LEVEL

LEVEL, an instrument which enables us to find a line or surface exactly level; that is, such as shall be everywhere parallel to the true horizon, or at right angles to the plumb-line or direction of gravity. It also enables us to find the difference between the heights of two or more places on the surface of the earth.

Amongst the great variety of instruments which have been invented for these purposes, and which are all guided by the agency of gravitation, the following are the most important and useful.

Levels in which the plummet or plumb-line forms the most essential part have been constructed in many different forms, and under as many names, to suit the purpose or fancies of the various artificers who employ them; such as masons, bricklayers, paviors, carpenters, and others. But the general principle of the construction is to attach a thread and plummet to the upper part of a board or flat frame of wood or metal, in such a manner that, when the thread of this plummet, hanging freely, coincides with a fiducial line marked on the frame, it is at right angles to the base of the instrument; so that when this occurs, the thing to which the base is applied must obviously be level; but if the thread declines from the mark, the thing is lower on one side than the other. In some of these levels, the fiducial line or mark is at the middle of the base, in others at or near the end. But though it is easy to see, that the longer the base, and the farther down the thread the fiducial line extends, so much the more accurate or sensible is the instrument likely to be; yet little or no regard is generally had to this in practice. The vibrations of the lead, which are troublesome in nice operations, may be in a great measure prevented by having it immersed

Level. in a vessel with water or other liquor. Sometimes the plummet is enclosed in a glass cover to protect it from the agitation of the wind, and sights or even a telescope are occasionally applied. For a telescope may be added to any kind of level, by applying it upon or parallel to the base of the instrument when there is occasion to take the level of remote objects. One of the simplest forms of a plumb-level is shown in fig. 1, Plate CCCXXII.; and the next we shall describe is another.

The Artillery Foot-Level is in the form of a carpenter's square, having its two legs or branches of equal lengths; at their juncture is a little hole, whence hangs a thread and plummet playing on a quadrant, which is divided into twice forty-five degrees from the middle, as shown in fig. 2.

By placing the feet or ends of its two branches on a plane, in such a manner that the thread may play perpendicularly over the middle division of the quadrant, the plane is assuredly level. To use it in gunnery, place the two ends on the piece of artillery, which may be raised to any proposed height, as indicated by the plummet, whose thread will give the degree above the level.

Gunners' Level, for levelling cannons and mortars, consists of a triangular brass plate about four inches high (fig. 3), at the bottom of which is a portion of a circle, divided only into forty-five degrees, as this number is sufficient for the highest elevation of cannons and mortars, and for giving shot the greatest range. On the centre of a segment of a circle turns a piece of brass AB, which may be fixed by a screw at pleasure. Its lower end B serves for a plummet and index, to show the different degrees of elevation of pieces of artillery. This instrument has also a brass foot, not shown in the figure, to set upon cannons or mortars; so that when those pieces are horizontal, the instrument is perpendicular. The foot is to be placed on the piece to be elevated, in such a manner that the point of the plummet may fall on the proper degree; this they call levelling the piece.

The Balance-Level is wholly suspended as a pendulum. It is furnished with sights, which, when the instrument hangs at rest, show the line of level, and sometimes even a telescope is applied to it. Levels of this sort have been made in various forms. They have but little sensibility, and are liable to be sadly disturbed by the wind.

Desagulier's level, described at considerable length in almost every treatise on this subject, is essentially the same with the method of measuring heights by the barometer, to which, as a mode of levelling, we therefore beg to refer.

Reflecting Level, that made by means of a pretty long surface of water representing the same object inverted which we see erect by the eye; so that the point where these two objects appear to meet, is on a level with the place where the surface of the water is found.

There is another reflecting level, consisting of a mirror of steel, or the like, well polished, and placed a little before the object-glass of a telescope, suspended perpendicularly. This mirror must make an angle of 45° with the telescope, in which case the perpendicular line of the telescope is converted into a horizontal line, which is the same with the line of level.

Water-Level, that which shows the horizontal line by means of a surface of water or other liquor, founded on this principle, that a liquid always places itself level.

The most simple water-level is made of a long wooden trough or canal, whose sides are parallel to the base; so that, being equally filled with water, its surface shows the line of level. This is the chorobates of the ancients, described by Vitruvius.

