Water LEVEL

Water LEVEL, that which shows the horizontal line by means of a surface of water or other liquor; found on this principle that water 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 chorebates 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 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, 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 Traite 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

Level. 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.

13
Couplet's
improvement on De
la Hire's
level.

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 mere 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 must arise in practice from the want of connection between the telescope and the two boxes upon which it floats.

14
Depar-
cieux's le-
vel.

Plate
CCXCIII.
fig. 1.

Fig. 2.

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 4$\backslash frac\{1\}\{2\}$ 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, enclosed on all sides, having 8$\backslash frac\{1\}\{2\}$ 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 a b to prevent the wind from agitating the water.

15
Method of
adjusting it.

In order to adjust the level, place the box ABCD 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 engraven 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.

16
The mercurial level lately invented by the ingenious Keith's
Alexander Keith, Esq. of Ravelston, 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 ob-
long 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 Fig. 4.
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 these cavities
without touching the sides. A quantity of mercury is
then introduced into the communicating vessels A, A till
they be 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

Level. Fig. 5. 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 case, 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.

17 Description of a level upon a new principle. Plate CCXCIV. Fig. 1. kg. 1. 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 Nm'.—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 so 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 eye-glass 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.

18 Method of adjusting it. In order to adjust the instrument, place its base CD, upon a table, and move the telescope of the index MN till the image NM' 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.

19 LEVEL of Mr Huygens's invention, consists of a telescope a, (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 bb, 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 screw 3, and by the other a pretty heavy weight is suspended, in order to keep the telescope in equilibrium. 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-glass 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 case 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 slaves, 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 equilibrium, will be horizontal, or in a level.

20 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 Sisson, 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 slaves. The telescope, ABC, is made 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 supporters 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

Level. ly adapted to the eye. If these cross wires are at any time out of their adjustment, which is discovered by their intersection 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 $b\; b\; b$, placed on the telescope about an inch from the end for the eye. These screws act in perpendicular directions to one another, by unscrewing 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 intersection of the wires brought exactly in the axis of the tube.

To the telescope is fixed, by two small screws $c\; c$, 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 intersection 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 $D\; E$, 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 touches it only at one place.

The lower ends of these supporters are inserted into a strong brass plate $F\; E$, 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 $90^\backslash circ$; 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 staves, having a ball and socket, with three bell-metal joints, two strong brass parallel plates $L\; L$, four screws $e\; e\; e\; e$ for adjusting the horizontal motion, a regulating screw $M$ to this motion, 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 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 staves, 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 $e\; e$, 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 $f\; f$ 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 $e\; e$, and the other half by the screws $e\; e$.

Now the intersection of the wires being directed to any distant object, it may be one of the vanes of the staves 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 intersection 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 described.

Let $A\; B\; C$, fig. 3. be a vessel of glass hermetically sealed, its upper surface $A\; B\; C$ 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 $N\; E\; N$. The line $V\; O\; T$ intersecting the arch $N\; n$ in $B$, and extending to $T$, which is supposed to be the centre of the earth. Therefore, (HYDRODYNAMICS, art. 36.) the surface $N\; E\; n$ is the arch of a circle whose centre is $T$. $X\; Y\; Z$ is a right line fixed with respect to the radius $B$, and consequently with regard to the vessel $A\; B\; C\; D$. Now let the radius $O\; n\; =\; r$, $T\; n\; =\; R$, and the arch $B\; b\; =\; m$.

In the present situation of the vessel the vertical line $V\; T$ coincides with the radius $B\; O$; but if the position of the vessel is altered till $B\; O$ takes the situation $b\; o$, it will then make with $V\; T$ an angle $O\; e\; T$, which we shall suppose $1\text{'}\text{'}$, and which may be supposed equal to the angle $O\; b\; T$, as $B\; T$ may be considered as parallel to $b\; T$. The angle $X\; V\; T$ will now become $X\; V\; T$, and will vary by a quantity equal to $O\; b\; T$. Then by taking $N\; N\text{'}$, and $n\; n\text{'}$ equal to $B\; b$, the points $N\text{'}$ and $n\text{'}$ 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 $A\; B\; C$.

