Page:The American Cyclopædia (1879) Volume VI.djvu/77

DIAL 69 north of the equator the staff will cast a shad- ow upon the circle, which will traverse it with uniform motion, passing through its 360 in 24 hours. Such a circle and style would form an equinoctial dial, which, if placed on the meridian a at the equator, with its gnomon or style parallel with the earth's axis, would, while the sun's rays fall upon it, measure the time in the same way as at the pole; that is, the gnomon would traverse corresponding degrees at the same time. Instead of the shadow traversing the whole circumference of the dial, as it would at the pole, it can only traverse the lower half, between 6 in the morning and 6 in the evening. Place a similar dial D on the meridian a, between the equator and the north pole, also with its gnomon parallel to the earth's axis, and therefore inclined to the horizon with an angle equal to the latitude of the place, and the shadow of the gnomon will travel the face of the dial precisely as in the case of the circle C, with the difference that when the sun's path is north of the equa- tor the shadow will be cast before C in the II FIG. 2. morning and after 6 in the evening, falling upon the northern face; and when the sun's path is south of the equator it will fall upon the south face, but not till after 6 in the morn- ing nor until 6 in the evening. In the hori- zontal dial the shadow does not traverse the hour circle with a uniform motion (except at the poles), but travels faster the further it recedes from the vertical position ; so that the lines which mark the hours require to be fur- ther apart near the morning and evening than near noon. The hour lines may be deter- mined by the following elementary method of plane trigonometry. Let A B C D, fig. 2, be a horizontal plane upon which the hour lines of a dial are to be described. Draw the meridian line E F, and from E erect E H parallel to the earth's axis, to represent the gnomon. Then, with a centre G- on E H as an axis, describe the equinoctial circle I F, its plane being parallel with the plane of the equator. Draw the meridian line I F, and also G M, G N, and G C, dividing the arc O F into equal segments. Draw E M, E N", and E C on the horizontal plane, meeting G M, G N, and G C in M, N, and 0, and with a radius E F and a centre at E describe the arc F P, meet- ing E C in P ; also with the same radius and centre describe the arc F K, meeting the line E H in K. The angle F G C is at the centre of the equinoctial arc, and has F C for its tan- gent, and the angle F E C is at the centre of the hour arc and also has F C for its tangent. The corresponding equinoctial and hour arcs have therefore a common tangent. As G F is always perpendicular to E H, it follows that the radius of the equinoctial arc will always be the sine of the latitude arc F K. It will moreover be observed that the radii of the lati- tude and hour arcs will always be equal, and proportional to the radius of the equinoctial arc, as the hypothenuse of a right-angled tri- angle to one of its sides ; and therefore that the common tangent F will always measure a larger equinoctial than hour arc. All these relations being constant, we derive the follow- ing equations : rad. equinoc. arc = sin. lat. arc. tan. equinoc. arc = tan. hour arc. Multiplying equals by equal ratios, we have tan. hour arc x rad. equinoc. arc=sin. lat. arc x tan.equinoc. arc; and tan. hour arc = a. lat. arc x tan. equinoc. arc. rad. equinoc. arc. It will be observed that if the line E H were a rod, the circle I O F a material plane, and the angles at G each equal to 15, the apparent motion of the sun would cause the rod's shadow to fall upon the lines G M, G N, and G- C at the hours 1, 2, and 3 respectively, and also upon the lines E M, E N", and E of the horizontal plane or hour arc. Let it be re- quired to find the first hour angle on either side of the meridional line for a horizontal sun dial in the latitude of New York, which is 40 42' 43". By logarithms : sin. lat. arc 40* 42' 43" = 9-814419 tan. equinoc. arc for 1 hour = 15* = 9-428052 19-242461 rad. equinoc. arc = 10-000000 tan. hour arc, 9" 54' 45" = 9-242461 The hour angles for 10 and 2 o'clock will be found by substituting 30 for 15 of the equi- noctial circle, and for 9 and 3 o'clock by substi- tuting 45, and so on till 6 o'clock, when the angles will decrease in the same ratios in which they increased. A dial with its hour plane in a vertical position is called a vertical dial, and may be regarded as the complement of the horizontal, because the angle of inclination of the gnomon to the plane of the dial is the com- plement of that angle in the horizontal dial if taken to a latitude which is the complement of that for which it is intended as a vertical dial. Vertical dials have north and south