intersect and bisect each other; and one of the points where the ecliptic intersects the equator is called the first point of Aries (the point from which we have said that the right ascension is measured); the other, the first point of Libra. For reasons which will afterwards be seen, the former of these is also called the vernal equinox, the latter the autumnal equinox. The first point of Libra is often marked thus, ✩. SECTION 4.-Variation of Sun's upparent diameter Sun's orbit an Ellipse, the earth being in the focus Variation of angular velocity. Equable description of areas by Sun's radius vector. We have already stated, that if any plane pass through the centre of the sphere, its intersection with the sphere is a great circle; and that the ecliptic, or sun's apparent path, is such a circle. As yet we know nothing of the distance of the sun from the earth; but it is clear that we have no reason to suppose his distance equal to that of the stars, or always the same: if we have no means of comparing his distance with that of other objects, we can only ascertain the direction in which he appears. Thus the sun, the moon, the stars all appear alike situated in the concave surface of the heavens, although we shall hereafter see that their distances are almost immeasurably different; and any one who has seen a bright meteor at night, such as those commonly spoken of as shooting, or falling stars, knows that they also appear to the eye to be similarly distant. The very name indeed of shooting stars is evidently deduced from this circumstance. In their case indeed we know them really to be meteors engendered in the atmosphere, and therefore know their distance to be comparatively small, or we may sometimes ascertain it to be so by comparing observations made at different places; and in the case of the moon, we may perhaps imagine that she looks less distant than the sun or stars: but well ascertained by knowing its longitude and latitude, as by knowing its right ascension and declination; but the latter are the best adapted for observation, and the former may always be deduced from them. The terms longitude and latitude, as applied to a heavenly body, are established; but it is to be regretted that they have been chosen, as their sense in that application differs widely from their meaning with respect to places on the earth, being, indeed, analogous measures in the two cases, but referred to different circles and planes. this notion, when it obtains, is probably really owing to our habitual knowledge that she is so, or to our being able to distinguish different parts of her surface from each other, by the different degrees of their brightness. If however we abstract ourselves from such considerations, we shall find ourselves alike unable to estimate by the eye the distances of any of these objects, and consequently unable to say which are furthest from us. All therefore that we can conclude, from finding the apparent path of the sun to be a great circle of the heavens, is, that his motion is in a plane passing through the earth; but what the shape of his circuit is, whether his distance be always the same, or be variable, and whether, if it vary, it vary gradually and regularly, or suddenly and without any fixed rule, we as yet are quite ignorant. We know only that we continually see him more Eastward at every successive observation, and consequently conclude that his motion is always in that direction, and not occasionally East, and occasionally West. To determine the form of his orbit, we need only know the proportions of his distances from the earth at differ. ent times. For this purpose the actual distances are unimportant; for instance, if his distance is always the same, his orbit is a circle, whatever be that distance, which is the radius of the circle. Now the apparent diameter of an object, or the angle which it subtends at the eye of the observer, increases in the same proportion as the distance diminishes, so long as the real diameter continues unchanged, and the apparent diameter is very small: * or in such cases, the apparent diameter varies inversely as the distance. Now the apparent diameter of the sun may be very accurately measured, and it is found to be continually varying: his distance therefore is so also, for we have no reason whatever to suppose that his ac tual magnitude changes; and the proportions of his apparent diameters at different times being ascertained by observation, the proportions of his cor [In strictness, the tangent of the apparent radius (i. e., of the angle subtended by the radius)= real radius divided by the distance: and the real radius being always the same in the same body, the tangent of the apparent radius varies inversely as the distance. Whenever, therefore, the apparent radius is so small, that the arc and its tangent may be treated as equal, the apparent radius varies inversely as the distance: and so of course does the apparent diameter, which is double the apparent radius.] responding distances may be so also. If then we conceive lines drawn from the earth to every point of his orbit, the observation of his right ascension and declination will always determine on which of these lines he is situated; and if we take portions of these lines always in the inverse proportion to each other of his apparent diameters, the curve joining the extremities of these portions will be similar to his orbit. This perhaps may be rendered clearer by a figure, and an instance. Thus let us suppose the sun to be observed on any day, and that his apparent diameter is 31; again, and that his place in the ecliptic is then 700 from that of the first observation, and his apparent diameter then of 32'; and a third time, and that his place in the ecliptic is then 180° from that of the first observation (or exactly opposite to it), and his apparent diameter then of 324. Take any line E Si to represent the direction of the sun at the first observation; and produce it, in