DIALLING. 



697 



horizontal dial in CM, and that of the equinoctial dial in 

 EM ; then the line PQ being the common intersection 

 of the equinoctial and horizontal planes, which are 

 perpendicular to the meridian, that line itself is per- 

 pendicular to the meridian. See Geometry. 



Let a plane passing through the sun's centre and the 

 common axis of the dials, meet their planes in the lines 

 EH, CH, these lines will manifestly be the positions 

 of the shadows on the two dials at the same instant of 

 time. 



39. Now, at any given time, we know the angle 

 HEM wliich the revolving shadow EH makes with 

 the meridian line EM on the equinoctial dial, because 

 it is the horary angle which the sun has to describe, or 

 has described about the earth's axis, between the given 

 time and noon, and wliich is always proportional to that 

 time, reckoning 15 degrees of the angle to an hour. 

 And in the triangle CEM, right angled at E, we know 

 the angle ECM, which is always equal to the latitude 

 of the place for which the dial is to be constructed ; 

 and from these we must find the angle HCM, which 

 the hour line HC, on the horizontal dial, makes with 

 CM, the meridian, or 12 o'clock line. 



Let us denote the horary angle HEM, which the 

 sun describes between the given time and noon, 

 by the letter E, and the angle HCM, wliich the hour 

 line on the horizontal dial makes with the meridian, 

 by C, and let the angle ECM, the latitude of the place, 

 be L ; then, by plane trigonometry, in the two right 

 angled triangles, EMH, CMH, 



HM : ME : : tan. E : rad. 

 and CH : HM : : rad. : tan. C ; 

 therefore, ex cequo inv. (see Geometry.^ 

 CM : ME : : tan. E : tan. C, 

 but CM : ME : : rad. : sin. L ; 

 therefore, rad : sin.L: : tan. E : tan. C. 

 Now, the first three terms of this proportion are 

 known, therefore the last is also known ; and we get 

 this general formula for constructing a horizontal dial. 



tan. C=sin. L. tan. E (1) 



in which radius is supposed z= 1. The logarithmic for- 

 mula, deduced from it, may be expressed in words at 

 length, thus : 



To the logarithmic tangent of the horary angle descri- 

 bed by the sun between noon and the given lime, add the 

 log. sine of the latitude, and the sum, abating 1 0, ft he 

 log. of rad.) is the logarithmic tangent of the angle which 

 the hour line on the dial makes with the meridian line. 



40. Example. Let it be required to calculate the 

 angles which the hour lines on a horizontal dial, for 

 Edinburgh, make with the meridian or 12 o'clock line : 

 The latitude of Edinburgh being about 56°, a calcula- 

 tion for the hour lines of XI in the forenoon and I in 

 the afternoon would be as follows: 



log. tan. horary angle 15° 9.42805 



log. sin. lat. 56° 9.91857 



log. tan. 12° 32' 9.34662 



Hence it appears, that the hour lines for XI in the 

 forenoon, and I in the afternoon, must each make with 

 the meridian an angle of 12° 32'. 



The angles which the remaining hour lines make 

 with the meridian may be found in the same way, and 

 will be as follows : 



Hour lines of X and II 25° 35' 



IX and III 39 40 



VIII and IV 55 8 



VII and V 72 5 



VI and VI 90 



VOL. VII. PART II. 



Plate 



The hour lines of V in the morning, and VII in the 

 evening, make the same angles with the meridian as 

 the hour lines of VII in the morning and V in the af- 

 ternoon ; but they lie on opposite sides of the VI o'clock 

 hour lines. In like manner, the horn lines of IV in 

 the morning, and VIII in the evening, make the same 

 angles with the meridian as the hour lines of VIII in 

 the morning and IV in the afternoon, and so on. 



The construction of the dial is now very easy, as it 

 requires nothing more than to make an angle of a given 

 number of degrees. Thus, draw the meridian line 

 CM (Fig. 9.) and cross it at right angles by the six ~^ X a VIir ' 

 o'clock hour line CG ; and as the style of the dial must '*»' * 

 have some thickness, it will be proper to draw two pa- 

 rallel lines CM, CM' for the meridian line, so that the 

 distance between them may be equal to that thickness. 



From the points C, C, draw the lines CI, CXI on 

 opposite sides of the meridian, so that the angles MCI, 

 M'CXI may be each 12° 32'; and these lines will be the 

 hour lines of I in the afternoon, and XI in the forenoon ; 

 the former lying on the east and the latter on the west 

 side of the meridian, when the dial is placed in its pro- 

 per position. In the same way, all the other hour 

 lines may be laid down on the plane of the dial, using * 



a scale of chords, or a protractor, such as is commonly 

 sold by mathematical instrument makers. Or a quadrant 

 of a circle p q may be described on C as a centre, and 

 divided into 90 equal parts, and the hour lines drawn 

 at once through the points of the arc indicating the 

 number of degrees and minutes they ought to make with 

 the meridian. The stile KCL (Fig. 8.) must be so con- 

 structed that the angle contained by CK and CL, the 

 edges of one of its planes, may be 56°, the latitude of 

 the place, and it may be fixed into the plane of the 

 dial by two tenons at C and L let into openings made 

 to receive them. The edge CK must stand directly 

 over the meridian line CM, and then the afternoon 

 hours will be shewn by the limit of the shadow of the 

 triangular plane KCL. 



The stile may have any shape, provided its edge 

 CK be a straight line. It may even be a cylindrical 

 rod, but in that case the hour lines ought to be tangents 

 to its section with the plane of the dial. The angles 

 they make with the meridian will, however, be the 

 same. 



41 . Instead of an axis directed to the pole, we may sub- 

 stitute a vertical pin; for if, from any point K in the axis, a 

 perpendicular KL be let fall on the meridian line, and the 

 axis be removed, leaving the vertical line KL, it is evi- 

 dent that the shadow of its top K will come to any hour 

 line at the same instant that the edge of the shadow 

 of the axis CK would have fallen on that line. 



To form this stile, we must, at any point L in the 

 meridian, erect a vertical pin of such a height, that a 

 line drawn from its top to the centre of the dial, may 

 make with the meridian an angle equal to the latitude. 

 In this case the meridian may be a single line if the 

 stile have a sharp point, and then the extremity of the 

 shadow will point out the hour of the day. This kind 

 of stile, however, cannot indicate the hour for some 

 time after sun-rise and before sun-set, because of the 

 shadow extending beyond the limits of the dial. 



The hours may also be indicated by the shadow of 

 any point whatever, provided a line drawn from it to 

 the centre of the dial pass through the pole of the world. 

 Hence the stile may be any ornamental or emblemati- 

 cal figure: for example, Time and the hour may be 

 shewn by the shadow of the point of his scythe, &c. 



42. We shall here give a Table, calculated by thefor- 



4 T 



