of the earth, supposing it to be a perfect globe or sphere, which in this case may readily be granted.

Let an observer at A observe the point B, the summit of the Peak of Tenerif just appearing along the surface of the sea, (the state of the atmosphere which renders this impossible is not here considered). The height of this mountain. he knows to be about 15,400 feet, and that the diameter of the earth, supposing it a perfect sphere, is about 7,944 English miles, or 41,944,320 feet; consequently the radius or semidiameter will be 20,972,160 feet. In the triangle ABC are given the right angle at A, for BA is a tangent to the circle at that point, and consequently perpendicular to the radius or semidiameter AC, (Geom. Prop. 31;) also the side AC=20,972,160 feet; and the hypothenuse CB is likewise known, for CD being a radius of the same circle, must be equal to CA, to which adding the space DB, the height of the mountain = 15,400 feet, the whole CB, will be 20,987,560 feet. Then knowing the two sides of a right-angled triangle, the remaining side AB may be found

729,040 feet, or about 140 English miles, which is the distance at which the summit of the Peak of Tenerif might be perceived just rising from or sinking into the sea, provided the atmosphere would admit of vision along the surface at so great a distance, which is not the case.

On the other hand, by knowing the distance AB, we could calculate the height of the mountain, BD, which is the difference between the hypothenuse BC and the semidiameter DC and by measuring the angle ABD, formed at the summit of the mountain by the line of vision BA with BD or BC, the vertical line, or that indicated by a plummet as tending to the centre of the earth, we might ascertain the quantity of the angle at the centre ACB, or of the arch AD, and, consequently, of the whole circumfer ence of the globe, to which this arch bears a determined proportion.

From

From the inspection of the figure (10) it will be evident, that if two places, on the surface of the earth, are on the same horizontal line, as A and B, these must be at different distances from the centre, as AC and BC: and the method of ascertaining this difference is termed levelling, where in practice the short distances from one station to another, although in fact portions of the spherical surface of the earth, may safely be considered as straight lines, or tangents, to the curvature of the globe: then supposing the horizontal distance AB to be one English mile, or 5280 feet, the line DB which represents the difference between the distances of the points A and B from the centre of the earth will be about ,665 of a foot, or nearly 8 inches, supposing no allowance to be made for the effect produced in the apparent altitude of objects in the horizon by the state of the atmosphere: and if these calculations were continued, it would be found that the difference between the horizontal line AB and the true level AD, might be assumed as sufficiently correct in the proportion of the squares of the distances.

The following rule is near enough the truth, to be used in the practice of levelling. "Multiply the number of chains contained in the distance between the objects, whose difference of level is required, by itself, and this product by 124, a common multiplier in cases of this sort, on account of the curvature of the earth's surface; then divide this last product by 100,000, or cut off five figures from the right hand, when whatever stands on the left of the division will be inches, and the figures cut off will be decimal fractions of an inch.

The following Table of the Curvature of the Earth points out the quantity of depression of the true level below the apparent, calculated for every chain's length, or the 80th part of a mile.

Chains

By applying the 1st Example of Practical Geometry, Fig. 1, Plate 4, the angle of elevation of the top of the tower at A, above the level of the spectator at E, was measured by means of a quadrant, or other proper instrument for taking such angles: but for the practice of levelling, this would require an instrument constructed with the greatest accuracy, because a very small error in the ascertainment of the angle might give a very erroneous result, when the distance between the objects is considerable. On this account another mode of levelling is usually adopted, by means of an instrument represented at Fig. 11, Plate 4. This is a tin pipe bent upwards to a right angle toward each extremity at A and B into these upright parts are secured glass pipes, and the instrument is filled with water until it appears in the glasses, when the imaginary line ABD passing over the surface in both pipes will be a true level or horizontal line, or a tangent to the surface of the earth, respecting the foot of the instrument C. The manner of using this instrument is shown in Fig. 12 of Plate 4, where the bending line ADBGCK represents the surface of

a

a rising ground, and A and K the lowest and highest points between which the difference of level is required. Having prepared a number of long straight poles, let them be placed perpendicularly at the stations AB and C, at such distances asunder as may suit the surface of the ground and the nature of the levelling instrunient, which is to be placed as nearly in the middle between each two poles as can be done, as at D and G. When the instrument is at D, the observer, looking along the horizontal line indicated by the surface of the water in the two glass pipes, directs his assistant to make a mark at E on the pole A, where that line falls on the pole: the distance of this point E above the surface of the ground at A is then accurately measured, and written down. Again, the observer from the other end of the instrument looks along the horizontal line, directing the assistant to mark the spot F where it falls upon the second pole planted at B; the distance between FB being also carefully measured and marked down. Now, supposing the height of the point upon the first pole to be 10 feet above the surface of the ground at A, while that of the point F upon the second pule is only 24 feet, it is evident that the difference between these two quantities, or 74 feet, is the difference of elevation of the point B above the point A, that is, B is elevated 7 feet above the level of A. When this is performed, the instrument is removed from the station at D, and placed at G, when the former operations are repeated to ascertain the point H on the second pole: in measuring its elevation, however, the distance is to be reckoned not from the surface of the ground where the pole is fixed, but from the point F formerly ascertained. Again, the observer, by means of the horizontal line of the instrument, determines the point I on the third pole planted at C. Lastly, removing the instrument to the top of the rising ground at K, the point E is marked on the pole C, and the distance between

L and the point I, formerly determined, will show the elevation of L above the last horizontal line HI.

When as many horizontal lines have been observed, and the distances between them have been measured, as may be requisite for the levelling required, the total of these distances in elevation added together, (deducting the height of the instrument 4 feet above the surface of the ground at the last station K,) will show the whole difference of level between the two extremities of the ground to be levelled.

: Should the ground consist of a succession of heights and hollows, the difference between the levels observed in the hollows, and those on the preceding heights, must not be added to, but subtracted from the amount of the observed elevations.

Besides the instrument here described for levelling, many others are employed, such as the air-level, which points out the horizontal line by means of a bubble of air inclosed with some liquor in a glass tube, of a convenient length, whose ends are so shut up that no air can be admitted or make its escape. When the bubble stands at a mark made exactly in the middle of the length of the tube, the plane or ruler to which the tube is applied is then truly horizontal.

: The liquor commonly employed is oil of tartar, which is not liable to freeze like water, nor to be expanded or condensed like spirit of wine. When the tube is not level, the air bubble being specifically lighter than the liquor, will rise to the highest end. The glass tube is set in one of brass, and at each end are placed sights, for the better observing the horizontal line shown by the air-bubble; and the whole is fitted with a ball and socket to a fulcrum, for the purpose of keeping it steady in practice.

But the most accurate levelling instrument is the spirit