To find the hour-arcs, or angular distances of the hourlines, for a horizontal dial, the latitude being given.

1. If the meridians be drawn on the globe through every 15°, as they generally are :—

RULE.-Elevate the pole for the given latitude, and bring the first meridian (or any other) to the brazen meridian; then note down the distances from the under semi-circle of the brass meridian, at which the horizon is intersected by the several meridians, taking so many only as there are hours in the half of the longest day. Thus in a horizontal dial for north latitude, the north point being XII, or noon, the points of intersection towards the east, and beyond it, will be the places of the afternoon hours, I, II, III, IV, &c.; and those at corresponding distances from the north towards the west* and beyond it, will be the places of the morning hours, XI, X, IX, VIII, &c. (See the figure.)

The places of the half-hours may be marked by turning the globe 740, and noting again the intersections of the several meridians; and the places of the quarter-hours, by turning the globe 320;-but this will require that the globe be in excellent condition.

2. If the meridians of the globe be drawn, as they sometimes inconveniently are, through every 10° :

It is evident that, in this case, the intersections of the meridian lines will be at every forty minutes, instead of being at every sixty minutes; that is, the arcs from N. to E., instead of being for I, II, III, &c., will be, respectively, those for forty minutes past XII,-twenty minutes past I,

*The intersections from north towards west, and beyond it, being at the same angular distances as those from north towards east, and beyond it, they need to be found only for one side.

CONSTRUCTION OF A HORIZONTAL DIAL.

383

-II,-forty minutes past II,-twenty minutes past III,IV, &c.; hence,

RULE. Having marked off the intersections which shew the places of the even hours II, IV, VI, VIII, &c., turn the globe 5° westward, and the intervening intersections will be at the odd hours, I, III, V, &c., which may be noted accordingly.

The places of the half-hours, &c., may be found as before.

Ex. 1. What are the arcs between the hours XII and I, or XII and XI,—and XII and II, or XII and X, on the face of a horizontal sun-c -dial for Belfast, lat. 54° north?

Here, elevating the pole 54°, I find, that when a meridian line is brought to coincide* with the brass meridian, the two intersections first on each side of the north point (or XII) on the horizon, (and which are the places of the hours XI and I,) are each at an angular distance of 12,-the second, on each side of the north point, (or the places of X and II,) are at 254°, &c., &c.

Ex. 2. Find the hour-arcs for the remaining parts of a day on the horizontal dial proposed above, the longest day at Belfast being a little more than seventeen hours.

Here it is evident that the utmost hour-lines will be those of IV A.M., and VIII P.M.

Ex. 3. Find the angular distances of the hour-lines and half-hour lines of a horizontal dial constructed for London or Blackheath.

Ex. 4. Describe a horizontal dial for the latitude of Petersburgh, or the south of Mainland, Shetland.

Ex. 5. What difference is there between the angle contained by the hour-lines of I and II, or XI and X, on a horizontal dial, constructed for Petersburgh, and the angle contained by those hour-lines on one constructed for the Bermudas?

* It will be found convenient to keep it coinciding by the under semi-circle.

Ex. 6. What difference of degrees on the same two dials, between the numbers VI and VII?

Ex. 7. Describe horizontal dials for Lisbon and Washington.

Conversely.

Ex. 8. For what latitude must that horizontal dial have been made, on which the hours of IV and V are separated 20°?

Ex. 9. Where might a horizontal dial be erected, which would shew correct time by the motion of the shadow of its stile or gnomon over 15o each hour, or one quarter of a degree each minute; and what inclination must its gnomon have?

A sun-dial the face of which is perpendicular to the horizon, and therefore in the plane of the zenith and nadir, is called a vertical dial. It is a convenient one, and is often placed against the south wall of a church; but as the sun, at no time of the year, shines upon a wall so situated longer than twelve hours, the hours from VI to VI are all that are marked upon such a dial.

M Since the distance of any two places on the earth's surface is insignificant when compared with the sun's distance, it is evident that a dial constructed for the horizon of any latitude will shew us the time at our own place of residence, provided that, first taking care to place it duly north and south, we also incline its plane or face to our horizon just so much as the horizon of those who live due north or south of us in that latitude inclines; that is to say, if, according as the latitude for which it has been made lies northward or southward of us, we depress the north or south points of the surface of the dial.

Thus, Mainland of the Shetland Isles lies in latitude 60o N., or 6o due north of the city of York. A horizontal dial constructed for latitude 60°, and consequently adapted to shew correct solar time in Shetland, would serve for that purpose also at York, (notwithstanding the difference of its hour-arcs and of the inclination of its stile,)

VERTICAL DIAL, ETC.

385

if, after being adjusted to the meridian, the XII or north point of its plane were depressed 6o; for the distance of York from Shetland, or any other place, is insignificant; and the dial being thus arranged, the sun may be considered as shining on it in Shetland.

The mouth of the Ebro has, very nearly, the longitude of Blackheath, and the latitude of Madrid, or 401° N.-A horizontal dial for that latitude would shew solar time at Blackheath, as well as at the mouth of the Ebro, if, after setting it north and south, we were to incline its plane 11° southward; for its gnomon would thus be brought parallel to the earth's axis, and its surface would be in position with an horizon for which it was made.

N The south point of the vertical dial used in north latitudes is, of course, depressed 90°; and thus it appears that such a dial, in use by any of us, is the horizontal dial of those who dwell 90° south of us, or just so far south of the equator as we are south of the pole. Hence, it is the southern half of the axis that is represented by the gnomon of such a dial, and the inclination of its gnomon is always equal to the complement of the latitude in which it is fixed up.

Let the adjoining cut represent, as before, a transparent globe, with its meridians; let c be its centre, and R, Z, Z, R, an horizon of lat. 51° N.

Then Zen, Z, D, R, will represent a plane vertical to this horizon, and S, c, the southern half of the axis depressed 514° below R, Z, Z, R; and therefore making an angle with the vertical plane equal to the complement of 5140, or 3840.

It may be seen, by turning the figure so as to bring D to the left, that this would be a horizontal dial in lat 384° S.

In like manner, if we turn our page 380, so that its n, or needle end of the axis, may be directed downward on our right hand, it will be perceived that our horizontal dial would serve for the vertical one of lat. 3810 S.

PROBLEM T.

TERRESTRIAL GLOBE.

To find the hour-arcs of a vertical dial facing the south in north latitude, or facing the north in south latitude, the latitude being given.

RULE. Find the complement of the latitude, and elevate the pole of contrary name. Bring the first meridian (or any other) to coincide with the brass meridian; then taking the point at which the under semi-circle of the brass meridian intersects the wooden circle of the globe for the place of the shadow at XII or noon,-note down the points of intersection of the wooden circle by the meridians towards the east, for the afternoon hours I, II, III, IV, V, VI, and the points of intersection of those towards the west, for the morning hours XI, X, IX, VIII, VII, VI. For half-hours, &c., the globe may afterwards be turned 71° &c., as directed in Problem S.

If the meridians of the globe be drawn through every 10°, instead of through every 15°, the even hours II, IV, &c., must first be marked, and the odd ones, after turning the globe 5o westward, as directed in the second part of the rule for Problem S.

Ex. 1. Find the hour-arcs of a vertical dial facing the south at Belfast, lat. 541° N.

Here, elevating the south pole to the difference between 541° and 90°, or 35°, I find that, when any meridian is brought to coincide with the brass meridian-of the five meridians intersecting the wooden plane between the points marked "south" and "east," the first, or