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To find the area of a piece of Ground by intersections only, when all the angles of the field can be seen from any two Stations on the outside of the ground.
PL. 12. fig. 1.
LET ABCDEFG be a field, H and I two
places on the outside of it, from whence an object at every angle of the field may be seen.
Take the bearing and distance between H and I, set that at the head of your field-book, as in the annexed one. Fix your instrument at H, from whence take the bearings of the several angular points A, B, C, D, &c. as they are here represented by the lines HA, HB, HC, HD, &c. Again fix your instrument at I, and take bearings to the same angular points, represented by the lines IA, IB, IC, ID, &c. and let the first bearings be entered in the second column, and the second bearings in the third column, of your field-book; then it is plain that the points of intersection, made from the bearings in the second and third columns of
every line, will be the angular points of the field, or the points A, B, C, D, &c. which points being joined by right lines, will give the plan ABCDEFGHA required.
Bear. 180 Dis. 28C. of the Sta. H and I.
No. Bear. Bear.
The same may be done from any two stations within-side of the land, from whence all the angles of the field can be seen.
This method will be found useful in case the stationary distances from any cause prove inaccessible, or should it be required to be done by one party, when the other in whose possession it is, refuses to admit you to go on the land.
To find the content of a field by calculation, which was taken
In the triangle AIH, the angles AHI, ATH, and the base I being known, the perpendicular Aa, and the segments of the base Ha, AI may be obtained by trigonometry: and in the same manner all the other perpendiculars Bb, Cc, Dd, Ee, Ff, Gg, and the several segments at b, c,d, e, f, and g: if therefore the several perpendiculars be supposed to be drawn into the scheme (which are here omitted to prevent confusion arising from a multiplicity of lines) it is plain that if from bBCD Eeb, there be taken bBAG Feb, the remaindcr will be the map ABCDEFGA.
As before half the sum of Bb, and Cc multiplied by bc, will be the area of the trapezium 6BCc; after the same manner, half the sum of Cc, and Dd, multiplied by cd, will give the area of the trapezium cCDd; and again, half the sum of Dd, and Ee multiplied by de, gives the area of the trapezium dDЕe ; and the sum of these three trapezia will be the area of the figure bBC Deb.
Again, in the same manner, half the sum of Bb and Aa multiplied by ah, will give the area of the trapezium Bola ; and half the sum of at, and gG, by ag, gives the trapezium_anGg ; to these add the trapezia gG Ff, and fFEe, which are found in the like manner, and you will have the figure BAGFEeb, and this taken from bBC Deb, will leave the map ABCDEFGA. 2. E. F.
It will be sufficient to protract this kind of work, and from the map to determine the area as well as in plate 10. fig. 3. to find the areas of the pieces, 3, 4, 5, 6, 3, and 6, 7, 7, 6, from geometrical constructions.
How to determine the station where a fault has been committed in a field book, without the trouble of going round the whole ground a second time.
From every fourth or fifth station, if they be not very long ones, or oftener if they are, let an intersection be taken to any object, as to any particular part of a castle, house, or cock of hay, &c. or if all these be wanting, to a long staff with a white sheet ,
a or napkin set thereon, to render the object more conspicuous, and let this be placed on the summit of the land, and let the respective intersections so
taken be inserted on the left hand side of the fieldbook, opposite to the stations from whence they were respectively taken.
In your protraction as you proceed, let every tersection be laid off from the respective stations from whence they were taken, and let these lines be continued ; if they all converge or meet in one point, we thence conclude all is right, or so far as they do converge; but if we find a line of intersection to diverge or fly off from the rest ; we may be sure that either a mistake has happened between the station the foregoing intersection was taken at, and the station from whence the intersection line diverges, or there must be an error in the intersection; but to be assured in which of these the fault is, protract on to the next intersection, and having set it off, if it converges with the rest, though the foregoing one did not, we may conclude the fault was committed in taking the last intersection but one, and none in any station, and that so far is true as is protracted ; but if this as well as
; the foregoing intersection diverge or fly from the point of concourse or converging point of the rest, the error must have its rise from some station or stations, at or after that, from whence the last converging intersection line was taken: so that by going to that station on the ground, and proceeding on to that where the next, or from whence the following diverging intersection was taken, we can readily and with little trouble set all to rights.
But in most tracts of land, one object cannot be seen from every station, or from perhaps one fourth of them; in this case we are under the necessity to move the pole after we begin to lose sight of it, to some other part of the land, where
it may be seen from as many more stations as possible'; which is easily done by viewing the boundary before it be surveyed: the pole then being fixed in an advantageous place, the first intersection to it is best to be made from the same station from whence the last one was taken, and then as often as may be thought convenient, as before ; in like manner the whole may be done by the removal of
When we here speak of stations, we do not mean such as are usually taken at every particular angle of the field : for it is to be apprehended, that every skilful zurveyor, particularly such who use caiculation, vill take the longest distances possible, not only tolessen the number of stations, for the ease of eithæ protraction or calculation, but with greater certainty to account for the land passed by, on the rght hand or on the left, which is taken by off-ses: and surely it will be allowed that any measure daken on the ground, and the content thence arithnetically computed, will be much more accurate than that which is obtained from any geometrical projection.
Fron what has been said it is plain, that from this method any fault committed in a survey can be readly determined, and therefore must be much preferible to the present method of taking diagonals, c the bearings and lengths of lines across land, o accomplish that end ; which last method is too 'requently used by surveyors to approximate or arrie near the content, which will ever remain uncertin, let these diagonals be ever so many, till the station or stations wherein the error or errors were committed, be found; and the fault or fault be corrected.