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and bottom, set off the hours I. II. III. IV. V., &c. for the purpose of working the Problems on this map.

13. In fine, complete the chart by faithfully indicating all the chequered and indented sea-coast, it being the chief thing which in this chart claims attention : for, in all charts, the inland country is totally neglected, and, except particular towns, harbours, forts, lighthouses, beacons, and rocks, directly on the margin of the sea, the interior country is left unadorned. But, on the water, the winds, cur. rents, shoals, rocks, anchorages, and, in short, all things connected with nautical geography, are carefully and truly laid down, according to their direction, variation, bearing, and distance.

Though what is generally called the Globular Projection of the Sphere is preferable to the Stereographic in the construction of maps, so far as regards the division of lines, it has not the advantage of representing the angles on a map equal to their originals ; nor can the originals of the measures of lines and angles be obtained from it as from the stereographic ; since it depends upon no other principle than the whim of dividing the primitive and vertical circles equally, and drawing lines to form the meridians and parallels of latitude through every three points of division. When the eye is taken about three quarters of the radius out of the surface of the sphere, then all the circles, except those which pass through the eye, or parallel to the primitive, are projected into ellipses, those which pass through the eye are represented by straight lines, and those which are parallel to the primitive are projected also into circles. This is really a true projection, and the circles divide each other into equal parts very nearly ; but it is difficult to project.

However, wberever accuracy is required it ought to be used, as being a more natural representation from which the original measures can be obtained.

DIALLING.

DIALLING, or the method of constructing sun-dials, is a branch of mixed mathematics, which depends partly on the principles of geometry, and partly on those of astronomy.

The General Principles of Dialling. 1. The principles of astronomy teach us that the earth moves in an orbit about the sun, and completes a revolution in a year ; while, at the same time, it revolves uniformly from west to east on its axis, which, although it changes its place, is yet always parallel to a fixed imaginary line, called the axis of the world. By the first of these motions, the sun appears to move round the heavens, completing a revolution in the course of a year ; and by the second, the sun, and all the heavenly bodies, have an apparent diurnal motion about the earth from east to west.

2. The motion of the earth in its orbit is not equable; and hence it happens, that the apparent motion of the sun in the heavens is not quite uniform : besides, the plane of that motion does not coincide with the plane of the diurnal motion. On these two accounts, the apparent diurnal motion of the sun differs a little from uniformity.

3. In the theory of dialling, however, we are to suppose that the sun's diurnal motion is always perfectly uniform, and that it moves throughout the day in a circle parallel to the equator ; but as neither of these hypotheses is strictly true, the time of the day shown by a dial will in general differ from that shown by an accurate clock. However, the difference admits of exact estimation, and tables bave beea calculated which show its amount for every day throughout the year.

4. In constructing dials, it is also usual to leave the effect of refraction out of consideration ; its effect might, indeed, be exactly appreciated, and tables formed, by which the time indicated by the dial might be corrected; or the dial might even be so constructed as to give the time cleared from the error. But this would be a degree of refinement which may very well be overlooked in the practice of what, since the invention of clocks and watches, is now little more than a scientific recreation.

5. If the earth's radius had any sensible proportion to its distance froin the sun, that ought to be taken into account in the constructions of dials. But the earth is almost a mere point, as seen from the sun ; and herce it happens that the diurnal motion of the sun about any

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line on the earth's surface, which is parallel to its axis, may be accounted uniform, exactly as if it were performed about the axis itself.

6. To understand the nature of a dial, let us suppose that eEF (fig. 1) is a straight rod or wire, parallel to the axis of the earth ; or which, if produced, w.ald pass through the pole of the heavens ; and let us suppose that one of its extremities terminates at e in a plane, abcd having any position whatever. Let us farther suppose, that the wire passes through E, the centre of a circle ABCD, described on some solid substance, and that it is perpendicular to the plane of that circle : then, as the wire passes through the poles of the heavens, the circle ABCD will be parallel to the terrestrial equator, and it will be in the plane of the equinoctial circle in the heavens, because on the earth's surface any plane whatever, parallel to the equator, may be considered as coincident with it, when produced to the celestial sphere.

Now, because the axis of the earth perpendicular to the plane of the circle which the sun appears to describe in the heavens by his diurnal motion, and passes through its centre, and that the same is almost exactly true of every line parallel to the earth's axis ; when the circle ABCD is illuminated by the sun, the wire EF will project a shadow upon it, which will revolve about E as a centre, passing over equal arcs of the circumference in equal intervals of time. If, therefore, we suppose the circumference of the circle to be divided into twenty-four equal parts, and the points of division to be numbered 1, 2, 3, 4, &c. to 12, and again 1, 2, 3, 4, &c. to 12, as in the figure, and the circle to have such a position, that the shadow falis upon E 12 at noon; then, at one o'clock, it will have the position El; at two o'clock, it will have the position E 2; at three, the position E 3; and so on. In short, the hour of the day, from sunrise to sunset, will be indicated by the shadow, just as it is shown upon a watch by the motion of the hour hand. And as we suppose the motion of the sun to be quite uniform, the shadow will always have the same position at the same hour every day throughout the year.