It is also made with two cups fitted to the two ends of a pipe three or four feet long, about an inch in diameter, by means of which the water communicates from the one

cup to the other; and this pipe being moveable on its stand by means of a ball and socket, when the two cups become equally full of water, their two surfaces mark the line of level.

This instrument, instead of cups, may also be made with two short cylinders of glass three or four inches long, fastened to each extreme of the pipe with wax or mastic. Into the pipe is poured some common or coloured water, which shows itself through the cylinders, by means of which the line of level is determined; the height of the water being the same in both cylinders, whether these are themselves perfectly level or not. This level, though very simple, is yet very commodious for levelling at small distances. Various instruments constructed on this principle with water, mercury, and other liquids, have aimed at great perfection; but, as refined levels, we consider them so inferior to the spirit-level, shortly to be noticed, as not to merit a particular description here.

Air-Level, that which shows the line of level by means of a bubble of air enclosed with some liquor in a glass tube which is slightly convex upward, and has its two ends hermetically sealed. When the bubble fixes itself at a certain mark in the middle of the tube, the plane or ruler whereon it is fixed is level. When it is not level, the bubble will rise to one end. This glass tube may be set in another of brass, or in a piece of wood, having an aperture in the middle, through which the bubble of air may be observed. The liquor with which the tube is filled is most commonly alcohol or spirit of wine. This application of a bubble of air is said to be the invention of Dr Hooke; but some ascribe it to Thevenot.

One of these instruments made with sights, is an improvement upon that last described, and, by a little additional apparatus, becomes more commodious and exact. It consists of an air-level AB (fig. 4), about eight inches long, and seven or eight lines in diameter, set in a brass tube 2, with an aperture in the middle C. The tubes are supported by a straight ruler a foot long, at whose end are fixed two sights, 3, 3, exactly perpendicular to the tubes, and of an equal height, having a square hole, subdivided by a cross of thin brass, in the middle of which is drilled a very small hole, through which a point on a level with the instrument is observed. The brass tube is fastened on the ruler by means of two screws, one of which, marked 4, serves to raise or depress the tube at pleasure, for bringing it towards a level. The top of the ball and socket is riveted to a little ruler that springs, one end whereof is fastened with a screw to the great ruler, and at the other end has a screw, 5, serving to raise and depress the instrument when nearly level.

We should consider wires greatly preferable to sights with holes, which, though very small, will take in too great a space to determine the point of level precisely. But the principle of the air-level is brought to an incomparably greater degree of perfection in the instrument next to be described.

Spirit-Level. The most accurate levelling instrument in use, and that possessed of the greatest essential advantages, is the spirit-level, which was first constructed by Mr Sisson, and to which various additions and improvements have been since made, particularly by Mr Ramsden. The following is a description of one of the best of these levels. Fig. 5 represents the instrument mounted on its staves, PQR. The telescope, AB, is made from fifteen inches to two feet in length, as may be required. It is achromatic, of the best kind, and shows the objects erect. By turning the screw $\backslash alpha$ at the side of the telescope, the object-glass is moved; and thus the telescope is exactly adapted to the eye. In the focus of the eyeglasses are exceedingly fine cross wires, the intersection of which should be so perfectly in the axis of the tube,

Level. that by turning it round on its two supporters D, E, and looking through the telescope, the intersection of the wires may constantly cut the same part of the object viewed. They are easily adjusted to this state by means of the four screws bbbb placed on the telescope, near the end for the eye. These screws act in directions perpendicular to one another, so that, by unscrewing one and tightening the opposite one, the wire perpendicular to them may be moved either way at pleasure; and in this manner the other wire may be moved, and thus the intersection of the wires is brought exactly into the axis of the tube.

To the telescope is fixed, by two small screws, the level tube cd, containing the spirits, with a small bubble of air: this bubble, when the instrument is well adjusted, will settle exactly in the same place, in or near the middle of its tube, whether the telescope be reversed or not on the supporters, which in this case are kept unmoved. To render the level truly parallel to the telescope, the screw at e adjusts it vertically, and that at d horizontally. The two supporters D, E, on which the level rests and turns, are shaped like the letter Y. The telescope rests within the upper part of them; and the inner sides of each of these Y's are tangents to the cylindrical tube of the telescope, which is turned to a true cylinder, and each side touches it only at one place. The telescope can be confined in the Y's by means of two jointed arches or rims, ff, which fasten with pins, gg, as shown in the figure.

