the sum of the two angles, e A B and EBA, will be greater than EAB+EBA (the angle C, or the contained arc) by the angle of elevation, e AD; but if from e AB + EBA, we take the depression OB a, there will remain e AB + a BA, the sum of the two refractions; the rule for the mean refraction then in this case is, subtract the depression from the sum of the contained arc and the elevation, and half the remainder is the mean refraction *.

The refraction thus found must be subtracted from the angle of elevation as a correction, each observation being previously reduced, if necessary, to the axis of the instrument, as in the following example, taken from the Trigonometrical Survey:-At the station on Allington Knoll, known to be 329 feet above low water†, the top of the staff on Tenterden steeple appeared depressed by observation 3′ 51′′, and the top of the staff was 3.1 feet higher than the axis of the instrument when it was at that station. The distance between the stations was 61,777 feet, at which 3·1 feet subtend an angle of 104, which, 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 Tenterden steeple, the ground at Allington Knoll was depressed 3′ 35′′; but the axis of the instrument, when at this station, was 5.5 feet above the ground, which height subtends an angle of 18"4: this, taken from 3′ 35′′, leaves 3′ 16′′-6 for the depression of the axis of the instrument.

* The formula given in the "Synopsis of Practical Philsosophy" is identical with this rule:

Refraction

(A + E)

D

=

·; E being the apparent elevation of any height; D the apparent reciprocal angle of depression; and A the angle subtended at the earth's centre by the distance between the stations.

† A difference of opinion exists as to the zero from which all altitudes should be numbered. What is termed "Trinity datum" is a mark at the average height of high water at spring-tides, fixed by the Trinity Board, a very little above low-water mark at Sheerness. A Trinity high-water mark is also established by the Board at the entrance of the London Docks, the low-water mark being about 18 feet below this. Again, some engineers reckon from low-water spring-tides; and as the rise of tide is much affected by local circumstances, this latter must, in harbour, and up such rivers as the Severn, where the tide rises to an enormous height, be nearer to the general level of the sea. One rule given for obtaining the mean level of the sea, by reckoning from low-water mark, is to allow one-third of the rise of the tide at the place of observation.

At 206,265 feet distant, 1 foot subtends 1"; or at one mile it subtends 39"-06 nearly.

which in this example is nearly of the contained arc.

This, added to the depression at Allington Knoll, 3′ 16′′-6, gives 4' 40" 6 for the angle corrected for refraction; which, being 22′′-4 less than 5′ 3′′, half the contained arc, the place of the axis of the instrument at Allington Knoll is evidently above that at the other station by 6-7 feet, the amount which this angle 22′′4 subtends. This, taken from 329, leaves 322.3 feet for its height when on Tenterden steeple, corrected both for refraction and curvature. The result would have been the same if these corrections had been applied separately, as before described.

Correction for curvature.

D = 61,777 feet = 11·7 miles, log. 1.0681859

By employing the observation from Tenterden steeple, and estimating the refraction at of the curvature, or using the expression D2 for both corrections, the difference of level between these stations would appear about 12 feet greater; which shows how necessary it is, when accuracy is required, to ascertain the re

the

fraction at the time by reciprocal angles of depression or elevation. In another example (page 178, vol. i. "Trigonometrical Survey"), where the depression was observed to the horizon of the sea, dip of the horizon is calculated from the radius of curvature, and the known length of a degree. The difference between this calculated depression and that actually observed is, of course, due to refraction.

To return to the subject of the different methods of taking sections of ground, either

By angles of elevation and depression with the theodolite.

By the spirit, or water-level; or the theodolite used as a spiritlevel.

By the old method of a mason's level and boning-rods, and also by the French reflecting level.

The relative altitude of hills, or their heights above the level of the sea, or other datum, can also be ascertained by a mercurial mountain barometer; the lately-invented Aneroid; or by the temperature at which water is found to boil at the different stations whose altitudes are sought.

