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of rotation North of the equator, as eq, and let us conceive a great circle described through the points R and T, and passing also through the poles P and p. By a known property of the sphere, this secondary to the equator E Q bisects it, and every circle parallel to it. If, therefore, u and v be the points where it cuts the circle eq, uevis half that circle. Now the part of the circle eq which is above the horizon H N,, is tier, which is evidently greater than uev, or more than half the circle of rotation eq is above the horizon; and the part of the circle eq which is above the horizon H. N2 is t, er, which is evidently greater not only than u ev, but also than ter. But the elevation of the pole above the horizon H, N1, is PN1, its elevation above the horizon H, N, is PN; the greater portion, therefore, of the circle eq is above that horizon above which the pole is most elevated, or the horizon of that place which has the greater latitude. Now eq being a circle of rotation North of the equator, may represent the sun's line of diurnal motion when his declination is North; and we consequently deduce the following general rule, that wherever the North Pole is above the horizon, and the sun's declination is North, the day is longer than the night; and that in different places, the day is longer where the North Pole is more elevated above the horizon, and shorter where it is less so. In the same manner, taking the case of a circle of rotation South of the equator, as e q', the circle PRpT divides it also into equal parts u'e' v', u' q' v', and t'e'r', the part above the horizon H, N, is evidently less than u' e v'; and ta' ere's the part above the horizon H, Ng, is evidently still less than 'e'r'. Whenever the North Pole, therefore, is above the horizon, and the sun's declination is South, the day is shorter than the night; and in different places, the day is shorter where the North Pole is more elevated above the horizon, and longer when it is less so.

As the days both lengthen and shorten more and more as the elevation of the pole increases, of course the inequality of their lengths increases. Thus at London, where the elevation of the pole is of 51° 31', the longest day is of 16h 34m; the shortest of 7h 44", and the difference of 8h 50m. At Paris, where the pole is only elevated 48° 50', the longest day is of 16h 7m; the shortest of 8h 11m,

d the difference of 7h 56m. And at

Edinburgh, where the elevation is of 55° 57', the longest day is of 17h 25m; the shortest of 6h 53m, and the difference of 10h 32m.

We have, in our figure, a third horizon, H3 N3, of which we have yet made no use, but which will furnish us with some very important observations. It will be seen from the figure, that neither the circle eq, nor the circle e' q', ever meet this horizon at all; the circle e being entirely above it, the circle eq entirely below it. A body moving in the former circle is then alway above, in the latter always below, thi horizon. The elevation of the pol P is greater above the horizon H, N than above either H, N, or H, N,: as thi elevation inereases therefore, some ci cles of rotation become entirely abov the horizon, which were not so before and others disappear entirely below i It is easy to ascertain when this tak place for any particular circle. Takin the case of the circle e q, the point mo elevated above the horizon is e, whe it cuts the meridian on one side of t pole; the point least elevated above, most depressed below, the horizon is where it cuts the meridian on the ot side of the pole. Whenever theref q is above the horizon, the whole cir is so. Now, the zenith is 90° from horizon, and the pole 90° from the eq tor; and these equal arcs of 90° made up in one case of the distanc the pole from the zenith, and the 1 tude of the place, or elevation of pole above the horizon, in the othe the same elevation of the pole above horizon, and the depression of the p Q of the equator below it. The ze distance of the pole, therefore, and depression of the equator at the p Q below the horizon are equal. N it is clear that whenever the North d nation of a body, as Q q, is less than depression of the equator below horizon, as it is in the cases of the zons H1 N1, H, Ng, the body whe comes to the meridian at q is low the horizon; but that when declination Q q is greater than depression, as it is in the case of horizon Hs Ns, the body at q is a the horizon; and then its whole c of rotation is so. When the dec tion of the body, and the depre of the point Q are exactly equal body just touches the horizon at q all the rest of its course is compl above it. Generally therefore, whe

the North declination of a body is not less than the depression of the point Qof the equator, or than the zenith distance of the pole, no part of its whole course of diurnal rotation is below the horizon.

In the same manner, taking the case of the circle e' q', we shall find that the most elevated point e' never rises above the horizon H, Na, when the South declination E e' exceeds the elevation of the point E of the equator above the horizon; and as the 90° from the horizon to the zenith are made up of this elevation and the zenith distance of the point E, and the 90° from the point E to the pole are made up of the same zenith distance of the point E and the zenith distance of the pole, the elevation of the point E, and the zenith distance of the pole are equal. Generally therefore, wherever the South declination of a body is not less than the zenith distance of the pole, no part of its whole course of diurnal rotation is above the horizon.

