106 LECTURE X. 



above the horizon; at sea, the apparent horizon, formed by the surfiice of the 

 water, affords the most convenient determination ; but since the spectator is 

 somewhat elevated above the convex surface of the sea, the apparent horizon 

 is necessarily lower than the true horizon, and a correction is therefore re- 

 quired according to the height. In the open sea this correction may be de- 

 termined by measuring the whole angle above and below the apparent horizon, 

 and taking one fourth of the difference for the dip or depression. On shore, 

 a plumb line is the simplest instrument for determining the situation of the 

 horizon, and its accidental vibrations may be prevented by suspending the 

 weight in water or in oil. For small instruments, a spirit level, of which the 

 operation depends on hydrostatical principles, is capable of greater delicacy 

 than a plumb line. It readily indicates, when well made, an error of a single 

 second, but it requires some attention to avoid inequalities of temperature, 

 which would, tend to disturb its figure. Well rectified ether is found, on ac- 

 count of its perfect fluidity, to be the best liquid for a spirit level. An arti- 

 ficial horizon is a reflecting surface, employed for obtaining an image, as much 

 below the horizon, as the object is above it, and for measuring the angular dis- 

 tance of this image from the object : sometimes a plane speculum of glass or 

 metal is used for this purpose, being previously adjusted by a spirit level ; and 

 sometimes the surface of mercury, treacle, or tar, protected from the wind by 

 a vessel with holes in it, or by a glass cover, either detached, or simply 

 floating on the mercury, when this liquid is employed. 



It is in many cases simpler and more convenient to estimate angles, not by 

 the arcs subtending, them, but by their sines, or the perpendiculars falling 

 from one leg on the other. Thus, it is usual among miners, to say that the 

 ground rises or falls one foot, or one yard, in ten, when the sine of the angle 

 of its inclination to the horizon is one tenth of the radius. Angles of different 

 magnitudes are indeed proportional to the arcs, and not to the sines, so that 

 in this sense the sine is not a true measure of the comparative magnitude of 

 the angle; but in making calculations, we are more frequently obliged to em- 

 ploy the sine or cosine of an angle than the angle or arc itself. It is, how- 

 ever, easy to pass from one of these elements to the others by means either of 

 trigonometrical tables, or of the scales engraved on the sector. 



The sines, tangents, and secants laid down on the sector, may be employed 



