any two directions in which the glass may be pointed. The glass may be turned toward any point on the horizon, and also in an upward or downward direction. The scale on the circle is read by means of two microscopes, diametrically opposite to each other. Spirit-levels, micrometric arrangements, etc., are provided, and the observer has to be very careful about placing his instrument in the right position before he can actually begin operations. Although the stations are not at the same altitude, the angles between any two directions have to be so measured as if the stations were all on the same level, on a perfect plane. This plane is supposed to be vertical to the earth's radius which crosses that particular station—that is, a plane tangent to the earth's circumference and perfectly horizontal. The movement of the field-glass has, therefore, to be such that, although two stations may be situated lower or higher, the angle between the two directions can be read as if each of them had been raised or lowered vertically to the level of the station from which the observations are being made. The circle of the theodolite represents this horizontal plane; its center is supposed to be mathematically "the point," and the theodolite has, therefore, to be so placed that the center of this circle is in a perfect perpendicular to the point, while the surface of the circle itself is perfectly horizontal. The field-glass is situated on a support in the form of a double column, and the central axis of this support is vertical to the circle, and passes through its center. Delicate and exact mechanical arrangements permit of the glass being turned toward all points of the compass, and also of its being turned in an upward or downward direction; but each movement is either in a horizontal or a vertical direction to the circle. This enables the observer to obtain the angle desired—that is, the angle which any two directions would give if all the stations were at the same level.
The length of the sides of the triangles varies according to the facilities for extending a good net which the ground offers. From twelve miles upward is a suitable distance, the distance being in some cases only limited by the visual power of the glass.
The scale on the circle is divided into degrees, minutes, and possibly seconds, the latter and their fractions being read with the microscope.
The angles are measured as follows: When the theodolite is placed in its exact position, and the circle is perfectly horizontal, the glass is pointed successively at each of the surrounding stations, and for every direction the scale on the circle is read and noted on the field-book. Supposing the scale reads 56° 18' 12·075" when the glass is pointed toward one station, and 115° 56' 18·850" when pointed in another direction, the angle between the two directions is equal to the difference between the two readings, which is in this case 59° 38' 6·775". These readings are repeated several times, the circle being every time moved around its center in order not to have all the readings on the same divisions of the scale. When a complete set of observations has been