The greatest North and South declinations of the sun are of 23° 28' each; for, although there is some little difference in the observations of the two, they may without sensible error, for this purpose, be treated as equal. These are his declinations at the solstices. Where therefore the zenith distance of the pole is of 23° 28', or the latitude of 66° 32', the sun at the summer solstices, where he is at his greatest North declination, will, at his lowest point, only just touch the horizon, and all the rest of his course for that day will be above it; or there will be one day of 24 hours, with no night: and, in the same manner, there will be one period of 24 hours, when the sun is at the winter solstice, during which he will, at his highest point, only just touch the horizon, and the rest of his course will be below it; or there will be a night of 24 hours, and no day. If the zenith distance of the pole be less than 230 28', or the latitude greater than 66° 32', (as, for instance, if the zenith distance be 15°, or the latitude 75°,) then, as soon as the sun attains a North declination equal to that zenith distance, (in this instance 15°,) his daily course will only just touch the horizon; and from that time forward till he attains his greatest North declination, and again till he returns to the same North declination (15°), his whole course will be above the horizon; or, for a considerable period (in the instance put, from the 30th of April to the 12th of August), there will be uninter

rupted daylight. In the same manner, as soon as the South declination is equal to this zenith distance (here 15°), the sun's course will only just touch the horizon; and from that time till he attains his greatest South declination, and again till he returns to the same South declination (15°), his whole course will be below the horizon; so that for a considerable period (in this instance from the 2d of November to the 8th of February), there will be uninterrupted night. These intervals of uninterrupted day and night are obviously longer as the latitude increases; for then the zenith distance of the pole, or the declination at which the sun begins to be continually above or continually below the horizon, diminishes, and the sun in consequence attains that declination sooner after one equinox, and does not return to it till a shorter time before the other; or the interval during which he has that or greater declination is longer.

When the pole coincides with the zenith, or the latitude is 90°, the equator coincides with the horizon. In this case therefore, every circle parallel to the equator, and North of it, is entirely above the horizon, and every point of it at the same elevation; and all circles parallel to the equator, and South of it, are entirely below the horizon. Here then all bodies which have North declinations are always above the horizon, and describe circles in their daily rotation parallel to it; and all bodies which have South declination are always below the horizon. At the vernal equinox therefore, the sun, which then comes upon the equator, just coincides with the horizon; he is then continually above it while his declination is North, or until he returns to the equator at the autumnal equinox, and from that time continually below it, while his declination is South, or until he returns again to the equator at the succeeding vernal equinox. In this case, therefore, there is uninterrupted daylight for half the year, and then uninterrupted night for the remainder.

The pole may also coincide with the horizon, which, in this case, will be represented by the circle Pp, which, as we have already observed, bisects, and is perpendicular to, every circle of rotation, as EQ, eq, and e' q'. In this case then every heavenly body is an equal time above and below the horizon; and the sun therefore is so what

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of rotation North of the equator, as eq, and let us conceive a great circle described through the points R and T, and passing also through the poles P and p. By a known property of the sphere, this secondary to the equator EQ bisects it, and every circle parallel to it. If, therefore, u and v be the points where it cuts the circle eq, uev is half that circle. Now the part of the circle eq which is above the horizon H N,, is ter1, which is evidently greater than u ev, or more than half the circle of rotation eq is above the horizon; and the part of the circle eq which is above the horizon H2 N is to er, which is evidently greater not only than u ev, but also than ter. But the elevation of the pole above the horizon H, N1, is PN, its elevation above the horizon H, N, is PN2; the greater portion, therefore, of the circle eq is above that horizon above which the pole is most elevated, or the horizon of that place which has the greater latitude. Now eq being a circle of rotation North of the equator, may represent the sun's line of diurnal motion when his declination is North; and we consequently deduce the following general rule, that wherever the North Pole is above the horizon, and the sun's declination is North, the day is longer than the night; and that in different places, the day is longer where the North Pole is more elevated above the horizon, and shorter where it is less so. In the same manner, taking the case of a circle of rotation South of the equator, as e' q', the circle PRpT divides it also into equal parts u' e' v', u'q' v', and t'er, the part above the horizon H, N, is evidently less than u' e '; and ta' e' r', the part above the horizon Hg Ng, is evidently still less than t'e' r. Whenever the North Pole, therefore, is above the horizon, and the sun's declination is South, the day is shorter than the night; and in different places, the day is shorter where the North Pole is more elevated above the horizon, and longer when it is less so.

