That part of practical mathematics which teaches the method of ascertaining the limits and extent of land or estates, and of representing these in maps or plans, is called surveying, or land surveying; but this term, in a more extended sense, includes the valuing of landed property, the buying and selling of estates, and the dividing or laying out of landed property to the best advantage.

Considered as a branch of practical mathematics, surveying depends for its principles on GEOMETRY and TRIGONOMETRY, and as far as it is confined to the mensuration of plain surfaces, has already been considered under the article MENSURATION. It is the object of the present article to explain and illustrate the most approved methods of applying these principles to practice, and in particular to point out the use of the field book, and the mode of surveying large estates, towns, counties, or similar extensive tracts of land. We shall also point out the most approved mode of surveying subterraneous works, as coal-pits, mines, &c., a subject which has hitherto been entirely neglected in works of this nature.

Before entering on the practical part of the subject Surveying, it may be proper to mention the previous knowledge which a surveyor ought to possess, and to notice the instruments which he is to employ in his operations.

As a surveyor has perpetual occasion for calculation, it is necessary that he be familiar with the four first per for rules of ARITHMETIC, and the rule of PROPORTION, both surveryor in whole numbers, and in FRACTIONS, especially Decimals, with the nature of LOGARITHMS, and the use of Logarithmic Tables; and with, at least, ALGEBRAIC Notation. As it is his business to investigate and measure lines and angles, and to describe these on paper, he should be well acquainted with the elements of GEOMETRY and TRIGONOMETRY, and with the application of these principles to the MENSURATION of Heights, Distances, and Surfaces. In particular, he should be familiar with the best practical methods of solving the ordinary geometrical problems, and should be expert in drawing lines and describing figures. He should be acquainted with the principles and practice of LEVELLING; should know something of the principles of OPTICS and MAGNETISM, and should possess at least a smattering of the arts of DRAWING and PAINTING.

The instruments usually employed in surveying, have been enumerated under MENSURATION, vol. xiii. pp. ments. 511, 519, and of these the chain, the plane-table, the cross, and the theodolite, are there sufficiently described, and the CIRCUMFERENCE, the COMPASS, LEVELS, the PERAMBULATOR, and PROTRACTORS, are described, and their uses explained under their proper heads in the general alphabet of this work.

The most simple methods of surveying, are those in which the chain or the plane-table are employed, and of these methods a general idea has been given under MENSURATION. It may be necessary in this place to describe a little more at large the use of the plane-table, as this instrument is one of the most convenient for surveying fields, or other small plots of ground.

In preparing the plane-table for use, a sheet of paper Practical that will about cover the plane-table, is to be wetted, then directions spread flat on the table, the marginal frame of which for using is to be pressed down on its edges, so as to keep it table smooth and even. On this paper, thus stretched, the plan of the field or other plot is to be traced in the following manner.

Suppose it be required to make a plan of a field that has the figure represented at A, B, C, D, E, F, fig. 1. Plate DXXV., and in such a situation, that all its angles 2. Fig. 1. and are accessible.

The plane-table is to be fixed at one of the angles, as at A, in the position represented at fig. 2. and its surface must be brought to a horizontal plane. A point is then to be made on the paper with a pencil, as at a, to represent the point A, where the plane-table is stationed. Fixing a needle perpendicularly at this point, the index of the table is to be applied to the needle, on that side which corresponds with the sight vanes, and is to be turned round this point, sliding on the table, till the eye looking through the sights, perceives a mark set up at the point B. A line is now to be drawn from a along the edge of the index. In the same manner a line is to be drawn from a, marking the direction of the side AF. Thus the angle b a f, (fig. 2.) will be similar to the angle BAF (fig. 1.): the plane-table is now to be removed from the point A, to another corner Surveying of the field, as B, and a pole, or other mark is to be left at A. The length AB is to be measured by the chain, and a proportional length marked off on the paper, in the direction a, b, from a plotting scale, or scale of equal parts. Proceeding as at first, a line is to be drawn from b towards c, in the direction of the side BC, and marking the measure of the angle CBA. In this manner, by placing the plane table successively at each corner of the field or plot of ground, the outline figure of the whole will be transferred to the paper, and a, b, c, d, e, f, will be the plan of the field A, B, C, D, E, F.

If it be not convenient to place the plane table at the corners of the ground to be surveyed, the plan may be taken by placing the instrument anywhere within the area, as at E (fig. 3.) in the middle of the field A, B, C, D. In this case we can readily find the direction of the lines EA, EB, EC, ED, and the angles which they form at the point E. By measuring the distances from E to the several angular points, and transferring the proportional distances from the plane scale upon the paper, and then joining the points thus found, there is easily traced the outline of the whole field.

It may happen that no part of the ground to be measured is accessible, except one line, as the line AE in the space A, B, C, D, E, F, G, (fig. 4.)

