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For both the projecting parallels being perpendicular to the groundline, they form with it two right angles ; and when the planes come into coincidence, these two lines are in the same plane, and making angles with it on the same side equal to two right angles, they form one straightline.

18. If, therefore, we draw the te ground-line x'ox, and in it take

(some fixed point from which the distances are estimated)

the part of, and draw pp” perx'

pendicular to the ground-line; if, also, we take g P the vertical projecting parallel, and p" the horizontal one; we shall then have the point p perfectly defined by the lines o G, GP and GP".

In this,

og is the distance of the point from the plane Y Z, G P is its distance from the plane x y, and GP" is its distance from the plane x z. Of these we shall have rarely occasion to mention og, and the other two are called the horizontal distance of the point P, and the vertical distance of the same point.

19. In conformity with the method of notation here employed, we shall hereafter designate the ground-line by x' x (or sometimes ox), the vertical projection of points (which in their own places in space are denoted by P, Q, R, &c,) by p', d', r', &c., and their horizontal projections by p", 2", R", &c.

20. The intersection of one plane with another, considered as being situated in the latter, is called the trace of the former plane upon the latter, or simply the trace of that plane.

21. When the traces of a plane upon the planes of projection are given, the plane itself is given; since no other plane different from this can make those traces.

22. Most of the mental and practical processes with respect to points and lines, are effected by means of these projections, whilst those respecting planes are effected by means of their traces upon the planes of projection.

23. The planes of projection, when brought into coincidence (as in art. 17,) being considered as one, we shall denominate it the compound plane, or picture plane.

Having now stated the objects we have in view, and explained the signification of the terms we employ, together with brief demonstrations of their propriety in a geometrical sense, we shall, in our next number, proceed to the solution of a few elementary problems respecting the straight-line, plane, and dihedral angle. We shall afterwards give a series of applications of these constructions to practical purposes before we enter upon the more intricate inquiries to which descriptive geometry is applicable.

* The line Z Z', drawn through O in this perpendicular to the ground-line, is identical with the trace (next def.) of the plane which has been omitted, or that which, in Art. 7, was called the plane, Y Z.

A POPULAR COURSE OF ASTRONOMY.

No. V.

ONE of the most involved and complicated problems ever proposed to the ingenuity of man, was the problem of the Heavens. A hollow concave above him, the whole of whose surface, go where he may, is apparently at the same comparatively small distance from him; the sun taking his journey across it, in a path which is not daily the same, returning day after day through some unknown region, to flood again the vast canopy of the heavens with light; stars seen in thousands at night, on this vast canopy, moving with one common motion slowly across it, between night-fall and day-break; this host of stars, different at different seasons of the year, but the same at the same season, preserving, in the general alteration of their position, their relative distances; except six of them, which wander about among the rest with a most devious motion, and are therefore called planets; the moon, too, moving with the common daily motion of the rest of the host of heaven, but, besides, revolving completely through it. every month; Winter, Spring, Summer, and Autumn, connecting themselves somehow with the variations of the daily path of the sun, and returning year after year at their appointed seasons; and eclipses of the sun and moon dependent by some inscrutable relation upon relative positions of the sun and moon;—all these things requiring, as they must have done and did, a great length of time, and much and patient observation to discover, constitute in their aggregate a relation of phenomena which as far surpasses any other yet unravelled by the higher researches of the human intellect, in its complication, and the vastness and dignity of the truths which it embraces, as in the simplicity of its results.

