frac{A}{2} \), wherefore \( \tan \frac{A}{2} = \frac{PO}{AP} \); and similarly, \( \tan \frac{C}{2} = \frac{PO}{PC} \), \( \tan \frac{B}{2} = \frac{PO}{CR} \). Also, the area of the triangle is equivalent to the rectangle under AR and OP, or \( ABC = \sqrt{\left( AP \cdot PC \cdot CR \cdot RA \right)} \). Hence, the three angles and the area can be found by a very concise operation.

**Example.**—Let \( CB = 517.7 \); \( BA = 789.5 \); \( AC = 904.6 \).

\[ \begin{align*} CB & = 517.7 \quad AP = 588.2 \quad \log = 2.7695250 \\ BA & = 789.5 \quad PC = 316.4 \quad \log = 2.5002365 \\ AC & = 904.6 \quad CR = 201.3 \quad \log = 2.3038438 \\ Sum & = 2211.8 \quad RA = 1105.9 \quad col = 6.9562841 \\ & = 2|43298894 \\ \log OP & = 2.2649447 \\ \frac{1}{2} A & = 17^\circ 22' 31.6'' \log \tan = 9.4954197 \\ \frac{1}{2} C & = 30^\circ 11' 13.5'' \log \tan = 9.7647082 \\ \frac{1}{2} B & = 42^\circ 26' 14.9'' \log \tan = 9.9611009 \\ Area & = 203.5451 \quad \log = 5.3086606 \end{align*} \]

the last four logarithms being obtained by subtracting each of the first four from the logarithm of the inscribing radius.

**Spherical Trigonometry.**

A spherical triangle is a portion of the surface of a sphere inclosed by three arcs of great circles, and represents the corner or solid angle formed by the meeting of three planes at the centre of the sphere. The sides of the triangle measure the angles, and the angles of the triangle measure the edges of the corner. Since the sum of the three angles of a spherical triangle exceeds half a revolution by a quantity proportional to the surface of the triangle, we have to consider the three angles and the three sides, any three of which being given, the other three may have to be calculated; and therefore the cases are more numerous than in plane trigonometry. But the number of investigations is reduced one-half by the consideration of what is called the supplemental or polar triangle.

From each of the corners A, B, C, of the spherical triangle ABC, as a pole describe a great circle, so as to form a new triangle PQR, then the corners of PQR are necessarily the poles of the sides of ABC.

Let the sides BA, BC, produced if necessary, meet PR in S and T, then the arc ST is homologous with the angle ABC; now PT and SR are quadrants, wherefore PR is the supplement of ST; wherefore each side of PQR is supplementary to an angle of ABC, and conversely. Hence, if one spherical triangle have its sides supplementary to the angles of another, its angles also are supplementary to the sides of that other.

**Right-angled Spherical Triangles.**

Let ABC represent a spherical triangle having a right angle at C; join A, B, C with O, the centre of the sphere, so as to trace a corner bounded by the three planes AOB, BOC, COA, the two last being perpendicular to each other.

In OC take any point D, thence draw DF perpendicular to OA, at F in the plane AOB raise FE also perpendicular to OA, and join DE; DE is evidently normal to the plane AOC, and the angle DFE measures the inclination of the two planes COA, BOA, and is thus equal to the angle A.

If we suppose OE to be the tabular radius, OD is the cosine of BC, DE its sine; OF the cosine of BA, EF its sine; but \( OD : OF :: R : \cos AC \), wherefore \( R : \cos AC :: \cos BC : \cos AB \) (1); and again, \( FE : ED :: R : \sin CAB \), wherefore \( R : \sin A :: \sin AB : \sin BC \) (2).

If we suppose OD to be the tabular radius, DE becomes the tangent of BC, DF the sine of CA; but FD : DE :: \( R : \tan DFE \), wherefore \( R : \tan A :: \sin AC : \tan BC \) (3). Lastly, if OF be made the tabular radius, FE becomes the tangent of AB, FD that of AC; now EF : FD :: R : cos EFD, whence R : cos A :: tan AB : tan AC (4).

As an analogous construction may be made for the angle B, we have similarly R : sin B :: sin AB : sin AC (5); R : tan B :: sin BC : tan AC (6); and R : cos B :: tan AB : BC (7).

By inverting the 7th proportion, and combining it with the second, we obtain cos B : sin A :: cos AB : cos BC, therefore 1 : cos AC :: sin A : cos B (8); and similarly, 1 : cos BC :: sin B : cos A (9); also, by combining the 1st, 3d, and 6th proportions, we have 1 : cot A :: cot B : cos AB (10). From these ten proportions we have the following solutions of the various cases:

**Case 1.**—Given A and B.

sec BC = sec A · sin B; sec AC = sin A · sec B; sec AB = tan A · tan B.

**Case 2.**—Given A and AB.

sin BC = sin AB · sin A; tan AC = tan AB · cos A; tan B = sec AC · cot A.

