H are fixed, the vertical plane drawn through them must be fixed also, and this meridian therefore is a fixed line. It will also be found by observation, that the points Q, q, where the paths of the different stars meet this meridian, are the most elevated points of those paths respectively, and consequently that a star which rises and sets attains its greatest height above the horizon, or culminates, when it is upon the meridian; and we may add, that the paths, TQR, tqr, of the stars observed, appear to be parallel to each other, and that each path is divided by the meridian into parts TQ, QR, or tq, qr, apparently equal and similar; that is to say, TQ equal and similar to Q R, and tq to qr.

In one part of the heavens the appearances are a little different. If we look to the Northern part we shall see many stars which never sink below the horizon. These stars pass the meridian twice, once at the lowest, and once at the highest point of their course. In moving from the lowest to the highest point, their course is entirely to the East of the meridian; in passing from the highest to the lowest point, it is entirely to the West of that line. These parts also in this case, as well as in that already mentioned, are similar and equal; and if the course of one of those stars be represented in the figure by the line UX V Y, every point in that line will be found to be equidistant from a certain point P, situated in the sphere, and UXV Y will be a circle, and the point P its pole. Besides this, it will be found that the path of every one of these stars which never set is a circle described at a given distance from the same point P, which itself appears stationary; and the position of this point being ascertained, it will further be found, that each of the paths TQ R, tqr, described by stars which alternately rise above the horizon and fall below it, and which appear, as we have already mentioned, to be parallel to each other, are themselves also portions of circles, every point of which is equidistant from the same point P, and which therefore are parallel to each other, and to the path UXV Y, described by a star which never sinks below the horizon. The point P therefore is the pole of all the parallel circles described by the stars; and if we suppose the sphere to be completed, as it is by the dotted lines in the figure, and p to be the point directly

HQqN, being his position. Let T be the spot where a star first rises, TQ R its path above the horizon, and R the point where it sets; and let the arch THR of the horizon, intercepted in one direction between the points T, R, be bisected in the point H, and the arch TNR intercepted in the other direction, be bisected in the point N. It is clear that H and N are at the two extremities of a diameter, for HR = HT, and RN=NT, and therefore, HRN (the sum of HR and RN)=H TN (the sum of HT and TN); or HR N, HTN are each of them semi-circles. Now, suppose tqr to be the path of another star above the horizon, its point of rising, r its point of setting; it will be found that the arch H = the arch Hr, and t N = Nr; that is to say, the points N and H in this case also bisect the arches of the horizon intercepted between the points of rising and setting. The same will be found to be true of every star (with some apparent exceptions to be presently noticed), and the points N and H are consequently fixed points, and independent of the particular stars by observation of which they were ascertained. Let HQqN be the intersection with the celestial sphere of the plane passing through NH and perpendicular to the horizon; this is the meridian of the observer at O, and as the points N and

opposite to P, p will be the other pole of the same circles; or there are two fixed points, one P found by observation, the other p deduced from it, to which the motion of the stars may be equally referred. O being the centre of the sphere, will be a point in the line which joins P, p. The stars which are seen to describe their whole circle round the pole are called circumpolar stars. We shall also find, that in the time in which any particular star describes a third or fourth part, or any given portion of its circle, all others describe the same portion of theirs; and consequently they all continue in the same positions with respect to each other, though their places vary with respect to the horizon and the observer.

Having thus far ascertained the appearances which the stars present, let us see if we can thence deduce any conclusions respecting the occasion of them. For this purpose let us suppose the hemisphere, HQ PN, to be made a complete sphere, as is done by the dotted lines in the figure. Let us also suppose that the whole sphere has a motion of rotation round a line joining P, p, which is called the axis; but that HRN T, the horizon, or the line in which the plane bounding the visible hemisphere meets the heavens, continues fixed: and let us see what would be the appearances presented in such a case. Let us take the case of a star upon the horizon at T. It is clear that, as by the rotation of the sphere it was transferred from T, it would appear to move in a line of which every point was equidistant from P, for every point in that line would be determined by the actual distance of the star from P. Its apparent path therefore would be a portion of a circle, every point of which is equidistant from P; and, in point of fact, we have already seen that it is so. In the same manner, if there be a star which never falls below the horizon, and whose distance from P is PV, its apparent path would be a circle, of which every point is at the same distance PV from P, or it would be represented by a circle, UXVY, which we have already seen to represent the apparent path of a star which never sets. Each of these circles, thus described by the motion of different stars,

