nishes as his declination increases, after it has once attained that value which makes it equal to 90°.

The greatest value of the declination is 23° 28': whenever, therefore, the zenith distance of the pole is less than 66° 32', the sun's meridian distance from the South point can never exceed 90°: this distance therefore is the sun's altitude at the time, and increases for all places as the sun moves Northward; and it is greatest on the same day, for different places, as the zenith distance of the pole increases, or the elevation of the pole diminishes. The influence of the sun then, as far as it is determined by his elevation, increases as the elevation of the pole diminishes, within these limits. We have already seen that the length of the day increases, when the sun's declination is North, as the elevation of the pole increases. These two causes of heat therefore are opposed to each other, and we cannot easily tell in what degree they may counteract each other; and in fact we know that in high latitudes, or places where the elevation of the pole is great, the summers are often very hot, although the elevation of the sun is small.

Where the zenith distance of the pole is greater than 66° 32', we are still less able to arrive at any satisfactory conclusion. In this case we have already seen that the meridian elevation of the sun may have to be measured Northward, and that it will then decrease as the declination increases, or as the length of the day increases. Thus the two causes which affect the solar power are here opposed, so as to prevent us from even saying when it is greatest at the place itself; and of course we have another difficulty, added to that already mentioned, in comparing it with the corresponding power in other situations. When the pole is in the horizon, the length of the day, as we have already seen, is always equal. In this case the power of the sun to communicate heat depends only on his elevation. Now here the equator passes through the zenith, and the meridian elevation of the sun is therefore greatest when he is in the equator, and continually less as he recedes from it, either towards the North or the South. Here therefore the equinoxes are the periods of the greatest, the solstices of the least solar power.

The periods of the least solar power, which correspond to the winter of differ

ent places, are more easily ascertained. The least distance from the South point is always the difference between the elevation of the equator where it crosses the meridian, and the greatest South declination: and this is necessarily the least meridian elevation, except where the poles are in the horizon, and there are equal elevations on the North and South side of the zenith. The least elevation therefore increases, as the elevation of the point of intersection of the equator and meridian increases, or as the zenith distance of the pole increases; and the greater this zenith distance the greater is the least meridian elevation of the sun. But we have already seen that, as the zenith distance of the pole increases, or its elevation diminishes, the length of the shortest days increases; and consequently on both accounts the influence of the sun, during the period of his South declination, is increased as the elevation of the North Pole is diminished. We should expect, therefore, to find the winters diminish in severity, as the elevation of the North Pole diminishes.

It may seem also, that, notwithstanding the difficulty of comparing the sun's extreme power at different places, his average power may be accurately compared. The whole time for which the sun in the course of a year is above the horizon, is everywhere the same or nearly so. If, in fig. 5, eq be a circle of rotation North of the equator, and e' q' be one South of it, and at the same distance from it, the portion t1 e r1 of the first circle which is above the horizon H, N1, and the part tq' r' of the second circle which is below it, are in all cases equal; and so, of course, are the parts remaining. The parts above the horizon therefore, ter and t1 e' r' are together equal to the whole of one of these circles; or the time for which the sun is above the horizon in the two days during which he describes the two circles e q, e' q', is equal to the time of describing one of the circles, or to a day, and the time during which he is below the horizon is so also; and, in this manner, as the sun in his course has equal North and South declination, two days may continually be found, in which, taken together, the whole time of the sun's being above the horizon is equal to that of his being below it; and the whole year may be divided into these pairs of days, or only not so, because the sun's motion being rather more rapid

in the Southern than the Northern part of his orbit, the days during which he is longer below than above the horizon, are not, in places North of the equator, quite so many as those when he is longer above it. In the same manner, if the circle eq is entirely above the horizon H, Na, the circle e' q' is entirely below it; and thus the period, during which the sun never sinks below the horizon, is counterbalanced by a period during which he never rises above it.

Taking the extreme case, when the pole is in the zenith, these periods are each of them from equinox to equinox, or they comprehend each half the sun's course, and they only differ therefore, like the others, by the small inequality occasioned by his variable rate of motion. When the pole is in the horizon we have already seen that the day and night are always equal. Neglecting therefore the slight inequality we have mentioned, (which makes the whole period during which, in the course of a year, the sun is above the horizon, somewhat longer when the North Pole is so, and shorter when the South Pole is, and thus tends to render the heat of the former greater than that of the latter climates) we may consider that everywhere the sun is half the year above, and half below the horizon; and his influence to produce heat throughout the year, and consequently the average heat, as far as he occasions it, will depend upon his average elevation above the horizon, and be greatest where that is greatest, and least where that is least. It would however lead to very complicated investigation, if we were to endeavour to determine the manner in which this average elevation differs at different places. It is sufficient to state generally that it is greatest when the pole is in the horizon, and continually diminishes as the elevation of the pole increases; or the influence of the sun for the whole year is greatest when the latitude is nothing, it gradually diminishes with the increase either of North or South latitude, and is least where the latitude is of 90°. And it is familiarly known to every one that these deductions actually correspond with the general results of observation; that, generally, the countries near the equinoctial line (for so the line where the latitude is nothing is called, the day and night being there always equal) are the hottest, those in high lati tudes the coldest. The ancients indeed put so much faith in these considera

