s an instrument which enables us to find a line parallel to the horizon, or concentric with the circumference of the earth, and to continue it to any distance—to form a surface exactly level, having all its points at equal distances from the earth's centre, or to find the difference of ascent between several places for the purpose of making roads, conducting water, draining low grounds, rendering rivers navigable, forming canals, &c., &c.

Among the great variety of instruments which have been invented for these purposes, the following are the most important and useful.

Air-Level, that which shows the line of level by air-level means of a bubble of air enclosed with some liquor in a glass tube of an indeterminate length and thickness, whose two ends are hermetically sealed. When the bubble fixes itself at a certain mark, made exactly in the middle of the tube, the plane or ruler wherein 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, having an aperture in the middle, through which the bubble of air may be observed. The liquor with which the tube is filled is oil of tartar, or aqua fecunda; these not being liable to freeze as common water, nor to rarefaction and condensation, as spirit of wine is. This application of a bubble of air was the invention of Dr Hooke.

There is one of these instruments made with sights, which is an improvement upon that last described, and which by a little additional apparatus, becomes more commodious and exact. It consists of an air-level, (fig. 1.) 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 ends are fixed two sights, 3, 3, exactly perpendicular to the tubes, and of an equal height, having a square hole, formed by two fillets of brass. brafs crossing each other at right angles, in the middle of which is drilled a very small hole, through which a point on a level with the instrument is observed. The brafs 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.

The instrument just described, however, is still less commodious than the following one; for though the holes be ever so small, they will take in too great a space to determine the point of level precisely.

The instrument alluded to consists of an air-level, with telescopic sights. This level (fig. 2.) is like the last; with this difference, that, instead of plain sights, it carries a telescope to determine exactly a point of level at a great distance. The telescope is a little brafs tube, about 15 inches long, fastened on the same ruler as the level. At the end of the tube of the telescope, marked 1, enters the little tube 1, carrying the eyeglass and a hair placed horizontally in the focus of the object-glass, 2; which little tube may be drawn out, or pushed into the great one, for adjusting the telescope to different sights; at the other end of the telescope is placed the object-glass. The screw 3, is for raising or lowering the little fork, for carrying the hair, and making it agree with the bubble of air when the instrument is level; and the screw 4, is for making the bubble of air, D or E, agree with the telescope; the whole is fitted to a ball and socket. M. Huygens is said to be the first inventor of this level; which has this advantage, that it may be inverted by turning the ruler and telescope half round; and if then the hair cut the same point that it did before, the operation is just.

It may be observed, that one may add a telescope to any kind of level, by applying it upon, or parallel to, the base or ruler, when there is occasion to take the level of remote objects.

Dr. Delaguliers contrived an instrument, by which the difference of level of two places, which could not be taken in less than four or five days with the best telecopic levels, may be taken in as few hours. The instrument is as follows. To the ball C (fig. 3.) is joined a recurve tube BA, with a very fine bore, and a small bubble at top A, whose under part is open. It is evident from the make of this instrument, that if it be inclined in carrying, no injury will be done to the liquor, which will always be right both in the ball and tube when the instrument is let upright. If the air at C be so expanded with heat, as to drive the liquor to the top of the tube, the cavity A will receive the liquor, which will come down again and settle at D, or near it, according to the level of the place where the instrument is, as soon as the air at C returns to the same temperature as to heat and cold. To preserve the same degree of heat, when the different observations are made, the machine is fixed in a tin vessel EF, filled with water up to gh above the ball, and a very sensible thermometer has also its ball under water, that one may observe the liquor at D, in each experiment, when the thermometer stands at the same height as before. The water is poured out when the instrument is carried; which one may do conveniently by means of the wooden frame, which is let upright by the three screws, S, S, S, (fig. 4.) and a line and plummet PP, (fig. 5.) At the back part of the wooden frame, from the piece at top K, hangs the plummet P, over a brafs point at N; MM are brackets to make the upright board KN continue at right angles with the horizontal one at N. Fig. 6. represents a front view of the machine, supposing the fore part of the tin vessel transparent; and here the brafs socket of the recurve-tube, into which the ball is screwed, has two wings at II, fixed to the bottom, that the ball may not break the tube by its endeavour to emerge when the water is poured in as high as gh.

