DIALLING. 



705 



Theory 

 and Con- 

 struction. 



To find the 

 declination 

 of a plane. 



Plate 

 CCXXIX. 



Fig. 9. 



Inclining 



dials. 



yig. ic. 



for E, will give the angles the hour lines make with 

 the substile on the other side. The reason of this is 

 too evident to require particular explanation. 



The ingenious diallist may verify the calculations of 

 the angles in the two examples of our first method by 

 this second method ; and he will find, that in the first 

 example, the angle made by the axis and substile is 

 28° 1', the angle made by the substile and vertical is 

 22° 4 V, and the difference of longitude 41" 4.5'. 



In the second example, the axis makes with the sub- 

 stile an angle of 24° 6', the substile with the vertical 

 an angle of 30° 59', and the difference of longitude is 

 55° 47'. 



The dial of Ex. 1 . would therefore be a horizontal 

 one for the lat. 28° 1', and that of Ex. 2. for the lat. 

 2i° 6'. 



Tojind the Declination of a Plane. 



68. We shall suppose that CO, the axis of the dial, 

 (Fig. 9-) has been fixed, according to the method ex- 

 plained in art. 32. and that the meridian line O XII has 

 been traced, as the foundation of all the other opera- 

 tions. From C, the extremity of the axis, let fall a 

 perpendicular CB on the meridian line ; and mark the 

 point B where it meets the line ; then through B trace 

 a horizontal line h BH on the plane of the dial, and 

 mark several points on it, such as H, //, equally dis- 

 tant from the point B. This done, measure very 

 accurately all the sides of the triangles CBH, CB h ; 

 and in each calculate the angle at B. These angles 

 ought to be the supplements of one another, and hence 

 we may prove the accuracy of the observation. If they 

 are both right angles, the plane is perpendicular to the 

 meridian, and has no declination ; but if they are un- 

 equal, the difference between each and a right angle is 

 the declination of the plane. 



The declination may also be easily found by placing 

 a mark at a great distance in the plane of the dial, and 

 then determining the position of this mark in respect 

 of the meridian line traced on the horizontal plane ; an 

 operation which may be performed by any instrument 

 for measuring angles. Or otherwise, the eye may be 

 placed in the direction of the plane, so as to observe 

 the instant when some one of the heavenly bodies passes 

 the plane, and the hour of this observation being no- 

 ted, the azimuth of the plane may be easily calculat- 

 ed. 



Inclining Dials. 



6*9. Inclining or oblique dials are traced on planes 

 which stand at oblique angles to the horizon. They 

 are either reclining or jrroclining. A reclining plane 

 is one that leans backwards from an observer, and a 

 proclining plane, which is also sometimes called an in- 

 clining plane, leans forwards. 



The reclination and proclination of a plane is the 

 angle it makes with a vertical plane, or it is the num- 

 ber of degrees that the plane leans from or to an ob- 

 server, reckoned from the zenith. But inclination is 

 properly the angle a plane makes with the horizon. 



70. Whatever be the situation of a plane on which a 

 dial is to be made, it is always possible to trace upon it 

 a meridian line OB, in the direction of twop'umb-lines, 

 as explained in art. 32 ; and to fix at O, a point in that 

 line, (Fig. 10.) an axis OC, pointing to the pole of the 

 world. From C, the extremity of the axis let a per- 

 pendicular BC be drawn to the meridian line, meeting it 



VOL. VII. PART 11. 



Fig. 10. 



in B, and forming the right angled triangle COB. As 

 the lines CB, BO, may be measured, the angle COB 

 may be found. 



Now, if we suppose the plane to be a vertical dial at 

 some place of the earth, then OB will be a vertical line ^p^xix 

 at that place, and the angle COB the complement of its 

 latitude, which will therefore be known. If, in addi- 

 tion to this, we knew the declination of the plane, we 

 might calculate the angles which the hour lines make 

 with the meridian line, by the formula of art. 62. But 

 it is easy to find the declination ; for the horizon ought 

 to be perpendicular to the vertical line OB. Now BC 

 is also perpendicular to OB ; therefore, if in //BH, a 

 perpendicular to OB, we take BH=B/;, and join CH, 

 Ck, the triangles CBH, CB/«, will lie in the hyj)otheti- 

 cal horizon, and the inclination of the plane to the hy- 

 pothetical meridian OBC may be determined, as shewn 

 in art. 66. 



71. If, besides the position of the meridian and the 

 axis, we know the declination and reclination of a plane, 

 then we may find the latitude of a place, where the 

 plane would be a vertical declining dial, and also the 

 declination of the dial at that place, and with these da- 

 ta find the angles which the hour lines make with the 

 meridian by the formula for vertical dials, given in art. 

 62. Or else we may find the latitude and longitude of 

 a place, where the plane would be a horizontal dial, and 

 thence find the position of the substile, and construct 

 the dial by the rules for a horizontal dial. 



72. To begin with the first of these methods. Let 

 SHNA be the meridian ; (Fig. 11.) Vp the axis of p;„ 

 the sphere; Z the zenith of the place where a dial is to 

 be made; SEN the horizon; S, E and N, the south, east 

 and north points respectively. Also, let AFH be a 

 plane or circle of the sphere on which the dial is to be 

 made, and which meets the horizon in F, and intersects 

 the plane of the meridian in the line H h ; then POp 

 will be the axis of the dial, and HO^ the meridian line. 

 Moreover, the arch EF between the east point of the 

 horizon and the plane will be its declination ; and the 

 spherical angle HFN its inclination to the horizon, or 

 the complement of its reclination from the vertical po- 

 sition. 



Let sEra be the horizon of the place which has H for 

 its zenith, (and of course where the plane HFA would 

 be a vertical declining dial,) and let it cut the plane in 

 f. Because the two horizons SEN, «Em are perpendicu- 

 lar to the meridian, their intersection E will be the east 

 point in both. 



Let L=PN, the given latitude of the place where the 

 dial is to be made. 

 D=FE, the given decimation of the dial. 

 R=Comp. of angleyFE, its given reclination. 

 *=zPw, the latitude of the place where the dial 



would be vertical, which is to be found. 

 A=:E/* its declination there, which is also to be 

 found. 

 Then x — L=Nn= measure of angle FE/* 



In the spherical triangle E/F, right angled at f, we 

 have, by spherics, 



Rad. : sin EF :: sin. F : sin. TLf, 

 Rad. : cos. EF : : tan. F : cot. E. 

 From these proportions, we get 

 cos. R sin. D \ 



Sin. A: 



rad. 



, . - cot. R cos 

 Cot.(*_L)= ^ 



■ »y 



(7) 



From these equations, we get the arcs A and A, which 

 4u 



