706 



DIALLING. 



Pl\tf. 

 CCXXIX. 



Fig. 12. 



Fig. 11. 



being substituted instead of L and D in the formula 

 art. 64, it will be a general expression for the angle O, 

 which the shadow makes with the meridian line on the 

 reclining dial. 



73. Example. — Let it be required to find the hour 

 lines on a south dial plane FH//, (Fig. 11.) that declines 

 westward 25°, and reclines 15° in latitude 54^. 



In this example, D=25°, 11=15", L=54° 30'. 

 To find A. To find x. 



Rad. . . . 10.00000 Rad 10.00000 



Cos. R . . . 9.98494 Cot. R . . . 10.57195 

 Sin. D . . . 9-62595 Cos. D. . . . 9-95728 



Sin.(A=29°6') 9-61089 Cot.(A— L)= 16° 27' 10.52923 

 Hence a=(a— L) + L=l6 o 27'-f-54»30'=70° 57'. 

 As the dial declines to the west, we have (art. 64.) 



Tan. 0= 



Tan. O: 



tan. a sin. E 



cos. (E — d) 



tan, a sin. E 



"cos. (E + cQ 



T . . sin. >. tan. A 



In these, tan. a= 



for the afternoon. 



for the forenoon. 



rad. 



cot. A sin. d 

 Tan. «= ■ — : . 



9111. A 



By proceeding as in the examples of art. 65, we shall 

 find d, and log. tan. a. 



The remainder of the calculation differs in no re- 

 spect from that of the first example in that article. The 

 Table exhibiting the values of the two series E and 

 Ez+r6?, and the angles O, which the hour lines make 

 with the meridian line, will be as follows : 



Given < .L.o mc hence ■) , 



}_A24 o'Jj (.log- tan - a 



=9-51759 



Hours. 



E 



(E=j=d) 



Angle O 



VII. P.M. 



105° 



82° 5' 



6t>° 34' 



VI. 



90 



67 5 



40 12 



V. 



75 



52 5 



27 21 



IV. 



60 



37 5 



19 39 



III. 



45 



22 5 



14 6 



II. 



30 



7 5 



9 25 



I. 



15 



7 55 



4 54 



XII. noon. 















XI. A.M. 



15 



37 55 



6 10 



X. 



SO 



52 55 



15 15 



IX. 



45 



67 55 



47 47 



VIII. 



60 



82 5.5 " 



GG 36 



The hour lines being drawn on the dial, so as to make 

 with the XII. o'clock line the angles in this Table, the 

 dial will be constructed. It is represented in Fig. 12. 



74. To determine the position of the hour lines by 

 the second method, or by finding the place of the earth 

 where the dial would be horizontal, we must determine 

 the position of the substile in respect of the meridian, 

 and also find the latitude and longitude of the plane. 

 To do this, retaining the construction of Fig. 11, let a 

 great circle PA, passing through the pole, and perpen- 

 dicular to the plane of the dial,' meet it in the line OA, 

 which -will be the substile; then the angle AOH is the 

 angle which the substile makes with the meridian, or 

 XII o'clock hour line, the angle AOP is the elevation 

 of the axis above the plane of the dial, and the spheri- 

 cal angle APH is its difference of longitude. 



In the spherical triangle AMP, right angled at A, 

 the angle AHP being equal to fH s, is measured by 

 the arc fx ; but this arc is the complement of the arc 

 /'E, which we have (art. 72.) expressed by A ; therefore 



the angle AHP is the complement of A. Again, the 

 arc HP is the complement of the arc Pn, which we-have 

 expressed by A. Now we have given formulas (art. 

 72.) for the computation of A and a ; therefore the an- 

 gle AHP, and the side HP of the spherical triangle, may 

 be considered as known. 



The remaining three parts of the triangle may be 

 found from the following analogies. (See Spherical 

 Trigonometry.) 



Rad. : cos. H : : tan. PH : tan. AH, 

 Rad. : sin. PH : : sin. H : sin. AP, 

 Rad. : cos. PH : : tan. H : cot. P. 

 By substituting A and A in these proportions, we find 

 Tan. AH (the angle made by? _ sin. A cot, a 



the substile and meridian) j " ra d. 



Sin. AP (the angle made by 7 cos. A cos. x 

 the axis and substile) '_ 



~ , ,.^> /. , .. Cot. A sin. 

 Cot. P (the diff. of long.) = 



Theory 

 and Con- 



rad. 



rad. 



Plate 

 CCXXIX. 



Fig. 12. 



If we apply these formulae to the example of last ar- 

 ticle, we shall find 



Angle made by sub. and merid. 8° 1' 



Angle made by axis and substile, 17 20 



Diff. of long, of dial plane, 25 20 



The dial may now be constructed as a horizontal dial 

 for lat. 17° 20', and as the difference of longitude is 25" 

 20', which corresponds to l h 41-J-" 1 , and the meridian of 

 the dial plane lies to the east of the XII o'clock hour 

 line, we must find the hour lines of 2 h 41i ra , 3" 41| m , 

 &c. reckoned from the substile of the dial, and consi- 

 der them as the hour lines of XI, X, &c. in the fore- 

 noon ; also we must find the hour lines of 41| m , and on 

 the same side of the substile as the others, and consider 

 it as the hour line of I. The hour line of II will lie 

 on the other side of the substile, and will correspond to 

 18| m . The hour line of III will correspond to l h 18|m 

 from the substile, and so on. See Fig. 12. 



75. We have, for the sake of brevity, given no geo- 

 metrical construction for declining or reclining dials, 

 because in making a dial, every tiling ought, as far as 

 possible, to be determined by calculation, and scales 

 ought to be employed no farther than in laying down 

 the angles. However, if any one should wish for ge- 

 ometrical constructions, he may readily derive them 

 from the formulas, (art. 64. and 72.) which are very 

 simple. 



Of the time at xvJiich the Sun begins or ceases to shine on 

 a given Plane on a given Day. 



76. This is a problem of some importance in dialling, Tune at 

 because by resolving it, we learn what hour lines we which the 

 ought to trace on a dial. Our limits will not allow us sun tegms 

 to enter minutely into the various cases, but we shall °o shinTon 

 briefly indicate how it is to be resolved in the general a K ; ven 

 case, supposing a south reclining plane declining to the plane on a 

 west. Retaining, therefore, the construction of Fig. given day. 

 1 1 , as described in art. 72, let PS be the hour circle pas- p lg . \ )# 

 sing through the sun, when he is in the plane of the 



dial on the afternoon of a given day. In the spherical tri- 

 angle HAP, we have already found AP, the measure of 

 the angle made by the axis and substile, and the angle 

 HPA the difference of longitude Now, in the sphe- 

 rical triangle SAP, right angled at A, besides AP, 

 we know SP, the distance of the sun from the pole on 

 the given day. Hence the angle SPA may be found by 

 this proportion, 



Tan. PS : tan. PA :: rad. : cos. APS. 

 The angle APS expressed in time, is half the period the 



