DIALLING. 



Theory 

 and Con- 



Vertical 



declining 



dials. 



Vertical 

 south dial 

 declining 

 westward. 



703 



Tan. O = 



cot. L sin. L 



cos. E cos. D-fsin. L sin. iTsin. D 

 Let d be such an arc, that 



Tan. d = sin. L tan. D, 



* vi...,,, 4.u * ™ sin.Lsin.Dcos.rf 



from which it follows, that cos. D = = = 



sin. a 



This value of cos. D being substituted in the formula, 



it becomes 



cot. L. sin. d 



sin. E 



Tan. O _ gin D ~ cos .. £ cos. d+ sin. E sin. d' 

 Now, the denominator of the fraction is evidently 

 cos (E — d), or cos. (d — E). Hence, we have the fol- 

 lowing very simple formula for the calculation of the 

 angle which the shadow of the axis of a vertical de- 

 clining dial describes in any time before or after noon. 

 Let L be the latitude of the place, 

 D the declination of the dial, 

 E the horary angle described by the sun, reckon- 

 ing from noon, 

 O the angle described by the shadow. 



t- i , , , sm - L tan. D. 

 •bind an angle d, such that tan. a= —3 > 



Also a line which we shall call the tangent of an angle 



co- tan. L sin. d. 

 a, such that tan. a =- 



Then, tan. O = 



tan. 



sm. 

 a sin. E 



D. 



(6) 



cos. (E — d) ' 

 for the afternoon hours, 

 tan. a sin. E 



and tan. O = „ „ /T? ", . N for the forenoon hours, 

 cos. (£j-\-a) 



In applying the formula, if the horary arch E be 

 less than the angle d, we may take d — E instead of E 

 — d, for cos. ( E — d) and cos. (d — E) are expressed by 

 the same quantity. 



65. We shall now give examples of the application 

 of the formula. 



Example 1. Let it be required to find the angles 

 which the hour lines make with the meridian in a ver- 

 tical south dial, that declines to the west 36°, the lati- 

 tude of the place being 54i degrees. 



In this example, the dial has the same aspect as that 

 from which the formula has been investigated; for the 

 half of the plane on which the afternoon hours are 

 drawn passes between the meridian and prime vertical, 

 making with it an angle equal to 36° ; hence we have, 

 L=54° 30' ; D=36°. 



Calculation of the angle d ; 



Tan. D 

 Sin. L 



Logarithms. 

 9.86126 

 9.91069 1 



Tan. («/=30°36') 9.77195 



Calculation of the quantity, tan. a. 



Co-tan. L 9-85327 



Sin. d 9.70675 



Sin. D Ar. Comp. 0.23078 



Tan. a 



9.79080 



As it is only the logarithm of the quantity tan. a that 

 we want, we have no occasion to seek for the angle it- 

 self. ° 



Next, to find the angle made by an hour line, as, for 

 example, I in the afternoon, we have E=15°, and d— 

 1i=zl5° 36', and the calculation may stand thus, 



Logarithms. 

 9.41300 

 9-79080 

 0.01630 



Sin. E 



Tan. a 



Cos. (d — E) Ar. Comp 



Tan. (0=9° 26') . . . 9.22010 



Hence it appears, that the hour line of I in the after- 

 noon makes an angle with the meridian of 9° 26'. 



For XI in the forenoon we have E=15°, and dA-E 

 =45° 36'. T 



Sin- E 9-41300 



Tan. a 979080 



Cos. (E + d) Ar. Comp. 0.15511 



Vertical 



declining 



dials. 



Tan. (0=12° 52') 



9.35891 



The angles made by the remaining hour lines, may 

 be found in the same manner ; and these, as well as the 

 data from which they are derived, are expressed in the 

 following Table, which extends from IX in the morn- 

 ing to VIII in the evening, the time during which the 

 dial is illuminated. 



Given L=54.° 30'7 , fd=30° 36' 



D=3G 5 n 



£log. tan. a=9.79080. 



Hours. 



E. 



E=fcrf. 



Angle O. 



IX A. M. 



45° 



75° 36' 



60° 21' 



X 



30 



60 36 



32 10 



XI 



15 



45 36 



12 52 



XII 







30 36 







I P.M. 



15 



15 36 



9 26 



II 



30 



36 



17 10 



III 



45 



14 24 



24 17 



IV 



60 



29 24 



31 34 



V 



75 



44 24 



39 52 



VI 



90 



59 24 



50 30 



VII - 



105 



74 24 



65 44 



VIII 



120 



89 24 



88 52 



This dial is represented in Fig. 6. And it is evi- 

 dent that if the hour lines be produced, and the axis Plate 

 continued on the other side of the plane, (considered p CX * IX ' 

 as transparent,) we shall have a north dial declinine to g " ' 

 the east 36°. 



Example 2. Suppose a vertical south dial to decline Vertical 

 east 49°, in the latitude 51* degrees. sout h dial 



In this case, the plane of the dial passes between the declining 

 north and east points ; therefore, if we reckon the de- east ward» 

 clination from the east towards the south, in the pre- 

 sent case, D=360°-— 49°. Let D'=49°, then D=360 

 — D' ; hence tan. D=— tan. D' ; and since the sign of 

 tan. d depends on that of tan. D, it follows that tan. d is 

 negative, and d between 270° and 360°. Let d'=360 

 — d; then tan. d'=( — tan. d= — sin. L tan. D) = sin. 

 L tan. D', and cos. (d— E)=cos. J3G0 — (rf'-f-E) 1 



=cos.(a?'-f.E);alsocos.(o!-|-E) = :cos. I36O— (d'— E)l 



== cos. (a" — E). Hence, in this case, our formula, (art. 

 64. ) becomes 



n tan. a sin. E 

 tan. 0=- 



: sin. (d' + E) 

 , tan. a sin 



and tan. 0= 



'sin. {tt— E) 



for the afternoon, 



E 



for the forenoon. 