**Case 3.**—Given A and AC.

tan BC = sin AC · tan A; tan AB = tan AC · sec A; sec B = sec AC · csc A.

**Case 4.**—Given A and BC.

sin AC = tan BC · cot A; sin AB = sin BC · csc A; sin B = sec BC · cos A.

**Case 5.**—Given AB and BC.

sec AC = sec AB · cos BC; sec B = tan AB · cot BC; sin A = csc AB · sin BC.

**Case 6.**—Given AC and BC.

sec AB = sec BC · sec AC; tan A = tan BC · csc AC; tan B = sec BC · tan AC.

**Quadrantal Spherical Triangles.**

When one side of a spherical triangle is a quadrant, it may be revolved by considering its conjugate. Thus, if ABC be a triangle, having AB = 90°, we may form another triangle A'B'C', the angles of which are the supplements of the sides of ABC, and therefore having C' = 90°. In this way, we obtain the ten following equations which contain the solution of every possible case.

cos C = - cos B · cos A; cos C = - cot BC · cot AC; sin A = + sin C · sin BC; sin B = + sin C · sin AC; tan B = - tan C · tan BC; tan A = - tan C · tan AC; tan B = + sin A · tan AC; tan A = + sin B · tan BC; cos AC = + cos B · sin BC; cos BC = + cos A · sin AC.

**Spherical Triangles in General.**

**Case 1.** Given the three sides.

Having made a construction analogous to that given in Case 9 of Plane Trigonometry, O is the pole of a small circle which touches the three sides internally, and Q that of another small circle which touches the same three sides externally.

In the right-angled triangle OPC, we have R : tan OCP :: sin Trigonometry. tan PO : tan QCR; but QCR is the complement of OCP, wherefore tan OQP :: R : R : tan QCR, wherefore tan PO : sin CP : sin CR : sin QR, or tan PO : tan QR = sin PC : sin CR. Again, in the triangles APO, ARQ, R : tan QAR = sin AR : tan RQ : sin AP : tan PO, wherefore sin AR : sin AP :: tan RQ : tan PO :: tan PO : tan RQ², or sin AR : sin AP :: tan AP :: tan PC · sin CR : tan PO², whence

\[ \tan PO = \sqrt{\frac{\sin AP \cdot \sin PC \cdot \sin CR}{\sin AR}}. \]

The tangent of PO being thus found, we readily obtain the tangent of the half angles by observing that \( \tan \frac{A}{2} = \tan PO \cdot \tan \frac{C}{2} = \frac{\tan PO}{\sin PC} \cdot \tan \frac{B}{2} = \frac{\tan PO}{\sin CR}. \) Hence a form of procedure quite similar to that for the analogous case of plane triangles.

**Example.**—Let BC = 84° 27' 48"; AB = 95° 44' 51"; CA = 53° 14' 17".

BC = 84° 27' 48"; AP = 32° 15' 40" log sin = 9·727 3611 AB = 95° 44' 51" PC = 20° 58' 37" log sin = 9·553 8733 CA = 53° 14' 17" CR = 63° 29' 11" log sin = 9·351 7397 233 26' 58" RA = 116° 43' 28" log cose = 0·049 0612 Log tan PO = 9·641 0178

A = 78° 41' 46" ; A = 39° 20' 29" ; log tan = 9·913 6567 C = 101° 25' 13" ; C = 50° 42' 36" ; log tan = 9·687 1442 B = 52° 06' 50" ; B = 26° 03' 25" ; log tan = 9·889 2781

If only one of the angles had been wanted, say that at A, we should have used the formula

\[ \tan \frac{A}{2} = \sqrt{\frac{\sin PC \cdot \sin CR}{\sin AP \cdot \sin AR}}. \]

**Case 2.** The three angles being given.

A, B, C, being the three given angles; let us construct a triangle A'B'C', having B'C' = 180° - A; C'A' = 180° - B; A'B' = 180° - C; then shall we have A'R' = 270° - \(\frac{1}{2}(A + B + C)\); A'R' = 90° - \(\frac{1}{2}(A + B + C)\); P'C' = 90° - \(\frac{1}{2}(A + B + C)\); C'R' = 90° - \(\frac{1}{2}(A + B + C)\); therefore tan \(\frac{A'}{2} = \sqrt{\frac{\cos(S - C) \cdot \cos(S - B)}{-\cos(S - A) \cdot \cos S}}\) in which S is put for \(\frac{1}{2}(A + B + C)\). But A' = 180° - BC, wherefore cot \(\frac{BC}{2} = \sqrt{\frac{\cos(S - C) \cdot \cos(S - B)}{-\cos(S - A) \cdot \cos S}}\), by help of which formulas any one of the sides may readily be found. When all the three sides are wanted, an arrangement similar to the last may be used.