The lower ends of these supporters are inserted into a strong brass bar F G, so as to stand perpendicularly on it. To one of these is applied a fine threaded screw, H, to adjust the tube, when on its supporters, to a true level; and to the same supporter D is sometimes applied a line of tangents as far as ten degrees, in order to take an angle of depression or elevation to that extent. Between the supporters is also sometimes fixed a compass-box I, divided into 360°, and again into four times 90°, having a centre pin and needle, and trigger to throw off the needle from the centre when not used; so as to constitute a perfect circumferentor, connected with all the foregoing improvements. This plate is fixed on a conical brass ferrule, which is adapted to the axis or bell-metal frustum of a cone at top of the brass head of the staves, having a ball and socket, with three bell-metal joints, two strong brass plates LL, four screws eeee, for adjusting the conical axis of the stand to be truly vertical, a regulating screw M to the motion on this axis, and a fastening screw N to tighten it on the cone when necessary. The fastening screw N, and the regulating screw M, by which the whole instrument is moved steadily round the vertical axis through a small space, were an addition of Mr Ramsden's. It is to be observed that, since the lower of the two plates LL generally partakes of the inclination of the ground, while the upper should always be level, they are almost never parallel, though usually called "the parallel plates."

Manner of adjusting the level at one station. The whole level being now placed steadily on its staves, we must gradually adjust the axis of the stand to be truly vertical, if it is not so already. For this purpose the telescope must be placed in a line with two of the screws e e, and then levelled by these, if necessary, till the bubble of air in the spirit-tube keeps its position in the middle. It is next to be turned into a line with the other two screws e e at right angles to the former, and to be also levelled in that position.

The horizontal motion on the vertical axis being thus far adjusted, the rims ff of the Y's are to be opened, the telescope is to be taken off and laid the contrary way upon these supporters. If the bubble of air then rests exactly the same as before, the level and telescope are already adjusted rightly to one another; but if the bubble does not remain the same, the end to which it goes must be noticed, and its position altered; correcting the

error partly by the screws e, d, and H, and partly by repeating the adjustments above described, upon the screws eeee.

The intersection of the wires being now directed to any distant object, it may be one of the vanes of the staves or poles hereafter described. (See LEVELLING STAVES.) If they continue to be against it precisely, while the telescope is turned round its axis on its Y's, it proves, as before mentioned, that the axis coincides with the intersection of the wires. In this easy mode of adjustment consists the excellence of the instrument; for it is hereby capable of being adjusted by only one station and one object, which will at the same time determine it to be in a true level. If, by change of weather, accident, or other cause, the instrument should have lost its level or adjustment, it may thus be readily restored and re-adjusted at the first station; which is an advantage many levelling instruments do not possess.

Since it is often of great importance to execute levelling with accuracy, and the best instrument when out of its adjustment being of little use, it is quite necessary that every person using such an instrument should have it readily in his power to correct it; and the above, which is Ramsden's construction, is well adapted for this purpose. There are no doubt simpler instruments, as, for instance, some of Troughton's; but none of these simple ones admit of anything like such perfect adjustments as the one just described, though they may suit fully as well in the hands of a person not capable of making the adjustments, and of course not capable of taking a nice level at all.

It is well known, that when a plummet or pendulum vibrates freely in a circular arc, the tendency of gravity to bring it to the resting position is everywhere as the sine of the angular distance from that position. Now, if the bubble in the spirit-level moves in a circle, it is obviously acted on by a force precisely similar to that which gravity exerts upon the plummet; and therefore it would seem that a spirit-level should measure small angles with the same accuracy as a sector whose radius is equal to that of the curvature of the glass tube, or a plumb-line of the same length; but there are some causes which diminish its accuracy. When the bubble of air has been brought to the middle of the glass tube, and when the tube, after being deranged, is restored to the very same position, we cannot be sure that the bubble of air will return to the very middle of the tube. This irregularity is produced by the friction of the included liquid against the sides of the tube, and depends on the magnitude of the bubble and the quantity of liquid. In a good level, where the bubble moves about five lines for a minute of inclination, this uncertainty does not exceed half a line, which may be ascertained by pointing the telescope to any object. The coincidence of a plumb-line with a particular mark will, on account of the insensible oscillation of the thread, leave about double the uncertainty which is left by the index of a sector, and which may be estimated at about a hundredth of a line. A variety of interesting researches on this subject by Mr Nixon will be found in the Philosophical Magazine for April and May 1827, March 1829, and June 1831.

