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also the only means by which they can be measured with any degree of minuteness by small instruments.
It is frequently necessary to refer to trigonometrical stations long after the angles have been observed; either for the purpose of fixing intermediate points, or of rectifying errors that may have crept into the work. Large marked stones should therefore be always buried under the principal stations which are not otherwise identified by permanent erections, and a clear description of the relative position of these marks with reference to objects in their vicinity should be always recorded. If however any station should be lost, and its site required to be ascertained for ulterior observations, the following method, which has been adopted by Colonel Colby, will be found to answer the purpose with very little trouble and with perfect accuracy.
Let D be the lost station, the position of which is required. Assume T as near as possible to the supposed site of the point in question, (in the figure the distance is much exaggerated, to render the process intelligible), and take the angles ATB, BTC; A, B, and C being corresponding stations which have been previously fixed, and the distances of which from D are known. If the angle ATB
be less than the original angle ADB, the point T is evidently without the circle in the segment of which the stations A and B are situated; if the angle be greater, it is of course within the segment. The same holds good with respect to the angles BTC and BDC.
Recompute the triangle ABD, assuming the angle at D to have been so altered as to have become equal to the angle at T, and that the angle at A is the one affected thereby.
Again, recompute the triangle, supposing the angle at B the one affected. In like manner in the triangle BDC recompute the triangle, supposing the angles at B and C to be alternately affected by the change in BDC. These computations will give the triangles ABE, ABÉ, BCF, BCF calculated with the values of T, as observed at the first trial station (in both the present cases greater than those originally taken at D), and the angles at A, B, and C, alternately increased and diminished in proportion. Produce AT and BT, making Ti and Tr' equal respectively to ED and E'D, the differences between the distances just found and the original distances from the point D; and through the points 11', which fall nearly, though not exactly, in the circumference of the circle passing through ABD, draw the line 0 0 which may be considered as a tangent to the circle. A repetition
' of the same process in the triangle BCD gives the points 22', through which draw the line NN', the intersection of which with 0 0 gives the point T which is approximately the lost station required. Only two
. triangles are shown in the diagram, to prevent confusion, but three at least ought to be employed to verify the intersection at the point T if the original observations afford the means for doing so; and where the three tangent lines do not meet, but form a small triangle, the centre of this is to be considered the second trial station, from whence the real point D is to be found by repeating the process described above, unless the observations taken from it prove the identity of the spot by their agreeing exactly with the original angles taken during the triangulation.
If the observed angle T be less than the original angle, the distances T1T1, T2 and T2 must be set off towards the stations A, B, and C, for the point T'; and these stations should be selected not far removed from D, and forming triangles approaching as near as possible to being equilateral, as the smallest errors in the angles thus become more apparent. If the observations have been made carefully and with due
attention to these points, the first intersection will probably give very near the exact site of the original station, or at all events a third trial will not be necessary.
To save computation on the ground, it is advisable to calculate previously the difference in the number of feet that an alteration of one minute in the angles ABC, &c., would cause respectively in the sides AD, DB, DC, &c. The quantities thus obtained being multiplied by the errors of the angle at T, will give the distances to be laid off from T in the direction AT, BT. And in order also to avoid as much as possible any operations of measurement to obtain the position of the point T, the distances from the trial station T should be laid down on paper on a large scale in the directions TA, TB, &c. (or on their prolongation), to obtain the intersection T of the lines 1 l'and 2 2' and from this diagram the angle formed at T with this point T and the line drawn in the direction of any of the stations A, B, or C, can be taken, as also the distance TT'; the measurement of one angle and one short line is all that is required on the ground.
The triangulation should never be laid down on paper until its accuracy
has been tested by the actual measurement of one or more of the distant sides of the triangles as a base of verification, and by the calculation of others from different triangles to prove the identity of the results.* Beam compasses, of a length proportioned to the distance between the stations, and the scale upon which the survey is to be plotted, are necessary for this operation; and when the skeleton triangulation is completed, the next step is the delineation of the roads, &c., and the interior filling of the county, either entirely or partially, by measurement, as has been already stated.