**Example.**—A = 89° 58' 43"; B = 76° 47' 19"; C = 69° 19' 48".

| 89° 58' 43" | 28° 04' 12" | log cose = 9·945 6524 | 76 47 19 | 41 15 36 | log cose = 9·976 0588 | 69 19 48 | 48 43 07 | log cose = 9·819 3844 | 236 05 50 | 118 02 55 | log (- sec) = 0·327 6984 | 29·968 7940 | | 9·984 3970 |

BC = 84° 53' 42" ; 42° 26' 51" ; log cot = 0·038 7446 CA = 75° 51' 11" ; 37° 55' 35" ; log cot = 0·108 3382 AB = 88° 44' 0" ; 34° 22' 0" ; log cot = 0·015 0126

**Case 3.** Two sides and the contained angle being given.

Let AB, BC, be the given sides, ABC the given angle. From A draw the arc AD of a great circle perpendicular to the side BC, then we have tan BA · cos B = tan BD, whence BD, and then DC can be found. Now, R : tan B :: sin BD : tan DA and tan C : R :: tan DA : sin DC, wherefore, compounding sin DC : sin BD :: tan B : tan C, which gives us C; and, again, R : cos AD :: cos In order to find the angle A, we observe that cot ABD = sec BA = tan BAD; and also, that, since tan BD = sin AD · tan BAD and tan DC = sin AD, tan DAC, tan BD : tan DC :: tan BAD : tan DAC.

Example.—Let AB = 61° 44' 14"; BC = 98° 22' 45"; ABC = 76° 40' 20".

| B = 76° 40' 20" | log cos = 9.362 7115 | | --- | --- | | AB = 61 44 14 | log tan = 9.209 5352 | | BD = 23 12 334 | log tan = 9.632 2467 | | BC = 98 22 45 | log tan = 9.675 3348 | | DC = 75 10 11-6 | log tan = 9.120 1471 |

These four proportions are known under the name of Napier's Analogies, in honour of their illustrious discoverer. By help of the first two we obtain half the difference and half the sum of the unknown angles, whence those angles themselves can be found; and then by help of either the third or the fourth we compute the half of the third side.

Taking the preceding example, we have

| AB = 61° 44' 14" | BC = 98 22 45 | | --- | --- | | CB + BA = 160 06 59 | CB - BA = 36 38 31 | | ½ (CB + BA) = 80 03 29-5 | sec = 0.762 8389 | | ½ (CB - BA) = 18 19 15-5 | cos = 0.977 4083 | | ½ B = 38 29 10 | cot = 0.101 9464 | | ½ (CB + BA) = ... | sin = 0.497 3096 | | ½ (CB - BA) = ... | csc = 0.606 5710 | | ½ (A + C) = 81 48 57-9 | tan = 0.842 1936 | | ½ (A - C) = 21 58 38-8 | tan = 0.605 9170 |

This process is convenient when only the third side, or only one of the angles, is wanted; for the complete solution, the following investigation is to be preferred.

AB and BC being the given sides, bisect the third side AC in D, and make DP perpendicular to AC, continuing it to meet AB produced in F, join CF, thus forming an isosceles spherical triangle AFC. Bisect the angle BCF by CG, and let fall the three perpendiculars GK, GL, GM; these three are evidently equal to each other, and also BK to BM, wherefore the arc BG must bisect the angle FBC. Also, since MP = PL, AM = CL = CK, so that CK is half the sum, KB half the difference of AB and BC, while ACG is half the sum, GCK half the difference of the angles A and C; moreover, since the whole revolution at G is made up of 2FGL, 2LGC, and 2BGK; the half revolution is composed of FGL, LGC, BGK, therefore CGD is equal to BGK; at the same time, it may be observed, that KBG is the complement of the half of the angle ABC.

Since BGK is a right angle tan KG = sin BK · tan KBG; and also tan KG = sin CK · tan KCG, wherefore sin BK : sin KC :: tan KCG : tan KBG, that is

\[ \frac{\sin \frac{CB + BA}{2}}{\sin \frac{CB - BA}{2}} : \cot \frac{B}{2} : \tan \frac{A - C}{2}. \]

(1)

Again, in the right-angled triangle GKC, tan GC · cos GCK = tan KC; and in GDC, tan GC · cos GCD = tan CD; wherefore cos GCK · cos GCD :: tan KC : tan CD, that is

\[ \cos \frac{A - C}{2} : \cos \frac{A + C}{2} : \tan \frac{CB + BA}{2} : \tan \frac{AC}{2}. \]

(2)

Lastly sin GK = sin GC · sin GCK; sin GD = sin GC · sin GCD; sin GK · tan BGK = tan BK and sin GD · tan DGC = tan DC; wherefore, since BGK = DGC; sin GCK : sin GCD :: tan BK : tan DC, that is

\[ \sin \frac{A - C}{2} : \sin \frac{A + C}{2} : \tan \frac{CB - BA}{2} : \tan \frac{AC}{2}. \]

(3)

AC in this example is the angular distance of the star α Andromedae from β Orionis (1850).