In this case, the plane table is to be fixed at the point A, of the base line AE, and a point made on some part of the paper at pleasure, to represent the station A, and the base line AE is in the usual manner to be ascertained and laid down. Then from the station A, the situation or direction of the points B, C, D, E, F, G, are to be observed through the sights of the index; and lines corresponding to the lines AB, AC, AD, AE, AF, AG, are to be laid down on the paper, but of an indefinite length. When this is done, great attention must be paid to preserve the table steady and perfectly horizontal. The length of the base line AE being determined, the table is now to be removed to the other extremity E, and so disposed that the base line on the paper may be exactly over the base line EA of the field; and proceeding as before, the directions of the lines EA, EB, EC, ED, EF, EG, are to be determined, and corresponding indefinite lines drawn on the paper. The points where these last lines cross those before traced, are to be carefully noted, and the outline joining all these points of section, will correspond to the outline of the plot to be surveyed.

The following general directions to be observed in using the plane table, are given by Dr Hutton. 1. Let the lines on which stations are made be directed towards objects as far distant as possible; and when any such object is set, go round the table and look through the sights from the other end of the index, to see if any other remarkable object be directly opposite; if there be none such, endeavour to find another forward object, such as shall have a remarkable backward opposite one, and make use of it, rather than the other; because the back object will be of use in fixing the table in the original position, either when you have measured too near to the forward object, or when it may be hid from your sight at any necessary station by intervening hedges, &c.

2. Let the said lines, on which the stations are taken, be pursued as far as conveniently can be done; for that will be the means of preserving more accuracy in the survey work.

3. At each station it will be necessary to prove the truth of it, that is, whether the table be straight in the line towards the object, and also whether the distance be rightly measured and laid down on the paper. To know whether the table be set down straight in the line, lay the index on the table in any manner, and move the table about, tilt through the sights you perceive either the fore or back object; then, without moving the table, go round it, and look through the sights by the other end of the index, to see if the other object can be perceived; if it be, the table is in the line; if not, it must be shifted to one side, according to your judgment, till through the sights both objects can be seen. The aforesaid observation only informs you if the station be straight in the line; but to know if it be in the right part of the line; that is, if the distance has been rightly laid down: fix the table in the original position, by laying the index along the station line, and turning the table about till the fore and back objects appear through the sights, and then also will the needle point at the same degree as at first. Then lay the index over the station point and any other point on the paper representing an object which can be seen from the station; and if the said object appear straight through the sights, the station may be depended on as right; if not, the distance should be examined and corrected till the object can be so seen. And for this very useful purpose, it is advisable to have some high object or two, which can be seen from the greatest part of the ground accurately laid down on the paper from the beginning of the survey, to serve continually as proof objects.

When from any station, the fore and back objects cannot both be seen, the agreement of the needle with one of them may be depended on for placing the table straight on the line, and for fixing it in the original position.

The foregoing examples are extremely simple, as the Method bounding lines are straight and regular. Here, therefore, it is not requisite to measure what surveyors call the offsets, or the perpendicular distances between a base line, and the several angles which it subtends. It seldom happens, however, that the work can be carried on in so regular a way, as the bounding lines, even of small pieces of ground, are generally more or less crooked.

Let us suppose A, l, m, n, o, p, q, r, (fig. 5.) to be a crooked hedge, or other boundary of a piece of ground, and A B the general base line subtending its several angles. In measuring along this base, when the surveyor comes opposite to any of the bendings or corners of the fence, as at c, d, e, &c., he measures the perpendicular offsets c l, d m, e n, &c., either with the offset staff, or, if they are of considerable length, with the chain. These offsets are to be noted down, as will be explained immediately.

When the offsets are not very large, their places may be determined pretty exactly by the eye, especially when assisted by laying down the offset staff in a direction perpendicular to the base, and opposite to the angles; but when the offsets are very large, their positions are best determined by the cross, or the plane table, in the following manner. In measuring along A B (fig. 5.), when Here \( O_1 \) is the first station, where the angle or Surveying bearing is \( 105^\circ 25' \). On the left, at 73 links in the distance or principal line, is an offset of 92; and at 610 an offset of 24 to a cross hedge. On the right, at \( O \), or the beginning, an offset 25 to the corner of the field; at 248 Brown's boundary hedge commences; at 610 an offset 35; and at 945, the end of the first line; the \( O \) denote its terminating in the hedge. And so on for the other stations. A line is drawn at the end of every station line, to prevent confusion.

Various improvements have been made on the field-Crocker's book, especially by Mr Abraham Crocker, and Mr field-book John Bodham. We shall give a specimen of each.

Fig. 6. represents a page of Mr Crocker's field-book, exhibiting a part of the survey of an estate called the Mill Estate; the outlines of which were surveyed with the theodolite, and the interior parts filled up with the chain. In this book the operations are noted down, so as to begin from the foot of the page, carrying them on upwards.

