PROBLEM LXIII.

To make a horizontal dial for any latitude.

Since

Definitions and Observations.-Dialling, or the art of constructing dials, is founded entirely on astronomy; and, as the art of measuring time is of the greatest importance, so the art of dialling was formerly held in the highest esteem, and the study of it was cultivated by all persons who had any pretensions to science. the invention of clocks and watches, dialling has not been so much attended to, though it will never be entirely neglected; for, as clocks and watches are liable to stop and go wrong, that unerring instrument, a true sun-dial, is used to correct and to regulate them.

Suppose the globe of the earth to be transparent (as represented by Fig. 4. Plate II.) with the hour circles, or meridians. &c. drawn upon it, and that it revolves round a real axis NS, which is opaque and casts a shadow, it is evident that, whenever the edge of the plane of any hour circle or meridian points exactly to the sun, the shadow of the axis will fall upon the opposite hour circle or meridian. Now, if we imagine any opaque plane to pass through the centre of this transparent globe, the shadow of half the axis NE will always fall upon one side or other of this intersecting plane.

Let ABCD represent the plane of the horizon of Lon don, BN the elevation of the pole or latitude of the place; so long as the sun is above the horizon, the shadow of the upper half NE of the axis will fall somewhere upon the upper side of the plane ABCD. When the edge of the plane of any hour circle, as F, G, H, I, K, L, M, O, points directly to the sun, the shadow of the axis, which axis is coincident with this plane, marks the respective hour line upon the plane of the horizon ABCD; the hour line upon the horizontal plane is, therefore, a line drawn from the centre of it, to that point where this plane intersects the meridian opposite to that on which the sun shines. Thus, when the sun is upon F, the meridian of London, the shadow of NE the axis will fall upon E, XII. By the same method, the rest of the hour lines are found, by drawing, for every hour a line from the centre

of the horizontal plane to that meridian, which is diame trically opposite to the meridian pointing exactly to the sun. If, when the hour circles are thus found, all the lines be taken away except the semi-axis NE, what remains will be a horizontal dial for the given place. From what has been premised, the following observations naturally arise:

1

1. The gnomon of every sun-dial must always be parallel to the axis of the earth, and must point diretly to the two poles of the world.

2. As the whole earth is but a point when compared with the heavens, therefore, if a small sphere of glass be placed on any part of the earth's surface, so that its axis be parallel to the axis of the earth, and the sphere have such lines upon it, and such a plane within it as above described; it will show the hour of the day as truly as if it were placed at the centre of the earth, and the body of the earth were as transparent as glass.

3. In every horizontal dial, the angle, which the style, or gnomon, makes with the horizontal plane, must always be equal to the latitude of the place for which the dial is made.

Rule for performing the Problem.-Elevate the pole so many degrees above the horizon as are equal to the latitude of the place; bring the point Aries to the brass meridian; then, as globes in general* have meridians drawn through every 15 degrees of longitude, eastward and westward from the point Aries, observe where the meridians intersect the horizon, and note the number of degrees between each of them; the arcs between the respective hours will be equal to these degrees. The dial must be numbered XII at the brass meridian, thence, XI, X, IX, VIII, VII, VI, V, IV, &c. towards the west, for morning hours; and I, II, III, IV, V, VI, VII, VIII, &c. for evening hours. No more hour lines need be drawn than what will answer to the sun's continuance above the horizon on the longest

*On Carey's large globes, the meridians are drawn through every ten degrees, an alteration which answers no useful purpose whatever, and is in many cases very inconvenient. To solve this problem, by these globes, meridians minst be drawn through every fifteen degrees with a pencil.

day at the given place. The style or gnomon of the dial must be fixed in the centre of the dial-plate, and make an angle therewith equal to the latitude of the place. The face of the dial may be of any shape, as round, elliptical, square, oblong, &c. &c.

Example. To make a horizontal dial for the latitude

of London.

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Having elevated the pole 51 deg. above the horizon, and brought the point Aries to the brass meridian, you will find the meridians on the eastern part of the horizon, reckoning from 12, to be 11° 50′, 24° 20′, 58° 3′, 58° 35', 71°6; and 90° for the hours I, II, III, IV, V, and VI: or, if you count from the east towards the south, they will be 0°, 18° 54', 36° 25′, 51° 57′, 65° 40', and 78° 10′, for the hours VI, V, IV, III, II, I, reckoning from VI o'clock backward to XII. There is no occasion to give the distance farther than VI, because the distances from XII to VI in the forenoon are exactly the same as from XII to VI in the afternoon; and hour lines continued through the centre of the dial are the hours on the opposite parts thereof.