Case 4.—Two angles and the intermediate side being given.

The solution of this case is quite analogous to that of the preceding; thus, if the two angles at A and C and the intermediate side AC be known, we may let fall from A a perpendicular AD upon the opposite side, so as to form two right-angled triangles.

Then cos AC · tan C = cot CAD, whence CAD and, by subtraction, DAB can be found. Now in the right-angled triangle ADB, tan AD = tan AB · cos DAB, while in ADC, tan AD = tan AC · cos CAD, therefore cos DAB : cos CAD :: tan AC : tan AB from which all is obtained. Again, cos B = cos AD · sin DAB and cos C = cos AD · sin DAC, wherefore sin CAD : sin DAB :: cos C : cos B which gives the angle B. Lastly, to find BC we have tan AC · cos C = tan DC · tan DC · tan DAB · cot CAD = tan DB. Example.—Let \( A = 107^\circ 23' 46'' \); \( C = 75^\circ 49' 28'' \); \( AC = 67^\circ 29' 32'' \).

\[ \begin{align*} AC &= 67^\circ 29' 32'' \\ C &= 75^\circ 49' 28'' \\ CAD &= 33^\circ 25' 03'' \\ A &= 107^\circ 23' 46'' \\ DAB &= 73^\circ 58' 42'' \\ CAD &= 33^\circ 25' 03'' \\ AC &= 67^\circ 29' 32'' \\ AB &= 82^\circ 11' 52'' \\ CAD &= 33^\circ 25' 03'' \\ DAB &= 73^\circ 58' 42'' \\ C &= 75^\circ 49' 28'' \\ B &= 64^\circ 41' 51'' \\ DC &= 30^\circ 34' 59'' \\ CAD &= 33^\circ 25' 03'' \\ DAB &= 73^\circ 58' 42'' \\ AD &= 72^\circ 13' 28'' \\ AC &= 102^\circ 48' 27'' \\ \end{align*} \]

Or thus for AC, and B:

\[ \begin{align*} \frac{1}{2}(A-C) &= 31^\circ 34' 18'' \\ (A+C) &= 183^\circ 13' 14'' \\ (A-C) &= 31^\circ 34' 18'' \\ \frac{1}{2}(A+C) &= 91^\circ 36' 37'' \\ \log \sec &= 1:551 2790 (-) \\ \log \cos &= 9:983 3038 \\ \log \tan &= 9:824 8288 \\ \log \sin &= 9:434 6355 \\ \log \cot &= 0:000 1715 \\ \frac{1}{2}(CB+BA) &= 92^\circ 30' 10'' \\ \frac{1}{2}(CB-BA) &= 10^\circ 18' 17'' \\ CB &= 102^\circ 48' 27'' \\ BA &= 82^\circ 11' 52'' \\ \frac{1}{2}(CB-BA) &= 10^\circ 18' 17'' \\ \frac{1}{2}(CB+BA) &= 92^\circ 30' 10'' \\ \frac{1}{2}(A-C) &= 31^\circ 34' 18'' \\ \log \cot &= 0:198 3437 \\ \end{align*} \]

Case 5.—Two sides and an angle opposite one of them being given.

This, like the corresponding case in plane trigonometry, admits of two solutions, and we can only discriminate between these by a knowledge of the circumstances with which the work is connected.

We have, in relation to the figure for case 3, \( R : \sin B : \sin A : \sin AD \) and \( R : \sin C : \sin CA : \sin AD \); wherefore \( \sin B : \sin C : \sin AC : \sin AB \), and in general \( \sin A : \sin BC : \sin B : \sin CA : \sin C : \sin AB \).

By help of this proportion we can compute the sine of the other opposite angle, and thence obtain the two supplementary angles of which it is the sine. Having ascertained which of the two angles is to be taken, we have now two sides and the two angles opposite to them, and can readily obtain the third side and the third angle by help of Napier's analogies.

Case 6.—Two angles and a side opposite one of them being given.

The very same remarks apply to this as to the preceding case, the modes of solution being so closely related that it is not worth while to give the details of both.

Example.—\( A = 78^\circ 21' 40'' \); \( C = 59^\circ 47' 30'' \); \( AB = 48^\circ 13' 20'' \); \( BC \) obtuse.

In this account of trigonometrical calculations, the leading cases and the most convenient modes of operating have alone been given. For fuller information on special cases, and the formulae applicable to them, the reader is referred to any of the thousand and one treatises on the subject.

FIELD OPERATIONS.

The operations of the surveyor may be treated under three heads: the measurement of lines; the measurement of angles; and the determination of direction.

