Page:Encyclopædia Britannica, Ninth Edition, v. 7.djvu/179

161 DIALLING 161 Itself, l.y subdivision of the small arcs Ar, rs, st, &c., we may draw the hour lines corresponding to halves and quarters, but this only where it can be done without confusion. Draw ASD making with AC an angle equal to the latitude of the place, and let it meet EG in D, through which point draw FDG at right angles to AD. With centre A, and any convenient radius AS, describe an arc of circle RST, and graduate this arc by marking degree divisions on it, extending from at S to 23 on each side at ft and T. Next determine the points on the straight line TDG where radii drawn from A to the degree divisions on the arc would cross it, and care fully mark these crossings. The divisions of RST are to correspond to the sun s declination, south declinations on RS and north declinations on ST. In the other hemisphere of the earth this would be reversed ; the north declinations would be on the upper half. Now, taking a second year after leap year (because the declina tions of that year are about the mean of each set of four years), find the days of the month when the sun has these different declina tions, and place those dates, or so many of them as can be shown without confusion, opposite the corresponding marks on FDG. Draw the sun-line at the top of the card parallel to the line ACB ; and, near the extremity, to the right, draw any small figure intended to form, as it were, a door of which a b shall be the hinge. Care must be taken that this hinge is exactly at right angles to the sun-line. Make a fine open slit c d right through the card and extending from the hinge to a short distance on the door, the centre line of this slit coinciding accurately with the sun-line. Now, cut the door completely through the card ; except, of course, along the hinge, which, when the card is thick, should be partly cut through at the back, to facilitate the opening. Cut the card right through along the line FDG, and pass a thread carrying a little plummet W and a very small bead P; the bead having sufficient friction with the thread to retain any position when acted on only by its own weight, but sliding easily along the thread when moved i &amp;gt;y the hand. At the back of the card the thread terminates in a knot to hinder it from being drawn through ; or better, because giving more friction and a better hold, it passes through the centre of a small disc of card a fraction of an inch In diameter and, by a knot, is made fast at the back of the disc. To complete the construction, with the centres F and G, and radii FA and GA, draw the two arcs AY and AZ which will limit the hour lines ; for in an observation the bead will always be found between them. The forenoon and afternoon hours may then be marked as indicated in the figure. The dial does not of itself dis criminate between forenoon and afternoon ; but extraneous circum stances, as, for instance, whether the sun is rising or falling, will settle that point, except when close to noon, where it will always be uncertain. To rectify the dial (using the old expression, which means to pre pare the dial for an observation), open the small door, by turning it about its hinge, till it stands well out in front. Next, set the thread in the line FG opposite the day of the month, and stretching it over the point A, slide the bead P along till it exactly coincide with A. To find the hour of the day, hold the dial in a vertical position in such a way that its plane may pass through the sun. The verti- cality is ensured by seeing that the bead rests against the card without pressing. Now giadually tilt the dial (without altering its vertical plane), until the central line of sunshine, passing through the open slit of the door, just falls along the sun-lii.e. The hour line against which the bead P then rests indicates the time. The sun-line drawn above has always, so far as we know, been used as a shadow-line. The uppar edge of the rectangular door was the prolongation of the line, and, the door being opened, the dial was gradually tilted until the shadow cast by the upper edge exactly coincided with it. But this shadow tilts the card one- il uarter of a degree mere than the sun-line, because it is given by that portion of the sun which just appears above the edge, that is, by the upper limb of the sun, which is one-quarter of a degree higher than the centre. Now, even at some distance from noon, the sun will sometimes take a considerable time to rise one-quarter or a degree, and by so much time will the indication of the dial be in error. The central line of light which comes through the open slit will be free from this error, because it is given by light from the centre of the sun. The card-dial deserves to be looked upon as something more than a mere toy. Its ingenuity and scientific accuracy give it an educa tional value which is not to be measured by the roughness of the results obtained, and the following demonstration of its correct ness will, it is hoped, usefully close what we have to say on this subject. Demonstration. Let II (fig. 9) be the point of suspension of the plummet at the tim-3 of observation, so that the angle DAH is the north declination of the sun, P, the bead, resting against the hour- line VX. Join CX, then the angle ACX is the hour angle from noon given by the bead, and we have to prove that this hour-angle is the correct one corresponding to a north latitude DAC, a north declination AH and an altitude equal to the angle which the sun-line, or its parallel AC, makes with the horizontal. The angle PHQ will be equal to the altitude, if HQ be drawn parallel to DC, for the pair of lines HQ,HP will be respectively at right angles to the sun-line and the horizontal. Draw PQ and KM parallel to AC, and let them meet DCE in M and N respectively. Let HP and its equal HA be represented by a. Then the follow- ing values will be rdadily deduced from the figure : AD = a cos. dccl., DH = a sin. decl., PQ = a sin. alt. CX =. AC = AD cos. lat. a cos. decl. cos. lat. PN = CV=CX cos. ACX = a cos. decl. cos. lat. cos. ACX. NQ =MH=DHsin.MDH=a sin. -led. sin. lat. {: the angle MDH - DAC =. latitude). And, since PQ = NQ + PN, we have, by simple substitution, a sin. alt. a sin. dccl. sin. lat. +a cos. de,cl. cos. lat. cos ACX ; or, dividing by a throughout. sin. alt. =;sin. dccl. sin. lat. -f cos. decl. cos. lat. cos. ACX . . . (At wuich equation determines the hour angle ACX shewn by the bead. To determine the hour-angle of the sun at the same moment, let Fig. .10. fig. 10 lepresent the celestial sphere, HR the horizon, P the ^olo, and Z the zenith, and S the sun. VII. 2I