64 MATHEMATICS. 
place by means of a plate, with screws. To use the instrument, it is placed 
upon the stand, which is set over the vertex of the angle to be measured, 
and turned, until the sight line of the telescope falls in a vertical plane with 
one leg of the angle to be measured. The position of the needle is then to 
be noted. Suppose A to be 364°: the compass is again turned, until the 
axis of the telescope lies in the vertical plane of the second leg. As the 
needle, DE, retains its parallel position, it will now intersect a second point 
on the limb, which must also be read off. Suppose this to be 1202°: then 
the angle measured will be 1202°—36}°= 843°. Any number of angles 
may be thus measured from a single station, and their legs measured with 
the chain. 
Although the compass has received various constructions, and is in 
general use, it is considerably behind the plane table in the accuracy of its 
results. The latter gives its results directly, instead of comparing them by 
the comparison of several measurements. In the compass, errors may 
occur in reading off the angles, which can never be determined to a greater 
degree of accuracy than 7°; this defect, added to several others, may intro- 
duce false results of no inconsiderable magnitude. As an illustration of 
this contingency, let us refer to pl. 4, fig. 11, where, by a slight error in 
determining a single angle, the defective figure A’B’C’D’E is obtained 
instead of the correct one ABCDE. 
The prismatic compass of Schmalkalder (pl. 5, figs. 35, 36), improved by 
Major von Decker in 1810, is well calculated for rapid military surveying. 
It consists of a plate, upon which is a small compass of about three inches 
in diameter, upon whose needle, 0, a disk of pasteboard with a graduated 
limb is so fastened as to turn with it. The arrester or stop, b, serves to lift 
this slightly from its pivot, when not in use. The sight vanes are attached 
in the direction of the diameter; the objective at h, and the ocular at f, 
with their axes in the same vertical plane. <A three-sided prism, ade, is 
attached to the ocular diopter, having a small mirror in its hypothenuse, 
which looks towards the graduation of the limb. To use this instrument, it 
is to be held horizontally before the eye in the vertex of the angle to be 
measured, and a sight taken through the upper part of the two vanes along 
one leg of the angle. Then, by the turning of the needle, a number on the 
limb will come under the prism, which, reflected by the mirror, can be read 
off through the lower part of the slit g. The second leg is then to be sighted 
in the same way, and the angle itself determined as in all compasses. The 
are m serves for the rectification of the instrument, with reference to the 
variations of the needle. The graduation on the limb is inverted, so as te 
be seen directly by reflection from the mirror. The diopters can be made 
to turn down when the instrument is to be packed up, in which case the 
stop b is to be set, the point » for regulating m, fixed, and then the whole 
may be packed in a box four inches square, and one and a halt inches deep. 
Its indications are sufficiently accurate for such purposes as military recon- 
noissance, &e. 
The theodolite, as represented in fig. 38, pl. 5, is a perfected form of the 
full circle astrolabe. ‘This instrument is calculated, not only for operations 
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