Though the preceding figure was drawn for a horizontal dial, yet any other plane might be substituted. The objections to a dial are, that the shadow of the style is not sufficiently well defined to give very accurate results, even for ordinary purposes : that refraction, which always makes the sun appear a little too high, throws the shadow a trifle towards noon at all times, that is, makes the time too fast in the morning, and too slow in the evening ; and that a correction is always necessary in order to find mean or civil time. Even if the first objec tion could be got over, the corrections requisite for the two latter would prevent persons in general from making use of the instrument. If the edge of the style be not very narrow, it is necessary to have the morning and evening halves of the dial separated by the breadth of that edge.
Those who understand spherical trigonometry will easily see that the general problem of a sundial consists in that of finding out where the hour-lines cut a given circle, as follows. Let n Q o be the circle In which the plane of the dial produced cuts the heavens, and let the angle c A e, which it makes with the horizon (h), and c n which it q is the continuation of what is called the suhstyle. Now in the right
angled triangle A NB, we have cos It cos NB= whence N n is found ; to which add the latitude of the place, r N, and r n is found. The equations tan r . cos = tan Q n, sin P . sin nt = sin P Q show how to place the substyle with respect to B, the point answering to noon ; and also how to place the style with respect to the substyle. To find the point v at which any given hour-line, r v, cuts the circle o 13, first find the angle Q r n from cot Q P13 = tan m. cos r ; and v P n, the hour-angle from noon of the sun (v being a point in the shadow). The difference of these angles, Q Pv, or their sum, is then known ; and o v is found from tan QV = tan QPV sin PQ It will be better for the beginner to verify these steps on a correctly' drawn figure, or to modify them, than to make purely algebraical alterations. Also it is to be remembered that the position of the dial may require both sides of it to be graduated, and the style to extend in both directions, to suit all times of the year and all hours of the day.