696 



DIALLING. 



and Con- 

 struction. 



mula of art. 3$. by which a horizontal dial may be con- 

 structed for «vy place in Great Britain. 



A Table of the Angles which the Hour-lines form tvitk 

 theMeridianon a Horizontal Dial/or every half Degree of 

 Latitude, from 50° to 59° 30'. 



Lati- 



A. M. 



A. M. 1 A. M. 



A. M. 



A.M. 



A. M 



tude. 



I. XI. 



II. X. 



11. IX. 



IV.VIII. 



V. VII. 



VI. VI. 



50° ' 



11°38' 



23°51' 



37°27' 



53° 0' 



70°<i3' 



90° 0' 



50 30 



11 41 



24 1 



37 40 



53 11 



70 51 



90 



51 



11 46 



24 10 



37 51 



53 24 



70 58 



90 



51 30 



11 51 



24 19 



38 4 



53 36 



71 6 



90 



52 



11 55 



24 27 



38 14 



53 46 



71 13 



90 



52 30 



12 



24 36 



38 25 



53 58 



71 20 



90 



53 



12 5 



24 45 



38 37 



54 8 



71 27 



90 



53 30 



12 9 



24 54 



38 48 



54 19 



71 34 



90 



54 



12 14 



25 2 



38 58 



54 29 



71 40 



90 



54 30 



12 18 



25 10 



39 8 



54 39 



71 47 



90 



55 



12 23 



25 19 



39 19 



54 49 



71 53 



90 



55 30 



12 28 



25 27 



39 29 



54 59 



71 59 



90 



56 



12 32 



25 35 



39 40 



55 8 



72 5 



90 



56 30 



12 36 



25 45 



39 50 



55 18 



72 12 



90 



57 



12 40 



25 51 



39 59 



55 27 



72 17 



90 



57 30 



12 44 



25 58 



40 9 



55 37 



72 22 



90 



58 



12 48 



26 5 



40 18 



55 45 



72 27 



90 



58 30 



12 52 



26 13 



40 27 



55 54 



72 S3 



90 



59 



12 56 



26 20 



40 36 



56 2 



72 39 



90 



59 30 



13 



26 27 



40 45 



56 10 



72 44 



90 



Use of the 

 Table. . 



In this Table, the angles formed by the lines for V 

 in the morning and VII in the evening, IV in the 

 morning and VIII in the evening,, &c. are not marked, 

 because, it has been already observed, they are the 

 same as those for VII in the morning and V in the 

 evening, VIII in the morning and IV in the evening, 

 only they he on opposite sides of the VI o'clock hour 

 lines. 



The use of the Table may be easily comprehended : 

 If the place for which a horizontal dial is to be made, 

 corresponds with any latitude in the Table, the angles 

 which the hour lines make with the meridian may be 

 seen at once. For example, it appears that the hour 

 lines of XI and I must, in the latitude of 56°, make 

 angles of 12" 32' with the meridian. If the latitude 

 be not contained in the Table, proportional parts may 

 be token without any sensible error. Thus, if the lati- 

 tude be 54° 15', and the angles made by the hour lines 

 of XI or I be required ; as it appears from the Table 

 that the increase of 30' in the latitude, viz. from 54° to 

 54° 30', corresponds to an increase of 4' in the hour 

 angle at the centre of the dial, we may infer, that an 

 increase of 1 5' will require an increase of 2' nearly ; 

 and therefore that the angle required will be 12° 16'. 



Geometrical Construction of Horizontal Dials. 



Geometri- 43. As every geometrical problem admits of various 

 cal con- constructions, so the hour lines on a horizontal dial may 

 struction of be determined in various ways, according to the view 



Sis* " that is taken of the sub J ect - The y ma y ah, however, 

 be deduced from the formula investigated in art. 39, 

 namely, that radius is to the sine of the latitude, as the 

 tangent of the horary angle described by the sun be- 

 tween any hour and noon, is to the tangent of the angle 

 which the hour line on the dial makes with the meri- 

 dian. From this formula we immediately derive, 



Method I. Theoiy 



and Coti- 



44. Let CMO, CMO' (Fig. 10.) be the meridian ^™tkm. 

 line on the dial, the space between CM, CM' being p7*te"""" 

 left for the thickness of the style, and CC its centre, ccxxvm. 

 and 6 C 6 the six o'clock hour line. Fig. lo. 



1 . Make a right angled triangle cmo, Fig. 1 1 . of any F - 

 magnitude, having one of its acute angles c equal to the a ^' 

 latitude of the place. 



2. In the meridian, take CM and CM' equal to c m, 

 the hypothenuse of the triangle, and MO and M'O' 

 equal to m o, the side opposite to the angle c. 



3. Through M, M' draw PQ perpendicular to CO. 



4. On O and O', as centres with OM as a radius, de- 

 scribe quadrants MH, M'H'. 



5. Divide each quadrantal arc into six equal parts. 



6. Through the points of division draw the lines O I, 

 O 2, O 3, &c. also O' 11, O' 10, O' 9, &c. meeting PQ 

 in v, u, x, &c. and in r, s, t, &c. 



7. From the points C, C draw lines C 1 v, C2u, C3 x, 

 &c. to the points v, u, x, &c. and C 1 1 r, C 1-0 s, C 9 1, 

 &c. to the points r, s, t, &c. and these will be the hour 

 lines of the dial, viz. C 1 and C 1 1 will be the hour lines 

 of I in the afternoon and XI in the forenoon, and C 2, 

 C 10 the hour lines of II and X, and so on. 



8. The hour lines before six in the morning, and af- 

 ter six in the evening, are to be found from the ad- 

 joining intermediate hours, as directed in art. 40. 



The demonstration of this construction is obvious ; 

 for in the right angled triangles OM v, CM v, we have 

 CM :Md:: rad. : tan. MC v, 

 and M v : MO : : tan. MO v : rad. 

 Therefore, ex wnuo inv. CM: MO: :ton. MO v : tan. MC v, 

 but CM : MO : : c m : m o : : rad. : sin. lat. ; hence, rad. : 

 sin. lat. : : tan. MO v : tan. MC v. 



Therefore, the angle MC v is rightly determined, 

 (art, 39-) and the demonstration applies alike to all the 

 hour lines. 



This construction, although very simple, is rather 

 inconvenient in practice, because the lines 04, 05, and 

 08, 07, may go off the surface on which the dial is 

 to be delineated, before they meet the line PQ. The 

 next construction has not this defect. 



Method II. 



45. Let CM, CM' be the double meridian line (Fig. Fig. 12. 

 12.), and 6C6 the six o'clock hour line, and let cmo 

 (Fig. 11.), be a right angled triangle, constructed as 

 directed in the first operation of Method I. 



1. On C, C the centres of the dial, with a radius 

 equal to c m, the hypothenuse of the triangle com 

 (Fig. 11.) describe semicircles on opposite sides of the 

 meridian. 



2. On the same centres, with a radius equal to o m, 

 (the side opposite to the angle which is the latitude) 

 describe other two semicircles also on opposite sides of 

 the meridian. 



3. Divide each quadrant of the two semicircles into 

 six equal parts, at the points of division 1, 2, 3, &c. 

 11, 10, 9, &c. and let the numbers be written at the 

 points of division, in the same order, in respect to the 

 meridian, as the characters for the hours are to be 

 placed on the dial. 



4. Then, to find the position of any hour line, as, 

 for example, that for three in the afternoon : Let D be 

 the third point of division on the inner circle, and E 

 the third point of division on the outer circle, reckon- 