In surveying after this method, Mr Crocker advises to choose two or more eminences, as principal stations, and measure a general base line from one station to the other, noting each hedge, brook, or other remarkable object as it is passed by; measuring also such short perpendicular lines to such bends of hedges as may be near the base. From the extremities of this base-line, or from any convenient parts of it, the surveyor must proceed with other lines to some remarkable object situated towards the sides of the estate, without regarding the angles they make with the base-line or with one another, remembering to note every hedge, brook, or other object by which he passes. These lines, when laid down by intersections, will with the base-line form a principal triangle on the ground to be surveyed; several of which, if necessary, being thus laid down, the surveyor may proceed to form other smaller triangles and trapezoids, on the sides of the former; and so on till the several enclosures are finished.

This principal triangle being completed, and laid down on the rough plan paper, the parts, exterior as well as interior, are to be completed by smaller triangles and trapezoids.

When the whole plan is laid down on paper, the contents of each part of the estate may be calculated by the methods already explained under Mensuration.

In countries where the lands are enclosed with high hedges, and where many lanes or roads pass through an estate, a theodolite may be employed with advantage, in ascertaining the angles of such lands; and by these means an outline of the estate may be obtained, and the lane lines serve as the bases of such triangles and trapezoids as are necessary to fill up the interior parts.

To illustrate this method, let us take AB in the plan of the estate, (fig. 8.) for the principal base line. From B go off to the tree at C, noting down in the field book every cross hedge as you measure on, and from C measure back to A, noting down every thing remarkable, as before directed. This figure also illustrates the method of measuring the cross lines, offsets, and interior parts and enclosures.

Fig. 7. represents a page from Mr Rodham's field Rodham's book. His method of procedure is as follows:—Like field book Mr Crocker, he begins from the bottom of the page, fig. 7, and writes upwards; denoting the crossing of fences, by s- lines.

| Offsets and remarks on the left. | Stations, Bearings, and Distances. | Offsets and remarks on the right. | |---------------------------------|----------------------------------|---------------------------------| | \( O_1 \) | \( 105^\circ 25' \) | 25 corner. | | Cross a hedge, 24 | 73 | Brown's hedge. | | | 24 | | | | 610 | | | | 934 | | | House corner, 51 | 25 | | | | 120 | | | | 34 | | | | 734 | | | A brook, | 30 | | | | 248 | | | | 639 | | | Footpath, | 16 | | | Cross-hedge, | 18 | | | | 973 | |

Of the three columns which compose this field book, the middle or principal column is for noting down the stations, angles, bearings and distances, as they are ascertained, and the columns on the right and left are for the offsets to the right and left of the principal course, which are placed against their corresponding distances in the middle column, as also for occasional remarks or memorandums, to which it may be useful to refer in drawing the plan of the surveyed lands. Surveying lines drawn across the middle column, or only a part of such a line on the right and left opposite the figures, to avoid confusion, and the corners of fields, and other remarkable turnings in the fences, towards which offsets are taken, by lines joining like the fences, as will be best seen by comparing the specimen at fig. 7, with the plan at fig. 9.

The marks called \(a, b, c, \&c.\) are best made in the fields, by making a small hole with a spade, and placing there a chip or small piece of wood, with the particular letter marked on it, to prevent one mark being taken for another, on any return to it, though in general the name of a mark is very easily seen, by referring in the book to the line in which it was made. After the small Italic letters have been gone through, the capitals may be next employed, and the Roman letters afterwards, and so on. Perhaps it would be preferable to distinguish the marks by figures.

The letters in the left hand corner at the beginning of each line, denote the mark or place measured from; and that at the right hand corner of the end, is the mark measured to. But when it is not convenient to go exactly from a mark, the place measured from is described such a distance from one mark towards another; and where a mark is not measured to, the exact place is ascertained by writing, turn to the right or left hand, such a distance to such a mark, it being always understood that those distances are taken in the chain line.

The characters used are for \(f\) turn to the right hand, \(l\) for turn to the left hand, and \(A\) placed over an offset, to show that it is not taken at right angles with the chain line, but in the line with some straight fence, being used chiefly when crossing their directions, and is a better mode of ascertaining their true places than by offsets at right angles.

When a line is measured whose position is determined, either by former operations (as in the case of producing a given line or measuring from one known place or mark to another) or by itself (as in the third side of a triangle) it is called a fast line, and a double line is drawn across the book at the conclusion of it; but if its position be not determined (as in the second side of a triangle) it is called a loose line, and a single line is drawn across the book. When a line becomes determined in position, and is afterwards continued, a double line is drawn half through the book.

When a loose line is measured, it becomes absolutely necessary to measure some line that will determine its position. Thus, the first line \(a b\), (fig. 9.) being the base of a triangle, is always determined, till the third side \(j b\) is measured; then the triangle may be constructed, and the position of both is determined.

At the beginning of a line to fix a loose line to the mark or place measured from, the sign of turning to the right or left hand must be added (as at \(j\) in the third line); otherwise a stranger, when laying down the work, may as easily construct the triangle \(h j b\), on the wrong side of the line \(a h\), as on the right side; but this error cannot be committed, if the sign above named be carefully observed.