The sphere of the heavens has been hitherto spoken of as fixed and immoveable in space; and as having in its centre the EARTH,—of dimensions infinitely small and evanescent with regard to it-rolling perpetually round one of its own diameters, but never moving its centre from that of the great quiescent sphere of the visible heavens. The reader is now about learn that this description of the position of the earth in space is incorrect :—that it does not occupy continually the same position in the centre of the sphere of the visible heavens-that its centre, and the axis within itself about which its revolution takes place, are not at rest—that these are in fact moving at the rate of about 19 miles in each second of time that this motion is not directly forward in space, but continually round in a curve which returns into itself, and which is very nearly a circle, whose radius is ninety-five millions of miles -that nevertheless this enormous circle of the earth's revolution is itself as nothing in its dimensions compared with the dimensions of the great sphere of the visible heavens, so that the motion of the earth from the centre of that sphere, may be considered evanescent as compared with the radius of the sphere, and everything which occurs with regard to the fixed stars, as occurring precisely as it would occur if the earth's centre were really quiescent in space. Thus, then, whatever has been

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argued from the appearances of the fixed stars, on the hypothesis of the quiescence of the centre and axis of the earth in space is accurate, although this hypothesis be false.

Besides the stars called fixed, because they retain always their positions with respect to one another on the sphere of the heavens, there are other bodies visible on it, whose position does not appear to be fixed, either in reference to one another or to the fixed stars; these are the sun, moon, and planets—these, it will now be shown, lie greatly nearer to us than the fixed stars, and thus within that great sphere which has been hitherto designated the sphere of the heavens—they describe enormous ellipses in space, which are yet so small in comparison with the dimensions of that sphere, that they may be considered scarcely to deviate from its centre, and that thus, infinitely great in themselves, but infinitely small in comparison with the distance of the stars, they are all described round one common focus, very near to which lies the centre of the sun.

The process of reasoning by which the complicated apparent motions of the sun, moon, and planets, are made to resolve themselves into these few real and elementary motions, is one of the highest and most successful efforts that has ever been made by the intellect of man.

If the heavens be watched from night to night, continual alteration of the positions of the planets among the fixed stars will, from such observations, continued for a few nights, be very plainly perceived; the planet Jupiter, for instance, being seen one night in the neighbourhood of a particular fixed star, will on the next be found slightly to have receded from it, the space of a week will produce a very marked separation, a month will have taken it completely away from it, and a year will probably have carried it into some opposite quarter of the heavens into the discussion of these apparent motions of the planets, which are very remarkable, we shall not at present enter—it is enough here to state the fact, that there are such motions, and that they do not take place irregularly and towards all parts of the heavens, but that they are, for the most part, confined to a certain zone or belt of it, about 18° in width. This zone, or belt of the heavens, is called the zodiac. A line drawn along its centre would be a great circle of the heavens, and would cut the equinoctial at an angle of 23° 28'.

The moon, too, takes her wandering solitary course eastward along this zone in the heavens. And her broad disc is continually seen covering and passing over the stars which lie along her path. Her motion, although somewhat irregular, is very rapid, being upwards of 13° in 24 sidereal hours, or nearly half a degree every hour, so that she

may almost be seen to move among the stars.

Now the question at once suggests itself, does the sun too move, or appear to move over the concave of the heavens in which he, as well as the moon occupies a place, or does he remain in a fixed position among the stars? This question cannot be determined in reference to the sun, as we determine it of the moon—we cannot see the sun's motion among the stars, for when the sun is up, the stars are to the naked eye invisible ;—how is it then determined Thus :-If the sun were apparently fixed like the stars, the time intervening between the passage of the meridian of any

particular place over the sun, and its return to the sun again, would evidently be precisely equal to the time of its passage over a fixed star, and its return to that fixed star again. Now this is not the case. One of these periods is called a solar, and the other a sidereal day, and the solar day is not of the same length with the sidereal day, it is always longer than it; that is, the meridian of any place on the earth's surface, always revolves from a fixed star to that star again, sooner than it revolves from the sun to the sun again. The sun then does not remain, or appear to remain, fixed like the stars, on the sphere of the heavens, it moves in the same direction in which the meridian moves, the meridian arriving at the place in the heavens where the sun was on the preceding day, before it arrives at the sun. Now the meridian revolves with the earth eastward, the apparent motion of the sun on the sphere of the heavens is therefore eastward.