Graduated rods, tapes, and chains are used for the measurement of distance, the mode of operating depending on the degree of precision which is desired; for ordinary field-work, chains of 100 links are used, some being 100, some 65, and, for lightness, some only 50 feet long. The chain of 66 feet was contrived by Gunter in order to obtain the advantage of decimal calculation, for ten squares of such a chain make one acre, and thus the acre consists of 100,000 square links; 80 of Gunter's chains make one mile.

When the tape is used, care must be taken to stretch it always to the same tension; it is not to be depended on for nice work; its lightness is its great recommendation.

For the measurement of base-lines, where the utmost attainable degree of precision is needed, rods of wood or metal are employed, and means are provided for ascertaining their temperature in order that their expansions and contractions may be allowed for; the readings also are made by help of elaborate micrometric apparatus.

The measurement of angles proper is accomplished by means of the sextant, reflecting circle, repeating circle, and analogous instruments. The repeating circle consists essentially of two telescopes, \( AB \), \( CD \), turning on either side of a graduated circle, and having the planes in which they move brought as closely as possible together. Each of these telescopes can be secured and adjusted to any part of the limb by means of a clamp and tangent screw; and one of them, at least, as \( AB \), carries a reader \( R \), to indicate its position. The whole instrument is so supported on a stand as that the plane of the circle can be brought into the plane of the angle which is to be measured. Having secured the telescope \( AB \) so that the reader attached to it may indicate zero, we bring both telescopes to point to one of the signals, and secure \( CD \) to the limb. Releasing \( AB \), we now bring it to point to the second signal, taking care that \( CD \) still point to the former; the reader now indicates the angular distance between the two signals. \( AB \) being now secured to the limb, and \( CD \) released, we again bring both telescopes to the first object, secure \( CD \) in that position, release \( AB \), and bring it once more to the second object, while \( CD \) points to the first; the reader now indicates the double of the angle.

By continuing this process, we can repeat the angle as often as we choose; then, dividing the ultimate reading by the number of the operations, we obtain the value of the angle with more precision than if we had made only one measurement, for the minute errors unavoidable in the graduation of the limb are subdivided. But no number of repetitions can give results true to a quantity less than that which is appreciable by the telescope, or than what the stability of the instrument would warrant on a single observation, because each observation is accompanied by its own error, so that the sum of all the readings—that is, the ultimate reading of the instrument—includes a quantity of such errors as there are observations, so that the quotient must still be affected by the average error to which the instrument is liable.

The measurement of angles is most conveniently made by help of the sextant, or the reflecting circle. The construction of the sextant, and its use in astronomy, are given in the article NAVIGATION. The objects observed are then supposed to be remote, and the instrumental parallax has not been noticed; when the objects are near, this parallax becomes considerable, and we must therefore examine its source and the manner of correcting for it.

The inclination OCL of the two mirrors is double of SHP the angle which the two objects subtend, not at C the centre of the instrument, but at a variable point H, so that, if CP be joined, CP'H is the parallax or the difference between the true and the apparent angle, when the object P' is close at hand.

If the index I be brought back a little behind the zero point O, until the mirror AE be equally inclined to the lines FC, CP, the image of P will be seen in the same direction with P; wherefore, if we first bring the direct and reflected images of that end of the signals which is to be looked at directly to agree, as when taking the index error of the instrument, and correct for that error in the usual way, we shall obtain the angle CPS freed at once of parallax and of index error.

The sextant and octant, like all other fragmental instruments, are liable to this objection, that there are no means for ascertaining whether the actual axis of motion coincides with the centre of the graduation, or for correcting the error of centering; the observer has to rely entirely on the accuracy of the workmanship. By using an entire circle we obviate this inconvenience, and obtain also the advantage of repetition when great precision is required. The management of the reflecting circle does not differ, in any essential particular, from that of the sextant. There is in it this great advantage, that the left or the right hand object may be viewed directly.

Almost all the operations of the surveyor have reference to the positions which objects have on the surface of the earth; and hence, he is far oftener occupied in determining the directions of lines than in measuring their angles. These directions are, in general, indicated by reverting them to the direction of the plumb-line, and to that of the meridian line; the first being obtained by processes purely mechanical, the second by the study of the motions of the stars. Frequently, indeed, the magnetic instead of the true north is used; but this is by no means satisfactory practice, since the direction of the magnetized bar is subject to hourly, monthly, and secular variations, and is liable to great changes by change of position, so that the north shown by the compass at one station often differs by several degrees from that shown at another.