Levels are commonly made of glass tubes in the state they are obtained at the glass-house. Of these, the straightest and most regular are selected and examined, by filling them nearly with spirit of wine, and ascertaining by trial that side at which the bubble moves most regularly, by equal inclinations of the instrument upon a stage called the bubble trier, which is provided with a micrometer screw for that purpose. The most regular side is chosen for the upper part of the instrument, the others being of little consequence to its perfection. Sp-

rit of wine is used, because if pure it does not freeze at natural temperatures, and is more fluid than water. Sometimes, indeed, though very rarely, ether is employed. The tube and the bubble must be of considerable length. The longer the bubble, the more sensible it is to a small inclination. A very small bubble is scarcely sensible, appears as if attached to the glass, and moves but slowly.

In the use of a level of this kind constructed by Langlois, it was remarked, that when it was properly set in the cool of the morning, it was no longer so in the middle of the day, or when the weather became hot; and that when it was again rectified for the middle of the day, it became false in the evening, after the heat had diminished. The bubble was much longer in cold than in hot weather, and when longer it was too much so, and could not be kept in the middle of the tube, but stood a little on the one or the other side, though the inclination was precisely the same. These defects were small, and such as claim the notice of careful observers only; but they appeared of too much consequence not to produce a wish to remedy them. It was observed that they arose from inequalities in the interior surface of the tube; and by examining a great number of tubes, selected for levels of the same kind, there was reason to conclude that all these levels would have more or less of the same defects, because there was not one tube of a regular figure within. They were at best no otherwise cylindrical than plates of glass can be said to be plane before they are ground. The irregularities were easily discernible.

It was therefore concluded that it would be advisable to grind the inner surfaces of the tubes, and give them a regular cylindrical or rather spindle form, of which the two opposite sides should correspond with portions of circles of very long radius. To accomplish this, a rod of iron was taken of twice the length of the glass tube, and on the middle of this rod was fixed a stout tube of brass of the same length as the tube of glass, and nearly filling the bore. The rod was fixed between the centres of a lathe, and the glass gently rubbed on the brass cylinder, with fine emery and water, causing it to move through its whole length. The glass was held by the middle, that it might be equally ground, and was from time to time shifted on its axis, as was also the brass cylinder, in order that the wear might be everywhere alike. The operation had scarcely commenced before the tube broke; and several others experienced the same misfortune, though they had been well annealed. It was supposed that the emery which became fixed in the brass might contribute to split the glass, each grain continuing its impression with the same point, in a straight line, and in some instances might be disposed to cut the glass as diamond. Yet it is curious that some artists find a wooden cylinder to suit pretty well, probably because it does not hold the emery so firmly as brass does. But when a cylinder of glass was substituted instead of the brass, the emery, rolling on the surface of the glass instead of fixing itself, had better success; so that the tube and cylinder touched each other through their whole length. The same operation was continued, using finer and finer emery to smooth the tube, and prepare it for polishing; after which the tube and cylinder having been well washed, thin paper was pasted round the cylinder, and very equally covered with a small quantity of Venice tripoli. The tube was then replaced and rubbed as before, till it had acquired a polish.

A level thus ground may be either of the proper sensibility, or be too much or too little sensible. It will be too sluggish, if, before grinding, exclusive of the inequalities of the tube, its diameter in the middle of the length should much exceed the diameter of the extremities; or it

will be too sensible if this diameter should not sufficiently exceed the other; or, lastly, if the middle diameter be smaller than that of the extremes, the bubble will be incapable of continuing in the middle, but will, in every case, either run to one or the other end, or be divided into two parts.

