In the Trigonometrical Survey of England, vol.ii., the allowance is from to of the intermediate arc. This difference of ailowance arises not from inaccuracy of observation, but rather from the inconstancy of the producing causes. The evaporation of rains and dews, and a change of temperature producing variable and partial refractions, the engineer or surveyor should find the exact amount from actual observations. The following method for finding the amount of refraction, is perhaps one of the best that could be employed.

Let C represent the centre of the earth, A and B the true places of two stations above the surface, which in the accompanying figure is represented by SS. AD and BO are horizontal lines at right angles to the radii AC and BC. And let a and b be the apparent places of A and B. Thus the angles DAB and OBA are called the reciprocal depressions,

S

B

and their sum would always be equal to the contained arc, if there was no refraction. But as a and b are the apparent places of the objects A and B, the observed angles of depression will be DAb and OBa; therefore 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 refraction. Hence the following rule, when both objects are reciprocally depressed. Deduct the sum of the two depressions

greater, which, in very delicate operations, should be allowed for, if the distance OP be considerable. The station P would likewise be differently affected by refraction; but in the common operations of levelling, the distance between the instrument and either station is in general so small, that no appreciable error can arise from the inequality of distance, being at most but a few yards. In the practice of levelling, however, it would be advisable to plant the instrument as near the middle point between the stations as you could guess. The assistant could very easily accomplish this, by pacing the distance between the back station and the instrument, and going forward an equal distance.

Hence it appears, that in the practice of levelling for roads, canals, or sections, curvature or refraction is not to be taken into account, as the work is not affected by either; but in extensive trigonometrical operations, when the stations observed might be many miles distant from the place of observation and from each other, due allowance must be made both for curvature and refraction, which, in such cases, amount to very considerable quantities. A correction must likewise be made for the effects of curvature and refraction, when only one observation is taken, in order to ascertain the relative height of a distant station as compared with the place of observation, as they are differently affected.

ON LEVELLING WITH THE THEODOLITE.

Levelling for sections is performed by the theodolite, by means of the angles of elevation and depression of any station, and its distance from the place of observation.

In this example it may be observed, that the mean refraction is of the contained arc, nearly.

The depression at 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 evident 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 diminished by 6.7, leaves 322.3 feet for its height when on Tenterdon steeple, corrected for refraction and curvature.

In the common practice of levelling for roads or canals, neither curvature or refraction is allowed for. Because when the levelling instrument is in the middle point between both stations, they are equally affected both by curvature and refraction.

Let us suppose the stations B and O to be equally distant from the point of observation A, it is manifest they would be equally affected both by curvature and refraction, and consequently would remain equally distant from the

B A

P

centre; but if we suppose the instrument at A, as before, and the stations to be at unequal distances from it, as B and P, then the amount of curvature at P would be

greater, which, in very delicate operations, should be allowed for, if the distance OP be considerable. The station P would likewise be differently affected by refraction; but in the common operations of levelling, the distance between the instrument and either station is in general so small, that no appreciable error can arise from the inequality of distance, being at most but a few yards. In the practice of levelling, however, it would be advisable to plant the instrument as near the middle point between the stations as you could guess. The assistant could very easily accomplish this, by pacing the distance between the back station and the instrument, and going forward an equal distance.

Hence it appears, that in the practice of levelling for roads, canals, or sections, curvature or refraction is not to be taken into account, as the work is not affected by either; but in extensive trigonometrical operations, when the stations observed might be many miles distant from the place of observation and from each other, due allowance must be made both for curvature and refraction, which, in such cases, amount to very considerable quantities. A correction must likewise be made for the effects of curvature and refraction, when only one observation is taken, in order to ascertain the relative height of a distant station as compared with the place of observation, as they are differently affected.

ON LEVELLING WITH THE THEODOLITE.

Levelling for sections is performed by the theodolite, by means of the angles of elevation and depression of any station, and its distance from the place of observation.

After the line to be levelled has been marked out by long pickets, placed at every change of inclination of the surface, you procure a levelling staff, and set the vane or some visible mark to the exact height of the optical axis of the telescope. Now the theodolite being fixed at the extremity of the line, and the assistant sent to the first of these marks in the line, the angle of elevation or depression, which the mark on the levelling staff makes, is read off on the vertical arc of the theodolite, and the distance between the instrument and the staff measured. If the vane or mark on the staff is intended to be permanent, the instrument must be set to one fixed height, by means of a rod marked for the purpose. When the height of the instrument is altered, the vane or mark on the staff must be changed, so as always to be of the same height from the ground as the optical axis of the telescope. It is sometimes the practice, in order to insure greater accuracy, to make reciprocal observations, by making the instrument and staff change places.

It matters not whether the distances are chained before or after the angles are observed, provided marks are left at every change in the surface to be levelled. Both operations may be performed at the same time, by first fixing the instrument at the extremity of the line, and raising or lowering the vane, till it is bisected by the horizontal wire of the telescope, taking care to note the height on the staff at each place.

When the instrument and staff change places, for the purpose of comparing the angles of elevation and depression, if there is any difference a mean is taken, to which the instrument must be set, and clamped. In planting the staff, the assistant is only to fix it on those