NOTE 5. If the stile be considered as a mere line, its foot will be the centre C, from which the hour lines are to be drawn.

NOTE 6. At a pole one dial of this kind will shew all the hours of the day; but in any other place the two, that are necessary for the two poles, must be used in connection to shew time through the year; the northern answering this purpose when the sun is north of the equator, and the southern when it is south of it. In northern latitudes the former is called the superior equinoctial dial, and the latter the inferior equinoctial dial; and the contrary in southern latitudes.

These two dials may be described on opposite faces of the same plane, and have for stiles the same pin passing through it. Figure 1 is the superior, and figure 2 the inferior equinoctial for our latitude.

PROBLEM V.

To draw geometrically a polar dial, or a horizontal dial at the equator.

1. Draw two parallel lines AB, DE, at a convenient distance for inscribing the hour lines.

2. Draw MCF, mcf, perpendicularly from AB across DE, making CF, cf, each equal to the height of the stile, and Cc equal to its thickness, for the XII o'clock line.

3. On the centres F, f, and with any convenient radius, as FC, describe the quadrants CH, cG. Divide each quadrant into 6 equal parts, at m, m; n, n, &c. and draw lines from the respective centres F, f, through all the points of division till they intersect the line DE in the points I, II, &c. numbered toward the east from the XII o'clock line. Then through the points I, II, &c. draw,

lines from DE to AB parallel to MC; and they will be the hour lines.

I.

NOTE 1. The stile of this dial may be a plate in form of a parallelogram, of the thickness Cc, length MC, and height MKCF; and it must be set on MCcm, perpendicular to the plane of the dial; and the shadow of the top points to the time.

NOTE 2. When this dial is used, its plane must be set parallel to that meridian, which is perpendicular to the meridian of the place; that is, with the XII o'clock line. parallel to the axis of the earth, and a line perpendicular to it, as AB, horizontal.

NOTE 3. If the stile be considered as a mere geometrical plane, the line MC may be taken for its base, and a semicircle must be described on F, and divided into 12 parts; and the rest of the process will be the same as beforc. A similar observation is applicable to other dials.

NOTE 4. At the equator one dial of this kind is sufficient to shew the time; but at any other place two, described on the opposite faces of the plane, are necessary; one, called the superior polar dial, for the time between 6 A. M. and 6 P. M. and the other, called the inferior polar dial, for the rest of the day. Figure 1 is a superior, and figure 2 an inferior polar dial for our latitude, only some divisions on the dial plane are omitted for want of room.

PROBLEM VI.

1. Describe a circle M, with a convenient radius, and assume a convenient point A out of the centre, for the centre of the dial, and through A draw the meridian, or XII o'clock line AB, and at the distance Aa, equal to the thickness of the stile, draw ab parallel to AB.

2. Make

2. Make the angle BAC equal to the latitude, suppose 42° 23′ 28". Assume a point D in the line CA, and draw DE perpendicular to AB, and it will represent a rod used for a perpendicular stile on the point E. Draw DF perpendicular to AC; and through F draw the contingent line GH perpendicular to AB, which here represents the equinoctial.

3. Make FI and fi each equal to DF, and with centres I, i, and any radius, as, FI, describe the quadrants FK, fk, and divide each into 6 equal parts. Through the points of division draw lines from the centres I, i, to intersect GH in the hour points m, m, n, n, &c. Then, from the centres A, a, draw lines through the points m, m; n, n, &c. VOL. II.

GG g

as

as Am, Am; and an, an, &c. and they will be the hour lines, to be numbered I, II, III, &c. from XII toward the east. The VI o'clock hour line must be drawn parallel to GH, or perpendicular to the meridian; and the hours before and after VI are found by producing the opposite hour lines through the centre; which may be done also on other dials, that shew more than 12 hours of the day.

NOTE 1. In order to find an hour line, when the line from the centre of the dividing circle does not cut the contingent line within the plane of the dial; draw a line gd near the boundary parallel to the substile, intersecting the line from the centre of the dividing circle in d. Then, by Prob. XVI. of Geometry, find a fourth proportional to if, fa, and de; and set it from c to h. A line, drawn from the centre of the dial through the point h, as aV in this example, will be the hour line required.

NOTE 2. The stile may be a triangular plate similar to AED, placed perpendicular to the plane of the dial on the substile ABba; and the edge AD will shew the time. This stile may be lengthened or shortened by producing or curtailing the side AD. Or the gnomon may be a perpendicular pin on E, and in height equal to ED; and then the shadow of the top will point out the time.

NOTE 3. When this dial is used, its plane must be horizontal, and the XII o'clock line on a meridian line, with the end B toward the north.

NOTE 4. This dial shews all the hours of the artificial day at any season of the year.

PROBLEM