In choosing a line to fix a loose one, care must be taken that it does not make a very acute or obtuse angle, as in the triangle \(p B r\); by the angle at \(B\) being very obtuse, a small deviation from truth would make the error at \(B\) when constructed very considerable; but by constructing the triangle \(p B g\), such a deviation is of no consequence.

When the words leave off are written in the field book, it is to signify that the taking of offsets is discontinued; and of course something is wanting between that and the next offset.

The general use of the theodolite in measuring separate plots, has been described under Mensuration. The following practical directions for the use of this instrument are given by Mr Crocker, and apply to his field book, exemplified at fig. 6. and the plan at fig. 10.

Suppose the surveyor to plant his theodolite in the road \(O i\), and having duly adjusted it, by placing its head exactly horizontal, by the levels; and setting the index part of the limb exactly at \(360^\circ\); and by moving the whole head about till \(360^\circ\) in the compass-box comes to the line in the north end of the needle; there fixing all fast, by the screw under the head, between the legs, he will have his instrument completely adjusted.

The theodolite thus adjusted, the surveyor sends one of his assistants forward as far as he can conveniently see how to measure a straight line, as at \(O 2\). Taking then his angle of observation, by his telescope, to the picket at that station, he finds it to be \(60^\circ\) from the north part of his magnetic meridian line towards the east, which he enters in his field book, noting it with \(NE\), as a memorandum on which side of the magnetic meridian it lies. He is now to fasten his limb to the other part of the head, by a screw for that purpose.

His chain-man having laid the chain in the direction to the picket \(O 2\), in order to measure the line, he makes such offsets to the right and left, in his first chain's length, as may be necessary. At his first station, he finds that on the right, the general road fence is 30 links, and also a nook of 40 links more, and 30 links broad; and that on the left of his station he has an offset of 10 links, all of which he must note in his field book. Proceeding forward on this line, he finds at 300 he has an offset of 25 on the right, where is a gate, which he has to notice; and, on the left 20, which determines the breadth of the road at that spot. At 400, he will find 10 on the right and 20 on the left to be the breadth; and at 700 (the end of the line) he will find 35 on the right and 15 on the left to be the breadth of the road; where also he will find a small road branching off to the right. Thus the first station line is finished.

To this spot (which is his second station) he brings the theodolite; and after setting it level, he unlocks the under screw, and turns the whole head about, till, through the telescope, he sees the back picket or station staff to be cut by the cross hairs. Here, again, locking the head of his theodolite firm by the under screw, he must unscrew the limb, and turn it about, till through the telescope, he has a view of the picket at \(O 3\); the bearing of which he will find to be \(253^\circ 10'\) from the north to the eastward, which he will enter in his field book. Measuring on from \(O 2\), towards \(O 3\), he will find at 130 links, that he is come to a turnpike, where the breadths at the right and left are 30 and 15. At 200, he has an offset of 15 on the left, and a break off at the right of another road, at 25 from his line, with two other offsets, as expressed in the field book. It Having thus taken the circuit of this estate, the measurer must proceed to plot the same on paper, with some convenient scale.

The scale usually employed for this purpose is that called the plotting scale, plane scale, or scale of equal parts, represented at fig. 11. and 12.

This instrument contains different scales or divided lines, on both sides. There are on one side a number of plane scales, or scales of equal divisions, each of a different number to the inch, and also scales of chords and 12. for laying down angles, and sometimes the degrees of a circle marked on one edge, answering to a centre marked on the opposite edge, by which means it also answers the purpose of a protractor. There are several diagonal scales on the other side, of different sizes, or different dimensions to the inch, serving to take off lines expressed by numbers to three dimensions, as units, tens, hundreds, as also a scale of divisions which are the 100th parts of a foot. The most useful of all the lines which can be laid down on this instrument, though not always done, is a plane scale on the two opposite edges, made thin for the purpose. This line is very useful in surveying; for by laying down the instrument on paper, with its divided edge along a line whereon several distances are to be laid off, for the places of offsets, &c.; these distances are all transferred at once from the instrument to the line on the paper, by making small points or marks against the respective divisions on the edge of the scale.

The business of plotting or laying down a plan of an estate from the memoranda of a field book, is a very important branch of the surveyor's office. This will best be understood by an example, which we shall take also from Mr Crocker. It is adapted to the page of his field book, already alluded to; and the plan, when completed, is seen at fig. 10.

The vellum or paper on which the plan is to be drawn, being smoothly laid on a drawing board, the magnetic meridian is to be represented by a line drawn from the bottom to the top.

A point is to be made about the middle of this line, on which is to be laid the centre of the circular protractor, placing the straight edge in such a manner as to coincide with the said meridian line: draw a pencil line around the edge of the protractor.

The protractor being thus placed, and firmly fixed by means of pins in that position, or by a lead weight, the field book is to be inspected for the quantity of the angle at O1, which, in the present case is stated at 69° north-easterly. This degree is then to be looked for on the circular edge of the protractor, and a mark made on the paper with a fine plotting-pin, at that number, which is to be marked 1, denoting O1.