the opposite direction, to Sa: then E Sa will represent the direction of the sun at the third observation; and if E S2 be drawn making an angle of 70° with E S, it will represent the direction of the sun at the second observation. Now the apparent diameter of the sun at the first observation is 31, at the second it is 32'; his corresponding distances, therefore, are in the proportion of 32 to 31, or of 64 to 63. If, therefore, we take in the line E S, a portion Es, of any magnitude we choose, and measure upon E S2 a portion Es2 which shall be to E 82 in the proportion of 63 to 64, the distances E81, E82 will be proportionate to the distances of the sun at those two observations. In the same manner, the apparent diameter of the sun at the third observation being of 32', his distance at that observation is to his distance at the first observation in the proportion of 31 to 324, or of 63 to 65, and it is to his distance at the second observation in the proportion of 32 to 324, or of 64 to 65. If, therefore, we take in the line E S., a portion Ess, which shall be to Es in the proportion of 63 to 65, or to E s in the proportion of 64 to 65, the distance Es, will be proportionate to the distance of the sun at the third observation. All the distances, therefore, E s,, E82, E83, will be proportionate to the distances of the sun at the different observations; and the points 81, 82, 83, will accurately represent his situations at those times. In the same manner his situations at other and intermediate times may be represented, and the line joining all these points (the curve 8, 8, 8) will represent his orbit. It is found, by careful observation, that this line is an ellipse, of which the earth is in the focus*. The ellipse is an oval curve, divided into two equal and similar parts by a line drawn through its foci, which is called its major axis, or transverse axis. If from one of the foci lines be drawn to every point in the curve, the greatest of these lines is that portion of the major axis which is drawn through the other focus to meet the curve, and the least is the other portion of the major axis; and the lines continually diminish from the greatest to the least, and continually increase from the least to the greatest. It follows therefore, if the sun moves in an ellipse round the earth in the focus, that his apparent diameter should continually increase from its least amount, which is when the sun is at his greatest distance, or in his apogee (from two Greek words signifying away from the earth) to its greatest amount, which is when the sun is at his least distance, or in his perigee (from two Greek words signifying about or near the earth); and as the two portions of the ellipse are equal and similar, the diminution of his distance in the one case will correspond with the increase of it in the other, and at the end of a complete revolution he will again be at his original distance. And so we actually find it. The least apparent diameter of the sun is of 31 nearly, and is at present observed about the 30th of June; he is therefore then in his The ellipse is one of the curves called conic sections. It is the curve made by cutting a cone by a plane, which passes through it without intersecting the base. Its fundamental property is this: there are two points within it, called the foci, such that the sum of the distances of any point in the curve from the two foci is always the same. This property furnishes an easy mode of drawing the curve. If a thread be fixed at the two foci, and a pencil carried round within the thread, so as always to keep it stretched, the pencil will describe an ellipse; for the whole length of the thread is always the same, and it constitutes the two distances from the foci. apogee: his apparent diameter then continually increases until (about the 30th of December) he arrives at his perigee, when his apparent diameter is of 32′ 35′′ nearly and then again it continually diminishes until (about the 30th of June) it is again of 31, from which value it again begins to increase. The variations of the apparent diameter being so small, those of the distance are so also, or the ellipse differs but little from a circle. Having thus ascertained that the sun appears to move in an ellipse round the earth, we next inquire what is the rule of his motion, whether it be regular or irregular, uniform or variable. We have already seen that we can, by ascertaining the right ascension and declination of a body, determine its place in the heavens, or the direction in which it is seen from the earth: we may therefore ascertain at each particular instant the point of the ecliptic in which the sun is, and, consequently, the arc through which he has moved in each particular interval, or the angles which the directions in which he is successively seen make with each other. We have thus the means of ascertaining his angular velocity, as referred to the earth. Now this angular velocity is found to be variable, and it is also found to be greatest when the apparent diameter is greatest or the distance least, and least when the apparent diameter is least or the distance greatest; and generally, the greater the distance the less we find the angular velocity. Its inequalities however are greater than those of the apparent diameter or distance. We have already seen that the least apparent diameter is to the greatest nearly in the proportion of 30 to 31; but the proportion of the least angular velocity to the greatest is nearly that of 30 to 32, and hardly so great. On accurate comparison of the different angular velocities with the distances, it is found that they vary inversely as the squares of the distances, that is, that they diminish in the same proportion as the squares of the distances increase, and increase in the same proportion as the squares of the distances diminish. For instance, if the distance is doubled, the angular velocity is reduced to a quarter of its former amount; if the distance is