As the days both lengthen and shorten more and more as the elevation of the pole increases, of course the inequality of their lengths increases. Thus at London, where the elevation of the pole is of 51° 31', the longest day is of 16h 34m; the shortest of 7h 44m, and the difference of 8h 50m. At Paris, where the pole is only elevated 48° 50', the longest day is of 16 7; the shortest of 8h 11m, and the difference of 7h 56m. And at

Edinburgh, where the elevation is of 55° 57', the longest day is of 17h 25m; the shortest of 6h 53m, and the difference of 10h 32m.

We have, in our figure, a third horizon, Ha Ns, of which we have yet made no use, but which will furnish us with some very important observations. It will be seen from the figure, that neither the circle eq, nor the circle e' q', ever meet this horizon at all; the circle eq being entirely above it, the circle eq entirely below it. A body moving in the former circle is then always above, in the latter always below, this horizon. The elevation of the pole P is greater above the horizon H, N3, than above either H, N, or H, N,: as this elevation inereases therefore, some circles of rotation become entirely above the horizon, which were not so before; and others disappear entirely below it. It is easy to ascertain when this takes place for any particular circle. Taking the case of the circle eq, the point most elevated above the horizon is e, where it cuts the meridian on one side of the pole; the point least elevated above, or most depressed below, the horizon is q, where it cuts the meridian on the other side of the pole. Whenever therefore q is above the horizon, the whole circle is so. Now, the zenith is 90° from the horizon, and the pole 90° from the equator; and these equal arcs of 90° are made up in one case of the distance of the pole from the zenith, and the latitude of the place, or elevation of the pole above the horizon, in the other of the same elevation of the pole above the horizon, and the depression of the point Q of the equator below it. The zenith distance of the pole, therefore, and the depression of the equator at the point Q below the horizon are equal. Now, it is clear that whenever the North declination of a body, as Q q, is less than the depression of the equator below the horizon, as it is in the cases of the horizons H, N1, H, N2, the body when it comes to the meridian at q is below the horizon; but that when the declination Q q is greater than that depression, as it is in the case of the horizon H, Na, the body at q is above the horizon; and then its whole circle of rotation is so. When the declination of the body, and the depression of the point Q are exactly equal, the body just touches the horizon at q, and all the rest of its course is completely above it. Generally therefore, wherever

the North declination of a body is not less than the depression of the point Q of the equator, or than the zenith distance of the pole, no part of its whole course of diurnal rotation is below the horizon.

In the same manner, taking the case of the circle e' q', we shall find that the most elevated point e' never rises above the horizon H, N,, when the South declination E e' exceeds the elevation of the point E of the equator above the horizon; and as the 90° from the horizon to the zenith are made up of this elevation and the zenith distance of the point E, and the 90° from the point E to the pole are made up of the same zenith distance of the point E and the zenith distance of the pole, the elevation of the point E, and the zenith distance of the pole are equal. Generally therefore, wherever the South declination of a body is not less than the zenith distance of the pole, no part of its whole course of diurnal rotation is above the horizon.

The greatest North and South declinations of the sun are of 23° 28' each: for, although there is some little difference in the observations of the two, they may without sensible error, for this purpose, be treated as equal. These are his declinations at the solstices. Where therefore the zenith distance of the pole is of 23° 28', or the latitude of 66° 32', the sun at the summer solstices, where he is at his greatest North declination, will, at his lowest point, only just touch the horizon, and all the rest of his course for that day will be above it; or there will be one day of 24 hours, with no night: and, in the same manner, there will be one period of 24 hours, when the sun is at the winter solstice, during which he will, at his highest point, only just touch the horizon, and the rest of his course will be below it; or there will be a night of 24 hours, and no day. If the zenith distance of the pole be less than 23° 28', or the latitude greater than 66° 32', (as, for instance, if the zenith distance be 15°, or the latitude 75°,) then, as soon as the sun attains a North declination equal to that zenith distance, (in this instance 15°,) his daily course will only just touch the horizon; and from that time forward till he attains his greatest North declination, and again till he returns to the same North declination (15), his whole course will be above the horizon; or, for a considerable period (in the instance put, from the 30th of April to the 12th of August), there will be uninter

rupted daylight. In the same manner, as soon as the South declination is equal to this zenith distance (here 15°), the sun's course will only just touch the horizon; and from that time till he attains his greatest South declination, and again till he returns to the same South declination (15°), his whole course will be below the horizon; so that for a considerable period (in this instance from the 2d of November to the 8th of February), there will be uninterrupted night. These intervals of uninterrupted day and night are obviously longer as the latitude increases; for then the zenith distance of the pole, or the declination at which the sun begins to be continually above or continually below the horizon, diminishes, and the sun in consequence attains that declination sooner after one equinox, and does not return to it till a shorter time before the other; or the interval during which he has that or greater declination is longer.