The field-book is then to be inspected for the ∠ at O2, which in this case is 253° 10', where a mark is to be made as before.

A similar process is to be followed with all the other angles, till the surveyor comes to the close on O1.

All the angles being thus marked off, the protractor is to be removed.

The place where the beginning of the work should be placed is then to be considered, that the whole may come within the compass of the paper laid down; where a mark is to be made, noting it as O1, the beginning of the plot.

The fore edge of the parallel ruler is then laid from the the central point where the protractor lay, to the mark on the pencilled circle denoting \( \theta_1 \). The fore edge of the parallel ruler is next moved till it touch the point determined on for the beginning of the plot; from which a pencil line in the direction from the north to the eastward, is drawn, about the length of the whole line of this \( O = 765 \).

A feather-edge scale is applied to this pencil or obscure line, the division of it at the beginning, marking off every progressive number where any offsets have been made, as at 300, 400, and 762.

The scale is then turned across the line (by some cross division), and the offsets on each side of the station line are pricked off. At 0, or \( \theta_1 \), the field book shows that on the left hand, at 10 links, is the boundary line of that side, where there is likewise a small road branching off. The offset on the right hand is 30, which, with +40, goes to the extent of a small corner, also 40 links in breadth. At 300 on the left there is an offset of 25, and on the right another of 25, where there is also a gate to be noticed. At 760 there is an offset on the left of 15; and on the right, one of 35, where a small roadway branches off. All these offsets are to be pricked off as the surveyor proceeds. The boundary lines are drawn through these offset points, and in this manner the first station is completed.

The parallel ruler is then laid from the centre to the angular point of \( \theta_2 \); the limb of it is moved till it touches the end of the last station line, from which another obscure line is drawn, from the north-easterly, as noted in the field book.

The edge of the scale is then applied as before, and the numbers 30, 200, and 265 are pricked off. There is a tall gate at 30 links, and a lane of 30 links broad, going off at an acute angle. At 265, the end of this station, the offsets are 30 and 10.

The line from \( \theta_3 \) is then laid off, as before directed, north-easterly, and the numbers 20 and 293 are pricked off. Opposite to 20 is a hedge branching off to the left, and at 293 the offsets are 35 and 5.

The line north-easterly is laid off from \( \theta_4 \), and the numbers on that line are pricked off as they appear in the field book, and the offsets are made as follows. At 120, 15 and 20 are set off; at 410 are 30 and 15, where two hedges branch off nearly in the direction of the side sketches. At 480 the offsets are 25 and 5, where there is a cross hedge on the left. At 750 on the left, is 30+15 with a cross hedge, and on the right 10. At 1050 on the left, is 20 with a cross hedge, and 20 on the right. At 1150 on the right, is 20+20, where stands a house. At 1300 on the left, is 5 with a cross hedge; on the right is 30, with a road branching from it; 1350 completes this line.

At \( \theta_5 \) the work takes another direction, and goes backward towards the west. The ruler is laid from the centre to this station, and an obscure line drawn in the direction mentioned. The distances and offsets are pricked off as in the field book. Here are offsets on one side only, not being in a road way.

At \( \theta_6 \) set off the line south-westerly, pricking off the distances and offsets as in the field-book.

This specimen is sufficient to give a complete idea of the practice of plotting; and more would be only a tedious repetition. It must, however, be observed, that the accuracy and facility of the work greatly depend on the judgment and care exercised in keeping a correct and clear field-book.

When a circuit is plotted off, the measurer must fill up the interior, by separately completing the measure of each field with the chain, so that they may be laid down on the plan in their proper situations and dimensions. The lines taken with the theodolite will here be of great service, as the base lines of a number of interior angles.

The surveyor having thus on paper, a representation of the estate, must draw such measuring lines on it, as will enable him to calculate the content of each field separately. Having made out a fair plot of his work, another line must be drawn for the true meridian, to the eastward of the former, according to the variation of the magnetic needle, where the estate lies. On this true meridian line may be placed any device whatever, as a north point. A title must also be given to the map, a scale drawn of the proportion used in the plotting, and a border to the whole.

Having thus explained the general practice of surveying according to the latest improvements, we shall shew how a surveyor is to proceed in measuring and planning counties and towns.

To survey a County or large Tract of Land.—1. Chuse Metho two, three, or four eminent places for stations, such as the tops of high hills or mountains, towers, or church steeplees, which may be seen from one another, and from which most of the towns, and other places of note, may also be seen. And let them be as far distant from each other as possible. On these places raise beacons, or long poles, with flags of different colours flying at them, so as to be visible from all the other stations.

2. At all the places which are to be set down in the map, plant long poles with flags at them of several colours, to distinguish the places from each other, fixing them on the tops of church steeplees, or the tops of houses, or in the centres of smaller towns.

