perpendicular to the plane; mark in the morning where the end of the shadow touches one of the circies; in the afternoon mark where the end of the shadow touches the same circle; divide the arch of the circle contained between these two points into two equal parts; a line drawn from the point of division to the centre of the plane will be a true meridian, or north and south line; and, if this line be bisected by a perpendicular, that perpendicular will be an east and west line: thus you will have the four cardinal points; but, to be very exact, the plane must be truly horizontal, the wire must be exactly perpendicular to the plane, and the extremity of its shadow must be compared not only upon one of the circles, as above described, but upon several of them.

Rule 2. Fix a strong straight wire sharp pointed at the top in the centre of your piane, nearly perpendicular; place one end of a wooden ruler on the top of the wire, and with a sharp pointed iron pin, or wire, in the other end of the ruler, describe an arch of a circle: take off the ruler from the top of the wire, and observe, at two different times of the day, when the shadow of the top of the wire falls upon the arch of the circle described by the ruler; mark the two points, and divide the arch between them into two equal parts, and draw a line from the point of bisection to the centre of your plane: this will be a meridian line.

Rule 3. Hang up a plumb-line in the sun-shine, so that it may cast a shadow, of a considerable length, upon the horizontal plane, on which you intend to draw your meridian line; draw a line along this shadow upon the plane, while at the same time a person takes the altitude of the sun correctly with a quadrant, or some other instrument answering the same purpose; then, by knowing the latitude of the place, the day of the month, and of course the sun's declination, together with his altitude; find the azimuth, from the north, by spherical trigonometry, and subtract it from 180°; make an angle, at any point of the line which was drawn, upon your plane, equal to the number of degrees in the remainder, and that will point out the true meridian. See Keith's Spherical Trigonometry, page

274.

PROBLEM LXIII.

To make a horizontal dial for any latitude.

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. Since 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, in 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 London, 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 BCD. 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 place 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 diametrically 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 obser vations naturally arise:

1. The gnomon of every sun-dial must always be parallel to the axis of the earth, and must point directly 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 shew 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 these meridians intersect the horizon, and note the number of degrees between each of them; the arches 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

* On Cary's 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 his globes, meridians must be drawn through every fifteen degrees with a pencil.

Hours.

will answer to the sun's continuance above the horizon on the longest 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.

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′, 38° 3′, 53° 35', 71o 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 distances far ther than Vi, because the distances from XII to VI in the fore noon 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 arches, but the halves and quarters from XII to VI.

The calculation of the hour arches 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 arch, and its vertical angle the hour angle. Hence,

As radius, sine of 90°

Is to sine of the latitude;

So is the angent of the hour angle,

To the tangent of the hour arch on the horizon.

It may be observed here, that if a horizontal dial, which shews 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 should 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 natural cause.

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 upon a pedestal, and as the sun always shines upon such a dial when he is above the horizon, provided no objects 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