PROBLEM XVII.

To draw geometrically an erect dial without a centre

far declining dial.

or a

If the pole have but little elevation above the plane of the dial, and the centre of the dial be within it, the hour lines, especially near the substile, will be too near to each other to be readily distinguished. To increase the distances of the hour lines, the stile may be raised higher." If the centre be thus removed beyond the boundary of the dial, the construction of the dial may be performed in the following manner.

EXAMPLE.

Suppose the plane to decline 70° eastward, in the latitude of 42° 23′ 28′′ north; required the construction of the dial.

1. Suppose the present centre of the dial to be A; through A draw the meridian AB, in which assume a point C, through which draw the horizontal line DCE.

2. Make the angle CAE equal to the complement of the latitude, cutting the horizontal line in E; the angle DCF equal to the complement of the declination on the right, if the declination be westward; but on the left, if it be eastward.

3. Make CF equal to CE; and from F draw FG parallel to AC, cutting the horizontal line in G. Then GF is the present height of the stile, and G the foot of it.

4. Through AG draw AGH, and agh parallel to AGH at the distance Gg, equal to the thickness of the stile, for the substile.

5. Through G draw IGK perpendicular to AGH; and also through some other point L the perpendicular MLN, cutting the meridian AB in N.

6. In GI take GO equal to GF, and draw AOP, for the stile.

7. From L let fall the perpendicular LQ on AP; set LQ from L to R, and draw RN.

8. Draw Ss parallel to AP at a convenient distance, for the new stile. Let fall the perpendicular LS on Ss; set LS from L to H, and from 1 to h; and draw Uh parallel to RN.

9. With centres H, h, and any convenient radius, as HL, describe the quadrants LV, Iv. Then, beginning at the point, where hU cuts lv and gives the hour point of XII, proceed, as in the last Problem, to find the hour points.

10. Let

10. Let fall the perpendicular GW on Ss; set GW from G, g, to X, x; and draw xn, making the angle gxn equal to the angle IhU.

II. With centres X, x, and any convenient radius, as XG, describe the quadrants GY, gy; and, beginning at n, as xn gives XII, proceed, as before, to find the hour points in IK. Then through the corresponding hour points in both contingent lines draw right lines, and they will be the hour lines required.

PROBLEM XVIII.

To draw a horizontal dial by means of the dialing scales; o the hour line and line of latitudes on the Plane Scale.

1. Draw AB, and ab parallel to AB at the distance Aa, equal to the thickness of the stile, for the substile and XII o'clock line.

2. Through

2. Through A draw CD perpendicular to AB, for the VI o'clock line.

3. Take the extent of the latitude of the place from the line of latitudes, and set it from A to C, and from a to D; with centres C, D, and the length of the hour line as radius, describe ares cutting AB in B, and ab in b; and join C, B, and D, b.

4. Take the extent of 1, 2, 3, &c. hours from the hour line, and set them successively on BC, bD, from B toward C, and from b toward D, for the hour points. Then lines, drawn from A through the hour points in CB, will be the hour lines from VI to XII; and those, drawn from a through the hour points in bD, the hour lines from XII to VI. The rest is obvious, being performed as before directed.

NOTE I. The line Inclination of Meridians, on the Plane Scale, may be used instead of the hour line, 15° being al lowed for one hour, 30° for two hours, &c.

NOTE 2. If it be required to draw a south vertical dial by means of these Scales, the complement of the latitude must be set from A to C, and from a to D. And then we may proceed as before.

PROBLEM

PROBLEM XIX.

To draw a dial on the surface of a sphere,

Assume two diametrically opposite points on the sphere, for the two poles; and let P be one of them. With cen tre P, and radius equal to the distance of a point on the surface in the middle between P and the other pole, describe a circle EQ, for the equator, and divide it into 24 equal parts for hours. These are to be numbered from east to west; with six on the top, the hours of the forenoon above the equator, and those of the afternoon be low it.

NOTE. The sphere is to be placed so, that the elevation of the pole P may be equal to the latitude, or the axis parallel to the earth's axis, and the line, or meridian, APB in the meridian of the place. Then the circle bounding the illuminated hemisphere shews the time at its intersection with the equator EQ.

PROBLEM