To correct these defects, and to give the instrument the required degree of perfection, it is proper to examine its figure before the grinding is entirely finished. For this purpose, after cleaning it well, a sufficient quantity of spirit of wine must be put into it, and secured by a cork at each end. The tube must then be placed on the forks or Y's of a bubble trier, and its sensibility, or the magnitude and regularity of the space run over by the bubble, by equal changes of the micrometer screw, must be ascertained. If the run or spaces passed over be too great, they may be rendered smaller by grinding the tube on a short cylinder; but if they be too short, they may, on the contrary, be enlarged, by grinding on a longer cylinder. It is necessary, therefore, to be provided with a number of these cylinders of the same diameter, but of different lengths, which it is advisable to bring to a first figure, by grinding them in a hollow half cylinder of brass. By means of these it will be easy to regulate the tube of the level to any required degree of sensibility, after which the tube may be very quickly smoothed and polished.

A level thus ground was one foot in length, and so was the cylinder on which it was first worked. When finished it was found to be too sensible. It was therefore worked on another cylinder between nine and ten inches long, which diminished its sensibility so far, that the bubble, which was nine inches and four lines long, at the temperature of 68° Fahrenheit, was carried from the middle of the tube exactly one line for every second of inclination. This sensibility was thought sufficient; but if greater is required, it may be obtained by the process here described. For more on this subject, we again beg to refer to Mr Nixon's papers already cited.

It may be remarked, that a glass tube is very subject to be split by grinding its inner surface; the same tube will not be endangered by grinding its external surface, even with coarse emery; and when once the polish of the inside is ground off, the danger is over, and coarser emery may be used with safety. Thick glass is more subject to this misfortune than thinner. The coarsest emery used in grinding the tube here spoken of was sufficiently fine to employ one minute in descending three inches in water.

LEVELLING may be defined the art which enables us to find a line or surface exactly level, as also to find how much higher or lower any given point on the surface of the earth is than another.

The practice of levelling, therefore, consists, 1. in finding and marking two or more points that shall be on a level, as already defined; 2. in comparing the points thus found with other points, to ascertain the difference in their heights or levels, for the purpose of making roads, conducting water, draining low grounds, rendering rivers navigable, forming canals, and the like.

With regard to the theory of levelling, we must observe, that a plumb-line, hanging freely in the air, marks the direction of gravity, and a line drawn at right angles to the direction of the plumb-line, and touching the earth's surface, is a true level only on that particular spot; but if this line which crosses the plumb be continued for any considerable length, it will rise above the earth's surface, and the apparent level will be above the true one, because the earth is globular; and this rising will be as the square of the distance to which the said right line is produced; that is to say, it is raised eight inches very nearly above the earth's surface at one mile's distance; four times as much, or thirty-two inches, at the distance

Levelling, of two miles; nine times, or seventy-two inches, at the distance of three, &c. This is owing to the globular figure of the earth; and this rising is the difference betwixt the true and apparent levels; the curve of the earth being the true level, and the tangent to it the apparent level. Hence it appears, that the less distance we take betwixt any two stations, the truer will be our operations in levelling; and so soon does the difference between the true and apparent levels become perceptible, that it is necessary to make an allowance for it if the distance betwixt the two stations exceeds two chains.

Let BD be a small portion of the earth's circumference, whose centre of curvature is A, and consequently all the points of this arch will be on a level. But a tangent BC meeting the vertical line AD in C, will be the apparent level at the point B; and therefore DC is the difference between the apparent and true level at the point B. The distance CD, therefore, must be deducted from the observed height, to have the true difference of level, or the difference between the distances of two points from the surface of the earth, or from the centre of curvature A. But we shall afterwards see how this correction may be avoided altogether in certain cases.

