710 



DIALLING. 



Theory 

 and Con- 



y- COS. 2 (s — X) a* COS. 2 (s — x) 



sin. 2 <p ; 



&■+* 



(D). 



Sin. 2 s sin. 2 ( t — * ) 



Hence, by adding we find, 



y* cos. 2 (e — x)_ a 1 cos. 2 (i — a) 

 sin. 2 £ sin. 2 (s — x) 



But this equation manifestly belongs to an ellipse, the 

 co-ordinates to the axes of which are x and y ; hence we 

 have an elegant property of the hour points on these 

 dials, namely, that they are all in the perimeter of an 

 ellipse, of which 



rm. -j . . a cos. (t — x)~i 

 lne mend, semi-axis = — . - -L I 



™'&? ) r • ■ • ( E ) 



The other semi-axis — — ; — — -' — — \ 

 sin. (s — x) J 



The Anahmmalic or Azimuth Dial. 



The ana. 86. The analemmatic dial is constructed on the prin- 

 lfmmatic ciples which Ave have explained, art. 83 — 86. Its stile 

 or azimuth j s ver tical ; therefore recurring to the notation and for- 

 mulas of these articles, we have £=90°; and hence the 

 equations of the dial are, 



xz=a tan. x cos. <p • 

 y-=.a sec. x sin. <p ; 

 vz=a tan. 2. 

 The dimensions of the ellipse which passes through 

 its hour points, are these, 



Semiconj ormerid. axis=« tan. x ; 

 Semitransverse axis . . =a sec. X ; 



Eccentricity rra. 



The values of x, y, v, are all the elements wanted to 

 Plate construct the dial, either arithmetically or geometricall y. 

 CCXXX. The geometrical construction may be as follows. See 

 Big. 1. Fig. 1. 



1. Draw two straight lines Aa, Bb, intersecting each 

 other at right angles in O. 



2. In OA, one of these lines, take OD of any suitable 

 length for the eccentricity of the dial ■ and at the point 

 D draw DB, so as to make with DO an angle equal to 

 the latitude of the place. Then OB shall be half the 

 lesser axis, and B the 12 o'clock hour point. 



3. In OD take OA and Oa, each equal to DB; and 

 Aa shall be its greater axis, and Aa the six o'clock hour 

 points. 



4. On O as a centre, with OA and OB as radii, de- 

 scribe circles, and divide the quadrants that are in the 

 same angle, each into six equal parts, as in the Figure. 



5. From K, any one of the points of division in the 

 outer circle, draw KL perpendicular to OA ; and from 

 ft, the corresponding point in the inner circle, draw 

 F& parallel to OA, meeting KL in N, which will be 

 one of the hour points, and in the same way may all 

 the others be found, as is shewn in the Figure. 



6. At the point D make angles ODE, ODe each 23| 

 degrees, the sun's greatest declination ; and E, e, shall 

 be the positions of the bottom of the stile, at the sum- 

 mer and winter solstices, the former lying on the north, 

 and the latter on the south side of O, the middle of the 

 dial. 



7. Describe a circle with DO as a radius, and find 

 the tangents of the series of arcs, 1°, 2°, 3°, &c. to 23" 

 of that circle, and lay them as a scale from O to E, 

 and e in each side of O. 



8. Find in the Tables, p. 715. the sun's declination on 

 the first day of every month, and mark the beginning of 

 the month on the scale Ee, opposite to its corresponding 

 degree of declination. As many of the intermediate 

 days as there may be room for, may in like manner be. 

 marked on the scale. 



Theory 

 and Con- 



Plate 

 CCXXX. 

 Fifr 2. 



9. The stile must now, by some mechanical contri- 

 vance, be placed over the scale so as to admit of being 

 moved along it, and set to any day, and the dial is 

 finished. 



To prove that the hour points have their proper po- 

 sition, let x=LN, y=OL., <p=angle HOK ; then be- 

 cause 



rad. : cos. <p : : KO : KL : : K) (=a tan. x) : NL(=i), 

 and rad. : sin. ? : : KO (=a sec. x) : OL (=y) 

 we have x—a tan. x cos. <f>, and y—a see. x sin. <p, as 

 they ought to be. 



This dial was given by Vaulezard, in 1644, in a 

 French work called Traite de I'origine, Demonstration. 

 Construction, et Usage du Quadrant Analemmalique. 

 It made also the principal object of Forster's Elliptical 

 Horologiography, published at London, in 1 654. It is 

 sometimes joined to horizontal dials, to which it is an 

 elegant appendage ; because the shadows on the two 

 dials can, in general, only indicate the same hour, when 

 both stiles are in the plane of the meridian ; and hence 

 the compound dial can be placed in a proper position, 

 without the help of a compass or meridian line. Fig. 2. 

 represents a dial of this construction. It has the ad- 

 vantage of not being subject to the error of refraction. 



LamberCs Dial. 



87. M. Lambert remarked in the Berlin Ephemeri- Lambert's 

 des for 1777, that a dial, with a moveable centre, dial. 

 might be constructed, in which the hour points should 

 stand at equal distances, on the circumference of a cir- 

 cle. It is easy to see that this is possible ; for the ge- 

 neral expressions (E), for the semiaxes of the ellipse in 

 which the hour points are situated in this kind of dial, 

 being, by art. 85, 



a cos. (i — x) a sin. s 



sin. (j — x) ' sin. (s — x) ' 



if we suppose i, the elevation of the stile, to be such 

 that cos. (s — A)=sin. t; the axes of the ellipse will be 

 equal, and it will become a circle. This condition re- 

 quires that s — A-f-£=90° ; from which we find, 

 t=l(90 + x); 

 v=.a tan. % ; 

 rad, of dial=a tan. ^ (90 -f. x). 

 In the latitude of Edinburgh, which is 55° 58', the 

 elevation of the stile of a dial of this kind would be 

 !(90°-|-55° 58')=72° 59'. The geometrical construction 

 of this dial is extremely simple. 



1. Take a straight line OD (Fig. 3.) of any length, F| ^ 

 and at O one of its extremities draw OB perpendicular 



to it. 



2. At the point D, make the angle ODB equal to 

 half the sum of 90°, and the latitude of the place ; and 

 OB will be the radius of the dial. 



3. Describe a circle on O as a centre, and divide each 

 quadrant into six equal parts, and the points of division 

 will be the hour points of the dial. 



4. Draw two lines DE, De, and make a scale of tan- 

 gents of the sun's declination from O to E and e, and 

 against the divisions of the scale write tlie days of the 

 month, as described in the analemmatic dial. 



5. Place the stile over the meridian, so that it may 

 admit of being adjusted to the time of the year exactly, 

 as in the analemmatic dial, and so that it may make 

 with the horizon an angle equal to BDO, and the dial 

 will be constructed. 



Note. The style must be on the north or south side 

 of the point O, according as the sun is on the north or 

 south side of the equator. 



If the shifting of the position of the stile in these di- 



