est day in summer augments, and the shortest in winter diminishes, as we approach the pole; and when the zenith is only distant from it by a quantity equal to the obliquity of the ecliptic to the equator, the Sun never sets on the day of the Summer solstice, nor rises at the winter solstice; still nearer the poles, the time of its presence and of its absence on the horizon exceeds several days and even months. Finally under the pole the horizon being the equator itself, the Sun is always above the horizon when on the same side of the equator, and always below it when on the opposite side; so that there is but one day and one night throughout the year. The intervals which separate the equinoxes and the solstices are not equal: that from the vernal equinox to the autumnal is about seven days longer, than from the autumnal to the vernal; the proper motion of the Sun, therefore, is not uniform ;-accurate and multiplied observations have taught us, that it is the most rapid in a cer tain point in the solar orbit, situated near the winter solstice, and slowest in an opposite point situated near the summer solstice; The Sun describes in a day * 1°1327, in the first point, and only † 1°0591 in the second; thus during the course of a year the Sun's daily motion varies from the greatest to the least, by three hundred and thirty-six thousandths of its mean value. To obtain the law of this variation, and in general that of all the periodical inequalities, the following consideration has been made use of, since the sines and cosines of angles become the same at every circumference to which they arrive, they are proper to represent these inequalities; in this manner, therefore, all the inequalities of the celestial motions have been expressed, and it only remains to separate these inequalities from each other, and to determine the angles on which they depend. In this manner it has been found that the variation of the angular velocity of the Sun is very nearly proportional o the cosine of its mean angular distance from the point where its velocity is the greatest. It is natural to think that the distance of the Sun from the Earth varies, with its angular velocity, and it has been proved to do so by the measures of its apparent diameter. This augments and diminishes in the same time and according to the same law as this velocity, but in a ratio only half as great; when the velocity is greatest the apparent diameter is *6035",7; when the velocity is least it is only +5836",3, thus its mean diameter is 5936"; this quantity should be diminished a few seconds, to allow for the effect of irradiation which dilates a little the apparent diameters of luminous bodies. The distance of the Sun from the Earth, being reciprocally as its apparent diameter, its increase follows the same law as the * 32′ 35′′ 5 + 28′ 49′′ 30. 42,25. diminution of its diameter. We name Perigee the point of the orbit in which the Sun is nearest to the Earth, and Apogee the opposite point in which it is the most remote. It is in the first of these points that the Sun has both the greatest apparent diameter, and the greatest velocity; at the second point both the diameter and the velocity are at a minimum. Toexplain the diminution of the apparent distance of the Sun it is sufficient to suppose it farther from the Earth, but if the variation in its motion arose from this cause alone, and if the real velocity of the Earth in its orbit was constant,its apparent velocity should be diminished in the same proportion as its apparent diameter, but it diminishes in a ratio twice as great; there is therefore an actual retardation in the motion of the Sun, as it recedes from the Earth. By the combined effect of this retardation of the velocity and augmentation of the distance, the angular motion in a day diminishes as the square of the distance increases, so that its product by this square All the measures of the apparent diameter of the Sun, compared with the observations of its daily motion, confirm this result. Let us imagine a straight line joining the centres of the earth and Sun, and call it the RADIUS VECTOR of the Sun, it is easy to see that the small sector, or area traced in a day by this radius round the Earth, is proportional to the product of the square of this area by the daily motion of the Sun: thus this area is constant, and the whole area traced by the radius vector, reckoning from a fixed radius, increases as the number of days elapsed from the epoch on which the Sun was upon this radius. From hence results this remarkable law in the motion of the Sun; namely that, The area described by its radius vector are proportional to the times. If, from these data, we every day set down the position and length of the radius vector of the solar orbit, and then make a curve pass through the extremities of these |