Now, calling the angle $B\; T\; b\; =\; \backslash phi$, we have (Euclid, book i. prop. 32.) $B\; O\; b\text{'}\; =\; \backslash phi\; +\; 1\text{'}\text{'}$, and $\backslash phi\; +\; 1\text{'}\text{'}\; :\; \backslash phi\; =\; b\; T\; :\; b\; O\; =\; R\; :\; r$, and consequently $r\; =\; \backslash frac\{R\; \backslash phi\}\{\backslash phi\; +\; 1\text{'}\text{'}\}$, and substituting instead of $1\text{'}\text{'}$ and $\backslash phi$ arcs of the same value, having unity for radius, the product $R\; \backslash phi$ will be equal to the arc $E\; e$, for which we may take $E\; b$ or $m$; and

Level. since (see Tables de Berlin, tom. iii. p. 270.) $1\text{'}\text{'}\; =\; 0.000004848137$, we shall have

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

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 $r\; =\; \backslash frac\{0.00833333\}\{0.000004848137\}\; =\; 1736$ 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 shown 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 $\backslash frac\{1\}\{30\}$ 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 $\backslash frac\{1\}\{30\}\; :\; \backslash frac\{1\}\{358\}\; =\; 358\; :\; 14.32$ feet, the length required.

On the Construction of Levels.

Levels are commonly made of glass tubes in the flat 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 Sieur 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, 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 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 (cuivre) 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, before

Level. 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 last, 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 flagrant, 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 extremities, 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 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 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.

LEVELLING may be defined, the art which instructs us in finding how much higher or lower any given point on the surface of the earth is than another; or, in other words, the difference in their distance from the centre of the earth.

The practice of levelling therefore consists, 1. In finding and marking two or more points that shall be in the circumference of a circle whose centre is that of the earth. 2. In comparing the points thus found with other points, to ascertain the difference in their distances from the earth's centre.

With regard to the theory of levelling, we must observe that a plumb line, hanging freely in the air, points directly towards the centre of the earth; and a line drawn at right angles, crossing the direction of the plumb line, and touching the earth's surface, is a true level only in 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, however much it is raised above the earth's surface at one mile's distance, it will rise four times as much at the distance of two miles, nine times 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 real 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 betwixt 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 in length.

Let BD, fig. 4. be a small portion of the earth whose centre is A, then (HYDRODYNAMICS, art. 36.) all the points of this arch will be on a level. But a horizontal line 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 always be deducted from the observed heights, before we can have the true differences of level, or the difference between the distances of two points from the surface of the earth, or from the centre of curvature A.

In order to find an expression of DC, we have (Euclid, book i. prop. 47.) $AC^2\; =\; AB^2\; +\; CB^2$, and calling $AB\; =\; R$, $BC\; =\; m$, and $CD\; =\; x$, and considering that $AC\; =\; R\; +\; x$, we have the equation $R^2\; +\; 2Rx\; +\; x^2\; =\; m^2\; +\; R^2$. But as the value of the arc DB is always sufficiently small, that CD may be regarded as sufficiently small when compared with AD or AB, we may safely consider $x^2$ as nothing in the preceding equation, which in that case becomes $x\; =\; \backslash frac\{m^2\}\{2R\}$. The

Levelling. mean value of R may be considered as 19630764 feet, and therefore the value of $x$ may be deduced from the equation $x\; =\; \backslash frac\{m^2\}\{2\; \backslash times\; 19630764\}\; =\; \backslash frac\{m^2\}\{39261528\}$, $m$ being expressed in feet. Hence it is obvious, that the depression of the true level is as the square of the distance; and if this distance be 6000 feet, we shall have $x\; =\; 0.91698$ of a foot = 11 inches.

The preceding formula supposes 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. This effect has been considered in the following table, which contains the difference between the apparent and true level, both when the refraction of the atmosphere is omitted, and taken into account.

Levelling. The following is a simple rule for determining the depression of the true level in the practice of levelling.

"Multiply the number of Gunter's decimal statute chains that are contained in length between any two stations where the levels are to be taken by itself, and the product arising therefrom again by 124, which is a common multiplier for all manner of distances for this purpose on account of the earth's curvature: then divide the second product arising therefrom by 100,000; or, which is also the same, with the dash of the pen cut off five figures on the right hand side of the product, and what remains on the left side is inches, and the five figures cut off decimal parts of an inch."

The following is A Table of Curvature of the Earth, and shows the quantity below the apparent level at the end of every number of chains to 100.

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 CCXCV. fig. 1. where AB are the station points of the level; CD the two points ascertained. Let the height

shows that B is three feet lower than A.

If the station-points of the level are above the line of sight, as in fig. 2. and the distance from A to C be six feet, and from B to D nine feet, the difference will still be three feet which B is higher than A.

As an example of compound levelling, suppose it were required to know the difference of height between the point A on the river Zome, and N on the river Belann, fig. 3. (As our author could find no satisfactory examples in any English author, he copied this and the following ones from M. le Febure). In this

operation stakes should be driven down at A and N, exactly level with the surface of the water; and these stakes should be so fixed, that they may not be changed until the whole operation be finished: a plan of the ground between the two rivers should then be made, by which it will be discovered, that the shortest way between the rivers is by the dotted line AC, CH, HN; from whence also 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, 12 stations are marked. Stakes ought to be driven in at the limits of each station, as A, B, C, D, &c. They ought to be about two or three inches above the ground, and driven 18 inches into 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; in another, the number of feet, inches, and tenths; with the points of sight indicated on the station-staff at A, viz. 7. 6. 0. In the third column, the second limit B; in the fourth, the height indicated at the station-staff B, viz. 6. 0. 0. Lastly, in the fifth 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; in the second 4. 6. 0.; in the third, C; in the fourth 5. 6. 2.; in the fifth 560, the distance between B and C.

