f Mr Huygens's invention, consists of a telescope a, (fig. 11.) in form of a cylinder, going through levels, a ferril, in which it is fastened by the noddle. This ferril has two flat branches 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 foresp 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 telephones 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 wherein 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 equilibrium, will be horizontal, or in a level.

SPIRIT-LEVEL. The most accurate levelling instrument, and that possessed of the greatest essential advantage in use, is the spirit-level; which was first constructed by Mr Sifton, 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 stays. 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 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 \( b \) 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 it 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 \( c \), 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 supports, 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 supports \( D \), 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 supports 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 supports, 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 supports 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 flaves, 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 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 flaves, 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 supports. 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 intersection of the wires being directed to any distant object, it may be one of the vanes of the flaves 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.

**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 \( N \) in \( B \), and extending to \( T \), which is supposed to be the centre of the earth. Therefore, (Hydrodynamics, art. 56,) 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 \( ABCD \). Now let the radius \( ON = r \), \( TN = R \), 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 \( b \), 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 \( BT = \phi \), we have (Euclid, book i. prop. [32]) \( BO = \phi + i'' \), and \( \phi + i'' : \phi = bT : bO = R : r \), and consequently \( r = \frac{R}{\phi + i''} \), 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 \( Ec \), for which we may take \( Bb \) or \( m \); and since Since (see Tables de Berlin, tom. iii. p. 270.) \( r = \frac{m}{0.00004848137} \), we shall have

\[ r = \frac{m}{0.00004848137 + \phi} = \frac{m}{0.00004848137} \]

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

Let us suppose for the sake of example that Bb or its equals NN', n n', is one-tenth of an inch or 0.0083333 feet, 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 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 \( \frac{1}{50} \) of a line, that is, about double the uncertainty which is left by the index of a sector, which may be estimated at about \( \frac{1}{100} \) 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 \( \frac{r}{\pi} : \frac{1}{50} = 358 : 14.32 \) feet, the length required.

(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.

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 Sieur Langlois, it was remarked, that when it was properly set, in the cool of the morning, it was no longer fo 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 fo, 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 (cuvire) 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 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 flaccid, 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 run or spaces passed over be too great, they may be rendered smaller by grinding the tube on a flatter 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 Réaumur 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.