any
given angle, may be ascertained without a protractor or other angular
instrument, by means of a Table of Chords. So, also, may any required
angle be protracted on paper, through the same simple means. In the first
instance, draw a circle on paper with its centre at the apex of the angle
and with a radius of 1000, next measure the distance between the points
where the circle is cut by the two lines that enclose the angle. Lastly
look for that distance (which is the chord of the angle) in the annexed
table, where the corresponding number of degrees will be found, where the
corresponding number of degrees will be found. If it be desired to
protract a given angle, the same operation is to be performed in a
converse sense. I need hardly mention that the chord of an angle is the
same thing as twice the sine of half that angle; but as tables of natural
sines are not now-a-days commonly to be met with, I have thought it well
worth while to give a Table of Chords. When a traveller, who is
unprovided with regular instruments, wishes to triangulate, or when
having taken some bearings but having no protractor, he wishes to lay
them down upon his map, this little table will prove of very great
service to him. (See "Measurement of distances to inaccessible places.")

[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 genera