And, moreover, the amount of this daily motion of the sun eastward may readily be found; we have only to subtract a sidereal day (that is, the time which the meridian occupies in revolving from the sun on one day to the same place in the heavens on the next) from a solar day, or the time of the revolution of the meridian from the sun on one day to the place which the sun actually occupies in the heavens on the next day. The difference will be the time which the meridian has occupied in revolving from the sun's place on the preceding day to its place on this day, this difference will be found different for different days in the year, but its average is 3 561" of sidereal time. This, then, is the mean sidereal time which the meridian occupies in revolving from the sun's place on one day in the heavens to its place on the next day. Now the meridian revolves through the whole 360° of the heavens in 24 sidereal hours, or over 15° of it in one sidereal hour; it revolves therefore, as may readily be found by the rule of three, over an arc of 59'8" in this 3' 56}" of time ; and therefore the sun's place on the second day is 59' 8" more to the east than on the first day, or its daily motion is 59'8" eastward. But an arc of 59 8" being multiplied by 365 will give us 360°. In 3654 days, therefore, the sun will have revolved eastward on the sphere of the heavens through 360°, that is, completely round it—this period is one

solar year.

the stars,

The sun, then, although we cannot see him moving on the heavens, there being no fixed object visible upon them when the sun can be seen to which we can refer his motion, does yet present the same phenomena as though he moved continually like the moon eastward among except that instead of completing his revolution, as the moon does, in one lunar month, his gyration takes him a whole year.

But what path does he describe in the heavens; he revolves round them, but in what route? As we cannot see him among the fixed stars, how shall we find out his course? Thus:—We may find out as is now to be shown, what declination circle he is on for every day of the year

at noon, and also we may find what is his declination, that is, we may find where he is on his declination-circle. Knowing these two elements, we shall know his exact position on the sphere of the heavens, and referring to a celestial globe or chart, we shall tell what stars are in his neighbourhood.

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The declination-circle on which the sun is, may be found thus:let the exact time of the meridian passing over the sun's centre on any day be noted—now that meridian we know will return to its place in the heavens, or revolve through 360°, in 24 sidereal hours, it will therefore revolve through 180° in 12 sidereal hours. Let us then observe what stars the meridian is passing over precisely 12 sidereal hours after our first observation; we shall know that these stars are 180° from the sun's place on the preceding noon: counting, therefore, off 180° westward, on the equinoctial of a globe from that declination-circle on which are these stars, we shall know that the sun was on the declination-circle which passes through the point which we thus find on the preceding

noon.

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Again, to determine the declination of the sun, we have only to find, by

observation, his meridional zenith distance, z s, and subtract it from the latitude, zi, the remainder will be the declination s.

Finding thus the declination of the sun, and the position of his declination-circle for the noon of every day in the year, we can mark on the celestial globe his position for every day, and joining these positions, we shall obtain his path on the sphere of the heavens. Now, all this has been carefully done and the apparent path of the

sun among the fixed stars is traced on all our celestial globes. The sun is thus ascertained to have, in common with the moon and planets, its path, called the Ecliptic, along that zone or belt of the heavens which we have called the zodiac; it is a great circle of the sphere, constituting in point of fact the centre of that belt which stretches go on either side of it. The ecliptic is inclined to the equinoctial, at such an angle, that at the greatest separation of these circles there are 23° 18' interval between them, measured on one of the declination-circles of the sphere. Along this path in the heavens the sun would appear to us to move, as the moon does, among the fixed stars, were it not that, by the superior brilliancy of the sun, the stars are invisible to the naked long as he is above the horizon.

The sun and moon both, then, have an apparent motion round the earth, the one in a year, and the other in a month. It is now to be shown that this apparent motion of the sun round the earth is not a real motion of the sun, but that it results from a real motion of the earth round the sun; and further, that the apparent motion of the moon is a real motion, resulting from an actual monthly revolution about the earth.

In the first place, then, it is asserted that a real motion of the earth round the sun would produce precisely those appearances observe in the heavens, and which have been attributed to an annual motion of the sun along that line which has been called the ecliptic.

Let A, B, C be positions of the earth, in an orbit which it is supposed to describe about the sun, at intervals each of a sidereal day. Also let AP, BP, CP, be lines drawn from a certain fixed star to a particular

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