It is not necessary to have these marks at many places at once, as suppose ten at a time. For when the angles have been taken at the two stations, to all these places, the marks may be removed to new ones, and so successively to all the places required. These marks being set up at a convenient number of places, and such as may be seen from both stations, go to one of these stations, and with an instrument for taking angles, standing at that station, take all the angles between the other station, and each of these marks, observing which is blue, which red, &c. and on which hand they lie; and set all down with their colours. Next go to the other station, and take all the angles between the first station, and each of the former marks, and set them down with the rest, each against those corresponding with the same colour. If practicable, the angles may also be taken at some third station, which may serve to prove the work, if the three lines intersect in that point where any mark stands. The marks must be allowed to remain till the observations are finished at both stations, and then they must be taken down, and set up at fresh places. The same operations must be performed at both stations, for these fresh places, and the like for others. The instrument for taking angles must be exceedingly accurate, made on purpose with telescopic sights, sights, and of three, four, or five feet radius. A circumferentor is reckoned a good instrument for this purpose.

3. Though it be not absolutely necessary to measure any distance; because a stationary line being laid down from any scale, all the other lines will be proportional to it; yet it is better to measure some of the lines, to ascertain the distances of places in miles; and to know how many geometrical miles there are in any length; and from thence to make a scale for measuring any distance in miles. In measuring any distance, it will not be exact enough to go along the high roads, on account of their turnings and windings, and scarcely ever lying in a right line between the stations, which would cause endless reductions, and create trouble to make it a right line, for which reason it can never be exact. But a better way is to measure in a right line with a chain, between station and station, over hills and dales, or level fields, and all obstacles. Only in cases of water, woods, towns, rocks, banks, &c. where one cannot pass, such parts of the line must be measured by the method of inaccessible distances; and besides, allowing for ascents and descents, when we meet with them. A good compass that shews the bearing of two stations, will always direct to go straight, when the two stations are not seen; but when a straight progress can be made, offsets may be taken to any remarkable places, likewise noting the intersection of the stationary line, with all roads, rivers, &c.

4. From all the stations, and in the whole progress, care must be taken to observe sea coasts, the mouths of rivers, towns, castles, houses, churches, windmills, watermills, trees, rocks, sands, roads, bridges, fords, ferries, woods, hills, mountains, rills, brooks, parks, beacones, sluices, floodgates, locks, &c. and in general every thing remarkable.

5. When the first and main station lines are done, which command the whole country, inner stations are then to be taken at some places already determined, which will divide the whole into several partitions, and from these stations may be determined the places of as many of the remaining towns as possible. If any remain in that part, more stations may be taken at some places already determined, from which the rest may be determined. Proceeding thus through all parts of the country, station may be taken after station, till all that are required be determined. In general, the station distances must always pass through such remarkable points as have been formerly determined by the preceding stations.

6. The position of the station line measured, or the point of the compass on which it lies, must be determined by astronomical observation. Hang up a thread and plummet in the sun over some part in the station line, observing when the shadow runs along that line, and at that moment take the sun's altitude; then having his declination, and the latitude, the azimuth will be found by spherical trigonometry. The azimuth is the angle which the station line makes with the meridian, and therefore a meridian may easily be drawn through it by hanging up two threads in a line with the pole star, when due north, which may be known from astronomical tables. Or thus: Observe the star Alioth, or that in the rump of the Great Bear, being that next the Surveying square; or else Cassiopeia's hip; observing by a line and plummet when either of these stars and the pole star comes into a perpendicular; and at that time they are due north. Therefore two perpendicular lines being fixed at that moment, towards these two stars, will give the position of the meridian.

A Town or City may be surveyed with any of the Method of instruments for taking angles, but best of all with the surveying plane table, where every minute part is drawn while in sight. It is also proper to have a chain of 50 feet long, divided into 50 links, and an offset-staff of 10 feet long.

Begin at the meeting of two or more of the principal streets through which the longest prospect may be had, to get the longest station lines. Having there fixed the instrument, draw lines of direction along those streets, using two men as marks, or poles set in wooden pedestals, or perhaps some remarkable places in the houses at the farther ends, as windows, doors, corners, &c. Measure these lines with the chain, taking offsets with the staff, at all corners of streets, bendings, or windings, and to all remarkable objects, as churches, markets, halls, colleges, eminent houses, &c. Then remove the instrument to another station along one of these lines, and there repeat the same process as before, and so on till the whole be completed.

Thus, in fig. 13. (part of the New Town of Edinburgh) fix the instrument at A, and draw lines in the direction of all the streets meeting in that place, and measure AB, noting the street on the left at m. At the second station B, draw the directions of the streets meeting there, and measure CD. Do the same at D, and measure DE, noting the place at the cross streets at p. In this manner go through all the principal streets. This being done, proceed to the smaller and intermediate streets; and lastly to the lanes, alleys, courts, yards, and every part which it may be deemed expedient to represent.

