66 MATHEMATICS. 
this purpose, when the angle has been measured, other things remaining the 
same, the clamp G is loosened, and the alidade telescope is brought again to 
the first object, without displacing the index: the clamp is to be applied, 
and the measurement gone over again. This operation may be repeated 
several times. The mean of these several measurements is then to be taken, 
which will in general be more accurate than any single one. In the com- 
mon alidade there are two indices with verniers, and in the circular alidade 
four, at right angles to each other, so that at each single measurement, the 
mean from two or four observations can be taken. 
The Graphometer (pl. 5, fig. 37) is essentially only a simplified theodo- 
lite, applicable to the minor geodetical operations. It is fixed on a stand by 
means of a socket and screw, K, and has the nut and screw arrangement, I, 
together with the correcting telescope, GH. Instead of the full circle, there 
is only a semicircle, A, upon which, besides the level, C, is attached an 
erecting compass, a. The clamp 0 serves to fix the instrument, as in the 
theodolite, and the alidade B has only one index with vernier. The upright, 
D, carries the telescope, EF, which has no graduated limb attached. It is 
used like the theodolite, although repetition is only practicable in the case 
of very acute angles. | 
Reflecting instruments, as a means of measuring angles, are next to be 
mentioned. In these, only one leg of the angle to be measured is observed 
directly, the signal of the other being attained by reflection into the field of 
view of the instrument. Reflecting instruments were first invented by 
Hadley, in 1740, for use at sea, where a fixed stand, or several telescopes, 
could not be employed; thence they were transferred to astronomical and 
geodetical operations. 
Hadley’s sextant (fig. 32) consists of a sector, containing the sixth part 
of the circle, or 60°. This arc, AB, forms the base of the instrument, and 
to it are attached two radii, and several cross pieces for the sake of addi- 
tional strength and stability. Upon one of these radii is a post with a ring 
for. attaching a small telescope, in the prolongation of whose axis on the 
other radius is attached the objective, H. The objective is divided into 
two equal parts, of which the lower, H,is a mirror, and the upper, G, a 
transparent plate of glass. A vertically central line passes through both 
halves. The central piece, C, consists of two superincumbent plates, 
turning on a common axis, and in the prolongation of this axis a plane 
mirror stands perpendicularly, so that when the index D, which moves over 
the limb on an arm in the prolongation of the mirror, stands at 0° of the 
scale, the two mirrors are parallel to each other. E is a small frame with 
colored glasses, which can be turned up to protect the eye during sunshine, 
or in observations in the sun. To use the sextant, the observer stations 
himself in the vertex of the angle to be measured, and directs the instrument 
in such a manner that one of the objects is seen through the telescope and 
the upper part of the objective. The arm with the index is then turned 
until the second object is reflected from the mirror C to the mirror in the 
lower part of the objective, and the fine adjustment made by means of the 
tangent screw, b. This adjustment consists in causing both objects to be 
66 
