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Contained arc 61,777 feet =
10'6" nearly. 7 18
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.
Curvature = 91.26
= log. 7.1336617
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
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, the 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 bisected by the wires of the telescope, and the height on the staff noted at each place.
* 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.
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 eye. piece 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.
owes its name to the supports upon which the telescope rests. This instrument, as well as Mr. Troughton's improved level, and the dumpy level introduced by Mr. Gravatt, are described at length in Mr. Simms' “Treatise on Mathematical Instruments." It is decidedly inferior to the two last mentioned, its only claim to notice when compared with them being the greater ease with which its adjustments are made; though this advantage is again partially negatived by the equal facility with which they are deranged.
The first adjustment in the Y level is for the line of collimation ; and the method is the same as that described in page 23 for the theodolite, half the error being corrected by the screws acting upon the diaphragm containing the cross hairs.
The second adjustment (that of the spirit level attached to the telescope) is also similar to that for the theodolite *. After the air-bubble has been brought into the centre by the plate-screws, the telescope is reversed in the supports, and if it has moved to either end of the level, it is brought back to its central position, one half by the screw at one end of the level, and the other half by the plate-screws, there being no vertical motion as in the theodolite. This correction will probably require two or three repetitions.
The third adjustment is for the purpose of bringing the Y supports exactly on the same level when the previous corrections have been made, so that the optical axis of the telescope may always revolve at right angles to the vertical axis of the instrument. This is effected by first levelling the telescope when placed over two opposite screws, and then turning it round so that the eye-piece and the object-glass may change places. If in this reversed position the bubble is no longer in the centre, it must be adjusted, one half being done by turning the milled headed screw A, placed directly below one of the Ys, which is thereby raised or lowered in its socket, and the other half
* Before adjusting the focus of the object-glass, that of the eye-piece should be always attended to, both in the spirit level and theodolite; it should be drawn out till the cross wires are clearly defined, and there is no instrumental parallax; so that on fixing their intersection on some distant object there may be no displacement of the contact on moving the eye sideways to the right or left,