These positions may be thus illustrated :-If you take a top, or any body which you can spin with great steadiness and accuracy, and place a spot of ink upon it, and then spin it with great velocity, so that the spot returns to the same place in less time than is necessary for the eye

having every point in it equidistant from the same point P, they all would, as before, be parallel circles. Again, as each of them is described in consequence of the same general motion of rotation of the whole sphere, each would be described in the same time; namely, the time of that rotation; and in the same manner, in any portion of that time, each star would describe the same portion of its own circle; namely, the same portion of that circle which the sphere describes of a complete revolution. All these are the appearances which we have already seen that the heavens, in fact, present.

The appearances presented by the motions of the stars may then be accounted for on the supposition that the sphere of the heavens revolves round an axis joining P, p. They cannot be explained however on this supposition, except by supposing that the sphere goes through a complete revolution. The motion of a star which is seen to rise and set, as that whose path is TQR, might be explained by imagining the sphere to make only a part of a revolution; and the magnitude of that part would depend on the proportion which the visible path TQR bore to the whole circle of which it formed a part; but the stars which never set, as that whose path is UXVY, are seen to describe the whole circle, and their motion therefore can only be thus explained on the supposition of a complete revolution of the heavens on the axis Pp. If however this be the case, the motions of the stars which sink below the horizon must also be continued below it, or they will describe below it the remaining parts (those represented by the dotted lines) of the circles TQ RB, tqrb. Let us see if we have any means of discovering, by observation or reasoning, whether they do so.

The first remark that occurs on this question, is that the supposition that they describe below the horizon the remainder of the circle, of which they are seen to describe part above it, at once accounts for one circumstance that seems

to lose the impression last made by it (See Treatise on Optics, p. 41), the effect produced will be that of a line encompassing the top, and forming a circle upon it, of which every point will be equidistant from the extremity of the axis upon which the top spins. The apparent track of the point is of course the same, whether its motion be quick or slow; but by the rapidity of its motion we gain the advantage of actually seeing the whole track at once.

to admit of no other explanation. We trace the path of a star from its rising at T, to its setting at R, and then lose sight of it; but on the next night we again see it appear at the same point, T. We know therefore that the star is in some way transferred from the point R, where it sets, to the point T, where it rises; and the most probable way in which we can suppose this transference effected, is the continuation below the horizon of the same motion which it had when above it, or the description of that circle, R BT, which it would describe on the supposition that the whole heavens revolve round the observer.

If however we take the case of a star rising just at the time when the stars begin to appear in the evening, and setting as day breaks on the following morning, it is evident that its path below the horizon, if it be described at all, must be described by day; or that the same motion of revolution continues by day, which we seem to have ascertained to exist by night. Does observation then confirm, or disprove this conclusion? The sun and moon are visible by day, but their motions, although they generally confirm it, are of a more complicated nature, and we there fore do not wish to draw our inferences on this point from them; and the stars are not visible to the unassisted eye when the sun is above the horizon. The telescope however, in the hands of a skilful observer, for only such a one can make the observations necessary for this purpose, removes this difficulty; with it he can, even when the day is brightest, ascertain the positions from time to time, and consequently the motions, of many of the brighter stars; and the result of these observations is, that the stars are ascertained to describe in the day-time the same courses which they are easily seen to trace in the night: and we consequently come to the conclusion that their motions may be accurately comprehended and explained, on the supposition that the whole heavens revolve about an axis, passing through the position of the observer, and carry the particular stars with them in their revolution.

If this be so, and the meridian of the place, HQ P N, be continued, as by the dotted line N Bp H, below the horizon, so as to complete the circle, this lower part of the circle will again intersect the circles T Q R B, tqrb,

in the opposite points, B, b, to those, Q, q, where the upper part of it met them above the horizon; and as Q, q, were the points most elevated above the horizon, B, b, will be those most depressed below it; or in other words, every heavenly body, which sinks below the horizon of a particular place, will be most depressed below it, when it passes the meridian of that place below the horizon, and of course below the pole. have already seen a corresponding result with respect to circumpolar stars, when they cross the meridian below the pole though above the horizon.

