Hipparchus was content to ascertain the position of the celestial bodies to within a third of a degree, and we are informed that Tycho Brahe, by a diagonal scale, was able to bring it down to something like ten seconds. Fig. 101 will show what is meant by this. Suppose this to be part of the arc of Tycho’s circle, having on it the different divisions and degrees. Now it is clear that when the bar which carried the pointer swept over this arc, divided simply into degrees, it would require a considerable amount of skill in estimating to get very close to the truth, unless some other method were introduced; and the method suggested by Diggs, and adopted by Tycho, was to have a series of diagonal lines for the divisions of degrees; and it is clear that the height of the diagonal line measured from the edge of the circle could give, as it were, a longer base than the direct distance between each division for determining the subdivisions of the degree, and a slight motion of the pointer would make a great difference in the point where it cuts the diagonal line. For instance, it would not be easy to say exactly the fraction of division on the inner circle at which the pointer in Fig. 101 rests, but it is evident that the leading edge of the pointer cuts the diagonal line at three-fourths of its length, as shown by the third circle; so the reading in this case is seven and three-quarters; but that is, after all, a very rough method, although it was all the astronomer had to depend upon in some important observations. [Illustration: FIG. 102.—The Vernier. ] The next arrangement we get is one which has held its own to the present day, and which is beautifully simple. It is due to a Frenchman named Vernier, and was invented about 1631. We may illustrate the principle in this way. Suppose for instance we want to subdivide the divisions marked on the arc of a circle, Fig. 102 _a b_, and say we wish to divide them into tenths, what we have to do is this—First, take a length equal to nine of these divisions on a piece of metal, _c_, called the vernier, carried on an arm from the centre of the circle, and then, on a separate scale altogether, divide that distance not into nine, as it is divided on the circle, but into ten portions. Now mark what happens as the vernier sweeps along the circle, instead of having Tycho’s pointer sweeping across the diagonal scale.