2. Make the angle ADE equal to the complement of the reclination or proclination downward, if it recline; but upward, if it incline, to cut the vertical line BC in E, And through E draw the meridian, or XII o'clock line, FEG perpendicular to BC.

3. Make EH equal to ED, and the angle EHI equal to the complement of the latitude on the left, when the plane reclines; but on the right, when it inclines; and I is the centre of the dial, which point and the meridian IC will be at the top, when the plane inclines. Through I and A draw IAK, and iak parallel to IAK at the distance Aa, equal to the thickness of the stile, for the substile.

4. Through A draw LM perpendicular to the substile for the contingent line, on which make AN AD, and draw IN for the stile.

5. Draw

5. Draw AO perpendicular to IN, and make AK, ak, each equal to AO. From k draw kQ to the intersection Qof the contingent and meridian.

6. With the centres K, k, and any convenient radius, as KA, describe the quadrants AR, ar; set one sixth of either successively on them each way from the point where kQ cuts ar, for the dividing points; and through these points draw lines from the centres K, k, to intersect the contingent in the hour points. Then lines, drawn from the centre of the dial I through these points, and numbered from IQ, the XII o'clock line, with XI, X, &c, upward on the north side, are the hour lines required,

NOTE. The stile may be a triangular plate similar to IAN.

PROBLEM XV,

To draw geometrically an inclining or reclining west dial.

The construction of this dial differs from that of the last only in the following particulars. The angle EHI, which is made equal to the complement of the latitude, must be taken on the right, when the dial reclines; but on the left, when it inclines. The meridian IG is at the bottom, if the plane recline; but at the top, if it incline. And the hours must be numbered the contrary

way.

PROBLEM

PROBLEM XVI.

To draw geometrically a south declining, reclining or inclin

ing dial.

EXAMPLE.

The declination of the plane westward from the south being 25°, its reclination 15°, and the latitude of the place 42° 23′ 28′′ north; required the construction of the dial.

1. Assume à point A for the foot of the stile; through which draw BC perpendicular to the horizon, and AD perpendicular to BC, and equal to the height of the stile.

2. Make the angle ADE equal to the reclination, or proclination of the plane upward, when it reclines; but downward, when it inclines; and make ADF equal to the complement of it. F is the nadir.

3. Through E draw the horizontal line GEH perpendicular to BC. In EB make EI equal to ED, and the angle EIK equal to the declination of the plane on the right, when it declines eastward; but on the left, when it declines westward; I being the dividing centre of the horizon GH.

4. Through K and F draw the meridian, or XII o'clock line, LKF. Through A draw AMN perpendicular to LF. Set AM from A to. O on the line AF; join O, D, and set OD from M to P in the line MN. Then P is the dividing

centre of LF.

5. Join P, K, and make the angle KPQ upward equal to the latitude; and draw PQ, intersecting LF in the pole Q for the centre of the dial.

6. Through Q and A draw QAR, and qar parallel to QAR at the distance Aa, equal to the thickness of the stile, for the substile.

7. Through A draw ST perpendicular to the substile QAR, intersecting the meridian in U, for the contingent line.

8. Make AV equal to AD, and draw QV for the stile.

9. From A let fall the perpendicular AW on QV; make AR, ar, each equal to AW, and draw rU.

10. With centres R, r, and any convenient radius, as RA, describe the quadrants AX, ax; set one sixth of ei

ther

ther successively on them each way from the point, where TU cuts ax, for the dividing points; and through these points draw lines from the centres R, r, to intersect the contingent in the hour points.. Then draw lines from the centre of the dial Q through these points, and they will be the hour lines of the dial.

NOTE I. In declining reclining dials, the horizontal line must be set parallel to the horizon.

NOTE 2. If PQ be parallel to the meridian LF, the substile QR and all the hour lines, drawn through the intersections of ST, will be parallel to LF. The stile may then be in form of a parallelogram.

NOTE 3. If the dial plane incline, the point E must be taken below A, and F above it; and the meridian LF must be drawn upward from K. In other respects the process is similar to the preceding for the reclining dial.

NOTE 4. A south-west reclining dial, when the upper side is turned downward, and the hours numbered the contrary way, becomes a north-west inclining dial, for the same degree of declination and proclination. And, in the same manner, a north-east inclining dial is drawn by means of a south-east reclining one; and also northwest and north-east reclining from south-west and southeast inclining dials; care being always taken to mark those lines, and those lines only, which will be useful on the face of the plane, for which the dial is intended.

PROBLEM

VOL. II.

Iri