We

As yet we have only considered the conclusions which an observer, confined to a single point on the earth's surface, would arrive at on this subject. We will now proceed to examine how they will be affected by a comparison with the results of other observations, made at a different place. The account which we have given of the observations made at one place, applies with equal correctness to all; that is to say, an observer situated anywhere upon the earth, finds that the apparent paths of the stars are circles, or portions of circles, each having every point in it equidistant from two fixed points, one in the observed heavens, and one in the other part of the sphere, supposed to be completed, and each bisected by a line passing through the visible fixed point, and dividing the visible heavens into two equal portions. In each case therefore, this line is what we have termed the meridian of the place of observation; and every place therefore has a meridian, passing through a fixed and immoveable point in the heavens. The position of this point may be ascertained by observation at each particular place, and it is found to be the same at all; the other extremity of the axis also is the same in each place.

We come therefore to this conclusion, that the axis Pp round which the revolution of the heavens takes place is a fixed and determined line, not depending on the situation of the observer: and this is one circumstance necessary to the establishment of our theory, that the apparent motions of the stars may be attributed to the revolution of the heavens round a fixed axis; for if observations made at each place gave a different axis, they would be inconsistent with such a supposition. The points, P, p, are only imaginary points, being

those where the axis Pp meets the imaginary sphere of the heavens; they are however important to be known, and go by the name of the poles of the hea vens. They are points, as we have already seen, in the meridian of every place, and therefore they no where appear either in the East or West side of the heavens; if however we conceive the heavens divided by a vertical plane, passing through the East and West point at any place, the points P and p will always be on opposite sides of this plane; that is to say, the one on the North side of it, the other on the South, and the same point P is always on the same side of the plane. If therefore (in fig. 1.) P represent the pole, which to an observer at O is on the Northern side of the heavens, P is always on the Northern side, and is called on this account the North Pole of the heavens, and in like manner p is the South Poler. There are however two points on the earth, (the poles of the earth) where the points P, p, are the one directly over the head, the other directly under the feet of the observer; here therefore there is no North or South point, and we shall hereafter see that the phenomena from which we deduced our definition of these points, namely, the rising and setting of stars, do not take place at these situations. We have already seen that P, p, are points in the meridian of every place; all these meridians therefore intersect each other at the two poles. If p, the South Pole, be above the horizon, P, the North Pole, will of course be below it.

One circumstance may here require explanation before we proceed farther. We have already seen that the centre of the heavenly sphere is a point in the axis P p, and that this centre appears to be the situation of the observer; and we have also said that the results of

The circle made by the intersection of such a plane with the sphere of the heavens is called the prime vertical.

A well-known star, a of the Little Bear, called also Polaris or the Pole Star, is at the distance of only one degree and fifty minutes (see the next note) from the North Pole P. Its motion, like that of all other stars, is in a circle, every point of which is equidistant from P. As P therefore is a fixed point, and the Polestar always very near it, the observation of the Polestar furnishes a very easy method of finding very nearly the fixed North Pole of the heavens. The meridian being a vertical circle, and passing through the pole to the horizon, the points where it intersects the horizon are those points of the horizon respectively nearest and most distant from the pole, and thus the North point of the horizon is that nearest to, and the South point of the horizon that most distaut from, the North Pole.

observation are the same, wherever on the earth's surface he be placed. If two observers be at situations, the one one thousand miles East of the other, the situation of both cannot be in the line Pp; but if the one is in it, the other must be nearly one thousand miles out of it: yet they both appear to be in it. We know from very simple reasoning, or we may easily satisfy ourselves by trial, that a small change of position in the observer does not affect the apparent position of a very distant object. Thus, if there be two trees, or two spires distant ten miles from each other, and two men stand half-way between them, the one precisely in the line joining them, and the other a yard on one side of it, each will alike feel that, to all common observation, he is exactly in the line which unites them. The angle between the two directions in this case, would be considerably less than half a minute*, and would not be observable except by instruments of some delicacy. In the same manner, if the distance to the points P, p, be excessively great in proportion to the distance between the situations of different observers, each observer will seem to be in the same position with respect to the points P, p, and the line joining them. There is therefore nothing absurd or contradictory in the apparent coincidence of each situation with the line Pp, if we only suppose the points P, p, so remote from the earth, that any line drawn on its surface is too small to be estimated in comparison with that distance; and we get therefore a notion of the vast distance of those points, instead of a difficulty affecting the notion of such a revolution, as we have supposed to take place. If however every point on the earth's sur

