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15. 111 whatever way a meridian is traced on the horizontal plane, it should be quite adjoining to the plane on which the dial is to he delineated ; and it ought to be so placed, that a vertical plane passing along the meridian line, may cut the dial at the point where the axis is to be fixed. To take a familiar example, we shall suppose that the dial plane is a vertical wall carefully smoothed and verified with a rule and plumb-line; and this being understood, it will be easy to suit the operation to any other plane.
Let BC (fig. 4) be the meridian line on a horizontal table, arad in. sq, two plumb-lines, which descend on the meridian line from a hori. zontal rod that has one end fixed in the wall, and the other supported. on a stand. If the table admits of being pierced with two holes, the plumb-lines may with advantage pass through them, and the plummets bang suspended in vessels filled with water. They will thus be more steady, and more easily adjusted. The eye is now to be die rected towards the wall, so that the visual ray may be in the plane of the plumb-lines; and then the line AK upon the wall, which they both appear to cover at once, will manifestly be the intersection of the plane of the meridian, and the plane of the dial; and, consequently, will be the twelve o'clock hour-line. A point A is now to be assumed, as the centre of the dial; and the axis AC must be fixed in the wall' in such a position, that it may lie in the plane of the threads in, sq, and make with a horizontal lipe AD, au angle equal to the latitude of the place, or with CR, a vertical line, an angle equal to the co-latitude ; and then it will manifestly be parallel to the axis of the world.
16. The stile may have any shape that admits of its being firinly fastened to the dial; and, before it is fixed, it may be convenient to fasten a piece of wood to the wall, so that it may have a plane surface exactly in the plane of the meridian, as indicated by the plumb-lines, and a line traced on its surface in the position of the axis or edge of the stile ; this board will serve to support the stile in its position, until it be fastened either with its plane in the plane of the meridian, or perpendicular to the plane of the dial; but it will look njost symmeirical in this last position. In whatever way it is fixed, the edge which projects the shadow must be in the plane of the meridian, and parallel to the earth's axis.
17. When the position of the plane of a dial in respect of the earth's axis is known, the determination of the bour-lines is a geometrical probleni by no means difficult. As at every hour the sun is in one or other of twelve great circles of the sphere, which intersect at the poles of the heavens, and which make equal angles with one anoiher, the general problem to be resolved is evidently this : Let there be twelve planıs, which intersect in a straight line, and make equal ungles with one another; and let these planes, indefinitely produced, meet another plane in any position whatever, to determine the lines in which they cut that plane. In resolving this problem, it will be convenient to begin with the more simple cases, and to reduce the others as much as possible to them.
Equinoctial Dial. 18. This dial, seen obliquely in its proper position, is represented by the upper part of fig: 1. Its plane is parallel to the equator, and is the same as the plane of the equinoctial circle in the heavens ; E is its centre, and EF its axis. As the hour circles in the heavens are perpendicular to the equinoctial circle, and divide it into twenty-four equal parts, the lines in which the plane of the dial cuts their planes, that is, the hour lines, will make twenty-four equal angles round the centre of the dial.
It appears, then, that to delineate a dial of this kind, nothing more is necessary than to describe a circle on its plane, and to divide its circumference into twenty-four equal parts ; and having drawn lines from the centre to the points of division, these will be the hour-lines against which the characters denoting the hours are to be written ; if the axis be now fixed perpendicular to the plane, the dial will be constructed.
In fixing this dial, the axis EF must be in the plane of the meridian, and must make with the horizontal meridian line, an angle equal to the latitude of the place, and then it will point to the pole of the heavens as it ought.
19. As the sun is one half of the year on the north side of the equinoctial, and the other half on the south side, it will be proper to trace hour-lines on both faces of the dial; and in north latitudes the hours will be shown on the upper face the dial in summer, and on the lower face in winter ; but on the equinoctial days, neither face will be illuminated.
The rays of the sun will always fall very obliquely on this dial in our latitudes, but to remedy this, a rim may be put round it, rising a little above the planes of its faces. The inside of the elevated part of the rim will be strongly illuminated by the sun's rays, and thus the hours will be more distinctly shown.
20. A dial of this construction, which admits of being adjusted to any latitude, is delineated at fig. 5. In this instrument, ABCD, and CDEF are two quadrangular pieces (which may be of ivory, wood, or metal), connected by means of a hinge C,D. An equinoctial dial is described on each side of ABCD, or on one of them, and in the centre I, a stile is placed at right angles to the planes of the dials. At G, in the middle of the piece EDCF, a magnetic needle is suspended. and covered with a plate of glass. At L, there is a quadrant fixes perpendicular to the plane of this piece, and divided into degrees. It passes through H, an aperture made to receive it in the upper piece.