When the pole coincides with the zenith, or the latitude is 90°, the equator coincides with the horizon. In this case therefore, every circle parallel to the equator, and North of it, is entirely above the horizon, and every point of it at the same elevation; and all circles parallel to the equator, and South of it, are entirely below the horizon. Here then all bodies which have North declinations are always above the horizon, and describe circles in their daily rotation parallel to it; and all bodies which have South declination are always below the horizon. At the vernal equinox therefore, the sun, which then comes upon the equator, just coincides with the horizon; he is then continually above it while his declination is North, or until he returns to the equator at the autumnal equinox, and from that time continually below it, while his declination is South, or until he returns again to the equator at the succeeding vernal equinox. In this case, therefore, there is uninterrupted daylight for half the year, and then uninterrupted night for the remainder.

The pole may also coincide with the horizon, which, in this case, will be represented by the circle Pp, which, as we have already observed, bisects, and is perpendicular to, every circle of rotation, as EQ, eq, and e' q'. In this case then every heavenly body is an equal time above and below the horizon; and the sun therefore is so what

ever be his declination, or at every period of the year.

We have now generally ascertained the manner in which the length of the day differs at different places. There are however two causes which make the length of the day greater and that of the night less than we have stated it. In speaking of the sun, as in the equator or horizon, we mean that his centre is so. At these instants therefore, his Northern limb, or the Northern part of his visible circumference, is North of the equator, his upper limb above the horizon; and as the mean apparent semidiameter of the sun is about 16', the sun begins to appear on the horizon, or the equator or any circle parallel to the equator, sooner than we have stated, by the time corresponding to that difference of elevation in the one case, and of declination in the other; that is to say, his upper limb appears on the horizon when his centre is 16' below it, and thus the length of each day is increased; and the greatest declination of the Northern limb being 16' greater than the declination of the centre, or being 23° 44', some part of the sun is always above the horizon at the summer solstice, where the zenith distance of the pole is 23° 44', instead of 23° 28'. So also, where the pole and zenith coincide, as the North limb of the sun is on the equator, when the declination of the centre is 16' South, some part of the sun is continually seen on the horizon, not merely from the time of his coming to the equinox, but from the time that his South declination is less than 16'. There is another cause also, which produces a similar effect, and to a greater degree. In speaking of the place of the sun, we have hitherto given the results of observation, as they are obtained after allowing for the operation of certain causes which materially complicate them in the first instance, and which we shall hereafter explain under the heads of parallax and refraction. Parallax tends to make the apparent place of a body lower than that which we consider it really to occupy, and which we call its real place; refraction to make it higher; and as the effect of refraction is much greater than that of parallax upon the sun, the joint effect of the two is to render the apparent higher than the real place. When the sun therefore is really upon the horizon he appears above it; and when he appears on it, he is really about 33' below it. The length of daylight is

increased by the time corresponding to this difference of elevation, and the zenith distance of the pole at the points where some part of the sun is first seen never to set at the solstice, and the South declination of the sun when first seen on the horizon where the pole and zenith coincide, are increased in exactly the same manner by this cause, as we have already seen that they are by the apparent magnitude of the sun himself.

A short notice of the manner in which the climates of different places differ will be sufficient. We have already seen that the sun's influence depends on the length of time during which he continues above the horizon, and the elevation he attains above it. The greatest elevation of the sun, like that of all other heavenly bodies in their daily rotation, is always when he is on the meridian; and his distance at that time from the Southern point of the horizon is always equal to the distance of the intersection of the meridian and equator from that point, increased by the declination, when North, and diminished by it when South*; or, as the distance of the intersection of the meridian and equator from the South point of the horizon is equal to the zenith distance of the pole, the sun's distance from that point when he is on the meridian is equal to the zenith distance of the pole increased by his declination when North, or diminished by it when South. When the declination is South, and greater than the zenith distance of the pole, this expression becomes negative; and the sun never rises. When it has a positive value, it continually increases as the South declination diminishes, or the North declination increases. Now, the whole arch of the meridian from South to North is 180°; and consequently when the meridian distance of the sun from the South point exceeds 90°, he is nearer the North point than the South point: his meridian elevation, therefore, in that case is his distance from the North point, and dimi

These conclusions will, perhaps, appear more plainly by a reference to a figure. In fig. 2, H being the South point of the horizon, and E the intersection of the equator and meridian, S, and S2 may represent two situations of the sun in the meridian, S, being a situation where his declination is South, S, where it is North. S, Hand S2 H are evidently his distances at those times from H, the South point of horizon; and S1 His the distance, EH, of the intersection of the me

ridian and equator from that point, diminished by E S1, the sun's South declination; S, H is the same distance EH, increased by E S2, the sun's North declination.