The simple plummet—that is, a fine thread with a small piece of lead attached so it—affords the most obvious, and, for many purposes, a sufficient indication of the direction of gravity; but it is superseded almost for ordinary work, and altogether for accurate purposes, by the spirit-level, which is a glass tube nearly filled with alcohol. When this tube is laid almost horizontally, the bubble or empty space comes to the higher end, and its position thus indicates that end which may be the higher. The upper inside surface of the tube is made slightly concave in the direction of its length, the curvature, in fine levels, being given by grinding; but, in those for ordinary use, the bend which the glass takes while being drawn out is held sufficient. The air-bubble sweeping along the upper surface of the tube is thus the counterpart of the plummet bob; the one rests exactly under, the other exactly above, the centre of curvature. If the tube be ground to a curve of 28 feet 6 inches radius, an inclination of one minute causes the air-bubble to move through one-hundredth part of an inch; such a level, which can be easily carried in the pocket, has as much precision as a plummet of 28 feet.

The tube is usually protected by a case, of which the under side is made accurately parallel to the surface at the middle, and divisions are marked along the glass to show the deviation from horizontality. In order, by help of such an instrument, to render a surface horizontal, or the normal to that surface vertical, it is necessary to examine the level when placed in two directions inclined to each other; hence, in many cases, two cross levels are used.

The spherical spirit-level which indicates horizontality at once, consists of a circular disc of glass having its under side ground concave; this disc is tightly fitted as the cover of a flat metal box, which is nearly filled with alcohol. The lower surface of the box, or the plane passing through the extremities of its three supports, is adjusted to be parallel to the spherical surface at the middle, and the air-bubble coming always to the highest point of the sphere, shows the deviation from horizontality, if there be any.

The directions of lines, in general, are ascertained by means of altitude-and-azimuth circles or theodolites. The essential parts of these instruments are as follows:—(1a). The telescope AB (fig. 15), having cross wires in its field-bar, and turning in a vertical plane upon (2a) the horizontal axis CD. This horizontal axis carries (3d) a graduated vertical circle E, which shows the inclination of the telescope to the horizon, its angular distance from the zenith, or its nadir distance, according to the taste of the observer; the nadir distance is the most convenient. In the older theodolites the telescope could only be raised or depressed 40 or 60 degrees; but in all good instruments the telescope may be turned completely round. The horizontal axis rests upon (4a) two pillars CF, DE, which are fitted with adjustments for rendering the axis truly horizontal. These again stand upon (5a) a plate or cross-bar PHG, fixed securely on the top of (6a) the vertical axis HI. This axis HI works in a hollow axis KL (7b), which ought to be perfectly concentric with it. By this arrangement the telescope BA is free to be pointed in any required direction. There is attached to the outer axis KL, a horizontal or azimuth circle M (8a), the divisions on which are read by verniers or micrometers attached to the cross-bar PHG; these show the bearing or azimuth of the telescope.

For the comfortable use of the instrument there are some secondary arrangements. The axis KL is made to work into a frame provided with screws for rendering the axis truly vertical, there being a pair of cross-levels, or else a spherical spirit-level, fixed to the plate PHG for assisting in this operation. By means of a strong clamp and tangent-screw the horizontal circle can be turned into and held in any required position. Since the reading of the horizontal angles would be vitiated by any displacement of the circle M, the better class of surveying circles carry a settler, that is, a telescope PQ fixed on the end of an axis NO, which rests in a frame that can be secured firmly in any required position on the axis KL. This settler, which ought to be quite as powerful as the principal telescope AB, is directed to any well-defined object, and then clamped firmly in its place; so that, if the circle M be disarranged, its position can be again put right by bringing the settler back to its signal by means of the lower tangent-screw. We have not room to detail the various adjustments of this useful instrument, and may only remark, that by reversing the position of the telescope, and taking the mean of the two sets of readings, the effect of any minute error in the horizontality of CD, or in the collimation of the telescope AB, is neutralised.

It is usual to reckon the bearings of objects from the north line, the degrees being numbered from north towards the east, so that a signal due east is said to bear 90°, one due south 180°, and one due west 270°; and therefore it is properly the first business of the surveyor to determine the true north.

For this purpose the most convenient proceeding is to establish a station for the theodolite near to head-quarters, and commanding a view of the district to be surveyed. At some distance from this a signal is to be set up in such a position that it may be illuminated by a lamp at night. Having then accurately estimated the bearing of that signal, we set the vernier to read that assumed bearing on the horizontal circle M, turn the whole instrument round until the telescope BA points to the signal, secure the outer axis KL in its place, and then bring the settler PQ to the same, or to some other well-defined signal capable of being illuminated at night. The Instrument is now in position for showing bearings as from the assumed meridian. The latitude of the plane has now to be ascertained by observing the least or greatest zenith distance of a star (see Astronomy, Practical, prob. ix, art 2, and Navigation, chap. iii, sect. 1), preferring those stars which pass near to the zenith, and using observations with the face of the instrument alternately east and west. This done, the readiest process for obtaining the error of the assumed meridian is to take the extreme east or west bearing of some star which passes between the pole and the zenith, those which are near the pole being preferred; the true bearing of the star when in this position is easily computed, and the difference between this and the assumed bearing gives the required error.