diminished by a third, the angular velocity is increased in the proportion of 1 to the square of twothirds, or of 9 to 4. These velocities however, and the distances themselves, may be considered for very short periods of time as constant, for the changes of distance in such periods are so small that they may be neglected. The whole variation of distance is only about a thirtieth part of the least distance; the greatest difference between the angular velocities only about a fifteenth part of the least angular velocity; and these differences are the accumulated differences of months: for an hour therefore, or a day, there will be no perceptible difference, and none which can at all affect the results which we shall deduce from the supposition that, during such a period, they are uniform. In fig. 3, let t be the place of the sun in its orbit, when one of these very short periods has elapsed since it was at s1. If we conceive a straight line drawn from the earth to the sun, and moving round the earth with the sun, as for instance, if they were joined by a wire, which we must suppose to be lengthened and shortened as the sun recedes from and approaches the earth, so as always to extend from the centre of the one to the centre of the other, and no farther, this line will originally have coincided with E s,, and its position, when the sun is at t, will be Et. While the sun has moved from s, to t,, therefore, or described the arc s, t, this line or wire will have described the small area Es, t. This line is called the radius vector, and it is of great importance to ascertain the areas which it describes, and these we shall find to be always equal in equal portions of time. [This area may be considered as a triangle, for the arc s, t, being very small, differs insensibly from the straight line joining the points sit; and the area of a triangle is equal to half the product of its base, and the perpendicular from its vertex. Again, the angle t E s, being very small, the perpendicular from the vertex t is indistinguishable from a small circular arc, whose centre is E, and radius Et, or Es1; and these also may be considered as equal. The magnitude of such an arc varies as the product of the angle t Es, and the radius Et, or Es. The area tEs, therefore (which is equal to half the product of its base and the perpendicular from its vertex) varies as the distance Es, multiplied by the product of the angle t Es, and the radius Es, (for its base is the distance Es1, and the perpendicular we have already seen to vary as the product of the two latter quantities); or it varies as the square of the distance Es, multiplied into the angle tEs. If, however, the area Est be supposed to be that described in some given portion of time, as an hour or a minute, and the angular velocity be uniform during the time of describing it, the angle t E s1, varies as the angular velocity; for it will increase or diminish exactly in the same proportion as that velocity increases or diminishes. The area t Es, therefore, described in such a period of time, varies as the square of the distance E 81, multiplied by the angular velocity. But we have seen that the angular velocity varies inversely as the square of the distance: the area t E s, therefore is not affected by the distance at all, or it is constant; for in whatever proportion it is increased by reason of the increase of the distance É si, in the same proportion it is diminished by the diminution of the angle t Es1.] If however the areas described by the radius vector in small equal portions of time be equal in every part of the orbit, the whole areas described by the radius vector in equal portions of time of any magnitude will also be equal; for they will be the sums of equal numbers of these small equal portions: and generally, the areas described by the radius vector in any times whatever will be proportional to those times. This therefore is ascertained to be a law of the motion of the sun round the earth, and it is one which we shall hereafter find of the very greatest importance. We have now ascertained these facts with respect to the sun : that its annual motion round the earth takes place in an ellipse, of which the earth is in one of the foci; that the plane in which he moves is inclined at an angle of 23° 28' to the plane of the equator; and that his radius vector describes areas proportional to the time. These are the principal phenomena of his motions: they require however some qualifications, which we shall hereafter mention; and we shall also see that we may explain his apparent motions on the supposition that he is at rest, and the earth moves round him, just as we have said that we shall be able to explain the apparent diurnal motions of the stars, by supposing them at rest, and the earth spinning on an axis. We may however at once proceed to explain how these facts produce some of the most important phenomena which we observe, especially the difference of the seasons in the same country, the different lengths of the day (as distinguished from the night) at different periods of the year, and the different climates of different countries; and also the equation of time, or the manner and degree in which the real differs from the mean solar time. SECTION 5.