tions, that they divided the whole earth into climates accordingly; and having no actual knowledge of the high Northern regions, nor of those near the equinoctial line, or South of it, they conceived the former to be uninhabitable from extreme cold, and the countries near the equinoctial to be equally so from extreme heat. The experience of modern times has proved that both suppositions were extravagant: and it has farther shewn that the heat of different places where the elevation of the pole is the same is very different; that it is indeed so much affected by local causes, as the elevation of the country above the level of the sea, its degree of cultivation, its geological constitution, the average moisture of the soil and atmosphere, the prevalent winds, the extent of continent with which it is connected, and other similar circumstances, that the mere consideration of the average influence of the sun, as deduced from his average elevation, is quite inadequate to give any information as to the comparative temperature of different places. Thus Edinburgh, Moscow, and Copenhagen, have all nearly the same latitude; but the Baltic is frozen up every year, while the sea near Edinburgh is unencumbered with ice, and the severities of a Russian winter, and the early period at which they commence, have too lately been the subjects of history, for us to want any other proof of the comparative mildness of the climate of Scotland*. It would therefore be useless to enter into any laborious research to discover the exact laws according to which a power, which we find not to have the importance once attributed to it, varies; nor should we have given to it even so much consideration as we have, unless the investigation had involved in it some facts and principles which are really of value; the explanations, we mean, of the manner in which the greatest and least solar power vary at different places, and the important general fact that the whole period of daylight during the year is equal, or very nearly so, at every place on the earth's surface.

Our intercourse with our North American possessions furnishes a still more striking example. The mouth of the river St. Lawrence is in the latitude of about 49° N., more than 20 sonth of Lonriver; yet all access to Canada is stopped by the don; and this is the most Northerly point of the

frost which closes the river at an early period of the winter, and the ice does not break up, and navigation recommence, until the month of May.

and of the use made of these mean values and results in treating of the equation of time, of which we have still to speak, and then the more obvious appearances of the sun, and their principal effects, will be for the present sufficiently explained.

The solar day is longer than the sidereal day in consequence of the motion of the sun Eastward in his orbit. It is evident that the degree of its excess above the sidereal day must be affected by the quantity of that motion, and must, when other circumstances are the same, be greatest when that motion is greatest, or when the sun is in his perigee, and least when that motion is least, or the sun is in his apogee. The motion of the sun goes through all its variations in the course of one revolution of the sun in his orbit; it admits, therefore, on the principles already explained, of a mean value. Let us call the sun S, and let us suppose a fictitious body, which we call S1, to move uniformly in the ecliptic, and to perform a complete revolution in the same time as S: the motion of S1, therefore, will be the mean motion, and its place, the mean place, of S. The difference between the places of S and S, will be an equation; it is called the equation of the centre. Let us suppose the two bodies, S, and S1, to be together when the sun is in apogee. As the revolution of the supposed body S1, is completed at the same time as that of S, they will be again together at the end of the year, or when the sun returns to his apogee. Besides this, the times of the sun's motion from apogee to perigee, and from perigee back to apogee, are equal; they are therefore each equal to half the time of his whole revolution, or to half the time of the revolution of S, or to the time taken by S1, which moves uniformly, to pass through half the ecliptic. But the sun's apogee and perigee are at the distance of half the ecliptic from each other; consequently, as S and S1, were together at the apogee, they are so also at the perigee, each taking the same time (half of the year) to pass through half of their orbits. We find therefore that the real and mean places of the sun coincide at the apogee and perigee.

