

704 D I A L L 



Theory 



;md Con- 



. struction. 



To find d' 



Tan. D' .... 10.06084 

 Sin. L 9-89354 



Vertical 



declining 



dials. 



Tan. (d'=42°) 9-95438 



To find tan. a, 



Cot. L 9.90061 



Sin. d' 9.82551 



Sin. D' Ar. Comp. 0.12222 



Theory 

 and Con- 

 struction;- 



Plate 

 CCXXIX. 



F>g. T. . 



Tan. a 



9.84834 



From the angle d a series of angles d-f" 15°, e?+ 30°, 

 &c. is to be formed for the afternoon hours, and ano- 

 ther d — 15, d — 30, &c. for the forenoon hours, and 

 from these the hour angles at the centre of the dial 

 may be calculated exactly, as in the last example. The 

 data and the results obtained are given in the following 

 Table: 



Given L=51° 



30'? . 

 0$ h< 



Cd'=42° 

 nee -J , 



{.log. tan 





D'=49 



, a— 9.84834. 



Hours. 



E. 



Ezfcd'. 



Angle 6. 



Ill A.M. 



135° 



93° 



95° 59' 



IV 



120 



78 



71 12 



V 



105 



63 



56 19 



VI 



90 



48 



46 30 



VII 



75 



33 



39 5 



VIII 



60 



18 



32 42 



IX 



45 



3 



26 32 



X 



so 



12 



19 49 



XI 



15 



27 



11 35 



XII 







42 







I P.M. 



15 



57 



18 32 



II 



30 



72 



48 47 



This dial is represented in Fig. 7. If the hour lines 

 were produced, and the axis continued through the 

 plane, we would evidently have a north dial declining 

 westward. 



In general, to make a north declining dial, we have 

 only to make a south declining dial, whose declination 

 is the same, arid lies the same way, and then turn it 

 upside down, and it will be a north declining dial ; but 

 the hours must be numbered the contrary way ; so that 

 the two examples we have given will apply to all the 

 varieties of declining dials, 



66, The formula which has been investigated in ar- 

 ticle 64, gives the position of the hour lines immediate- 

 ly, when the latitude of the place and declination of the 

 plane are known. But there is another method of con- 

 structing a declining dial, by considering it as a hori- 

 zontal dial for some place of the earth, as explained in 

 art. 52, and finding the latitude of that place, and the 

 difference between its longitude and that of the place 

 where the dial is to shew the hours. The first of these 

 is equal to the angle which the axis of the dial makes 

 with its plane, that is, the angle which the axis makes 

 with the substile ; and the second is the time which 

 the shadow takes to pass from the XII o'clock hour 

 line to the substile, from which the angles contained 

 by these lines may be found. These three elements 

 being known, viz, the latitude and longitude of the 

 place where the dial would be horizontal, and the angle 

 made by the twelve o'clock hour line and substile, the 

 construction of the dial 19 reduced to that of a horizon- 

 tal dial, 



ING. 



67. The elements for constructing the dial in this 

 way, may be found by spherical trigonometry, as fol- 

 lows : "*— "" Y"""* ' 



Let SN z be the meridian, in which Z and 2 are the Vertical 

 zenith and nadir, and P,p the poles. Let SEN be the declining 

 horizon, S and N being the south and north points, dials - 

 and E the east. Let ZF z be any vertical plane, or Plate 

 great circle, on which the dial is to be drawn ; let this £ CX * IX- 

 plane meet the horizon in F, and the meridian in the Bl S- 8 - 

 vertical line ZOz, and let it be cut perpendicularly in 

 A a by the plane of a great circle PA p passing through 

 the poles. Then, from the construction of the figure, 

 it appears that O p or OP is the axis of the dial, (ac- 

 cording as it faces towards the south or north), O z or 

 OZ the twelve o'clock hour line ; and O a or OA the 

 substile, (Art. 58.) 



In the right angled spherical triangle ZAP, (of which 

 A is the right angle) PZ is the latitude of the place 

 where the dial is to indicate time ; the angle AZP, 

 which is measured by the arch of the horizon FN, is 

 the complement of EF, the declination of the plane ; 

 and these are both given to find AP the measure of the 

 angle contained by the axis OP and substile OA, also 

 AZ the measure of the angle which the substile makes 

 with the vertical or twelve o'clock hour line ; and, last- 

 ly, the angle ZPA, which is the difference of longitude 

 of the plane, (Art. 60.) 



By the principles of spherics, (see Trigonometry, 

 Spherical), in the triangle ZAP, 



Pad. : sin. ZP : : sin. Z : sin. AP 

 Rad. : cos. Z : : tan. ZP : tan. AZ 

 Rad. : cos. ZP : : tan. Z : cot. P. 



Hence, putting L for the latitude of the place where 

 the dial is to be constructed, and D for the declination 

 of the dial, we get 



Rad. : cos. L . : cos. D : sin. lat. of dial, 



Rad. : sin.D : : cot.L : tan.of angle made by Sub.&Ver. 



Rad. : sin, L : ; cot. D : cot. dif. of long, of dial. 



Thus, it appears, that the dial would be horizontal at 

 a place of which the sine of the latitude is cos. L cos. D, 

 and the cotangent of its difference of longitude equal 

 to sin. L cot. D ; and, moreover, that the tangent of 

 the angle contained by the meridian line traced on the 

 dial for that place, and the vertical or twelve o'clock hour 

 line, where it is to be fixed, is equal to Sin. D cot. L ; 

 and from these three data the dial may be constructed, 

 either arithmetically or geometrically, by the formu- 

 la of art. 39, or the rules of art. 43—47, for a ho- 

 rizontal dial. And the hour lines must be so calcu- 

 lated, that one of them, to be taken as the twelve o'clock 

 hour line, may coincide with the vertical line. To do 

 this, we must form a series of arcs, by the repeated 

 addition of 15° to the difference of longitude, on the 

 one hand, and by the repeated subtraction of 15° 

 on the other, until at last there be a remainder less 

 than 15*. Then we must take the difference of lon- 

 gitude and these arcs as the successive values of E 

 in the formula of art, 39, and the corresponding va- 

 lues of C will be the angles which the hour lines on 

 one side of the substile make with it, and the angle 

 corresponding to the difference of longitude, will 

 evidently be equal to the angle the vertical line 

 makes with the substile. Again, by adding repeat- 

 edly 15 degrees to the difference between 15°, and the 

 remainder left in forming the first series j this diffe- 

 renee and the succeeding terms of the series being put 

 1 



