DIALLING. 



TO I 



Theory 

 and Con- 

 struction. 



cing directly eafet and west ; their planes, therefore, co- 

 incide with the plane of the meridian, and pass through 

 the poles of the world. 

 Vertical - T° explain the nature of these dials, let us suppose 

 east and that NS, Fig. 18. is a straight line traced upon their 

 west dials, planes in the direction of the earth's axis, and that it is 

 Plate crossed at right angles by a straight line EQ, which 



ccxxviu. will be the intersection of the planes of the ecruinoctial 

 Fig. 1?. anc i the meridian. Let us also suppose, that AB is a 

 thin cylindrical rod, held directly over the line a b, and 

 parallel to it, by two supports A a, B b ; and that this 

 rod passes through C the centre of a circle, which lies in 

 the plane of the equinoctial circle, and which touches 

 the plane of the meridian in c, the bottom of a perpen- 

 dicular C c : this circle will evidently be an equinoctial 

 dial, of which AB is the axis. 



Let CK be the shadow which the axis projects on 

 the plane of the circle, and let it be produced to meet 

 the vertical plane in k ; then a line drawn through /.-, 

 perpendicular to EQ, will evidently be the direction of 

 the shadow which the rod AB projects on the vertical 

 plane, at the same instant of time that it projects on the 

 equinoctial dial the shadow CK ; and as the hours are 

 indicated on the equinoctial dial by the position of the 

 revolving shadow CK, they will also be shewn on the 

 vertical plane EQNS, by the successive petitions of the 

 rectilineal shadow F k F, winch will always be parallel 

 toNS. 



Now, as the plane a AB b is perpendicular to the 

 plane of the meridian, and passes through the poles, it 

 must be the plane of the six o'clock hour circle, or that 

 circle in the heavens, passing through the poles of the 

 world, in which the sun is always seen at six in the 

 morning and six in the evening. Therefore the arc c K 

 of the equinoctial dial, intercepted between the perpen- 

 dicular C c and C k, the position of the shadow at any 

 time will be the measure of the horary angle described 

 by the sun in the heavens, between six o'clock and 

 that time ; and the straight line c k, the distance of the 

 shadow of the rod AB from the line a b immediately 

 under it, will be the tangent of that arc to the radius 

 Cc. 



57. Let the horary angle from six o'clock be denoted 

 by E', and let ck, the distance of the hour line from a b, 

 be x ; also let C c, the height of the rod above the plane 

 of the dial, be denoted by cl, then because rad. : tan. E' 

 : : d : x, the general formula expressing the position 

 of the hour lines on an east or west dial, in respect of 

 the line a b, will be (supposing radius=l ) 



x=d tan. E' (3) 



from which it appears, that, in these dials, the position 

 of the hour lines in respect of each other is altogether 

 independent of the latitude of the place. Indeed the 

 same thing might have been inferred from what has 

 been said in art. 51. and 52, for a vertical east or west 

 dial for any place whatever would manifestly be an 

 horizontal dial at the equator. 



Geometrical construction of Vertical East and West 

 Dials. 



Geometri- 

 cal con- 58. The geometrical construction of these dials will 

 struction of be as follows : 



vertical j Q n t j ie east or west ver ti C al plane, draw the hori- 



west dials. zontai hne HR, (Figs. 1. and 2.) and assume in it 



Plate any point c for the bottom of the style, the upper ex- 



ccxxix. tremity of which is to project a shadow on the hour 



Fig. i , 2. lines. 



Plate 

 CCXXIX. 



Fig. 3i 



2. Through the point c draw the hne NS, so as to 

 make with HR an angle Nc R, equal to the latitude of 

 the place. The angle NcR must be towards the right 

 hand on an east dial, but towards the left in a west 

 dial, and the line NS will point to the poles of the 

 heavens. 



3. Through c draw EcQ perpendicular to NS, and 

 EcQ will be the equinoctial. 



4. In c S take c b equal to the intended height of 

 the stile, and on b for a centre, with b c as a radius, de- 

 scribe a semicircle. 



5. Divide the semicircle into 12 equal parts. 



6. From b draw lines through the points of division 

 to meet the line EQ. 



7. Through the points of intersection draw lines per- 

 pendicular to EQ, and these will be the hour lines on 

 the dial against which the hours are to be written, as 

 in the Figure. 



S. At the points a, h, the style is to be erected (see 

 Fig. 3.) so that its height A a may be equal to b c, 

 which is also the distance between the hour lines of 6 

 and 9, and the dial is finished. 



The east dial will she\v the morning hours until it be 

 nearly noon, and the west dial will shew the afternoon 

 hours ; but neither can indicate the time of noon other- 

 wise than by the sun being in the plane of the dial. 



The truth of the construction follows too obviously 

 from the formula, (art. 54.) to require any formal de- 

 monstration. 



Polar Dial. 



5Q. A polar dial is that which is traced on a plane* p olar , ]a | 

 perpendicular to the meridian, and passing through the 

 poles ; therefore, like east and west dials, the axis of the 

 sj)here lies in its plane • and to shew the hours, its stile 

 must be formed like theirs, and fixed over the meridian' 

 line. 



The construction of this dial, which is represented- 

 at Fig. 4. will differ in no respect from that for an east fig. 4. 

 or west dial, except that in these, the line NS, which 

 passes through the pole, makes with the .horizontal line 

 HR an angle equal to the latitude; but in the polar 

 dial, the lines NS and HR are perpendicular to one 

 another : for if an east or west dial were to be turned 

 about the six o'clock hour line as an axis, so that the 

 plane of the dial were perpendicular to its former posi- 

 tion, it would then become a polar dial, and what was 

 before the hour line for six would be in its new position 

 the hour line for noon. 



This kind of dial may shew time from a little after 

 six in the morning to a little, before six in the evening,, 

 provided it be of sufficient extent to admit of the sha- 

 dow meeting its plane. At the hours of six in the morn- 

 ing or evening, its plane passes through the sun, and 

 therefore is not illuminated. 



If E denote the horary angle from noon, and x and 

 d represent the same things as in the formula for east 

 and west dials, the formula for constructing the polar. 

 dial will be, 



xzzd tan E (4) 



Vertical Declining Dials. 



60. Any dial described on a vertical plane that does w ert :. ,j 

 not directly face one of the cardinal points, is called a declining 

 vertical declining dial ; and of these there may be four dials. 



