the dial, round which the hour lines would be radți drawn from the foot of the style to make angles of 15°, 30°, 45°%, &c. with the meridian, or hour line of 12.

PROBLEM II.

22. To describe a horizontal dial for any proposed la titude.

This is the simplest dial to draw next to the polar dial, just mentioned; and it is the most useful, be cause, if it be posited where the sun's rays meet with no obstructions, that luminary will shine upon it from his rising to his setting. The theorems from which the construction is to be deduced are,

tan мb sin L tan P, and sin E = sin L.

Here мb is the measure of the angle H, between the 12 o'clock and any other hour line on the dial, and P the hour angle from the meridian, as it varies at the poles of the heavens; the latter, therefore, varies uniformly, while the former only varies uniformly in certain limited cases; as, for example, in the horizontal dial at the latitude of 90°. In any other latitude, making the terms of the equation homogeneous, we have sin 90° tan Hsin L tan P.

Hence, if two radii be assumed in the ratio of sin 90° to sin L, tan H referred to the former radius will always be equal to tan L referred to the latter. From this consideration the following construction is deduced:

On the proposed

plane assume the right line 12 нs for the meridian, or 12 o'clock hour line, parallel to which draw the line 12hs at a₫ distance

equal to the proposed 6 thickness of the style, Perpendicularly to these draw 6H6, for

5

the east and west line of the dial, or the 6 o'clock hour line. Make the angle 12HF the latitude of the place, and from 12 let fall the perpendicular 12F upon HF. Make 12P, upon н12 prolonged, equal to 12F. From P draw lines P1, P2, P3, &c. (to terminate in the line 12-5, perpendicular to 12H) and to make angles 12p1, 12P2, 12P3, &c. equal to 15°, 30°, 45°, &c. Then from the centre H draw нl, н2, н3, н4, н5, for the hour lines of I, 2, 3, 4, and 5, in the afternoon. Take, on the other side of the substylar line 12-11, 12-10, 12-9, &c. respectively equal to 12-1, 12-2, 12-3, &c. and from h. draw the lines h7, h8, h9, &c. Produce the lines 4H, 5H, for the lines of 4 and 5 in the morning; and produce the lines 7h, 8h, for the hour lines of 7 and 8 o'clock in the afternoon.

The truth of this construction is manifest from the remarks which precede it. For 12p = 12r is evidently the sine of the latitude to the radius H12. And while 12-1, 12-2, 12-3, &c. are the tangents of 15°, 30°, 45°, &c. respectively, to the radius P12; the same lines are tangents to the angles 12H1, 122, 12н3, &c. Conse quently, while the former are hour angles at the pole, the latter are the corresponding hour angles at the centre of the dial.

The quarter and half hour lines are drawn by setting off angles of 3° 45′, 7° 30′, 11° 25′, &c. from the meridian line: but they are omitted in the diagram to prevent confusion.

The angle 12HF is also equal to the elevation of the style; for EL.

As to the gnomon, it should be a metallic triangle of the thickness hн, and having one angle L the latitude. It It must be fixed perpendicularly to the plane, on the space left for it in the figure from hн towards 12, and having its angle L at hн: then will the style of the dial be parallel to the earth's axis.

PROBLEM III.

23. To describe an erect direct south dial for any proposed north latitude, or an erect direct north dial, for any proposed south latitude.

Here the formulæ are those numbered 13, 14, and 15, of which the two first are

tan мb cos L tan P, and sin E = cos L. Substituting H for мb in the former of these, and making the terms homogeneous, we have

sin 90° tan H = cos L tan P.

This equation is similar to the equation of the hour angles in the horizontal dial; and the construction will therefore be similar, except that cos L is here to be employed instead of sin L.

On the proposed plane draw the right line 6H6, for the east and west line of the dial, or the hour line of 6. In the middle of this line set off hн

6

hH

10.11.12

P

6

5

equal to the proposed thickness of the gnomon, and through h, A, draw lines perpendicular to 6н6 to terminate in the line 10-5, parallel to 6н6 at а convenient distance. Draw HF to make the angle 12HF equal to the latitude of the place for which the dial is made, and let fall upon HF the perpendicular 12F. Make 12P HF, the cosine of the angle 12HF to radius H12: draw from P lines P1, P2, &c. to make angles of 15°, 30°, &c. with p12: draw lines from i to meet these in the line 10-5; set off corresponding lines on the left side of the dial, and the construction is completed. The demonstration is the same as in prob. 2.

The angle made between the style and substyle is here equal to the complement of the latitude. When

the dial is placed vertically to face the proper cardinal point, the line 6H6 will be horizontal, and the style sloping downwards from H at an angle of 90°- L will again be parallel to the earth's axis, as it ought to be.

Note. An erect north dial for a place in north latitude is constructed in exactly the same manner as an erect south dial; but the position of the dial must be reversed. In the case of the north dial for north latitude the line 646, instead of being the top will become the bottom of the dial.

The same may be observed in reference to an erect south dial for a south latitude.

PROBLEM IV.

24. To describe an erect direct east dial, for any proposed north latitude, or an erect direct west dial for any south latitude.

It appears from art. 19, that the substyle in this dial will make an angle with the horizon equal to the latitude, that the hour lines will be all parallel to the substyle, and at the distances indicated by a cot P, a being the height of the gnomon, and P the hour angle from the meridian at the pole of the sphere.

2

10

Let AB be assumed as the horizontal line on the proposed dial. From the corner в draw BD to make an angle ABD the comD 1 2 3 4 5 6 7 8 plement of the latitude, and about the middle, H, of that line draw perpendicularly to it the line 6н6 for the 6 o'clock. hour line: this will, also, be the substylar line, and will evidently make with the horizon an angle B6н the latitude of the place. Assume any point, as that marked 11, for the point where the 11 o'clock

618

11

10

A3 4 5 6 7 8 9 10 11 B..

M

hour line is to intersect the line BH; and draw 11p to make the angle нllP = 15°: so shall PH be the height of the gnomon. Set off angles HP7, HP8, HP9, &c. respectively equal to 15°, 30°, 45°, &c. and through the points where the lines P7, P8, &c. intersect the line BD, draw lines parallel to PH: also, set off, from H towards D, H5 = H7, H4 = н8, &c. and through the points 5, 4, 3, &c. draw other lines parallel to PH, these shall be the hour lines required.

The truth of the construction is evident from what has preceded.

A rectangular gnomon of altitude = PH, being set up perpendicularly to the plane of the dial, will, when the dial itself is posited vertically to face the east, have its upper edge parallel to the earth's axis. It is here supposed that the upper edge of the gnomon is reduced to a mere line: if it have any measureable thickness, allowance must be made for it in the construction, as in the preceding problems.

Note. An erect direct west dial for any place in north latitude, may be constructed exactly in the same manner as is just taught, except that, instead of beginning the construction from the right hand, B, of the plane, the operation must commence at the left hand, A, and that the figures expressing the hours of 1, 2, 3, &c. in the afternoon, be placed between A and the hour line of 6.

The like may be observed with regard to an erect direct east dial for any south latitude.

PROBLEM V.

25. To describe an erect south dial for any proposed north latitude, to have a given declination from the west; or an erect north dial for a south latitude, to decline from the east.

The formula in the general problem which are applicable to the present, are those numbered (9), (10), and (11), viz.