- Comparative length of time during which bodies at different Declinations continue above the hori -Variation and length of Day at the same place-Tropics-EquiSeasons produced by the noxes Sun. WE have already seen that the apparent diurnal motions of the heavenly bodies are performed in circles, every point of which is equidistant from the pole: of these circles the equator is one, and is a great circle of the sphere; the others are all small circles parallel to the equator. The horizon, being a great circle as well as the equator, bisects that circle. It will immediately appear, also, from inspection of any of the figures, as fig. 1, or fig. 2, that of the parallel circles a larger portion is continually above the horizon, and a smaller continually below, as they approach nearer that pole which is above the horizon. Thus, in fig. 1, Pq being less than PQ, tqr is a larger portion of the circle tqrb than TQR is of TQRB: and the portions continually go on increasing, until, finally, the whole circle is above the horizon, as in the case of e a as as q in fig. 2, or of X VYU in fig. 1, when the distance, Pq or P U, of the body from the pole is less than the elevation of the pole above the horizon. In the same manner, if a body move in a circle distant from the other pole, p, less than its depression below the horizon, or than Hp, it will never appear above the horizon at all. We have also seen that the diurnal motions of the heavenly bodies are equable; that is, that the times of passing through every successive portion of their circles of motion bear the same proportion to the whole time of rotation, that the portion described bears to the whole circle; and that the times of rotation in every circle are the same. The larger, therefore, the portion of each circle of rotation which is above the horizon, or the nearer that circle approaches to the pole which is above the horizon, the longer the body is above it. If then the North Pole be apogee: his apparent diameter then continually increases until (about the 30th of December) he arrives at his perigee, when his apparent diameter is of 32′ 35′′ nearly and then again it continually diminishes until (about the 30th of June) it is again of 31, from which value it again begins to increase. The variations of the apparent diameter being so small, those of the distance are so also, or the ellipse differs but little from a circle. Having thus ascertained that the sun appears to move in an ellipse round the earth, we next inquire what is the rule of his motion, whether it be regular or irregular, uniform or variable. We have already seen that we can, by ascertaining the right ascension and declination of a body, determine its place in the heavens, or the direction in which it is seen from the earth: we may therefore ascertain at each particular instant the point of the ecliptic in which the sun is, and, consequently, the arc through which he has moved in each particular interval, or the angles which the directions in which he is successively seen make with each other. We have thus the means of ascertaining his angular velocity, as referred to the earth. Now this angular velocity is found to be variable, and it is also found to be greatest when the apparent diameter is greatest or the distance least, and least when the apparent diameter is least or the distance greatest; and generally, the greater the distance the less we find the angular velocity. Its inequalities however are greater than those of the apparent diameter or distance. We have already seen that the least apparent diameter is to the greatest nearly in the proportion of 30 to 31; but the proportion of the least angular velocity to the greatest is nearly that of 30 to 32, and hardly so great. On accurate comparison of the different angular velocities with the distances, it is found that they vary inversely as the squares of the distances, that is, that they diminish in the same proportion as the squares of the distances increase, and increase in the same proportion as the squares of the distances diminish. For instance, if the distance is doubled, the angular velocity is reduced to a quarter of its former amount; if the distance is diminished by a third, the angular velocity is increased in the proportion of 1 to the square of twothirds, or of 9 to 4. These velocities however, and the distances themselves, may y be considered for very short periods of time as constant, for the changes of distance in such periods are so small that they may be neglected. The whole variation of distance is only about a thirtieth part of the least distance; the greatest difference between the angular velocities only about a fifteenth part of the least angular velocity; and these differences are the accumulated differences of months: for an hour therefore, or a day, there will be no perceptible difference, and none which can at all affect the results which we shall deduce from the supposition that, during such a period, they are uniform. In fig. 3, let t be the place of the sun in its orbit, when one of these very short periods has elapsed since it was at s1. If we conceive a straight line drawn from the earth to the sun, and moving round the earth with the sun, as for instance, if they were joined by a wire, which we must suppose to be lengthened and shortened as the sun recedes from and approaches the earth, so as always to extend from the centre of the one to the centre of the other, and no farther, this line will originally have coincided with E s,, and its position, wher the sun is at t, will be Et. While the sun has moved from s, to t,, therefore or described the arc s, t, this line of wire will have described the small are Es, t. This line is called the radiu vector, and it is of great importance t ascertain the areas which it describes and these we shall find to be always equa in equal portions of time. [This area may be considered as triangle, for the arc s, t, being ver small, differs insensibly from the straigh line joining the points sit; and the are of a triangle is equal to half the produ of its base, and the perpendicular fro its vertex. Again, the angle t E s, beir very small, the perpendicular from th vertex t is indistinguishable from a sm circular arc, whose centre is E, a radius Et, or Es; and these also m be considered as equal. The magnitu of such an arc varies as the product the angle t Es, and the radius Et, Es. The area t Es, therefore (whi is equal to half the product of its ba and the perpendicular from its verte varies as the distance Es, multiplied the product of the angle t Es, and radius Es (for its base is the distan Es, and the perpendicular we have ready seen to vary as the product of |