DIALING.

145

line of It is to be drawn. The tangent of 2° 8' (the difference be-tween 45° and 50° .Vi') placed on the other side of CL, will deter-mine the point through which the hour-line of 111 is to be drawn:to winch 2° 8’, if the tangent of 15° be added, it will make 17° S';and this set off from K towards Q on the line EQ, will give the

f ioinl for the hour-line of IV': and so of the rest. —The forenooniour-lines are drawnahe same way, by the continual addition ofthe tangents 15°, 30°, 45°, &c. to 42° 52' (=lhe tangent KM) forthe hours of XI, X, IX, & c. as far as necessary; that is, untilthere be live hours on each side of the substile. The 6th hour,accounted from that hour or part of the hour on which the substilefalls, will be always in a line perpendicular to the substile, anddrawn through the centre C. 4. In all erect dials, CM, the hour-line of XII, is perpendicular to the horizon of the place for whichthe dial is to serve; for that line is the intersection of a verticalplane with the plane of the meridian of the place, both w hich areperpendicular to the plane of the horizon: and any line HO, or ho,perpendicular to CM, will be a horizontal line on the plane of thedial, along which line the hours mav he numbered ; and CM beingset perpendicular to the horizon the dial will have its true position.5. If the plane of the dial had declined by an equal angle towardthe east, its description would have differed only in this, that thehour-line of XII would have fallen on the other side of the sub-stile CL, and the line IIO would have a subcontrarv |>osition towhat it has in this figure. 6. And these two dials, with the upperpoints of their stiles turned tow ard the N. pole, will serve forother two planes parallel to them; the one declining from the N.toward the E. and the other from the N. toward the W. by thesame quantity of angle. The like holds true of all dials in general,whatever be their declination and obliquity of their planes to thehorizon.

RecuNF.RS. If the plane ot the dial not only declines, butalso reclines. Suppose its declination from fronting thesouth S be equal to the arc. SI), tig. 20, on tile horizon; and itsredination be equal to the arc Dri of the vertical circle DZ: thenit is plain, that if the quadrant of altitude Zril) on the globe, cutsthe point D in the horizon, and the redination is counted uponthe quadrant from 1) to ri; the intersection of the hour-circlePRri, with the equinoctial VVQE, will determine lid, the latitude■of the place d, whose horizon is parallel to the given plane Z/i atZ; and HQ will be the difference in longitude of the places at dand Z. Trigonometrically thus; Let a great circle pass throughthe three points, \V, d, E; and in the triangle WDri, right-angledat I), the sides WD-and Off are given; and thence the angle I) Wriis found, and so is the hypotlienuse Wri. Again, the difference,or the sum, of DWri and 1)W K, the elevation of the equinoctialabove the horizon of Z, gives the angle riW K ; anil the bypothe-uuse of the triangle A'Kri was just now found; whence the sidesHri and VV R are found, the former being the latitude of the placed, and the latter the complement of HQ, the difference of longi-tude sought. Thus, if the latitude of the place Z be 52“ 30'north;the declination SI) of the plane /h (which would be horizontal atd) he 3b°, and the redination be 15°, or equal to the arc 1) d; thesouth latitude of the placed, that is, the arc Kd, will he 15° 9';and RQ, the difference of the longitude, 30° 2'. From these data,therefore, let the dial, tig. 22, be described, as in the former ex-ample. 8. There are several things requisite in the practice ofdialing ; the chief of which shall be given in the form of arithme-tical rules, simple and easy to those who have learned the ele-ments of trigonometry. For in practical arts of this kind, arith-.mctic should he used as far as it can go; and scales never trustedto, except in the final construction, where they are absolutely ne-fcessaryin laying down the calculated hour-distances on the planeof the dial. The latitude of the place, the Sim’s declination, andhis hour-distance from the meridian, being given, to find, 1st, hisaltitude, 2d, his azimuth. 1. Let d, iig. 20, be the sun’s place,d R, his declination ; and in the triangle PZri, I’d the sum, or thedifference, of rill, and the quadrant PR, being given by the sup-position, as also the complement of the latitude PZ, and the angledPZ, which measures the horary distance of d from the meridian;tve shall (by spheric trigonometry) find the base Zri, which is thesun’s distance from the zenith, or the complement of his altitude.And 2. as sine Z d : sine Prf : : sine riPZ : riZP, or ef its supple-ment DZS, the azimuthal distance from the south. Or the practi-cal rule may be as follows: Write A for the sine of the sun’s alti-tude, L and l for the sine aad co-sitie of the latitude, 1) and d for▼ol. it.— no. 65.