After the doctor had contrived this machine, he considered, that as the tube is of a very small bore, if the liquor should rise into the ball at A (fig. 3.) in carrying the instrument from one place to another, some of it would adhere to the sides or the ball A, and upon its descent in making the experiment, so much might be left behind, that the liquor would not be high enough at D to show the difference of the level; therefore, to prevent that inconvenience, he contrived a blank screw, to shut up the hole at A, as soon as one experiment is made, that, in carrying the machine, the air in A may balance that in C, so that the liquor shall not run up and down the tube, whatever degree of heat and cold may act upon the instrument, in going from one place to another. Now, because one experiment may be made in the morning, the water may be so cold, that when a second experiment is made at noon the water cannot be brought to the same degree of cold it had in the morning; therefore, in making the first experiment, warm water must be mixed with the cold, and when the water has stood some time, before it comes to be as cold as it is likely to be at the warmest part of that day, observe and set down the degree of the thermometer at which the spirit stands, and likewise the degree of the water in the barometer at D; then screw on the cape at A, pour out the water, and carry the instrument to the place whose level you would know; then pour in your water, and when the thermometer is come to the same degree as before, open the screw at top, and observe the liquor in the barometer.

The doctor's scale for the barometer is ten inches long, and divided into tenths; so that such an instrument will serve for any heights not exceeding ten feet, each tenth of an inch answering to a foot in height.

The doctor made no allowance for the decrease of density in the air, because he did not propose this machine for measuring mountains (though, with a proper allowance for the decreasing density of the air, it will do very well), but for heights that want to be known in gardens, plantations, and the conveyance of water, where an experiment that answers two or three feet in a distance of 20 miles, will render this a very useful instrument.

Artillery Foot-Level is in form of a square, having its two legs or branches of an equal length; at a juncture, whereof is a little hole, whence hangs a thread and and plummet playing on a perpendicular line in the middle of a quadrant. It is divided into twice 45 degrees from the middle. Fig. 7.

This instrument may be used on other occasions, by placing the ends of its two branches on a plane; for when the thread plays perpendicularly over the middle division of the quadrant, that plane is assuredly level. To use it in gunnery, place the two ends on the piece of artillery, which you may raise to any proposed height, by means of the plummet, whose thread will give the degree above the level.

Carpenters and Paviors Level consists of a long ruler, in the middle of which, is fitted, at right angles, another somewhat larger. At the top of this is fastened a thread, which, when it hangs over a fiducial line at right angles with the base, shows that this base is horizontal. Sometimes this level is composed of one board. See fig. 8.

Gunners Level, for levelling cannons and mortars, consists of a triangular brass plate, about four inches high, (fig. 9.) at the bottom of which is a portion of a circle, divided only into 45 degrees; as this number is sufficient for the highest elevation of cannons and mortars, and for giving that the greatest range. On the centre of this segment of a circle is screwed a piece of brass, by means of which it may be fixed or screwed at pleasure. The end of this piece of brass is made so as to serve for a plummet and index, in order to show the different degrees of elevation of pieces of artillery. This instrument has also a brass foot, to set upon cannons or mortars, so that when those pieces are horizontal, the instrument will be perpendicular. The foot of this level is to be placed on the piece to be elevated, in such a manner, as that the point of the plummet may fall on the proper degree; this is what they call levelling the piece.

Masons Level, is composed of three rules, so joined as to form an isosceles triangle somewhat like a Roman A. At the vertex of this triangle is fastened a thread, from which hangs a plummet, that passes over a fiducial line, marked in the middle of the base, when the thing to which the level is applied is horizontal; but declines from the mark, when the thing is lower on the one side than on the other.

Plumb or Pendulum Level, that which shows the horizontal lines by means of another line perpendicular to that described by a plummet or pendulum. This instrument, (fig. 10.) consists of two legs or branches, joined together at right angles. The branch which carries the thread and plummet is about a foot and a half long; and the thread is hung towards the top of the branch, at the point 2. The middle of the branch where the thread passes is hollow, so that it may hang free everywhere: but towards the bottom, where there is a little blade of silver, on which is drawn a line perpendicular to the telescope, the said cavity is covered by two pieces of brass, making as it were a kind of cage, lest the wind should agitate the thread. For this reason the silver blade is covered with a glass G, in order, that it may be seen when the thread and plummet play upon the perpendicular. The telescope is fastened to the other branch of the instrument, and is about two feet long; having an hair placed horizontally across the focus of the object-glass, which determines the point of the level. The telescope must be fitted at right angles to the perpendicular. It has a ball and socket, by which it is fixed to the foot, and was invented by M. Picard.

