the dial is placed vertically to face the proper cardinal point, the line 6н6 will be horizontal, and the style sloping downwards from H at an angle of 90° again be parallel to the earth's axis, as it ought to be. L will 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 6A6, 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 3 4 5 6 7 8 9 > 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 complement of the latitude, and about the middle, H, of that line draw perpendicularly to it the line 6H6 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 A3 4 5 6 7 8 9 10 11 B H hour line is to intersect the line BH; and draw 11p to make the angle Hllp=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 = H8, &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 formulæ in the general problem which are applicable to the present, are those numbered (9), (10), and (11), viz. cot мb, or, cot Htan L sin D + for the determination of the hour lines; cot scot L sin D, Cob D cot P, COS L for, that of the angle between substyle and horizon; and sin E COS L COS D, for the elevation of the style. From these and a table of natural sines and tangents, the hour lines, &c. may easily be drawn, thus: Let ABCD be the plane on which the dial is to be constructed. Assume ы for the centre of the hour lines, and the vertical line 8 n n 6 n ነገ n n" 5 22/11 4 may, by taking that = 0), get a fixed hour line to which the others may be referred. when the hour angle at the becomes Thus, the expression, pole is 90° 6 × 15°, cot мH6 tan Aн6 tan L sin D; an equation by which the position of the 6 o'clock hour line may be readily determined. For, if Hm be taken between H and A, equal to the unit of any measure, as one inch, one foot, &c. we shall have mntan L sin D, in the same measure; the product being at once determinable by means of a table of sines and tangents. Through и and n draw a line both ways, to meet AD and the prolongation of EB; then that portion of this right line which falls on the left hand side of the plane will be the hour line of 6 in the morning, while the other portion to the right of H, and above the plane, would be the hour line of 6 in the afternoon: that, however, is useless in this dial. To ascertain the position of the other hour lines, we have only to estimate the remaining term of the equation by the several corresponding values of cot P for the different hours from 12 or 6, and set those values from n to n', n ta n", n to n"", &c. on the line mnn'n" perpen dicular to AB. These values are successively cot 75° = cos D sec L tan 15° cot 60° cos D sec L tan 30° cot 45° cos D sec L tan 45° = cos D sec L cot 30° cos D Sec L tan 60° cot 15° cos D sec L tan 75° (see chap. iv. 41.) These being computed and set off, the several right lines Hn'7, Hn"8, нn""9, &c. drawn from н through the points n', n", n", &c. will be the morning hour lines required. The afternoon hour lines may be computed and set off by the same formula. Indeed, taking the other vertical line mn'n'n"", &c. between H and в, (нm Hm) the several values of mn, nn', nn", &c. will be numerically the same, because the corresponding values of cot P are so; the only change being in the sign from +to, on account of passing the meridian. 1 Hence, since the values of nn', nn", nn'", &c. on the two parallel vertical lines are respectively equal, lines joining their extremities will form a series of parallelograms: and hence, the morning hour lines being drawn, the evening hour lines equally distant from 6 o'clock, may be determined by drawing n'n', n′′n", n'"'n' &c. parallel to 6н6, and then drawing through n', n", n", to the right, the lines Hn'5, Hn"4, Hn""3, &c. Thus, it appears, as has been observed by Ozanam, Emerson, and Delambre, that in all plane dials, one half of the dial being traced, determines the other by the several distances from the hour line of 6, measured on two vertical lines equidistant from the meridian. Thus these distances are equal between the lines from 6 to 7 in the morning and from 5 to 6 in the evening, 6 to 8 ..... morning..... 4 to 6...., evening, &c. The position of the substyle, in this instance between HM and HA, is determined by the preceding equation for cot s, and the elevation of the style by the preceding theorem for sin E. 26. But this kind of dial, as well as horizontal and direct dials, may be constructed independent of computation. Thus : On the proposed plane ABCD assume н for the centre from which the hour lines shall diverge, and draw the vertical line Hм for the hour line of 12. Produce HA to G, so that HA shall be to AG, in the ratio of sin 90° to sin D; draw G6 to make the angle AG6 L, the latitude of the place, and join H6, which will be the 6 |