DIALLING. 



099 



Construc- 

 tion of 

 dialling 



scales. 



Plate 

 ccxxviii. 

 Pig. 13. 



ed from the meridian on the quadrant through which 

 the afternoon hour lines are to pass. Draw EBA per- 

 pendicular to the meridian, and DB parallel to it, meet- 

 ing the perpendicular in B. 



5. Draw a straight line from C through B, and the 

 line CB will be the hour line for III in the afternoon, 

 as required. 



And in the very same way may all the other hour 

 lines be drawn on the dial. 



To prove the truth of this construction, let EB meet 

 the meridian in A, and join EC, which will evidently 

 pass through D. Because BD is parallel to AC, CE : 

 CD : : AE : AB ; but by construction, CE : CD : : rad.: 

 sin. lat. ; and, by trigonometry, AE : AB : : tan. ACE : 

 tan. ACB ; therefore, rad. : sin. lat. : : tan. ACE : tan. 

 ACB ; now ACE is equal to the horary angle which 

 the sun describes in three hours ; therefore CB is the 

 hour line for three in the afternoon, (art. 39.) 



Construction of Dialling Scales. 



46. There is another very elegant geometrical con- 

 struction for the hour lines, by which scales may be 

 made for the construction of dials, which save the la- 

 bour of dividing circles. 



To construct these scales, divide AB a quadrant of 

 a circle, into six equal parts. Draw the line b a to 

 touch the middle of the arc at G. Draw lines from 

 the centre through A and B, the extremities of the arc, 

 to meet the tangent in a and b, and also through the 

 divisions, to meet the tangent in the points against 

 which ih p numerals VI, V, IV, &c. are placed. Then 

 the line between ni£ fxtreme points a and b is the scale 

 of hours. 



Next, divide EF, a quadrant of the same uiTC;?; into 

 90 equal parts, (only every tenth division is marked 

 in the Figure). From the points of division draw per- 

 pendiculars to OF, the radius. Draw lines through E 

 and the bottoms of the perpendiculars, and produce 

 them, until they meet the circumference again in the 

 points 10, 20, 30, &c. Transfer the chords of the 

 arcs D 10, D 20, D 30, &c. (also the chords of the in- 

 termediate arcs not distinguished in the Figure) to a 

 straight line df, numbering them as in the Figure ; and 

 the line df will be the scale of latitudes. 



If the chords of all the arcs from 0° to 90° of the 

 quadrant EF be transferred to another straight line ef, 

 a scale of chords will be formed, which is frequently 

 wanted in making dials. 



Construction of a Horizontal Dial by the Scales. 



Construe- ^' Let CM ' C ' M ' be the meridian > «°^ 6 C ' C ' 6 the 

 tion of a" s i x o'clock hour line. (Fig. 14-.) 



horizontal 1. From the scale of latitudes take the extent from 

 dial by the the beginning of the scale to the division correspond- 

 scales. m g to t h e latitude of the place for which the dial is to 



J'lg. 14. jj e mac j Cj an j se t ^ ff f rom c to a, and from C to a'. 



2. From the points a, a', place lines a b, a' b', each 

 equal to the whole length of the scale of hours, to ter- 

 minate at b and b' in CM, CM', the meridian line. 



3. Transfer the divisions of the scale of hours to the 

 lines a 6, a' b', numbering them as in the Figure. 



4. From the points C, C draw the lines C 1, C2, 

 C 3, &c. also C 1 1, CIO, C 9, &c. and these will be 

 the hour lines of the dial. 



The morning hours before VI, and evening hours 

 after VI, are found as explained in the other construc- 



tions. And the stile is to be formed in all respects as Theory 

 described in art. 40. a » d (; ? n ~ 



To demonstrate the truth of this construction, let ^ uct1 ""' 

 the latitude for which the dial is made be equal to the p LAT £""™ 

 number of degrees in the arcEp, (Fig. 13.) Then, ccxxviii. 

 p q being drawn perpendicular to OF, and E q drawn fig. 13, u. 

 meeting the circle in r, and D r joined ; it is mani- 

 fest from the construction of Fig. 13. and Fig. 14. that 

 the triangle DrE ( Fig. 1 3. ) is in all respects equal to 

 the triangle a C 6 (Fig. 14.) so that D r=C a, rE=Cb, 

 and DE=« b ; and since in Fig. 13. rad. : sin. lat. :: EO 

 : O q : : E r : r D ; therefore, in Fig. 14. rad. : sin. lat. : : 

 b C : C a. 



Let H (Fig. 14.) be the point in which any one of the 

 hour lines (for example that for IV in the afternoon) 

 meets a b. In the six o'clock line, place CN equal to C b ; 

 join b N, and through H draw KHL parallel to CN, 

 meeting the meridian in K, and the line b N in L ; and 

 join CL. And because N b and a b are similarly divid- 

 ed at L and H, and a H and H b in Fig. 14. are re- 

 spectively equal to a IV and IV b in Fig. 13 ; there- 

 fore N b in Fig. 14. and a b in Fig. 13. are similarly di- 

 vided at L and IV. Now the triangles NC b (Fig. 14.) 

 and a O b ( Fig. 1 3.) are manifestly similar ; therefore 

 it is easy to see that the angle b CL in Fig. 14. must be 

 equal to b OIV in Fig. 13 ; and hence bCL in Fig. 14. 

 must be equal to the horary angle described by the sun 

 between noon and IV in the afternoon. 



Now LK = b K : HK : : tan. LC b : tan. HC b. But 

 b K : HK : : b C : C a : : rad. : sin. lat. ; therefore rad. : 

 sin. lat. : : tan. hor. ang : tan. HC b. Hence it follows, 

 (art. 39.) that the angle which the hour line HC, or 

 IVC, makes with the meridian, is of the proper mag- 

 nitude : and the same may be proved in like manner 

 of all the others. 



Construction of Jio?i:? ntal Dials ty a Glohe - 



48. The construction of a horizontal dial, and f"" c °n s! ™<^ 

 deed of any dial whatever, as will appear farther on, M 00 01 ■*- 

 may be very naturally deduced from the doctrine of the au'bva 

 sphere. For, let aVBp (Fig. 15.) represent the earth, gi be. 

 which we may suppose transparent, and let its equator Fig. 15. 

 be divided into 24 equal parts by meridian circles a, b, 

 c, d, e, &c. one of which is the geographical meridian of 

 any given place, as Edinburgh, which we may sup- 

 pose at the point a. If now the hour of 12 were 

 marked at the equator, both upon that meridian and 

 the opposite one, and all the rest of the hours in order 

 on the other meridians, they will be the hour circles of 

 Edinburgh, and the sun will move from one of them 

 to another in an hour. 



Now, if the sphere had an opaque axis, terminating at 

 the points P p, the shadow of the axis, which is in the 

 same plane with the sun and each meridian successively, 

 would fall upon every particular meridian, and hour, 

 when the sun came to the opposite meridian, and would 

 therefore shew the time at Edinburgh, and all other 

 places on the same meridian. If the sphere were now 

 cut through the middle, by a plane ABCD, in the ra- 

 tional horizon of Edinburgh, one half of the axis would 

 be above the plane, and the other half below it ; and if 

 straight lines were drawn from the centre of the plane 

 to those points where its circumference is cut by the 

 hour circles of the sphere, those lines would be the 

 hour circles of an horizontal dial for Edinburgh ; for 

 the shadow of the axis would fall upon each hour line 