Every circle is conceived to be divided into 360 parts called degrees, each degree into 60 minutes, each minute into 60 seconds, each second into 60 thirds, and so on. One degree is written 1°; one minute l', one second 1"; and so on; though the divisions beyond seconds are quite as frequently expressed in decimal parts of a second. Thus nine degrees, fifteen minutes, twelve seconds and twenty-four thirds, are written 9° 15′ 12′′ 24′′′′, or, as 24 thirds are, or of a second, they are also written 9° 15′ 12′′. 4. A degree therefore is the 360th part of a circle, a minute the 21600th part of it; half a minute, the quantity mentioned in the text, the 43200th part of a circle. The degrees, minutes, &c., of one circle, will of course occupy more space than those of another, exactly in the same proportion as the one circle is longer than the other, for they are in each the same proportional part of the whole circle. They correspond to the same angle in every case, but differ in linear magnitude, as the circles on which they are measured differ.

face be apparently in the line Pp, so must its centre also, which lies in the midst between these points. The axis Pp therefore may be considered to pass apparently through the centre of the earth; and we shall hereafter see the strongest physical reasons for believing that it actually does so.

There are several reasons which convince us, on very slight examination, that the earth itself is, speaking loosely, of a globular figure. They are collected and explained in the first pages of the "Treatise on Mathematical Geography;" and it is therefore unnecessary here to go into any account of them. But it is important here to point out how we are enabled by this globular figure of the earth to verify the conclusions we have already formed respecting the motions of the stars. The earth being convex at all points, the horizons of different places, which are always planes touching the earth at those points, will be inclined to each other at all different angles; and the height of the pole above the horizon will vary in consequence. The height of the pole above the horizon measures what is called the latitude of the place. We shall hereafter explain why it is sufficient now to state, that whenever we speak of the latitude of a place, as having a particular value, for instance 50°, we mean that the pole is there at the height of 50° above the horizon. We have already seen that there are certain stars which never sink below the horizon at a particular place, and which at that place are called circumpolar stars. If we take another situation, so that the height of the pole above the horizon be greater there than at the former place, some of those stars which before rose and set, will now (if the supposition, that they continue to describe below the horizon the remainder of the circle which they were seen to describe in part above it, is a true one) have their whole course above the horizon: in other words, they also will be circumpolar stars at the second place of observation; and we may there ascertain, by complete observation, whether they do actually describe the whole circle, as we have supposed

The argument in that treatise deduced from the appearances of heavenly bodies cannot properly be applied to the purposes of this treatise, as the reader must be supposed hitherto ignorant of the nature of these appearances. Omitting these, we have sufficient evidence, in p. 3 of that Treatise, of the globular figure of the earth in a very general sense, and that is all which we want for the present purpose.

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they would and in every case it is found that they do so. Again, the earth being globular, there are points on its surface from which an observer will actually see the opposite part of the heavens from that presented at any given places; and the question, whether the same motions be continued below the horizon which we observe above it, may in this way also be brought to the test of complete trial, and the truth of the doctrine will be established, if we find that in every part of the earth the same appearances of circular motion are observed in the paths of the stars above the several horizons of each observer. And this we find to be universally the case. In every way therefore the original conclusions, which we draw from observations at a single place, are confirmed by a comparison of those made at several.

We shall hereafter see that there is another and simpler mode of accounting for these appearances, by ascribing a motion of rotation to the earth itself. The appearances themselves however are exactly those which would result from the rotation of the heavens; and our only proof that the other theory will account for them, will be by shewing that it must necessarily produce the same appearances as the actual rotation of the heavens would do. We may therefore properly, for the present, treat the rotation of the heavens, not as an established fact, but as a supposition enabling us to account for, and represent all these appearances.

SECTION 2.-First Observations on the apparent Motions of the Sun-Diurnal Motion-Annual Motion. Having thus ascertained the apparent rotation of the heavens and the stars round an axis, we may proceed to consider the more complicated appearances presented by the sun. The first notion which we gain from observation is, that he also follows the same course of revolution as the stars; for he is seen to rise in the Eastern, and set in the Western, part of the heavens, culminating in the meridian, and rising and setting at points apparently equidistant, or very nearly so, from the North point of the horizon. So far the appearances of a single day seem to correspond with those already noticed in the stars. But when we register the observations of a long period, a year for example, we find a striking difference between the two cases. The star always