To find an expression for CD, we have (Euclid, iii. 36) $BC^2\; =\; CD(2AD\; +\; CD)$. But since in all cases of levelling CD is exceedingly small compared with 2 AD, we may safely neglect $CD^2$, and then $BC^2\; =\; 2AD\; \backslash times\; CD$, or $CD\; =\; \backslash frac\{BC^2\}\{2AD\}$. Hence the depression of the true level is equal to the square of the distance divided by twice the radius of the curvature of the earth. If we take the mean radius of the earth as the mean radius of its curvature, and consequently $2AD\; =\; 7912$ miles, then 5280 feet being one mile, we shall have CD the depression in inches

$=\; \backslash frac\{5280\; \backslash times\; 12\; \backslash times\; BC^2\}\{7912\}\; =\; 8.008\; BC^2$. But when BC is in yards, the value of CD in inches becomes .000002585 $BC^2$. From these data was computed the following

A table sufficiently large to embrace every case, by simply taking proportional parts between its numbers, would

occupy several pages; but the formula is of general application. However, when the distance is in miles, eight times its square will be very nearly the depression in inches, or two thirds of the same square the depression in feet. When the distance is in chains, a convenient rule for many ordinary purposes is to divide its square by 800; the quotient is very nearly the depression in inches. In this manner was computed the following table of the curvature of the earth:—

The preceding formulae and tables suppose the visual ray CB to be a straight line; whereas, on account of the unequal densities of the air at different distances from the earth, the rays of light are incurvated by refraction. The effect of this is to lessen the difference between the true and apparent levels, but in such an extremely variable and uncertain manner, that if any constant or fixed allowance is made for it in formulae or tables, it will often lead to a greater error than what it was intended to obviate. For, though the refraction may at a mean compensate for about a seventh of the curvature of the earth, it sometimes exceeds a fifth, and at other times does not amount to a fifteenth. We have therefore made no allowance for refraction in these tables or formulae; but we shall presently see how its effects may frequently be obviated.

Levelling is either simple or compound. The former is when the level points are determined from one station, whether the level be fixed at one of the points or between them. Compound levelling is nothing more than a repetition of many simple operations.

An example of simple levelling is given Plate CCCXXII. Simple fig. 7, where A, B are the station-points of the level; C, D the two points ascertained. Let the height from A to C be six feet, and from B to D nine feet, the difference is three feet which B is lower than A.

Had the station-points of the level been above the line of sight, and the distance from A to C been six feet, and from B to D nine feet, the difference would still have been three feet which B was higher than A.

But when the distance between the stations is consi-

derable, we must provide for the effects of the curvature of the earth and of the refraction of the atmosphere. As to the former, it may be readily ascertained by the tables and formulae given above, provided we know the distance; and we shall presently show how it may be frequently compensated for, or avoided altogether. But the refraction is so extremely variable, that it cannot be disposed of with the like certainty, though there are several ways in which its effects may often be avoided or compensated in a great measure along with those of curvature, as we shall now explain. If the levelling instrument, instead of being placed at the one station and directed to the other, were set either in the middle between or at equal distances from both stations, such instrument, properly adjusted, being directed first to the one station and then to the other, would, by means of the intersection of the wires, mark upon the poles at these stations two points, which would either be exactly on a level, or would differ only in a small degree, owing to some slight inequality in the refractions. For if, on account of the joint effects of curvature and refraction, the point observed on the one pole were, for example, six feet above the level of the instrument itself, the same must obviously hold of the point observed on the other pole, so far at least as the curvature of the earth is concerned; and unless the refraction be different on the different sides of the instrument (which may sometimes happen in a small degree), the points marked on the two poles must be on a level. In this way, it is obvious, that were it not for the figure of the earth being slightly spheroidal, the effect of curvature would be entirely avoided, and that of refraction very nearly.

Sometimes, in place of using two poles and one levelling instrument, two of the latter are employed simultaneously at equal distances from the same pole. In that case the difference in the heights of the points which they mark upon it should be equal to the difference in the heights of the instruments themselves, were it not for the source of slight inaccuracy already stated.

To facilitate the operations just mentioned, there is in the telescopes of some levels a semi-transparent micrometer scale instead of wires. This having its graduated edge crossing the axis at right angles, readily affords the means of approximately estimating equal distances from the stations or from the levelling instrument, without actual measurement. For this purpose an assistant at some distance holds up a pole so as to be parallel to said scale, while the observer, viewing the pole through the telescope, notices how many divisions it subtends on the scale; and, of course, in whatever manner the positions of the assistant or telescope are varied, if the same length of pole held parallel to the scale subtends the same part of it, the distance between them must be the same.