7. If the two planes ABCD, abcd are illuminated at once by the sun, the rod eEF will project a shadow on them both. Let us suppose that at the instants the line EF projects its shadow in the directions of the lines E 12, E 1, E 2, &c, on the upper plane, the shadow of eE falls in the lines e 12, e 1, e 2, &c. respectively on the lower plane; and let other cotemporaneous positions of the shadows be found for every hour the sun can shine on the planes ; then, as the shadow will always come to the same position on each plane at the same hour of the day, the hours will be indicated also by the shadow on the plane abcde.

8. Each of the planes ABCD, abcd is a dial: we have supposed the upper plane to be perpendicular to the axis of the world, and in this particular position, the shadow will describe equal angles on it in equal times. The plane of the dial may, however, have any position; but if it is not perpendicular to the earth's axis, the motion of the

shadow projected on it will not be uniform, as it is on the plane of the equinoctial.

9 The rod EF, which projects the shodow, is called the Stile; also sometimes the Aris of the dial.

The lines E 12, E1, &c. which indicate the position of the shadow at the different hours, are called Hour Lines. The bour-lines are evidently the common section of the plane of the dial, and a plano passing through its axis and the sun.

The point in which the axis of a dial meets its plane, which is also the common concourse of the hour-lines, is called its Centre. There are other technical terms belonging to this subject, but these we shall explain as we proceed.

10. The latitude of the place for which a dial is to be made, is an important element in their construction. This may be known by good maps, or it may be determined by astronomical observatioi.s.

How to trace a Meridian Line on any Plane. 11. In constructing a dial, it is always necessary to determine the line in which the plane of the meridian meets the plane of the dial. If the plane of the dial is not horizontal, it will be convenient, in the first place, to trace a meridian line on a horizontal plane near it. A meridian may also be found by any three shadows of an upright pin or stile. Let OV (fig. 2) be the stile which stands at right angles to the plane in O, and ÖA, OA', O'A", its shadows at three different times of the day. Then, if AV, A'V, A"V be joined, the angles AVO, A'VO, A'VO, are the sun's distances from the zenith at the times of noting the positions of the shadows; and these are known, because in the right-angled triangles AOV, A'OV, AWOV, the sides about the right angles at O are known, from which the angles at V may

be found.

Let us now suppose that the sphere is projected stereographically on the horizontal plane AA'A", so that is the centre of the primitive, the eye being in the nadir, then the lines AO, A'O, AKO, produced, will be the projections of azimuth circles ; if the projections of the sun's places, in these circles, at the times of observation, be now found, a circle traced through them will evidently be the projection of the circle of declination, which the sun describes in the heavens that day; and the position of the meridian may now be found, because it will pass through the centre of that circle, and O, the centre of the horizon. Hence we derive the following construction.

Make three right-angled triangles AOV, AOV, APOV (fig. 2), which have each VO=VO, in fig. 3, the height of the stile; and bisect the angles at V, by the lines Va, Va', Vo". Produce the shadows AO, AO, AO", so that Da, Oal, Oa", of fig. 5, may be respectively equal to Oa, Od, Oa", of fig. 4. Describe a circle through the points a, ', a", and from X its centre, draw a line through C; this will be in the direction of the meridian. For, by the principle of the stereographic projection of the spbere, if we take the horizonta'

plane A A A", for the plane of projection; the lines Oa, Od, Ol", will be the projections of circles passing through the zenith and the sun, at the times when the shadows have the positions OA, OA', OA"; and as by construction, Oa, Oa', Oa", are the tangents of half the zenith distances AVO, A'VO, A"VO, the points a, a, a", are the projected places of the sun; and the circle a, a', a", is the projection of the parallel it describes in the heavens on the day of obser. vation, and OX, which passes through its centre, is the projection of the meridian. See Projection of the Sphere.

12. We may even find the latitude of the place of observation : for if P, the projection of the pole of the circle, be found, then OP will be the tangent of half the distance of the pole from the zenith (OV being taken as radius), that is, the langent of half the complement of the latitude.

13. In this construction, no allowance is made for refraction, or change of declination. The zenith distances may, however, be core rected for refraction by the proper tables. And if the observation be made on the solstitial days, the error from change of declination will hardly be any thing. This method of tracing a meridian line was proposed by a very old author on dialling, named Mutio Oddi da Urbino, in a work called Gli Horologi Solari Nelle Superficie piane.

14. Another method of tracing a meridian linc is, by observing when two stars which have the same right ascension, or whose right ascensions differ by 180°, come into the same vertical plane ; for then they are both on the meridian. The observation may be made by means of a pline surface, kept in a vertical position by its own weight, or by any other suitable contrivance, and which is moveable about a vertical line. The pole star and the first s of the tail of the Great Bear, are applicable to this purpose. In the beginning of 1911, their mean right ascensions were,

Star ,
Pole Star,

1910 251

3N 13 41 41

177 43 22

This difference, although not exactly 180 degrees, is yet sufficiently near; because when , is on the meridian, the arc of 2° 16' 38'', by which the pole star has advanced in the small circle it describes, subtends an angle of about of only. The stars a of Ophiuchus, and ß of the Dragon, are well adapted to the saine purpose, the right ascensions and declinations are.

R. Ascen.

Declin.
• of Ophiuchus, 261° 32' 26" 12° 42' 29N'.
B of Dragon,

261 32 33 52 26 47N. As these have almost the same right ascension, and differ 40° in their declination, they are very proper for determining the position of che meridiau.

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