Thus, in latitude 60° 23' north, the assumed bearing of a signal bearing 283° 36', the greatest easting of the star β Cephei was observed to be 41° on March 11, 1860. On consulting the Nautical Almanac, we find that the north declination of the star at that date was 69° 59' 39", whence

\[ \begin{align*} \log \cos \text{decl. star} &= 9.535 2127 \\ \log \sec \text{lat. plane} &= 0.256 7774 \\ \log \sin 38° 16' 23'' &= 9.791 9901 \\ 24 41 &= \text{assumed bearing of star.} \\ 13 35 28 &= \text{error of assumed meridian.} \\ 283 30 &= \text{assumed bearing of signal.} \\ 297 05 23 &= \text{true bearing of signal.} \end{align*} \]

Having now obtained the true bearing of the signal, the position of the theodolite can be corrected, and the bearings of those station-points which are within sight determined.

In surveys of limited extent, embracing, say, only a few square miles, we may regard the surface of the earth as flat, and the meridian lines passing through the different points as all parallel to each other. In this case it is easy to transfer the theodolite to another station; for, if the bearing of B from A be 297° 05', that of A from B must be 117° 05', the difference being 180°. Hence, if the theodolite be carried to B, and so placed as that the bearing of A be 117° 05', the zero, or north line, will be parallel to what it was at A; in this way the instrument may be carried from station to station, until the directions of all the principal lines of the survey be ascertained.

When the ground is uneven, it is necessary to notice the inclinations of the lines: this is conveniently done by painting the signal-staves alternately black and white at each link or foot, and by directing the telescope to the division corresponding to the height of the instrument. On the common theodolite, the zero on the vertical circle shows the telescope to be horizontal, and we must distinguish elevation (+) from depression (-).

The conductor of the survey, after having obtained a general idea of the configuration of the district, proceeds to arrange the stations, choosing these so that the lines joining them may pass near to the boundaries of the fields, at the same time having regard to their being conspicuous. He then ascertains, in the manner above described, the directions of these as seen from each other, entering the observations in his Theodolite-Book; and afterwards he proceeds to measure such of their distances as may be required, and the offsets or deviations of the actual boundaries from the measured lines, recording these also in his Chain-Book.

The theodolite-book is commonly divided into five columns, the first for the name or number of the station to which the instrument is planted; the second for the actual readings on the horizontal limb; the third for the readings on the vertical arc; the fourth for the measured distances (to be extracted from the chain-book); and the fifth for the name or number of the signal. The following extract will sufficiently exemplify this matter:

| Station | Bearing | Inclination | Distance | Signal | |---------|---------|-------------|----------|--------| | F | 299° 31' | ... | ... | A | | | 17° 02' | ... | ... | B | | | 65° 16' | ... | ... | C | | | 99° 30' | +12° 54' | 633 | E | | | 256° 30' | -4° 20' | 529 | G | | A | 119° 31' | ... | ... | F | | | 73° 50' | +1° 18' | 708 | B | | | 177° 50' | -2° 19' | 422 | G | | C | 245° 16' | ... | ... | F | | | 275° 00' | -1° 55' | 780 | B | | | 171° 00' | +7° 43' | 515 | D | | | 269° 49' | ... | ... | E | | E | 279° 30' | -2° 54' | ... | F | | | 29° 49' | ... | ... | C | | | 85° 50' | +8° 15' | 390 | D |

The page of the chain-book is divided into three columns, the middle one to receive the distances measured along the traverse lines, and those to the right and left for offsets to either side. The beginning of each page is at the bottom, and the writing proceeds upwards, so as to imitate the progress on the field; and at the commencement of each line its bearing, as extracted from the theodolite-book, is given. The following is the chain-work for the field ABCDEFG.

| Corner 15 | 527 | Corner 10 | 429 | |-----------|-----|-----------|-----| | | 515 | Station D. | 0 | | | 500 | | 404 | | | 400 | | 370 | | | 300 | | 334 | | | 200 | | 300 | | | 100 | | 260 | | | 90 | | 200 | | | 80 | | 180 | | | 70 | | 160 | | | 60 | | 140 | | | 50 | | 120 | | | 40 | | 100 | | | 30 | | 80 | | | 20 | | 60 | | | 10 | | 40 | | | 0 | | 20 |

In making the plan, many surveyors use the protractor, and imitate on paper the operations which have been performed in the field; this plan, again, they re-measure for the area. This proceeding is liable to the very serious objection that the errors on paper far exceed those to which the field operations are liable. It is easier and preferable to compute the co-ordinates of the stations, and by help of these to lay down the plan and compute the areas. If the ordinates \(x\) be directed northwards, the \(y\)s eastwards, and the \(z\)s towards the zenith, the difference between the ordinates of one station, as A, and those of another station, as B, are given by the equations,

\[ \begin{align*} z_B - z_A &= \text{sine elevation} \times \text{oblique distance AB}, \\ \text{hor. distance AB} &= \text{cosine elevation} \times \text{oblique distance AB}, \\ y_B - y_A &= \text{sine bearing} \times \text{horizontal distance AB}, \\ x_B - x_A &= \text{cosine bearing} \times \text{horizontal distance AB}. \end{align*} \]