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

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. This is the invention of M. Cassini.

Water Level, that which shows the horizontal line Water level by means of a surface of water or other liquor; founded, or chorobates of the ancients.

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. See Chorobata.

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 to the other cup; 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 malic. 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, with respect to the centre of the earth, being always the same in both cylinders. This level, though very simple, is yet very commodious for levelling at small distances.

De la Hire's level consists of two vessels filled with water, and communicating with each other by means of one or more tubes. A small cylindrical box made of thin copper or planished tin, and terminating below in an obtuse cone, floats in each of these boxes, which are kept in a vertical position by introducing into the cones a ball of lead or a quantity of mercury. One of the boxes carries the object-glass; and the eye-glass along with the cross wires are fastened into the other, but in such a manner as to be elevated or depressed by sliding in two grooves, in order that the axes of the lenses may be exactly level, which is effected by measuring a base. See Traité du Nivellement par M. Picard. The inconveniences attending this instrument arise from the difficulty of bringing the floating eye-glass into the same line with the axis of the object-glass, and of making the boxes settle in such a position that distinct vision may be procured through the telescope; for if the wires in the focus of the eye-glass be out of the axis, or at the smallest distance from the focus of the object-glass, the image will be both indistinct and deformed. In order that De la Hire's level may may be perfect; it is necessary that the boxes should be of the same weight and magnitude, that the boxes which contain the water should be put nearly on a level by means of a plummet, that the same quantity of water should be introduced, and that the object-glass should be kept at the same height with the eye-glass. These conditions, which are requisite to the perfection of the level, are too numerous and too difficult to be attained, to render this instrument of any use where accurate results are required.

These defects in De la Hire's level were partly remedied by M. Couplet, by inserting the object-glass and eye-glass into the same tube, and by placing this telescope loosely on two boxes which formerly floated at random on the fluid. He equalized the weight of these boxes by means of a quantity of small shot, and verified the instrument by putting one of the boxes beneath the object-glass, and the other beneath the eye-glass of the telescope. It is evident, however, that the accuracy of Couplet's level depends upon the equal distribution of the small shot contained in the boxes; for if it is distributed unequally, the box will be more depressed on one side than another, and consequently the intersection of the cross wires in the focus of the eye-glass will either recede from, or approach to the surface of the water, according as the small shot is unequally distributed in the box which supports the eye-glass, or in that which carries the object-glass. Besides this source of error, considerable inconvenience might arise in practice from the want of connexion between the telescope and the two boxes upon which it floats.

The level of Deparcieux is properly an improvement upon that of Couplet. It consists of two parts, a box ABCD of light wood, in which are placed two vessels of tin EFG, EFG filled with water. These vessels are each 10 inches long, 7 inches wide, and 4½ deep, and communicate by one or more tubes GE. The other part is composed of three tubes M, M, M, and of two boxes L, L, encloped on all sides, having 8½ inches of length, 6 of breadth, and 4 of depth, and above these are soldered the three tubes. (Fig. 1. is a vertical section, and fig. 2. a horizontal section of the instrument). The two outermost tubes are telescopes from 18 to 36 inches long, pointed in opposite directions to prevent the necessity of turning the level, and are necessary for its adjustment and verification.—A piece of lead weighing about two pounds is soldered to the bottom of each box L, L, and a weight P of half a pound is made to move towards Q or R by the screw RQ, in order to adjust the level by making one of the floating boxes sink deeper in the water than the other. This weight should be fixed to a small tin tube which can move easily within the greater one, and the screw is turned by means of a handle similar to that which is used for winding up a clock. The whole instrument is thus covered with a case ab to prevent the wind from agitating the water.

In order to adjust the level, place the box ABCD adjusting it upon a table, and elevate one end or another by means of wedges till the intersection of the two cross wires in the focus of the eye-glass of one of the telescopes seems to fall upon a very remote object, each of these wires being moveable by screws so that their point of intersection can be varied. Then take the level out of the box ABCD, and invert its position, so that one of the tin boxes EF may occupy the position which the other had before, and look through the other telescope. If the intersection of the wires falls upon the same object, their position is correct, and the axes of the telescopes are parallel; but if it falls at a distance from the object, the point of intersection must be shifted one-half of that distance towards the object, and the same operation repeated till the intersection of the hairs of one of the telescopes covers the same point of the object that is hid by the intersection of the hairs of the other telescope. When this happens, the axes of the telescopes will be exactly parallel.

