to prove it so, move the eye up and down and if the wires and object appears not to move, all is complete; otherwise you must repeat the adjustment, till the motion of the eye will no longer detect the least movement. It is plain the axis of the two glasses forms a straight line, which is called the line of collimation. In the interior of the telescope are fixted across, fine threads or a spider's web, that their intersections may coincide with the axis, and cross it in the point O, the common focus of the two glasses, where the image of the object is formed. Note. This adjustment of the telescope will answer for the level as well as the theodolite. SECTION VI. PRACTICE OF LEVELLING. Levelling is the art of tracing a line at the surface of the earth, which shall cut the directions of gravity everywhere at right angles: it enables us to find the exact difference of level between any number of places, through which it may be necessary to run a railroad or canal. It is necessary to premise, that two or more stations on the surface of the earth, equally distant from its centre, are on the same true level. Also, that one place is higher than another, when it is farther from the centre of the earth, and a line equally distant from the centre in all its points, is called the line of true level. If the earth were an extend plane, all lines representing the direction of gravity at every point on its surface would be parallell to each other; but, in consequence of the figure of the earth being that of a globe, the direction of gravity invariably tends towards the centre of the earth. Hence, from what has been said, it is evident the true line of level must be a curve, and make a part of the earth's circumference, or at least parallel to it. But the line of sight BDH (fig. 14,) given by the operation of levels, is a tangent at the point of contact B, always rising higher above the true level, the farther the distance is, this being called the apparent line of level. Thus: CD, is the height of the apparent level above the true level, at the distance BC or BD; also, MN is the excess at N, HO at O. It is evident the difference is equal to the excess of the secant of the arch of distance above the radius of the earth. As the points D, M, and H, are higher than the point B, though all situated in the same line of sight, it is plain that a fluid cannot rest at any of those points, but will rush towards the point B, where it will remain. It is also evident, that the curvature of the earth, has the effect of depressing the apparent place of every object on its surface, and that the quantity of depression, has reference to the distance of the objects. Now to calculate the amount due to this curvature of the earth; put X for the excess of AH, above the radius AO, which is the excess of the apparent above the true level. Then R+X' R'+BII', that is R2+2RX+X'=R'+BH', whence 2RX+X2=BH2= X+(2R+X). But now as X is so small, when compared with the radius of the earth, we may substitute 2R, for (2R+X); and in short distances, we substitute the arc BO for BH. Hence 2RX=BO', and X=B02. In words: the difference (X) between the true apparent level, is equal to the square of the distance (BO) divided by the diameter of the earth, (2R,) and consequently is always proportional to the square of the distance. The mean diameter of the earth is 7916 miles, and the excess of the apparent above the true level for one mile BO2; 16 of a mile, or 8. 004 inches, or of a foot nearly. In words: two-thirds of the square of the distance in miles, will be the amount of the correction in feet. 2R. BO 2R The last figure shows that though a great number of places be situated in the same line of sight, no two places of them have the same level. The top of a castle at M, is in the same line of sight as B, but is higher than it in point of level, by the altitude MN. With a knowledge of this fact, connected with the well known tendency of every fluid to descend from a higher to a lower place, when al lowed to act freely, is founded the practice of conveying water from one place to another. Suppose a spring one side of a town, and that said spring seen from the town, appears by a levelling instrument, to be on a level with one of the streets to which it were required to bring the water for the use of the town. Suppose the spring one mile from the town; it is evident from what has been said, the water is 8 inches above the level of the town, and therefore the water may be conveyed to the town through pipes. But the effect of curvature is modified by another cause, arising from optical deception, which is called refraction. No object is seen in its true position, but is always seen in the direction of the ray of light which conveys to our senses the impression of said object. The luminous particles in passing through atmosphere, which is composed of an infinite number of concentric strata, increasing in density towards the earth, are refracted or bent continually towards the perpendicular, and therefore they must describe a curve in their passage, of which the last portion, or rather its tangent, indicates the apparent elevated situation of a remote object. This curve may be considered as nearly an arc of a circle, having for its radius 6 or 7 times the radius of our globe. Hence, to correct the error arising from refraction, we have only to diminish the effect of the earth's curvature, by one-sixth part of itself. It must be observed, that as the refraction varies with the change of the atmosphere, no rule can be given for finding the exact amount of refraction, which will answer in every case. It is, however, usual to estimate the effect of refraction at one-seventh of the curvature. In the Trigonometrical survey of England, the allowance is from to of the intermediate arc. The following method for tion is a very good method. Let C (fig. 15,) represent the centre of the earth, A and B the true places of two stations above the surface, which is represented by SQ. AP and BO, are horizontal lines at right angles to the radii AC and BC. Let A and B, be the apparent places of A and B. Thus the angles PAB and OBA, are called the reciprocal depressions, and the sum would always be equal their contained arc, if there was no refraction. finding the amount of refrac But as A and B are the apparent places of the objects A and B, the angles observed will be the depressions PAB and OBA; consequently, their sum, taken from the contained arc of distance, will leave the angles BAB and ABA, the sum of the two refractions, half of which will give the true refraction. Hence comes the following rule, when both objects are reciprocally depressed. Deduct the sum of the two depressions from the contained arc, and half the remainder is the mean refraction. But if the point B, be elevated to the point X, instead of being depressed, then the rule is, substract the depression from the sum of the contained arc, and the elevation and half the remainder is the mean refraction. The angles should be taken at the same time. The amount of refraction found by these rules must be substracted from the angles of elevations as a correction; each of the observations must be reduced when necessary, to the axis of the instruThe following example taken from the survey of England, will illustrate the whole. ment. At the station on Allington Knoll, which was known to be 329 feet above low water mark, the top of the staff on Tenterdon steeple was depressed by observation 3' 51", and the top of the staff was 3.1 feet higher that the axis of the instrument, when it was at the station. The distance between the stations was 61,777 feet at which 3.1 feet subtend and of 10".4, which being added to 3'.51", gives 4'. 1". 4 for the depression of the axis of the instrument, instead of the top of the staff. On Tenterdon steeple,` the ground at Allington Knoll was depressed 3'.35′′: the axis of the instrument when at this station, was 5.5 feet above the ground, which subtend an angle of 18".4; this deducted from 3'.35", leaves 3.16".6, for the depression of the axis of the instrument. Now, to convert the intercepted distances in feet to degrees, &c., we say, as the number of miles in a degree is 60', so is the number of feet in the contained arc to the corresponding degrees, &c. By this rule, contained arc 61.777 feet, 10'.66" nearly. Sum of depression, 4'.1".4+3'.16".6. { 10'.6" 7.18 2)2.48 1.24 Mean refraction, 1'.24". In this it appears the mean refraction is of the contained arc, nearly. The depression of Allington Knoll was 3'.16".6, which being added to the mean refraction, gives 4'.40.6 for angle corrected for refraction: now 4'.40".6 being less than half the contained arc (5'.3".) by 22".4, it is plain that the place of the axis of the instrument at Allington Knoll is above that at the other station by 6.7 feet, which is the amount that this angle 22′′.4 subtends. Then 329 less by 6.7, gives 322.3 feet for its height when on Tenterdon steeple, corrected for refraction and curvature. If the instrument be placed midway as at B, fig. 14, be |