half of forty, or twenty divisions, is the measure of the distance itself between the two points to which our attention has been directed, whether stars, craters in the moon, spots on the sun, and the like. Let us consider what is gained by this method over a measure taken by coincidence of the wires as a starting-point, and opening out the wires until they cut the points. In the method we have just described there are two chances of error in taking the measurements—the direct and indirect; but the result obtained is divided by two, so that the error is also halved in the final result. Now by taking the coincidence of the wires as the zero, or starting-point, the measure is open to two errors, as in the last case—the error of measurement of the points, _plus_ the error of coincidence of wires, an error often of considerable amount, especially as the warmth of the face and breath causes considerable alteration in the parts of the instrument, making a new reading of coincidence necessary at each reading of distance. As the result is not divided by two, as in the first case, the two errors remain undivided, so we may say that there is the half of two errors in one case and two whole errors in the other. Here then we use the micrometer to measure distances; but from a very short acquaintance with the work of an equatorial it will at once be seen that one wants to do something else besides measure distances. For instance, if we take the case of the planet Saturn, it would be an object of interest to us to determine how many turns, or parts of a turn, of the screw will give the exact diameter of the different rings; but we might want to know the exact angle made by the axis with the direction of the planet’s motion, across the field, or with, the north and south line. If we have first got the reading when the wires are in a parallel of declination, and then bring Saturn back again to the middle of the field and alter the direction of the wires until they are parallel to the major axis of the ring, we can read off the position on the circle, and on subtracting the first reading from this, we get the angle through which we have moved the wires, made by the direction of the ring with the parallel of declination, which is the angle required. We are thus not only able to determine the various measurements of the diameter of the outer ring by one edge of the ring falling on one of the fine wires, and the other edge on the other wire, but, by the position circle outside the micrometer we can determine exactly how far we have moved that system, and thus the angle formed by the axis of the ring of the planet at that particular time. [Illustration: FIG. 160.—The Determination of the Angle of Position of the axis of Saturn’s Ring. ]