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A right angle can often be laid off when no means


measuring other divisions of the circle are at hand. The distance AB can then be thus obtained :


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Of course with a sextant, or other means of observing the angle ACB, AB becomes simply the tangent of that angle to the radius BC: a table of natural sines and tangents engraved on the lid of any portable reflecting instrument is often of great service, particularly in sketching ground without any previous triangulation, and in obtaining the distance to an enemy's batteries, &c., on a military reconnaissance. The height of a point on an inaccessible hill may also be obtained without the use of instruments, thus:

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* A perpendicular can always be thus laid off with the chain :-suppose a the point at which it is required to erect a right-angle : fix an arrow into the ground at a, through the ring of the chain, marking twenty links; measure forty links on the line ab, and pin down the end of the chain firmly at that spot, then draw out the remaining eighty links as far as the chain wil stretch, holding by the centre fifty-link brass ring as at c; the sides of the triangle are then in the proportion of three, four, and five, and consequently cab must be a right angle.





An angle equal to any other angle can also be marked on the ground, with the chain only, by measuring equal distances on the sides containing it, and then taking the length of the chord the same distances, or aliquot parts thereof, will of course measure the same angle.





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the angle QSP is observed 35°, and PSO=40°, describe a circle

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passing through Q, S, and P, which is thus done :-Double the angle QSP which=70°; subtract this from 180, leaving 110°; lay off half of this, or 55° at PQR and QPR, and the angle at R is evidently 70°, or double QSP; now the angle at the centre being double that at the circumference, a circle described from R as a centre with the radius RQ, or RP, will pass through the point S. In like manner a circle described from V, with the radius VP, will also pass through S, and their intersection gives the spot required.

For the analysis of the calculation of this problem, vide "Puissant, Géodesie," vol. i. p. 233.

The method of surveying any tract of country through which a line of railway is projected or has been determined upon is so similar to that of measuring roads or other continuous lines by "traversing" with the chain and theodolite, that it does not require any peculiar directions. The lines, however, being generally very long, must be measured with the greatest exactness, and the angles be observed. with proportionate care. Where practicable also, the work should, whilst in progress, be tested by reference to known fixed points near which it in which can passes, most cases be obtained from good maps. The existing Standing Orders of Parliament regulate the scale upon which these surveys are required to be plotted in England; and the lateral deviation

allowed from the proposed line of rails, with other local causes, determine the breadth required to be embraced in the survey.

For the methods of laying out the lines of railways; the levels of the different portions; determining the curves, gradients, and slopes of embankments and cuttings, &c., every information can be obtained from the works of Mr. Hascoll and many others; and it would be out of place here to attempt any description of subjects which belong to a most important branch of civil engineering, and embrace such a multitude of details. A few remarks, however, upon the method of taking sections for railways, and the scales upon which they should be plotted, will be found in the chapter upon Levelling.






THE sketch of any portion of ground for military purposes should, in all cases, be accompanied by an explanatory statistical report, and the combination of these two methods of communicating local information constitutes what is termed a Military Reconnaissance, in which the importance of the sketch, or the report, predominates according to circumstances.

The object for which a reconnaissance is undertaken naturally suggests the points to which the attention of the officer should be principally directed; if for example, it is merely to determine the best line of march for troops through a friendly or undisputed country; the state of the communications, the facilities of transport, and possibility of provisioning a stated number of men upon the route, are the first objects for his consideration. If the ground in question is to be occupied either permanently, or for temporary purposes, or if it is likely to become the seat of war; his attention must be directed to its military features, and a sketch of the ground, with explanatory references, together with a full and correct report of all the intelligence he can collect from observation, or from such of the inhabitants as are most likely to be well acquainted with the localities*, and most worthy of credence, will demand the exertion of all his energies: upon the correct information furnished by this reconnaissance may depend, in a great measure, the fate of the army.

* It is almost needless to point out the incalculable advantages of being a good modern linguist to an officer employed on duty of this nature in an enemy's country.

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