When the dial is to be used, it must be placed on a horizontal plane, so that the needle may be in the magnetic meridian. The upper piece must now be turned round the hinge, so that the planes of the two pieces may make with each other an angle equal to the latitude,
measured by the graduated quadrant. The hour of the day will
then be shown by the axis I, on one or other of the two facon, except on the day of the equinox.
21. A dial, traced on the horizontal plane, is called a Horizontal Dial. This is the most common and most useful of any, because it admits of being always illuminated when the sun shines. A dial of this kind is represented in perspective in fig. 6. The point C is the centre, and CK, which is directed to the pole of the heavens, and makes with the plane of the dial an angle equal to the latitude of the place, is the axis.
To understand its nature and construction, let ABD be an equi. noctial dial, whose axis EF is the prolongation of the axis of the horizontal dial; and let the planes of the two dials meet in the line PQ, and suppose the plane of the meridian to cut the plane of the horizontal dial in CM, and that of the equinoctial dial in EM; then the line PQ being the common intersection of the equinoctial and horizontal planes, which are perpendicular to the meridian, that line itself is perpendicular to the meridian.
Let a plane passing through the sun's centre and the common axis of the dials, meet their planes in the lines EH, CH, these lines will manifestly be the positions of the shadows on the two dials at the same instant of time.
22. Now, at any given time, we know the angle HEM which the revolving shadow EH makes with the meridian lide EM on the equinoctial dial, because it is the horary angle which the sun has to describe, or has described about the earth's axis, between the given time and noon, and which is always proportional to that time, reckoning 15 degrees of the angle to an hour. And in the triangle CEM, rightangled at E, we know the angle ECM, which is always equal to the latitude of the place for which the dial is to be constructed ; and fron these we must find the angle HCM, which the hour-line HC, on the horizontal dial, makes with CM, the meridian, or 12 o'clock line.
Let us denote the horary angle HEM, which the sun describes between the given time and noon, by the letter E, and the angle HCM, which the hour-line on the horizontal dial makes with the meridian, by C, and let the angle ECM, the latitude of the place, be L; then, by plane trigonometry, in the two right-angled triangles, EMH, CMH,
HM : ME :: tan. E: rad.
and CH : HM :: rad. : tan. C; therefore, ex æquo inv. (see Geometry.)
CM : ME :: tao. E. : tan. C,
but CM : ME :: rad. : sin. L;
therefore, rad. : sin.L: : tan. E : tan. C. Now, the first three terms of this proportion are knowi, therefore the last is also known; and we get this general formula for constr.ct. ing a horizontal dial.
tan. C=sin. L. tan. E.
in which radius is supposed = 1. The logarithmic formua, deduced from it, may be expressed in words at len th, thus :
To the logarithmic tangent of the horary angle described by the sun between noon and the given time, add the log. sine of the latitude, and the sum, alating 10, (the log of rad.) is the logarithmic tangent of the angle which the hour-line on the dial makes with the meridian line.
23. Ex. Let it be required to calculate the angles which the hourlines on a horizontal dial, for Edinburgh, make with the meridian or 12 o'clock line : the latitude of Edinburgh being about fifty-six degrees, a calculation for the hour-lines of XI in the forenoon, and I in the afternoon, would be as follows:
log. tan. horary angle 15° 9.42805
9.34662 Hence it appears, that the hour-lines for XI in the forenoon, and I in the afternoon, must each make with the meridian an angle of 12° 32'.
The angles which the remaining hour-lines make with the meridian may be found in the same way, and will be as follows : Hour lines of X and II
25 ° 35' IX and III
39 40 VIII and IV
55 8 VII and V
72 5 VI and VI
90 0 The hour-lines of V in the morning, and VII in the evening, make the same angles with the meridian as the hour-lines of VII in the morning and V in the afternoon; but they lie on opposite sides of the VI o'clock hour-lines. In like manner, the bour-lines of IV in the morning, and VIII in the evening, make the same angles with the meridian as the hour-lines of VIII in the morning and IV in the afternoon, and so on.
The construction of the dial is now very easy, as it requires nothing more than to make an angle of a given number of degrees. Thus, draw the meridian line CM (fig. 7), and cross it at right angles by the six o'clock hour-line CG ; and as the style of the dial must have some thickness, it will be proper to draw two parallel lines CM, C'M' for the meridian line, so that the distance between them may be equal to that thickness.
From the points C, C', draw the lines CI, C'XI on opposite sides of the meridian, so that the angles MCI, M'C'XI may be each 12° 32'; and these lines will be the hour-lines of I in the afternoon, and XI in the forenoon; the former lying on the east and the latter on the west side of the meridian, when the dial is placed in its proper position. In the same way, all the other hour-lines may be laid down on the plane of the dial, using a scale of chords, or a protractor, such as is commonly sold by mathematical instrument makers. Or a qua