When the co-ordinates of any one of the stations are either known or assumed, those of each of the other stations can be easily deduced by means of these formulae, using either the ordinary trigonometrical tables or the traverse tables. Thus, if we assume for the station A, \(x_A = 500\), \(y_A = 100\), \(z_A = 150\), we have,

| Station | \(x\) or Lat. | \(y\) or Long. | \(z\) or Alt. | |---------|---------------|----------------|---------------| | A | 500-0 | 100-0 | 150-0 | | | +197-1 | +679-8 | 16-1 | Having proceeded from A successively to B, C, D, E, F, and G, we return from G to A. If all the operations have been absolutely exact, the co-ordinates of A thus obtained should have agreed exactly with those from which we first set out. Actually there are small errors—viz., -1 in the direction x, -1.1 in the direction y, and +5 in the direction z. These may naturally be attributed to the unavoidable minute errors in the measurements; and the close coincidence shows that no serious error has been committed. The positions of the several points may be taken as under, rejecting the fractional parts of a link:

| Point | x or Lat. | y or Long. | z or Alt. | |-------|-----------|------------|----------| | A | 500 | 100 | 150 | | B | 697 | 789 | 166 | | C | 629 | 1557 | 192 | | D | 125 | 1637 | 261 | | E | 97 | 1232 | 205 | | F | 202 | 629 | 173 | | G | 79 | 116 | 133 |

From these results, it is easy to mark on the plane the positions of the stations, and then to lay down the contour of the figure from the chain-book.

The area of the polygon is readily computed from the formula:

\[ \text{Area} = \frac{1}{2} \sum (x_i y_{i+1} - x_{i+1} y_i) \]

This formula is easily kept in mind: we proceed round the polygon from corner to corner, and multiply the x or latitude of the point at which we are by the y or longitude of the point before us minus the y of the point behind us. The aggregate of these products, attention being paid to the algebraic signs, is the double of the area.

The computation for the preceding example stands thus:

\[ \begin{align*} 500 \times 100 &= 50000 \\ 697 \times 789 &= 546073 \\ 629 \times 1557 &= 980553 \\ 125 \times 1637 &= 204625 \\ 97 \times 1232 &= 119304 \\ 202 \times 629 &= 127558 \\ 79 \times 116 &= 9026 \\ \end{align*} \]

Area of polygon = 739709

The areas included between the traverse lines and the actual boundaries of the field have now to be computed and added to or subtracted from the area of the polygon according to their positions; the subjacent computations of the offsets from A to B and from G to A exemplify sufficiently well the manner of carrying on the work. Strictly, the results have to be multiplied by the cosines of the inclinations of the respective traverse lines, in order to give the areas of the projections on a horizontal plane.

A to B:

\[ \begin{align*} -8(+0+9) &= -72 \\ 35(+9+13) &= +770 \\ 93(+13+28) &= +3813 \\ 273(+28+52) &= +21840 \\ 117(+52+92) &= +16848 \\ 33(+92+89) &= +6335 \\ \end{align*} \]

When a surveyor has carried his operations with the theodolite over a considerable distance, he is naturally led to inquire whether the instrument now show a north line exactly parallel to what it did at the outset. In order to avoid uncertainty in this respect, he seeks to reduce as far as possible the number of transferences, or even uses a more powerful instrument for connecting the bearings of the distant stations. He can readily obtain a verification of his work by appealing again to the astronomical method; but here he meets with this difficulty, that the true meridians of two places are not parallel to each other, the north lines converging on the north side and diverging on the south side of the equator.

In order to compute the convergence of two meridian lines, we may observe that these would meet each other on the prolongation of the earth's axis; hence the easting or westing divided by the cotangent of the terrestrial latitude must measure the convergence: now, one minute of longitude on the earth's equator measures 6086 feet or 9221 links; hence, if we divide the difference between the y's of two stations by 6086 or by 9221, according as our measurements are in feet or in links, and multiply the quotient by the tangent of the latitude, we obtain the convergence of the meridians in minutes. Thus, in our example of a very small survey made in north latitude 56°, the convergence of the meridians at the extreme east and west stations is computed as under:

\[ \begin{align*} y_1 &= 100 \\ y_2 &= 1636 \\ \log_{10} &= 3.18639 \\ \cot &= 0.63522 \\ \tan &= 0.17101 \\ 0'2469 &= \log_{10} = 0.39262 \\ \end{align*} \]

which gives 0'2468 or 14°8, about a quarter of a minute.

When the survey extends over many square miles, we can no longer neglect the curvature of the earth in any one of the operations; and even the oblateness must be taken into account.