The following Table, calculated by spherical trigonometry, contains not only the hour arcs, but the halves and quarters from XII to VI.

The calculation of the hour arcs by spherical trigonometry is extremely easy; for while the globe remains in the position above described; it will be seen that a right-angled spherical triangle is formed, the perpendicular of which is the latitude, its base the hour arc, and its vertical angle the hour angle. Hence,

Radius, sine of 90°

Is to sine of the latitude;

As tangent of the hour angle,

Is to the tangent of the hour arc on the horizon.

It may be observed here, that if a horizontal dial, which shows the hour by the top of the perpendicular gnomon, be made for a place in the torrid zone, whenever the sun's declination exceeds the latitude of the place, the shadow of the gnomon will go back twice in the day, once in the forenoon, and once in the afternoon; and the greater the difference between the latitude and the sun's declination is, the farther the shadow will go back. In the 38th chapter of Isaiah, Hezekiah is promised that his life shall be prolonged 15 years, and as a sign of this, he is also promised that the shadow of the sun-dial of Ahaz shall go back ten degrees. This was truly, as it was then considered, a miracle; for, as Jerusalem, the place where the dial of Ahaz was erected, was out of the torrid zone, the shadow could not possibly go back from any natu ral cause.

PROBLEM LXIV.

To make a vertical dial, facing the south, in north latitude.

Definitions and Observations.-The horizontal dial, as described in the preceding problem, was supposed to be placed on a pedestal, and as the sun always shines upon such a dial when he is above the horizon, provided no object intervene, it is the most complete of all kinds of dials. The next in utility is the vertical dial facing the south in north latitudes; that is, a dial standing against the wall of a building which exactly faces the south.

Suppose the globe to be transparent, as in the foregoing problem (see Figure 5, Plate II.) with the hour circles or meridians, F, G, H, I, K, L, M, O, &c. drawn upon it; ADCB an opaque vertical plane perpendicular to the horizon, and passing through the centre of the globe. While the globe revolves round its axis NS, it is evident that, if the semi-axis ES be opaque and cast a shadow, this shadow will always fall upon the plane ABC, and mark out the hours as in the preceding problem. By comparing Fig. 5 with Fig. 4, in Plate II, it will appear that the plane surface of every dial whatever, is parallel to the horizon of some place or other upon the earth, and that the elevation of the style or gnomon above the dial's surface, when it faces the south, is always equal to the latitude of the place whose horizon is parallel to that surface. Thus it appears that SP, which is the co-latitude of London, is the latitude of the place whose horizon is represented by the plane ADCB: for,

let the south pole of the globe be elevated 38 degrees above the southern point of the horizon, and the point Aries be brought to the brass meridian; then, if the globe be placed upon a table, so as to rest on the south point of the wooden horizon, it will have exactly the appearance of Fig. 5, Plate II. the wooden horizon will represent the opaque plane ADCB, the south point will be at B, and the north point at D under London, the east point at C, and the west point at A. Hence, we have the following:

Rule for performing the problem.-If the place be in north latitude, elevate the south pole to the complement of that latitude; bring the point Aries to the brass meri dian; then, supposing meridians to be drawn through every 15° of longitude, eastward and westward from the point Aries (as it is generally the case ;) observe where these meridians intersect the horizon, and note the num ber of degrees between each of them; the arcs between the respective hours will be equal to these degrees. The dial must be numbered XII at the brass meridian, thence, XI, X, IX, VIII, VII, VI, towards the west, for morn. ing hours; and I, II, III, IV, V, VI, towards the east, for evening hours. As the sun cannot shine longer upon such a dial as this than from VI in the morning to VI in the evening, the hour lines need not be extended any farther.

Example. To make a vertical dial for the latitude of London.

Elevate the south pole 384 degrees above the horizon, and bring the point Aries to the brass meridian; then the meridians will intersect the horizon, reckoning from the south towards the east, in the following degrees; 9° 28', 19° 45', 31° 54′, 47° 9′, 66° 42′, and 90o, for the hours I, II, III, IV, V, VI; or, if you count from the east towards the south, they will be 0°, 23° 18′, 42° 51′, 58° 6′, 70° 15′, 80° 32′, for the hours VI, V, IV, III, II, I. The distances from XII to VI in the forenoon are exactly the same as the distances from XII to VI in the after

noon.

The following table contains not only the hour arcs, but the halves and quarters from XII to VI; it is calculated exactly in the same manner as the table in the preceding problem, using the complement of the latitude instead of the latitude.