Levelling for sections by angles of elevation and depression with the theodolite is thus performed +:-The instrument is set up at one extremity of the line, previously marked out by pickets at every change of the general inclination of the ground; and a levelling-staff, with the vane set to the exact height of the optical axis of the telescope, being sent to the first of these marks, its angle of depression or elevation is taken; by way of insuring accuracy, the instrument and staff are then made to change places, and the vertical arc being clamped to the mean of the two readings, the cross wires are again made to bisect the vane. The distances may either be chained before the angles are observed, marks being left at every irregularity on the surface where the levelling-staff is required to be placed, or both operations may be performed at the same time, the vane on the staff being raised or lowered till it is

The dip of the horizon would be equal to the contained arc, when seen from objects on the spherical surface, if there were no refraction; which is therefore equal to the difference between the depression and the contained arc.

In taking sections across broken irregular ground intersected by ravines, this system of operation is recommended, as being much more easy and rapid than tracing a series of short horizontal datum lines with the spirit level. Where, however, this latter instrument can be used with tolerable facility, it should always be preferred.

bisected by the wires of the telescope, and the height on the staff noted at each place.

The accompanying sketch explains this method:-A and B are the places of the instrument, and of the first station on the line where a mark equal to the height of the instrument is set up; between these points the intermediate positions, a, b, c, d, for putting up the levelling-staff, are determined by the irregularities of the ground. The angle of depression to B is observed, and if great

accuracy is required the mean of this and the reciprocal angle of elevation from B to A is taken, and the vertical arc being clamped to this angle, the telescope is again made to bisect the vane at B. On arriving at B, after reading the height of the vane at a, b, c, &c., and measuring the distances A a, &c., the instrument must be brought forward, and the angle of elevation taken to C; the same process being repeated to obtain the outline of the ground between B and C. In laying the section down upon paper, a horizontal line being drawn, the angles of elevation and depression can be protracted, and the distances laid down on these lines; the respective height of the vane on each staff being then laid off from these points in a vertical direction, will give the points a, b, c, &c., marking the outline of the ground. A more correct way of course is to calculate the difference of level between the stations, which is the sine of the angle of depression or elevation to the hypothenusal distance AB considered as radius, allowing in long distances for curvature and refraction, which may be ascertained sufficiently near by reference to the tables.

The distances, instead of being measured with the chain, may, if only required approximately, be ascertained by means of a micrometer, attached to the eye-piece of the telescope*.

* Dr. Brewster's micrometrical telescope is described in Dr. Pearson's "Practical Astronomy," vol. ii. p. 235.

Mr. Macneil states that he has frequently used a scale of this kind attached to the eyepiece of his level.

Instead of only taking the single angle of depression to the distant station B, and noting the heights of the vane at the intermediate stations, a, b, c, &c., angles may be taken to marks the same height as the instrument set up at each of these intermediate points, which will equally afford data for laying down the section; but the former method is certainly preferable.

The details may be kept in the form of a field-book*; but for this species of levelling, the measured distances and vertical heights can be written without confusion on a diagram, leaving the corrections for refraction and curvature (when necessary) to be applied when the section is plotted.

Where a number of cross sections are required, the theodolite is particularly useful, as so many can be taken without moving the instrument. It is also well adapted for trial sections, where minute accuracy is not looked for, but where economy both of time and money is an object.

The theodolite is likewise used in running check levels to test the general accuracy of those taken in detail with a spirit level. Reciprocal angles of elevation and depression, taken between bench marks + whose distances from each other are known, afford a proof of the general accuracy of the work; and if these points of reference are proved to be correct, it may safely be inferred that the intermediate work is so likewise.

Instead, however, of observing reciprocal angles of elevation and depression between marks at measured distances, levelling for sections, where minute accuracy is required, is performed with a spirit level, or some instrument capable of tracing horizontal lines. The different instruments used for the purpose, and their adjustments, will be first described; and the most approved methods of using them, and keeping the field-book, as well as plotting the detail on paper, will be afterwards explained.

The species of level formerly in general use, termed the Y level,

* Bruff's "Engineer Field Work," page 122.

Marks on stumps of trees, mile or boundary stones, &c., or any convenient permanent object on which the staff is placed to obtain the comparative level of these intermediate points of reference. They are useful either for the subsequent laying out of the detail of work, or for comparison in running check or trial sections. Bench marks should be conspicuously marked and clearly described in the field-book, that no doubt may arise as to their identity.