It is also plain that they correspond nowhere else. The sun's distance from the earth continually decreases from apogee to perigee, and his angular velocity continually increases during the same period. It is evident then that as

his whole real angular motion for that period is equal to his mean angular motion for the same time, the real motion will at first be less, and afterwards greater than the mean motion. His real place therefore will at first fall behind his mean place, and the distance between them will increase day by day until his real motion becomes equal to his mean motion; the distance will then diminish as the real motion becomes greater than the mean motion, and this excess will finally bring them together again. It will not however do this until they reach the perigee, for we have already seen that they are then together, and this could not be the case if they had been so before; for as they are only brought together by the real motion exceeding the mean motion, and as the real motion continually increases from apogee to perigee, if at any period before the perigee S had come up with S1, at the following instant S would have passed S, by the excess of its motion, and would have continued from day to day to increase the distance between them by the continuing and growing excess of the real above the mean motion; and the consequence would be, contrary to the fact, that S would arrive at the perigee before S1. We arrive therefore at this conclusion, that the real place of the sun is behind his mean place, as he passes from apogee to perigee, the distance between them continually increasing for a certain time, then continually diminishing till the two places again coincide at the perigee. Exactly in the same manner we find that the real and mean place never coincide from perigee to apogee, only with this difference, that as in this half of the orbit the real motion is at first greater, and afterwards less, than the mean motion, the real place is always before the mean place, until, on the return to the apogee, they both again correspond. The greatest difference in the time of the approach of S and S, to the meridian cannot exceed 8m 248.

It is not however only by the variation in the sun's rate of motion, that the length of the solar day, or the interval between his successive appearances on the meridian, is affected; it is also influenced by the inclination of the ecliptic to the equator. The motion of rotation of the heavens is uniform; and consequently, if we suppose another fictitious body, S2, to be at the point at the same time with the supposed body S, to

move uniformly Eastward in the equator, and to pass completely round the heavens in that circle in the course of a solar year, as this body will every day have moved Eastward by an equal portion, it will always come to the meridian later by an equal period than the point which it occupied on the preceding day; and as this point returns to the meridian in a sidereal day, which we have seen to be always of the same length, and the excess of time above this sidereal day before Sreturns to the meridian is also equal, the intervals be. tween the successive arrivals of Są on the meridian are themselves equal; and each of them, as S, performs its whole revolution in a solar year, must be equal to a mean solar day. We now then have two fictitious bodies, S, and S2, each moving uniformly, the first in the ecliptic, the second in the equator; and, as these circles are equal, the actual motions of each are equal. Still we have to inquire whether their periods of arriving at the meridian are the same. For this purpose let us suppose (in fig. 6) that P represents the pole of the heavens, Y half the equator, Z half the ecliptic, (for these circles bisect each other in the points and,) and let Y and Z be the points of bisection of the arcs op Y, Z, respectively. Each of the arcs, o Y, ∞ Z, therefore is 90°, or they are equal; and, consequently, the fictitious bodies, S, and S2, which were together at op, and whose

90° also, or Z Y, the arc of a great circle which joins the points ZY, is perpendicular to the equator op Y at Y, and therefore is part of a meridian, PZ Y, passing through those points. Z therefore, being a point upon the same meridian as Y, comes upon the meridian of the place at the same time with Y, and, as the bodies S, and S2 are at Z and Y at the same time, they will there come on the meridian of the place together. But Z, being the middle point of the arc Z, is the sun's place when in the tropic.

From the time when the body S, leaves the point, until its arrival at Z, we shall find that it will always come upon the meridian of the place before S. To show this, let S, represent any intermediate position of that body, and let PS, s be a meridian drawn through it; of course therefore the points S1,s, come upon the meridian of the place together. If therefore s be the place of the supposed body S, at that time, the bodies S1,S2, will then come on the meridian of the place together; but if the body S have advanced beyond the point s, then S, will come on the meridian of the place later than s, and consequently than S, for s and S, arrive there together. But the actual motion of Sis equal to that of S1, or to S1. Whenever therefore S, is greater than s, the body S, is more advanced than the point s, or comes on the meridian later than S. Now it is a general property of all spherical triangles, as well as plane ones, that the greater side is opposite to the greater angle; and in the spherical triangle S, s, as long as the side ops is less than 90° or than op Y, the right angle S, s is greater than the angle S1s. Until S, and S. therefore arrive at the points Z,Y, S1, which is opposite to the right angles S1, is always greater than ops, and conse

[The arcs Z, Y, being each 90°, the point O is the pole of the circle Z Y, and consequently the arcs Z, Y, are secondaries to Z Y, or perpendicular to it.]

[Cos. S, s=sin. S, s, cos.s, by Napier's rules. Whenever, therefore, s is less than 90°, or at all points between Y and Z or Y, cos. Y s is positive, and consequently cos. S, sis so likewise, ors, is less than 90°; for the other factor, sin. S, s, is positive, since S, s is 23° 28'. When s is 90°, or Y, cos. s = 0, consequently cos. S1 s=0, or PS,s (in this case 2Y) = 90°, as we have already seen. When s is greater than 90°, cos. s is negative; therefore cos. S, s is so, or PS, s is greater than 90° or

than S, and consequently S, is less than Cs, or the body S, comes later on the meridian than S, as is afterwards independently shown in

the text.]

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