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..sion for cot H, the cotangent 7

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for the determination of the hour lines; cot s = cot L sin D, 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 II for the centre of the hour lines, and the vertical line .6 HM for the meridian, or 12 o'clock hour line. Now, * since the only variable quantity in the preceding expres

of the angle between the me- s ridian of the plane and any hour line, is cot P, the cotan- 9 gent of the corresponding D hour angle at the pole, we - * may, by taking that = 0, get a fixed hour line to which the others may be referred. Thus, the expression, when the hour angle at the pole is 90° = 6 x 15°, becomes * cot MI16 = tan A116 = tan L sin D : an equation by which the position of the 6 o'clock hour line may be readily determined. For, if Hin be taken between H and A, equal to the unit of any measure, as one inch, one foot, &c. we shall have mn = tan L sin D, in the same measure; the product being at once determinable by means of a table of sines and tangents. Through II 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,

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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 to n”, n to n”, &c. on the line mnn'n" perpendicular to AB. These values are successively

*** cot 75° = cos D sec L tan 15°

COS L

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cos D Q — o 2. 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, Hn”9, &c. drawn from H through the points no, no, n”, &c. will be the morning hour lines required. The afternoon hour lines may be computed and set off by the same formulae. Indeed, taking the other vertical line mn'n"n”, &c. between H and B, (Hm = 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.

Hence, since the values of mn', 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 non', n"n", n”n”, &c. parallel to 6H6, 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. &c. &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:

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On the proposed plane ABCD assume II for the centre from which the hour lines shall diverge, and draw the vertical line HM 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 AGG = L, the latitude of the place, and join H6, which will be the G o'clock hour line. From H draw HE parallel to 66 to meet a line of parallel to AB in E. From o the intersection of or and HM, draw of to make the angle Eor = complement of the given declination; make of = or, draw Fs perpendicular to oe, make os = os, and join Hs which will be the substylar line. Perpendicular to Hs draw the right line 5s 1, intersecting the meridian in 12, and the 6 o'clock hour line in 6. On 6-12 as a diameter describe a semicircle to cut His produced in P. With centre P and radius Ps describe a semicircle, which divide into arcs of 15° each, both ways from the line P6, or, which amounts to the same, set off angles 6P7, 7P8, 8P9, 9P10, &c. each = 15°. Through the points 5, 6, 7, 8, &c. where the lines bounding these angles cut the lines 5s.1, draw from H the lines H44, H55, H66, H77, &c. they will be the hour lines required. For, first, since the expression for the 6 o'clock hour line, when the terms are rendered homogeneous, is sin 90° tan AH6 = sin D tan L., . we shall have tan L to the radius sin D equal to tan AH6 to the radius sin 90°; which is obviously the case with regard to the triangles GA6, HA6. Consequently, the 6 o'clock hour line is rightly determined. Again, since EH is parallel to 6G, the angle of H is equal to the latitude, and EHo the co-latitude; therefore of = or, is the cotangent of the latitude to the radius oil ; and since Eof is equal to the co-declination, os = os is the sine of the declination to the radius of = ori or os = os = sin D cot L. But os = tan ohs = cots (substylar angle with horizon) to radius Ho: therefore cot s = sin D cot L, as it ought to be; and the substyle is rightly determined. But the substylar line is evidently a portion of the right line between the centre of the dial and the apparent pole: therefore, the apparent pole, P, lies in the prolongation of Hs. At the apparent pole, P, also the horary angle between 6 o'clock and 12, is 6 x 15°, or a right angle; and the angle in a semicircle is a right angle: therefore, the semicircle described upon 6sl2 as a diameter, will intersect the prolongation of Hs in P, the apparent pole. .

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The truth of the remainder of the construction is manifest”. 27. For astronomical methods of determining the meridian, and the declination of any vertical plane, the student may turn to prob. 2 of the next chapter, examples 3, 4, and 5.

CHAPTER x.

Astronomical Problems.

‘l. SINCE the science of astronomy has given birth to spherical trigonometry, it is to be expected that at least some elementary problems in astronomy may be admitted into an introduction like the present. To determine the position of points in the apparent heavens, astronomers first referred to two planes, the horizon and the meridian (see chap. viii. § 1), which are fixed in reference to any one place on the surface of the earth. But the necessity of comparing observations at different

* The problems given in this chapter will suffice to show the application of the principles of dialling to the most useful cases. They who wish to pursue either the theory or the mechanical part of dialling, through all its modifications, may consult Leadbetter's Mechanic Dialling, Emerson's Dialling, the treatise on dialling in the 5th vol. of Ozanam's Course of Mathematics, and that in the 3d vol. of Dr. Hutton's edition of Ozanam’s Recreations. There is also an elegant essay on dialling by M. de Parcieux, at the end of his Trigonometrie Rectiligne et Spherique; and a neat and sinple deduction of the practice of dialling from the principles of perspective, at the end of S’Gravesande's Essay on Perspective.

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