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 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 instrument, 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 È 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 pole is only 2 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 74 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 L is marked on the pole C, and the distance between

SL

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 diference 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 truly horizontal.

which the tube is applied is then

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 bail and socket to a fulcrum, for the purpose of keeping it steady in practice.

But the most accurate levelling instrument is the spirit

level, such as it is now made in London, which is fitted up with a telescope, a mariner's compass, and a set of screws, and other contrivances, by which the instrument can be adjusted and employed in the field with the greatest care as well as correctness.

MENSURATION OF SURFACES,

In the introduction to this tract on geometry (page 350) it was remarked, that all bodies consist of three dimensions, viz. length, breadth, and depth or thickness, but that although all these sorts of measure were absolutely inseparable from the idea of a body, yet each of them might be considered by us as existing independently of the others thus, for instance, if we say that the Thames is deep at Woolwich, we are not considering the breadth of the river; and when we say that it would take 27 yards of carpeting to cover a room, we think only of the length and breadth, but not of the thickness of the carpet. Hence, in treating of the mensuration of surfaces, we are to take into consideration the length and the breadth only of the body to be measured.

Fig. 13 of Plate 4 represents a square, ACDB, the measurement of which is required. Let each side be 5 inches in extent, as divided in the points numbered 1, 2, 3, 4, and 5: through the points in the side AC draw right lines parallel to the sides CD and AB, and upon the points in the side AB erect perpendiculars, which being parallel to the sides AC and BD, will, with the first drawn lines, form a series of squares whose sides will be equal to the

small divisions of the sides of the great square: and as the square of any number or quantity is obtained by multiplying that number or quantity by itself, if we multiply into itself the number of small parts contained in the side of the great square, which is 5, the product 25 will give the number of squares, whose sides are equal to these small parts, contained in the whole surface of the great square ACDB; but the sides being supposed to contain 5 inches, the superficial area of the great square will be 25 square

inches.

From this it is evident, that whatever be the denomi nation of the equal parts by which the sides or other dimensions of a surface are ascertained, the area or superficial content will be composed of squares whose sides are equal to one of such equal parts. If the side AB had been supposed to consist of 5 feet, yards, chains, miles, leagues, &c. the area would have consisted of 25 square feet, yards, &c.

The same reasoning will apply to other plane surfaces besides squares, for in Fig. 14, where the area of a rightangled parallelogram ACDB is required, if the side AC be divided into 4 equal parts, the side AB into 8 of the same parts, and lines be drawn through these divisions respectively parallel to the different sides of the figure, a series of small squares will be formed, equal in number to the product of the number of parts into which one side is divided by those of the other side: thus, by multiplying 8, the number of divisions in the base AB, into 4, the number of parts in AC, the product 32 is the number of small squares continued in the given rectangle ACBD, as may be observed by inspecting the figure.

As this mode of ascertaining the superficial area of any figure, supposes it to be subdivided into a number of small squares, and as squares require to have their sides equal and their angles right, it is evident that the dimensions

must