* During the progress of the triangulation of the Ordnance Survey of England, the latitude and longitude of several of the most conspicuous stations were found with the greatest care; the former by means of observations on stars near the zenith, with a large zenith sector of 8 feet radius; a description of which instrument is given in the “ Transactions of the Survey,” the “ Philosophical Transactions,” 1803, and in page 211 of Mr. Airy's Figure of the Earth, in the “ Encyclopædia Metropolitana.”
INTERIOR FILLING-IN OF SURVEY, EITHER ENTIRELY OR PARTIALLY,
The more minutely the triangulation has been carried on, the easier and the more correct will be the interior filling up, whether entirely by measurement with the chain, or only partially so, and the remainder completed by sketching; the former of these methods will be first explained.
Small triangles are formed by actual measurement with the chain between the nearest trigonometrical points (upon the accuracy of which they depend), the directions of the lines forming the sides of which are to be selected with reference to the ultimate objects of the delineation of the boundaries, woods, estates, parishes, &c.* Where it is practicable these lines should connect conspicuous permanent objects, such as churches, mills, &c.; and in all cases the old vicious system of measuring field after field, and patching these separate little pieces together, should be most carefully avoided.f The method of keeping the field-book in measuring the interior with the chain, and plotting from its contents, is of course similar to the usual mode of surveying estates, parishes, &c.; and, as stated in the preface, this preliminary knowledge is supposed to have been already acquired. But on an extensive survey one general system must of necessity be vigorously enforced, to insure uniformity in all the detached portions of detail; and it has been deemed useful to introduce some of the forms and make abstracts from some of the rules laid down in Colonel Colby's lithographed “Instructions for the Interior Survey of Ireland,” as being applicable to the accurate filling-in of all topographical surveys on a very extended scale.
* Great assistance is derived from a rough diagram representing the proposed method of proceeding, with references to the marks left on the measured sides of the triangles to be subsequently connected by check lines, either joining two sides, or extending from one side to the opposite angle; this may appear at first to be a waste of time, but it will soon be found to be the contrary, as the lines will be all run in directions advantageous to the filling up of the interior. These marks should be made on the ground, so as be easily recognised, and should be copied in the margin of the field-book.
+ Very excellent instructions for the guidance of surveyors employed in forming plans of estates and parishes are to be found in the report from Captain Dawson, Royal Engineers, to the Tithe Commissioners of England and Wales, November, 1836, from which report Mr. Bruff, in his “ Engineering Field-book," has extracted a number of valuable directions.
Previous to commencing any measurement, the ground should be carefully walked over for the purpose of laying out the work, and marks set up at the average height of a theodolite, on the highest parts of the different hills, on the necks of the ridges jutting out from them, and at the level of lakes and rivers in various parts of their course, as well as on the site of permanent objects, such as churches, &c. These levelling marks should be all numbered and entered in a separate book, termed a field levelling book, intended to contain reciprocal angles of elevation and depression, afterwards taken between them, for the calculation of the horizontal values of the measured lines and of their comparative altitudes; which quantities are subsequently reduced to their actual heights above the level of the sea.* During the measurement of the principal lines, suitable points are selected at which to connect them by check lines, † or on which to base minor triangles, and of course with a view to the determination of the natural and artificial boundaries, that, measured lines running near them, the whole of the interior content may be computed from the “Register," made out directly from the field-book, the calculation from the plot being afterwards made simply as a check upon the other. All trigonometrical points and levelling marks should be measured up to with the chain during the progress of the survey, and their distinctive letters or marks entered in the field-books. Allowance may be made for short distances, by holding up one end of the chain till it appears horizontal, and dropping a pointed plummet on the ground, in mea
* Among the advantages of connecting a well-arranged series of levels with the plan of any portion of country, is that of rendering it at once available to the engineer in selecting the best trial lines for railroads or canals. Captain Dawson, in his Report, alludes to the railroad between Derry and Enniskillen, which was decided upon from the data furnished by the Ordnance Survey of Ireland, though only plotted on the scale of six inches to one mile.
+ When the length of any line is not checked in any way by triangulation, it should be measured with most especial care.