the sine and co-sine of the sun’s declination, and II for the sine ofthe horary distance from VI. Then the relation of II to Avlillhave three varieties. 9. When the declination is toward the ele-vated pole, and the hour of the day is between XI1 and VI; it isA —LI),

A = LD-f-II/ri, and 11 =-. 2. When the hour is after

Id

LI) — A,

VI, it is A = LD — Hid, and II =-. 3. When the de-

Id

clinatiou is toward the depressed pole, we have A = Il/ri — LD,A-f LD,

and 11=-. These theorems will be found useful, and

Id

expeditious enough for solving those problems in geography anddialing, which depend on the relation of the sun’s altitude to thehour of the day.

Example 1. Suppose the latitude of the place to be 5 ly degreesnorth: the time five hours distant from XII, that is, an hour afterVI in the morning, or before VI in the evening; and the sun’s de-clination 20° north. Required the sun’s altitude?

Then to log. L = !og. sin. 51° 30' — 1.89354add log. D = log. sin. 20° 0' — 1.53405

Their sum — 1.42759 givesLI) = logarithm of 0.267664, in the naturaUinesAnd, to log. 11 = log. sin. 15° 0' — 1.41300,, ( log. I = log. sin. 38° 0' — 1.79414* U (log. ri = log. sin. 70° O'— 1.97300

Their sum — 1.18014 gives

If/ri = logarithm of 0.151408, in the natural sines. And thesetwo numbers (0.267664 and 0.151408) make 0.419072 = A;which, in the table, is the nearest natural sine of 24° 47', the sun’saltitude sought. In these calculations the radius is considered asunity, and not 10,00000, by which, instead of the index 9, wehave—1, which only makes the work a little easier. The samehour-distance being assumed on the other side of VI, then LD —■11/ri is 0.116256, the sine of 6° 401'; which is the sun’s altitude atV in the morning, or VII in the evening, when hisN. declinationis 20°. But when the declination is 20° S. (or towards the depress-ed pole) the difference 11/ri — Ll) becomes negative; and there-by shews, that an hour before VI in the morning, or past VI inthe evening, the sun’s centre is 6° 40£' below the horiz.on.

Examp. II. From the same data, to find the sun’s azimuth. IfII, L, and 1), are given, then from II living found thealtitude and its complement Zri: and the arc Pri (the distancefrom the pole) being given ; say, as the co-sine of the altitude isto the sine of the distance from the pole, so is the sine of the hour-distance from the meridian to the sine of the azimuth distance fromthe meridian. Let the latitude be 51° 30'.north, the declination15° 9' Si, and the time 11 h. 24 m. in the afternoon, when the surbegins to illuminate a vertical wall, and it is required to find theposition of the wall. Then, by the foregoing theorems, the com-plement of the altitude will he 81° 32Jf', and Pri, the distance fromthe pole being 109° 5', and the horary distance from the meridian,or the angle riPZ, 36°.

To log. sin. 74° 51' - — 1.984(54

Add log. sin. 36" 0' - - — 1.7692'’

And from the sumTake log. sine b 1° 32'-£

1.75386

1.99525

Remains — 1.75S61 = log. sin;35°, the azimuth distance sought. When the altitude is given,find from thence the hour, and proceed as above. 'Phis praxis isof singular use on many occasions; as, 1. in finding the declinationof vertical planes more exactly than in the common way, especi-ally if the transits of the sun’s centre are observed by applying aruler with sights, either plain or telescopical, to the wall or planewhose declination is required: 2. In drawing a meridian-line, andfinding the magnetic variation: 3. In finding the hearings of placesin terrestrial surveys; the transits of the sun over any place, or hishorizontal distance from it, being observed, together with the alti-tude and hour; and thence determining small differences of longi-tude: 4. In observing the variations at sea, &c.

9 p O*