T68 



DIALLING. 



Plitb 

 CCXXIX. 



Fig. 14. 



then it may be traced by the methods for describing a 

 conic section. But perhaps in practice it may be suffi- 

 cient to find the points in which the curve crosses the 

 hour lines of the dial, and then to trace it nearly cor- 

 rect by some mechanical contrivance. The intersection 

 of the curve and any hour line may be found by the 

 following geometrical construction. 



Let O be the centre of the dial, and O XII be the 

 meridian line (Fig. 14.). Take any two lines OM, ON, 

 having to each other the ratio of the cosine of the lati- 

 tude to radius ; and on O as a centre, with these lines 

 as radii, describe circles. Take OF in the meridian line 

 equal to the length of the axis of the dial, and at the 

 point F make the angle OFH equal to the sun's dis- 

 tance from the pole. Let OY, any hour line, meet the 

 lesser circle in K. Through K draw KL perpendicular 

 to the meridian, meeting the greater circle in G. Draw 

 OG, meeting FH in H. In the hour line OY, take 

 OB equal to OH, and B is the point in which the hour 

 line meets the path of the shadow. 



For, by trigonometry, OG : OK : : sin. OKL : sin. 

 OGL : : cos. KOL : cos. GOL ; that is, (because the 

 angle KOL or Y T OL=zC)rad. :cos.L:: cos. C : cos.HOL; 

 hence HOL is the angle we have denoted by v in the 

 first of the two formulas, art. 77- Now the angle F=5, 

 and OF=a ; and sin. H : sin. F : : OF : OH ; that is, 

 sin. (? + t>) : sin. 5 : : a : OH, hence OH has the value 

 of r in the second formula; and consequently OB=OH 

 is the length of the shadow. Whatever has been said 

 respecting the shadow on a horizontal dial will apply 

 to any dial whatever, if L be put for the latitude of the 

 place where the dial would be horizontal, and if we 

 take the substile as the meridian. 



Retrogradalion of the Shadow. 



Retrogra- 81. The shadow which is projected on a dial by the 

 dation of edge of a stile or axis directed to the pole, revolves about 

 the shadow. t ne ce ntre always the same way. The hour may also be 

 indicated by the shadow of a single point of the axis, 

 which may be the summit of an upright wire, as ex- 

 plained in art. 41, and then the shadow will proceed, 

 not from the centre, but from the bottom of the wire. 

 The shadow on a horizontal dial for any latitude out of 

 the torrid zone, with a stile of this construction, will al- 

 so move always the same way ; but at any place be- 

 tween the equator and tropics, there is a period of the 

 year when the shadow moves forward, until it reaches 

 a certain limit, and then it moves back again. This is 

 what is called the retrogradalion of the shadow. 

 Fij». 15. To see how this may happen, let O (Fig. 1 5.) be the 



centre of a horizontal dial, and OF the axis ; and let FA 

 be a vertical rod, which meets theaxis atF. The shadows 

 of the material lines OF, AF, will form a triangle Of A 

 on the surface of the dial, the vertex of which, f, will 

 trace a curve f f f" _/'", which in general will be an 

 hyperbola. In the temperate zone, the point A always 

 falls within the curve, but at any place between the 

 equator and either tropic, when the sun passes between 

 the zenith and the elevated pole, the point A falls with- 

 out the hyperbola, so that straight lines Af, Af" may 

 be drawn from it to touch the curve. Now at the in- 

 stant of sun-rise, the two shadows Of, A /will be pa- 

 rallel to one of the asymptotes ; and as the day advan- 

 ces, their intersection /"will move along the curve, arri- 

 ving first at/ 7 , the point in which the tangent meets the 

 curve, next at the vertex^", and then proceeding along 

 f" f", the other branch. Now by this motion of the 

 point f, the shadow A/ will turn about the point A in 

 a direction from O to f ; but when it has arrived at 



the position Af, it will be for a moment stationary; as Theory 

 the intersection /'advances to/'", the vertex, the shadow and Con ' 

 will evidently turn back; and when f arrives at/", the structlon ' 

 shadow will coincide with the axis. A like pheno- _ * 

 menon will be exhibited in the afternoon. The sha- 

 dow will at first recede from the axis, until it coincide 

 with the tangent A / "' ; and then it will again approach 

 the axis until it vanish at sun-set, and at that instant it 

 will be parallel to the other asymptote. 



All this may be observed on a dial in any latitude, 

 provided its plane pass between the sun and the equa- 

 tor at noon, and the hour be shewn by a stile perpen- 

 dicular to its plane. 



The phenomenon we have been describing will ap- 



Eear very simple to the mathematician, and perhaps 

 ardly worthy of particular notice ; it seems, however, 

 to have excited attention, probably because of the ap- 

 parent solution it affords to the return of the shadow on 

 the sun-dial of Ahaz. But this explanation is not ad- 

 missible, because in that case, the return of the shadow 

 must have been a thing altogether miraculous, other- 

 wise it would not have excited attention. 



Meridian of Mean Time. 



82. The hour of the day indicated by a good sun-dial M e ridianof 

 will agree with that shewn by a clock only on certain mean time, 

 days of the year ; in general, there will be a difference, 

 which is called the Equation of lime. The nature and 

 quantity of this correction has been fully explained in 

 Astronomv, p. 652; and a Table given in page 797, 

 by which it may be accurately found. 



A line may be traced upon the plane of a dial, in 

 such a manner, that the extremity of the shadow of the 

 stile shall fall on it at the instant of mean noon, which 

 will thereby be indicated just as the time of apparent 

 noon is shewn by the shadow of the edge of the stile 

 falling on the meridian line. The line which shews the 

 time of mean noon is called the meridian of mean time. 

 Its figure and position on a horizontal dial is shewn in 

 Plate CCXXVIII. Fig. 9. Plate 



To trace the meridian of mean time, the hour lines ccxxvin, 

 for 16 minutes before, and the same period after noon, Fi g- 9 - 

 should be found, and lines drawn from the centre to di- 

 vide the angles which they make with the meridian in- 

 to as many equal parts as may be thought necessary ; 

 for example, 96' lines may be drawn, and these will 

 correspond, with sufficient accuracy, to portions of 

 time, differing by 10 seconds one from another. But 

 in some declining dials, the angular motion of the sha- 

 dow at noon may deviate considerably from uniformity, 

 and then it may be necessary to find the correct posi- 

 tion of the hour lines for every fifth minute between 

 apparent and mean noon. 



The path of the shadow is next to be traced for as 

 many different days of the year, equally distant, as may 

 be thought necessary ; and the point in which the path 

 of the shadow cuts the hour line corresponding to the 

 equation of time of any day will be a point in the meri- 

 dian of mean time for that day. In this manner may any 

 number of points be found, and a line traced through 

 them ; and the days of the year being marked opposite 

 to them, the meridian of mean time will be finished. 



The first who has spoken of a meridian of mean time 

 was M. De Fouchy, of the Academy of Sciences, be- 

 fore the year 1740. The republic of Geneva had one 

 traced in 1 780 ; and the instant of mean noon was 

 made known by a signal from the church of St Peter, 

 to enable the watch-makers to regidate their instru- 

 ments to mean time. 