The level is then placed upon its stand, which is fixed to the box at K, and a very remote object is examined with one of the telescopes, so as to find the point of it which is hid by the intersection of the wires. The level is then inverted, and the object examined with the other telescope. If the intersection of the wires covers the same point of the object as before, the level is adjusted, and the object is in the line of apparent level passing through the intersection of the wires. But if this is not the case, the weight P towards Q or towards R, according as the point of the object first examined is above or below the intersection of the wires, in order to make the image of the object rise or fall one-half of the distance between the points that are covered by the intersection of the wires in each observation. The operation is then repeated, till the intersection of the wires in both telescopes falls upon the same point of the object, in which case the axes of the telescopes will be exactly level, and the instrument properly adjusted. It is obvious that by moving the weight P from the position which it has when the level is adjusted, the axes of the telescopes will be inclined to the line of the level either above or below it according as the weight is moved to one side or another. Hence, by measuring a base with a vertical object at its remote extremity, it may be easily found how many minutes or seconds correspond with a given variation in the position of the weight, merely by measuring the tangents on the vertical object; so that a scale may be engraved on the tube TT which will exhibit the angles of inclination to the line of apparent level, formed by the axes of the telescopes when the weight P has different positions.

The mercurial level lately invented by the ingenious Keith's Alexander Keith Esq. of Ravelton, is founded on the same principle as the levels of De la Hire, Couplet, and Deparcieux, with this difference, that mercury is employed instead of water. A section of the mercurial level is represented in fig. 3, where A, A are two oblong square cavities communicating by means of the channel MN. BB are two grooves hollowed out of the wood which contain the sights D, D', fig. 4, when the instrument is not in use. The sight D has a small hole in it, and the other is furnished with a cross hair. They are fixed into two pieces of ivory or hard wood, which are nearly of the same form as the cavities A, A, but a little smaller, so that they may go into the cavities without touching the sides. A quantity of mercury is then introduced into the communicating vessels A, A till they are about half full. The two sights are then placed in the cavities, and float on the horizontal surface of the mercury; consequently (Hydrodynamics, art. 34, 37) if the sights be of the same dimension and weight, a line joining the cross hair in D' and the small hole in D will be level or parallel with the horizontal surface of the mercury. The instrument completely fitted up is represented in fig. 5, where D, D' are the sights, D being the sight to which the eye is applied. When there is a strong wind the level is covered with a cage, in which two holes are left opposite to the sights.—The preceding level might be improved by making the cross hair move up and down with a screw, and by engraving a scale on the side of the square aperture at D', whose divisions being subdivided by a scale on the circumference of the nut that moves the screw, would indicate to great accuracy the angle of inclination.

The following mode of constructing a level upon a new principle has occurred to the writer of this article. Let AB be a reflecting surface either of glass or water, and let MN be a straight ruler held above this surface; thus it follows from optical principles that the line MN will be perpendicular to the plane AB when the object MN and its image NM' appear in the same straight line to an eye placed at M. Hence, by the bye, we may ascertain the error of a square, by placing one of its sides upon the surface of a looking glass, and applying the eye to its extremity M; for if it is inaccurate, the image of the side MN will form an angle with MN, thus if mN be the side of the square, its image will be N'm'.

Now let VV be a vessel containing either water or mercury, and let VV be the surface of the fluid. This vessel must be firmly connected with the base CD and also with the vertical plane EF (perpendicular to CD) by means of the cross bars a, b, c, d. The telescope AB is fastened to MN, another plane which rises perpendicular to the plane EF, and the plane MN is connected with EF by means of screws, that its side MN may be made to vary its angle with the horizon, in any direction. The vessel VV, therefore, and the planes EF, CD remain fixed, while the telescope AB and the plane MN can vary their position relative to the other parts of the level. The telescope AB should be so constructed as to answer the purpose of two telescopes. It has an object-glass both at A and B, and also an eyeglass with cross wires at A and B; and these are so fitted into the tube that when the eye is applied to the end B, the object glass at B, and the eye-glass at A with its cross hairs, may be turned to one side so as to have distinct vision with the remaining eye-glass at B and the object-glass at A. When the eye is applied to A, the eye-glass at B and the object-glass at A are moved out of the axis of the telescope for the same reason. This contrivance is for the purpose of avoiding the necessity of having two telescopes. The cross hair in the focus of each eye-glass must be made capable of varying their position, so that the point of intersection may be shifted for the purposes of adjustment.