We shall conclude this article with a few practical remarks on subterraneous surveying, or the method of surveying mines, and other works below ground, taken chiefly from Mr Fenwick's work on subterraneous surveying, lately published.

The instruments employed in surveying under ground, are the circumferentor, the chain (in coal mines) containing 100 links, and an instrument for taking the angles of elevation or depression, to reduce the measurements to horizontal distances, where the lines are not level. In lead mines, they sometimes employ a cord, divided into 10 feet, instead of a chain.

In conducting a subterraneous survey, the instrument used is placed where the survey is intended to commence, and a person goes forward in the direction of the line to be surveyed, holding a lighted candle in his hand, to the remotest point at which his light can be seen through the sights of the instrument; its bearing is then taken by the circumferentor, and noted down in the survey book. The surveyor then proceeds to take the distance of the light, or object, from the instrument, which is afterwards removed, and a person stands on the spot where it stood, holding one end of the chain in his hand, while another, going towards the object, holds the other end, together with a lighted candle, in the same hand, and being directed by the former, till the hand holding the candle and the chain is in a direct line with the object or light whose bearing was taken. At that place, the first chain is marked. The person who stood where the instrument was placed then comes forward to the mark at the end of the first chain, the other advancing forward another chain, with the chain and candle in the same hand, as before directed: here the second chain is to be marked. Proceeding in this manner till the distance of the object be determined, which being noted down in chains and links in the survey book, opposite to the bearing, the first bearing and distance is completed. Fixing the instrument again where the light as an object stood, or at the termination of the foregoing bearing and distance, and taking the second bearing, by directing the person to go forward as before, so far as his light can be seen, or at any convenient distance, the surveyor is to proceed as before, till the whole is completed.

Such surveys would require five people to be employed, that the work may be expeditiously performed; viz. one to carry forward the survey, and make the requisite observations and remarks; another to carry the instruments employed; another to direct the chain; a fourth person to lead it, and a fifth to go forward with a light, as an object, from one station to another. During the time of making the survey, care must be taken not to admit any iron or steel within four feet of the instrument, for fear of attracting the needle, which has been known to be affected at nearly three times that distance, by a massy piece of iron. If the glass of the instrument should require cleaning, it must be rubbed as gently as possible, and not with any silken substance, by means of which electric matter may be excited, and prevent the needle from traversing. Should such matter be excited, it may be discharged by touching the surface of the glass with a wet finger.

To render this system of surveying familiar to the young miner, it would be necessary for him to put up a number of marks on the surface, taking afterwards their bearing and distance from each other, according to the method before directed; but to make a nearer approach to the form of subterraneous surveying, it would be better to perform it at night, by the assistance of candles; and many evenings might be found favourable for this method of practising. Lanterns may be employed, if the current of air should be too strong for the flame of a candle.

The method of surveying and recording bearings is as follows.

Suppose the bearing of ABC (fig. 14.) is required. Set the circumcenter on A (the north being represented by N, and the south by S); then turning that part of the instrument having the fleur de lis, or other device, from you, or towards B, turn the instrument till the object B is seen through, and cut by the hair in the sights; and the angle NAB being the angle that the sights and line AB make with the magnetic meridian, NS will be the bearing of B from A, suppose 30°; which also being to the right side of the north meridian, will be north 30° east. Then bring the instrument forward to B, fixing it there, and directing the same sight at B towards C, as was directed at A, towards B; then observe the angle that BC makes with the magnetic meridian, which suppose 25° NBC; and being to the left of the meridian, will be north 25° west. To prove the work, and try the accuracy of the instrument when it is standing at B, apply the eye to that sight which was next B when it stood at A; then take the bearing of A from B, which, if found to be the reverse of B from A, shows the work to be so far true. The bearing of B being taken in like manner from C, will prove the truth of the survey. The degrees of each bearing must always be taken by the same end of the needle.

Suppose the bearing of B from A, C from B, and D from C, (fig. 15.) be required. Fix the instrument at A, with the fleur de lis, or other arbitrary device, towards B; then take the bearing of B, as before described, which suppose to make an angle of 30° NAB to the right with the magnetic meridian, or north 30° east; let the instrument be removed to B, and take the bearing of C, which suppose =30° NBC to the left, or north 30° west; then remove the instrument to C, and take the bearing of D, which suppose =65° SCD to the left, or south 65° east. Thus,

From A to B north 30° east. B to C north 30° west. C to D south 65° east.

This survey may be proved in the same manner as the preceding.