It being 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, and you will find that from 3 to C is 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. 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 every thing in its proper column: and when the whole is finished, add the sums of each column together, and then subtract the lesser from the greater; the difference, which in the present case is 5. 4. 8, shows the ground at N to be thus much lower than the ground at A.

To obtain a section of this level, draw the dotted line oo, fig. 4. 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 line from A to a: for the height of the level-point determined on the staff at this place, draw a line through

Levelling. $a$ parallel to the dotted line $oo$, 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 $oo$, which will cut the fifth perpendicular at $d$, the third station-staff. From this point set off 5 feet 6 inches $\backslash frac\{1\}\{2\}$ downwards to $C$, which will be our second limit with respect to the preceding one, and the 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 it is necessary to have an exact detail of the ground between the limits, we must then 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 marked $O,\; P,\; Q,\; R,\; S,\; T,\; U,\; X,\; Y$. Draw at pleasure a line $Z,\; \backslash bar\{Y\}$, fig. 5. to represent the level, and regulate the rest; then let fall on this line perpendiculars to represent the stakes 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. 5 feet 4 inches $\backslash frac\{1\}\{2\}$. Set off this measure upon the perpendicular $o$ the first limit; and from $o$, prolonging the perpendicular, mark off at $a$ the height determined at the first station-staff; then do the same with the second and third, and so on with the following, till this part of the work is finished; there remains then only to delineate in detail the ground between the station-stakes, the distances in this example being assumed larger on account of the detail.

To obtain the section of the ground between $O$ and $P$, place your instrument at one of the limits, as $P$, fixing it so that the cross hairs may answer to the point $C$; then look towards the first limit $o$, raising or depressing the vane till it coincides with the intersection of the cross hairs; and the line of sight from one point to the other will mark the level or horizontal line.

To set off the height of the brink of the river above the first limit, drive a stake down close to the ground at $a$; and place your station-staff upon it, observing where the hairs intersect the vane, which will be at 4 feet 10 inches; then laying off upon the line $oo$ the distance from the first to the last stake, let fall from thence a perpendicular, and set off thereon 4. 10. 0. to $a$, which gives the height at the first stake; or, which is the same, the height from the edge of the river above the surface of the water, as is evident from the section. Drive a second stake at $6$, in a line between the limits; place the station-staff upon this stake, and observe the height 4. 6. intersected by the cross hairs, the instrument still remaining in the same situation. Set off on the level line the distance from the first stake $a$

to the second $b$: and then let fall a perpendicular, and mark upon it 4. 6. to $b$, which gives the height of the ground at this place. Levelling.

The small hollow $c$ is marked out by driving down a third stake even with the ground, in the middle of it at $c$; but the exact distance of the second stake $b$ from the third $c$, must be marked upon the level line: then let fall a perpendicular from $c$, and set off upon it 6. 8. 0, pointed out by the cross hairs on the staff, which determines the depth of the hollow, as appears from the figure. As the distances between the stakes are now very short, they can easily be marked by the operator, who can settle any little inequalities by a comparison with those already ascertained. Proceed thus with the other stations till you arrive at the last, and you will always obtain an accurate section of your work; by which it is easy to form a just estimation of the land to be dug away, in order to form the canal, by adding the depth to be given to it.

Fig. 6. gives an example of compound levelling, where the situation is so steep and mountainous, that the stakes cannot be placed at equal distances from the instrument, or where it is even impossible to make a reciprocal levelling from one station to the other.— Thus suppose the 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 high the point $A$ is above $K$.

In great heights such as this, it will be necessary to proceed by small descents, as from $A$ to $D$. The instrument must be adjusted with all possible care; and it will 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, it having been taken from M. le Febure's practice.

In this case only two levellings are made between $A$ and $D$, though more would have been necessary; but they are omitted to avoid confusion. In the fourth station the height found was 16 feet 8 inches; but on account of the great length, it was requisite to reduce the apparent level to the true one, which is always necessary where the length is considerable. At the last limit we get the height from $N$ to $o$; then from $o$ to $I$; from $I$ to $K$, fig. 7. &c.; all which added together, and then corrected for the curvature, gives 47 feet 3 inches. Now, by adding each column together, and subtracting one from the other, we have 51 feet 9 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 about 45 feet. The general section of this operation is shown at fig. 7. 8. but