700 



DIALLING. 



Theory 

 and Con- 

 struction, 



of the dial when i t tell on the like hour circle of the 

 sphere. 



49. It appeal's, then, that to construct a horizontal 

 dial by the terrestrial globe, we must place the globe 

 in such a position, that the arc of the brazen meridian 

 between the pole and horizon may be equal to the la- 

 titude of the place, and that any one of the meridians 

 on the globe may coincide with the brazen meridian ; 

 and then the arcs of the horizon between its north 

 point and its intersections with the 24 meridians on the 

 globe will be the measures of the angles which the 

 hour lines on the dial must make with the meridian 

 line. 



. 50. From the same principles we may derive imme- 

 diately the formula which was investigated at art. 39. 

 For, let PHjo be any hour circle which cuts the hori- 

 zon in H, then in the Tight angled spherical triangle 

 PBH, there are given PB, one of its sides adjacent to 

 the right angle B, equal to the latitude, and the angle 

 HPB at the pole, which is equal to the hour angle from 

 noon, to find HB, the arc of the horizon between the 

 meridian and hour circle, passing through the sun, 

 which arc is the measure of the angle at the centre of 

 the dial contained by the meridian and hour line cor- 

 responding to that hour circle. 



By the principles of spherics, (see Spherical Tri- 

 gonometry), in any right angled spherical triangle, 

 radius is to the sine of either of the sides about the 

 right angle, as the tangent of the adjacent angle to 

 the tangent of the other side about the right angle ; 

 that is, in the present case, as radius to the sine of PB, 

 the latitude; so is the tangent of HPB, the horary 

 angle in the heavens, to the tangent of HEB, the angle 

 made by the hour line and the meridian at the centre 

 of the dial. 



Vertical South or North Dials. 



"Vertical 51. These dials are described upon vertical planes, 



5,011th or facing directly to the south and north. They are re- 

 north di- presented in Fig. 16. and Fig. 17. 



Plate ^ s tne P^ anes °f these dials coincide with the prime 



ecxxvni vertical, that is, the great circle of the sphere which 

 rig. 16 17„ passes through the zenith and the east and west points 

 of the horizon, their intersections with the meridian or 

 the XII o'clock hour line, will be a vertical line. The 

 theory of these dials might be investigated exactly in 

 the same way as that of the horizontal dial, and par- 

 ticular rules formed for their construction; but this is 

 not necessary ; for the geometrical constructions which 

 have been investigated for a horizontal dial, may be 

 made to apply to all dials whatever, by considering, 

 that if a horizontal dial were transferred from the place 

 for which it was made, to any other place on the earth's 

 surface, and fixed there in a position parallel to its ori- 

 ginal position, that is, with its plane parallel to the ho- 

 rizon of' the place for which it was made, and its axis, as 

 before, pointing to the pole of the heavens ; then, in its 

 new position, it mill indicate the hour of the day at its 

 original position, precisely as it did before it was remo- 

 ved. This proposition, although not exactly, is almost 

 exactly true, because of the great distance of the sun 

 from the earth in comparison to the distance of one 

 place on the earth from another. 



52. From the above principle we may infer, that any 

 plane dial whatever, at a given place, -will be a horizontal 

 dial for some place or other of the earth ; and, there- 

 fore, to construct a dial on a given plane, we have on- 

 ly to find what place of the earth lias its horizon paral- 



lel to that plane, and then on the given plane to con- 

 struct a horizontal dial for that place, and it will shew 

 the hour of- the day there. This, however, may not 

 be the hour of the day at the place where the dial is 

 intended to shew time, but then it will differ from the 

 true hour there always by the same given quantity, 

 namely, by the difference of the longitudes of the two 

 places reckoned in hours and minutes of time. For ex- 

 ample, if it should be found that a certain plane at 

 London was parallel to the horizon of St Petersburg ; 

 then a horizontal dial constructed on the plane for the 

 latter place, would show the hour at St Petersburg. 

 But as the difference of longitude between London and 

 St Petersburg is about 30 degrees, corresponding to two 

 hours in time, the dial would indicate noon when it 

 was only ten in the morning at London ; and it would 

 shew one o'clock when the true time at London was 

 eleven, and so on. However, the dial would be adapt- 

 ed to London if we wrote the character for the hour 

 ten on the St Petersburg meridian line, and that for 

 eleven on the. one o'clock hour line, and so with the 

 other hours. 



53. The zenith of any place being in a line passing 

 through the plane perpendicular to its horizon, it is 

 easy to see that two places on the earth's surface which 

 have their horizons perpendicular to one another, must 

 have their zeniths 90 degrees asunder. Hence it fol- 

 lows, that a vertical south or north dial at any place, 

 would be a horizontal dial to a place 90° south or north 

 from that place, and on the same meridian. A verti- 

 cal south or north dial at Edinburgh, for instance, the 

 latitude of which is 56°, would be a horizontal dial- sc 

 a place on the same meridian, and in 3-i of south la- 

 titude, and it would shew the same hour of the day at . 

 both places, because the time of noon happens at both 

 at tile same instant. 



54. Hence, if we put E for the. horary angle from, 

 noon, and C for the angle which the corresponding 

 hour line makes with the meridian of a north or south 

 dial, and L for the latitude of the place, we get im- 

 mediately, from the formula of art. 39, 



Tan. C=Cos. L tan. E (2) 



a general formula for constructing a north or south 

 dial; and by this, the angles which the hour lines 

 make with the meridian may be computed. The other, 

 three methods of constructing a horizontal dial, (art. 

 44 — 4.8), apply equally to vertical south or north dials, ( 

 only substituting the complement of the latitude, or 

 what it wants of 90° for the latitude, observing that . 

 the axis must make with the plane of the dial an angle 

 equal to the complement of the latitude, and must be 

 in the meridian, and then it will point to the pole of the 

 world, as must be the case in all dials. 



55. In north latitudes, a north dial is only illumi- 

 nated when the sun is on the north side of the equa- 

 tor ; and the nearest times of the day to noon that can 

 be shewn by it, are those at which the sun passes the 

 prime vertical on the day of the summer solstice. A 

 south dial can never be illuminated before six in the 

 morning, nor after six in the evening ; because, when 

 the sun rises earlier, and sets later, he does not pass 

 the prime vertical so early as six in the morning, and 

 he crosses it again before six in the evening. It will 

 be unnecessary, therefore, to describe upon either more 

 hour lines than can be wanted. 



Vertical East and West Dials. 



56. These dials are traced upon vertical planes, fa- 



Theory 

 and Coir* 