In order to adjust the instrument, place its base CD, upon a table, and move the telescope of the index MN till the image N'M' is in the same straight line with MN. Then look through the extremity B at a distant object, and mark the point of it which is covered by the intersection of the wires. Insert the whole instrument so that the end A may be at B, adjust the index MN as before, and look through the telescope at the same object. If the intersection of the wires falls upon the same point of the object as formerly, the instrument is properly adjusted. But if not, the intersection of the cross wires in one of the eye-pieces must be varied, as in the adjustment of Deparcieux's level, till it covers the same point of the object that was covered at the first observation. When this happens, the instrument is duly adjusted, and may be used by placing the base CD upon a stand, and adjusting the index MN; for when this is done, the axis of the telescope will be in a line accurately horizontal.

Level of Mr Huygens's invention, consists of a telescope o, (fig. 11,) in form of a cylinder, going through a ferril, in which it is fastened by the middle. This ferril has two flat branches b, b, one above, and the other below: at the ends whereof are fastened little moving pieces, which carry two rings, by one of which the telescope is suspended to a hook at the end of the ferril 3, and by the other a pretty heavy weight is suspended, in order to keep the telescope in æquilibrio. This weight hangs in the box 5, which is almost filled with linseed oil, oil of walnuts, or other matter that will not easily coagulate, for more aptly settling the balance of the weight and telescope. The instrument carries two telescopes close and parallel to each other; the eye-glasses of the one being against the object-glass of the other, that one may see each way without turning the level. In the focus of the object-glass of each telescope must a little hair be strained horizontally, to be raised and lowered as occasion requires by a little screw. If the tube of the telescope be not found level when suspended, a ferril or ring, 4, is put on it, and is to be slid along till it fixes to a level. The hook on which the instrument is hung is fixed to a flat wooden cross; at the ends of each arm whereof there is a hook serving to keep the telescope from too much agitation in using or carriage. To the said flat cross is applied another hollow one, that serves as a cage for the instrument; but the two ends are left open, that the telescope may be secured from the weather and always in a condition to be used. The foot of this instrument is a round brass plate, to which are fastened three brass ferrils, moveable by means of joints whereon are put stays, and on this foot is placed the box.

Fig. 12, marked I, is a balance-level; which being suspended by the ring, the two sights, when in æquilibrio, will be horizontal, or in a level.

Spirit-Level. The most accurate levelling instrument, and that possessed of the greatest essential advantages in use, is the spirit-level; which was first constructed by Mr Siffon, and to which some small additions and improvements have been since made. The following is a description of one of the best of these levels, as made by the principal mathematical instrument makers.

Fig. 13, is a representation of the instrument mounted on its complete stand. The telescope, ABC, is made graphical from 15 inches to two feet in length, as may be required. It is achromatic, of the best kind, and shows the objects erect. In the focus of the eye-glasses are exceedingly fine cross wires, the intersection of which is evidently shown to be perfectly in the axis of the tube; for by turning it round on its two supports DE, and looking through the telescope, the intersection of the wires will constantly cut the same part of the object viewed. By turning the screw a at the side of the telescope, the object glass at g is moved; and thus the telescope is exactly ly adapted to the eye. If these cross wires are at any time out of their adjustment, which is discovered by their interference not cutting the same part of the object during the revolution of the telescope on its axis, they are easily adjusted by means of the four screws \( b \) placed on the telescope about an inch from the end for the eye. These screws act in perpendicular directions to one another, by uncrewing one and tightening the other opposite to the wire, so that if connected with it, it may be moved either way at pleasure; and in this manner the other wire perpendicular to it may be moved, and thus the interference of the wires brought exactly in the axis of the tube.

To the telescope is fixed, by two small screws \( c \), the level tube containing the spirits, with a small bubble of air: This bubble of air, 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.

It is evident, that the axis of the telescope, or the interference of the wires, as before shown, must in this case be truly level. In this easy mode of adjustment consists the improvement of the instrument; and 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 any other cause, the instrument should have lost its level or adjustment, it may thus be readily restored and readjusted at the first station; which is an advantage possessed by none of the instruments formerly made. The two supporters \( D \), 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 cylindric tube of the telescope, which is turned to a true cylinder, and each touches it only at one place.

