Imágenes de páginas
PDF
EPUB

eot мb, or, cot H = tan L. sin » +

for the determination of the hour lines;

cot scot L sin D,

COS 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 for the centre of the hour lines, and the vertical line

6

7

Hм for the meridian, or 12
o'clock hour line. Now, A
since the only variable quan
tity in the preceding expres-
sion for cot H, the cotangent
of the angle between the me-
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
the others may be referred.
when the hour angle at the
becomes

a

6

[ocr errors][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

cot MI6 tan A16 tan L sin D ; an equation by which the position of the 6 o'clock hour line may be readily determined. For, if нm 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 11 and ʼn 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 to n", n to n'", &c. on the line mnn'n" perpendicular to AB. These values are successively

COS D

COS L

COS D

COS L

COS D

COS L

COS D

COS L

COS D

COS L

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, нn"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 formulæ. 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.

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 нn'5, Hn"4, нn""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:

[merged small][merged small][merged small][merged small][graphic][ocr errors][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed]

On the proposed plane ABCD assume I 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 н6, which will be the

=

o'clock hour line. From н draw HE parallel to 06 to meet a line or parallel to AB in E. From o the intersection of OE and Hм, draw or to make the angle Eor complement of the given declination; make or OE, draw FS perpendicular to OE, make os os, and join нs which will be the substylar line. Perpendicular to Hs draw the right line 5s1, 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 HS 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 5s1, draw from H the lines H44, н55, н66, н77, &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 Ан6 sin D tan L,.

we shall have tan L to the radius sin D equal to tan АHỔ 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 66, the angle оEH is equal to the latitude, and EHO the co-latitude; therefore OE OF, is the cotangent of the latitude to the radius OII; and since EOF is equal to the co-declination, os = os is the sine of the declination to the radius of = OE; or os os sin D cot L. But ostan оHS cot s (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 нS. At the apparent pole, P, also the horary angle between 6 o'clock and 12, is 6 × 15o, or a right angle; and the angle in a semicircle is a right

angle: therefore, the semicircle described upon 6s12 as a diameter, will intersect the prolongation of HS in P, the apparent pole.

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, ex-. amples 3, 4, and 5.

CHAPTER X.

Astronomical Problems.

1. 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.

« AnteriorContinuar »