[Table of Chords to Radius of 1000].

Triangulation.—­Measurement of distance to an inaccessible place.—­By similar triangles.—­To show how the breadth of a river may be measured without instruments, without any table, and without crossing it, I have taken the following useful problem from the French ‘Manuel du Genie.’  Those usually given by English writers for the same purpose are, strangely enough, unsatisfactory, for they require the measurement of an angle.  This plan requires pacing only.  To measure A G, produce it for any distance, as to D; from D, in any convenient direction, take any equal distances, D C, c d; produce B C to b, making c B—­C B; join d b, and produce it to a, that is to say, to the point where A C produced intersects it; then the triangles to the left of C, are similar to those on the right of C, and therefore a b is equal to A B. The points D C, etc., may be marked by bushes planted in the ground, or by men standing.

The disadvantages of this plan are its complexity, and the usual difficulty of finding a sufficient space of level ground, for its execution.  The method given in the following paragraph is incomparably more facile and generally applicable.

Triangulation by measurement of Chords.—­Colonel Everest, the late Surveyor-General of India, pointed out (Journ.  Roy.  Geograph.  Soc. 1860, p. 122) the advantage to travellers, unprovided with angular instruments, of measure the chords of the angles they wish to determine.  He showed that a person who desired to make a rude measurement of the angle C A B, in the figure (p. 40), has simply to pace for any convenient length from A towards C, reaching, we will say, the point a’ and then to pace an equal distance from A towards B, reaching the point a ae.  Then it remains for him to pace the distance a’ a” which is the chord of the angle A to the radius A a’.  Knowing this, he can ascertain the value of the angle C A B by reference to a proper table.  In the same way the angle C B A can be ascertained.  Lastly, by pacing the distance A B, to serve as a base, all the necessary data will have been obtained for determining the lines A C and B C. The problem can be worked out, either by calculation or by protraction.  I have made numerous measurements in this way, and find the practical error to be within five per cent.

Table for rude triangulation by Chords.—­It occurred to me that the plan described in the foregoing paragraph might be exceedingly simplified by a table, such as that which I annex in which different values of a’ a” are given for a radius of 10, and in which the calculations are made for a base = 100.  The units in which A a’, A a”, and B b’, Bb”, are to be measured are intended to be paces, though, of course, any other units would do.  The units in which the base is measured may be feet, yards, minutes, or hours’ journey, or whatever else is convenient.  Any multiple or divisor of 100 may be used for the base, if the