Suppose the subterraneous working ABCDA (fig. 16.) to be surveyed, beginning at the pit A. Fix the instrument at the centre of the pit A; then let a person hold a lighted candle at B (being the utmost distance at which it can be seen through the sights of the instrument), the bearing of which being taken from A, suppose due south, or in the direction of the magnetic meridian of A, and its distance from A suppose 6 chains 57 links, which is placed in the survey book as under: Remove the instrument to B, where the candle stood, and direct the person to place the lighted candle at C; then take its bearing from B, which suppose it to make an angle CBS = 80° with the magnetic meridian, or to bear south 80° west, and its distance being found 7 chains 10 links, remove the instrument to C, the candle being removed to D; then take its bearing and distance as before, which suppose north 10° west 5 chains; remove the instrument to D, and direct the candle to be placed at the centre of the pit A, where the survey commenced; then take its bearing from D, north 70° east 8 chains 35 links, and the survey is finished.

| Chains | Links | |--------|-------| | AB south | 6 | | BC south 80° west | 7 | | CD north 10° west | 5 | | DA north 70° east | 8 |

This survey may be proved by adding together the degrees contained in the interior angles, which, if they amount to 360°, the work will be right.

The proof may be made by finding the northing, southing, easting and westing of all the bearings and distances. If the southings are equal to the northings, and the westings equal to the eastings, then will the work be right.

Thus, The southings and northings therefore being equal, as also the eastings and westings, the work is thus proved to be right.

Mr Fenwick gives the following directions for planning subterraneous surveys, and for determining errors that may arise in plotting, through inattention to the magnetic variation.

As the magnetic meridian is always changing, the bearings of the same object, taken by such a meridian at different times, must also vary from each other, except reduced to bearings with the true meridian. Let NS (fig. 17.) represent the meridian of a plan, which is also supposed to be the true meridian; and if a subterraneous excavation is to be plotted on it from the pit A, and this excavation is found to form a bearing of north 10° west 10 chains, by an instrument whose needle had 20° of west variation; now if the excavation north 10° west 10 chains be plotted on the plan by its meridian NS, which is the true meridian, it will be represented by AB; but the bearing being taken by a needle having 20° of west variation, it should form a bearing of north 30° west with the meridian NI, as represented by A b; then A b will be the true direction of the excavation from the pit A, and b B will be the magnitude of the error. Or, instead of reducing the excavation to its bearing with the true meridian NI, it will be equally as true if n s is drawn on the plan, and made to represent the magnetic meridian of the needle by which the bearing was taken, with which AB will form a bearing of north 10° west.

We shall add a few examples illustrative of the error arising from plotting a subterraneous survey on a plan, without attending to the variation of the magnetic meridian, and also how its magnitude can be ascertained.

Example I.—The following is a subterraneous survey, commencing at a pit called the B pit, north 30° west 6 chains, north 70° east 10 chains, north 30° east 5 chains, and north 25° west 8 chains, which was surveyed by an instrument whose needle had 24° of west variation; under what bearings must the survey be plotted on a plan whose delineated meridian has 15° of west variation?

Reduce the bearings, as taken by a meridian having 24° of west variation, to bearings with a meridian having 15° of west variation; thus,

| With a meridian of 23° of west variation. | With the true meridian. | |------------------------------------------|------------------------| | N. 9° W. | N. 32° W. | | N. 30° E. | N. 7° E. | | N. 21° W. | N. 44° W. |

| With a meridian of 15° of west variation. | With the true meridian. | |------------------------------------------|------------------------| | N. 9° W. | N. 14° W. | | N. 30° E. | N. 21° E. | | N. 21° W. | N. 26° W. |

The survey must be plotted under bearings with a magnetic meridian having 15° of west variation, as above, commencing at the B pit.

Example II.—If the following subterraneous survey, north 9° west 8 chains, north 30° east 7 chains, and north 21° west 8 chains be made by an instrument whose needle has 23° of west variation, and plotted on a plan by a meridian having 5° of magnetic variation, without being reduced thereto; what will be the magnitude of the error resulting from such neglect?

Suppose A (fig. 18.) the point of commencement of the survey on the plan, and let the meridian of the plan be represented by N s, having 5° of west variation with the true meridian NS; then the first bearing, north 9° west 8 chains, will be represented by AB; the second, north 30° east 7 chains, by BC; and the third bearing, north 21° west 8 chains, by CD; then ABCD will represent the survey plotted, without attending to the magnetic variation: But as the survey was made by an instrument whose needle had 23° of west variation, therefore each bearing, when truly plotted, must be set off from a meridian of that variation, which, let n s represent; then, north 9° west 8 chains will be represented by A b, north 30° east 7 chains by b c, and north 21° west 8 chains by c d; then A b c d will represent the survey truly plotted, and d D will be the magnitude of the error.

Or the survey may be plotted by reducing the bearings, as taken by a meridian of 23° of west variation, to bearings with a meridian of 5° of variation, as represented by N s, and plotted from it accordingly; which will exactly coincide with A b c d, as before.

To discover, by calculation, the magnitude of the error, reduce the bearings of the survey, as taken by a magnetic meridian having 23° of west variation, to bearings with the true meridian; and also the same bearings, as if taken by a meridian having 5° of west variation, to bearings with the true meridian; then determine the northing and easting of D from d: thus, Therefore, the amount of the error, or the bearing and distance of D from d, will be north, 74° 15' east 6 chains 70 links with the true meridian.