The lower ends of these supporters are inserted into a strong brass plate \( F \), so as to stand perpendicularly on it. One of these is kept fast by a tightening screw \( G \), and to the other is applied a fine threaded screw \( H \), to adjust the tube, when on its supporters, to a true level. To the supporter \( D \) is sometimes applied a line of tangents as far as 12 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 degrees, and again into four 90°; having a centre pin and needle, and trigger, at \( d \), 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 \( K \), which is adapted to the bell-metal frustum of a cone at top of the brass head of the slaves, having a ball and socket, with three bell-metal joints, two strong brass parallel plates \( L \), four screws \( e \) for adjusting the horizontal motion, a regulating screw \( M \) to this motion, and a tightening 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 with accuracy through a small space in a horizontal direction, was an addition of Mr Ramsden's.

The manner of adjusting the spirit-level at the first station.—The whole level being now placed steadily on its slaves, it must be rendered parallel to the axis of the telescope before you adjust the horizontal motion. For this purpose the telescope must be placed in a line with two of the screws \( e \), and then levelled by these till the bubble of air in the spirit-tube keeps its position in the middle, while turned about to three points, making nearly right angles at the centre to one another.

The horizontal motion being thus adjusted, the rims \( f \) of the \( Y \)'s are to be opened, the telescope taken off and laid the contrary way upon the supporters. If the bubble of air then rests exactly the same, the level and telescope are adjusted rightly to one another; but if the bubble does not remain the same, the end to which the air bubble goes must be noticed, and the distance of it from the telescope altered; correcting one half the error by the screws \( c \), and the other half by the screws \( e \).

Now the interference of the wires being directed to any distant object, it may be one of the vanes of the slaves hereafter described; if they continue to be against it precisely while the telescope is turned round on its \( Y \)'s, it proves, as before mentioned, that the axis of the telescope coincides with the interference of the wires, and that the instrument will give the true level direction.

The operation of levelling being of a very accurate and important nature, 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 one above described appears to be the best adapted for that purpose of any hitherto contrived.

**Theory of the Spirit Level.**

Let \( ABC \), fig. 3, be a vessel of glass hermetically sealed, its upper surface \( ABC \) being the arch of a circle whose centre is \( O \). This vessel contains a quantity of spirit of wine or alcohol, whose level or surface is \( NEN \). The line \( VOT \) intersecting the arch \( Nn \) in \( B \), and extending to \( T \), which is supposed to be the centre of the earth. Therefore, (Hydrodynamics, art. 36.) the surface \( NE \) is the arch of a circle whose centre is \( T \). \( XYZ \) is a right line fixed with respect to the radius \( B \), and consequently with regard to the vessel \( ABC \). Now let the radius \( OB = r \), \( TR = R \), and the arch \( BB' = m \).

In the present situation of the vessel the vertical line \( VT \) coincides with the radius \( BO \); but if the position of the vessel is altered till \( BO \) takes the situation \( bO \), it will then make with \( VT \) an angle \( OeT \), which we shall suppose \( i'' \), and which may be supposed equal to the angle \( ObT \), as \( BT \) may be considered as parallel to \( bT \). The angle \( XVT \) will now become \( XV'T \), and will vary by a quantity equal to \( ObT \). Then by taking \( NN' \), and \( n'n' \) equal to \( BB' \), the points \( N' \) and \( n' \) will be determined, which in the new position of the vessel become the points in which the superior surface of the fluid meet the arch \( ABC \).

Now, calling the angle \( BTb = \phi \), we have (Euclid, book i. prop. 32.) \( BOb = \phi + i'' \), and \( \phi + i'' : \phi = bT : bO = R : r \), consequently \( r = \frac{R\phi}{\phi + i''} \), and substituting instead of \( i'' \) and \( \phi \) arcs of the same value, having unity for radius, the product \( R\phi \) will be equal to the arc \( Ee \), for which we may take \( Bb \) or \( m \); and since Since (see Tables de Berlin, tom. iii. p. 270.) \(1'' = \frac{0.00004848137}{m}\), we shall have

\[ \frac{m}{r} = \frac{0.00004848137 + \phi}{0.00004848137}, \quad \text{for BO} \]

will be very small compared with ET, and therefore the angle ET may be neglected in relation to the angle OAT.