But where great accuracy is required, a somewhat different and greatly preferable mode of avoiding the effects of curvature and refraction is by reciprocal levelling; which consists in two observers simultaneously taking the levels from the opposite ends of the same range, each having a levelling instrument placed close by the station-pole of the other. Thus, supposing the effects of refraction to be alike in both directions, it is evident that the difference in the heights of telescopes placed horizontally at T and t must equal the difference in the heights of the points P and p, to which these telescopes are respectively directed on the opposite station-poles PS and ps; so that p is as much below the level of P as t is lower than T, and therefore $Pt\; -\; pT$ is double the difference in the levels of T and t, or of P

and p. A great recommendation to this method is, that it does not require the distance between T and t to be known. But it readily affords the means of ascertaining the effect of refraction separately when the distance is known, because that is equal to the excess of the effect of the earth's curvature over the difference in the levels of T and P.

When the levelling instruments in the reciprocal method admit of their telescopes being elevated or depressed from a state of parallelism with the spirit-levels, and are likewise provided with the means of measuring such deviation; then, if each telescope be directed exactly to the other, the two angles which the axes of the telescopes make with the respective vertical lines, together with the horizontal angle (or that which these verticals form with each other), must obviously exceed $180^\backslash circ$ by the sum of the refractions in both directions. This, however, is more properly applicable to the mensuration of great differences of level by a trigonometrical process than to ordinary levelling. But neither this nor any other method yet known gives the exact values of the two refractions separately. It only gives their sum, still leaving some little uncertainty as to the separate or proper value of each refraction.

As an example of compound levelling, suppose it were required to know the difference of height between the points A and N, fig. 8. In this operation stakes or pegs should be driven down at A and N, nearly level with the surface; and should be so fixed, that they may not be changed until the whole operation is finished. A plan of the ground between the proposed points A and N should then be made, by which will be discovered the shortest way between them, and whence, too, the number of stations necessary to be taken will be determined. The operator will also be able to distribute them properly according to the nature and situation of the ground. In the figure twelve stations are marked. Stakes or pegs ought to be driven in at the limits of each station, as A, B, C, D, &c. They ought to be two or three inches above the ground, and driven firmly into it; but where rock occurs, it may be sufficient to make a mark on it. Stakes should also be driven in at each station of the instrument, as 1, 2, 3, 4, &c.

The operation may be begun in the following manner. Let the first station be at 1, equally distant from the two points A and B, which themselves are distant 166 yards. Write down then in one column the first limit A, with the number of feet, inches, and tenths, which the point of sight indicated on the station-staff at A, viz. seven feet six inches; in the second column, the second limit B, with the height indicated at the station-staff B, viz. six feet; lastly, in the third column, the distance from one station-staff to the other, which in this case is 166 yards. Remove now the level to the point marked 2, which is in the middle between B and C, the two places where the station-staves are to be held; observing that B, which was the second limit in the former operation, is the first in this. Then write down the observed heights as before; in the first column B with four feet six inches; in the second, C with five feet six inches; in the third, 560 yards, the distance between B and C.

Should it be impossible, on account of the inequality of the ground at the third station, to place the instrument in the middle between the two station-staves, find the most convenient point, as at 3; then measure exactly how far this is from each station-staff, as, for instance, from 3 to C 160 yards; from 3 to D 80 yards; and the remainder of the operation will be as in the preceding station.

In the fourth operation, we must endeavour to compensate for any error which might have happened in the last, from the instrument not being in the middle between, or

Levelling, at equal distances from the stations. Mark out, therefore, 80 yards from the station-staff D to the point 4, and 160 yards from 4 to E; and this must be carefully attended to, as by such compensations the work may be much facilitated. Proceed in the same manner with the eight remaining stations, observing to enter everything in its proper column. If all the ascents are not in one column by themselves, and all the descents in another, regard must be had to the proper signs of these quantities, which is rather more troublesome. And when the whole is finished, sum each column separately, and then subtract the less from the greater; the difference, which in the present case is 5 feet 4 inches $\backslash frac\{1\}\{2\}$ths, shows the ground at N to be thus much lower than at A.