Let us suppose for the sake of example that Bb or its equals NN', nn', is one-tenth of an inch or 0.0083333 of a foot, thus we shall have the length of the radius BO or \(r = \frac{0.00833333}{0.00004848137} = 1736\) feet nearly; thus a derangement of the vessel ABC which makes the radius BO, or the line XZ, vary a minute of a degree, will make each of the points N, n describe a space of 60 tenths or 6 inches, along the arc ABC, that is the same space which the extremity of a plumb line 1736 feet long, would describe when it moved through one minute of a degree. Hence we are able to render extremely sensible the smallest changes of position in the line XZ. The vessel ABC is nothing more than a spirit level, the line XZ representing the axis of the telescope which is attached to that instrument, as shewn in fig. 13, where CC is the level, and CA the telescope. The glass vessel, which is ground in the inside so as to be a portion of a circle of considerable radius, is almost entirely hid by the cylinder of brass which contains it, excepting a small part which appears in the centre of the cylinder; and the instrument must be so adjusted that when the bubble of air is in the middle of the glass tube, the axis of the telescope, the line XZ, is truly horizontal.

From these remarks, it would seem that a spirit level will measure small angles with the same accuracy as a sector whose radius is equal to BO, fig. 3, the radius of the curvature of the glass tube or of 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 centre of the glass tube, and when the tube, after being deranged, is brought to the very same position, we cannot be sure that the bubble of air will return to the very centre of the tube. This irregularity is produced by the friction of the included fluid against the sides of the tube, and depends on the magnitude of the bubble and the quantity of fluid. 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 an uncertainty of about \(1/5\) of a line, that is, about double the uncertainty which is left by the index of a sector, which may be estimated at about 100th of a line. But the radius of a tube, whose bubble moves five lines for a minute of inclination, will be found by a preceding formula to be about 358 feet; and therefore to know the length of a plumb line which will give the same precision, we have \(r : \frac{1}{2} = 358 : 14.32\) feet, the length required.

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. Spirit of wine is used, because it does not freeze, and is more fluid than water. Ether is better, because still more fluid (a). The tube and the bubble must be of considerable length. The longer the bubble, the more sensible it is to the smallest 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 Steur Langlois, it was remarked, that when it was properly set, in the cool of the morning, it was no longer so 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 irregularities 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 from the glass-house 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 copper (copper) of the same length as the tube of glass, and nearly equal in diameter to the bore. The rod was fixed between the centres of a lathe, and the glass gently rubbed on the copper cylinder, with fine emery and water, causing it to move through its whole length. The glass was held by the middle, in order that it might be equally ground, and was from time to time shifted on its axis, as was also the copper cylinder, in order that the wear might be everywhere alike. The operation had scarcely commenced,

(a) If the ether be not well rectified, it is subject to two great inconveniences in this use. If the tube be very slightly agitated, the ether divides itself into several bubbles, which employ a considerable time before they unite. In the second place, as this ether is decomposed in the course of time, it deposits very small drops of oil, which adhere to the tube, stop the motion of the bubble, and render the level very faulty. The ether is besides more fluid when rectified and freed from a saponaceous matter which causes its bad effects. 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 copper might contribute to split the glass, each grain continuing its impression with the same point, in the same right line, which in some instances might be as well disposed to cut the glass as diamond. A cylinder of glass was substituted instead of the copper, and the emery rolling itself on the surface of the leaf, instead of fixing itself, had better success; so that every part of the circumference of the tube and the 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 the paper was 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 irregularities of the tube, its diameter should much exceed in the middle of the length 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 runs or spaces passed over be too great, they may be rendered smaller by grinding the tube on a shorter 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.

The level which was thus ground is one foot in length; and the cylinder on which it was first worked is of the same length. When it was finished it was found to be too sensible. It was therefore worked on another cylinder of between nine and ten inches long, which diminished its sensibility so far, that the bubble, which is nine inches and four lines long, at the temperature of 16° of Reaumur above freezing, is carried from the middle of the tube exactly one line for every second of a degree of inclination. This degree of sensibility was thought sufficient; but any greater degree which may be required may be obtained by the process here described.

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 Levelling 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 without fear. Thick glass is more subject to this misfortune than thinner. The coarsest emery made use of in grinding the tube here spoken of was sufficiently fine to employ one minute in descending through the height of three inches in water.