To obtain a section of this level, draw the dotted line O O, fig. 8, either above or below the plan, which may be taken for the level or horizontal line. Let fall then perpendiculars upon this line from all the station-points and places where the station-staves were fixed. Beginning now at A, set off 7 feet 6 inches upon the vertical line from A to a; for the height of the level point determined on the staff at this place, draw a line through a parallel to the dotted line O O, which will cut the third perpendicular at b, the second station-staff. Set off from this point downwards six feet to B, which shows the second limit of the first operation; and that the ground at B is one foot six inches higher than at A: place your instrument between these two lines at the height of the level line, and trace the ground according to its different heights. Now set off, on the second station-staff B, four feet six inches to C, the height determined by the level at the second station; and from C draw a line parallel to O O, which will cut the fifth perpendicular at d, the third station-staff. From this point set off five feet six inches $\backslash frac\{1\}\{2\}$ths downwards to C, which will be our second limit with respect to the preceding one, and third with respect to the first. Then draw your instrument in the middle between B and C, and delineate the ground, with its inequalities. Proceed in the same manner from station to station, till you arrive at the last N, and you will have the profile of the ground over which the level was taken.

This method answers very well where only a general profile of the different stations is required; but where, for some special purpose, it is necessary to have an exact detail of the ground between the limits, we must go to work more particularly. Suppose, therefore, the level to have been taken from A to N by another route, but on more uniform ground, in order to form a canal. Draw at pleasure a line to represent the level, and regulate the rest; then let fall on this line perpendiculars to represent the staves at the limits of each station, taking care that they be fixed accurately at their respective distances from each other. The difference between the extreme limits in this case ought to be the same as in the former, viz. five feet four inches $\backslash frac\{1\}\{2\}$ths. Set off this measure upon the perpendicular of the first limit; and from it, prolonging the perpendicular, mark off the height determined at the first station-staff. Do the same with the second and third, and so on with the rest, till this part of the work is finished; there remains then only to delineate in detail the ground between the station-staves. Sometimes a series of stakes is fixed in the ground, of such various lengths that all their tops are on a level. This, however, can only be done where the difference of level is small; and is used rather with the view of regulating some works to be executed or erected along the line, than for the mere purpose of levelling.

Fig. 9 gives an example of compound levelling, where the situation is so steep and mountainous, that the staves cannot be placed at equal distances from the instrument, or where it is even impossible to make a reciprocal level-

ling from one station to the other. Thus, suppose the level point K to be the bottom of a basin where it is required to make a fountain, the reservoir being at A; so that, in order to know the height to which the jet d'eau will rise, it is necessary to know how much the point A is above K.

In great heights such as this, it will be necessary to proceed by small descents or ascents, as from A to B, or C. The instrument must be adjusted with all possible care; and it may even be proper, in some part of the work, to use a smaller instrument. The following is a table of the different operations used in making this level.

In this case only two levellings are supposed to be made between A and D, though more were necessary; but they are omitted to avoid confusion. In the fourth station the height found was sixteen feet eight inches; but, on account of the great length, it was requisite to reduce the apparent level to the true one. At the last limit we get the height from n to o; then from o to I; from I to K; all which added together, and then corrected for the curvature, give forty-seven feet three inches. Now, by summing each column separately, and subtracting one from the other, we have fifty-one feet three inches for the height which the point A is above the bottom of the basin, and which will cause the jet d'eau to rise forty-five feet. The figure shows only the general section of this operation, but an exact profile of the mountain is more difficult, as requiring many operations; though some of these might be obtained by measuring from the level line without moving the instrument.

When the principal limits of the levelling have been determined and fixed, it only remains to find the level between the limits, according to the methods already pointed out, using every advantage that may contribute to the success of the work, and at the same time avoiding all obstacles and difficulties that may retard or injure the operations. The first rule is always to take the shortest possible route from one limit to another, though this rule ought not to be followed if there are considerable obstacles in the way, as hills, woods, marshy ground, or if, by going aside, any advantage can be obtained. It may sometimes be useful to deviate very considerably from the general rule, in order to take in ponds, the surfaces of which, except during storms, might all be taken as perfectly level; and thus levels are frequently taken across the country for a considerable way.

Further information connected with the practice of levelling will be found under the articles relating to Canals, Inland Navigation, Railroads, Surveying, &c.; as also in the various accounts which have been published of the Trigonometrical Survey, and in other works on similar subjects.