This value of d, found from astronomical tables, and substituted for it, will give the two corresponding values of t, or the two moments at which these circumstances take place. Here it must be remarked that, when the values of t are imaginary, or cos a greater than d, no eclipse can take place. When they are just equal to each other, the quantity under the radical becomes equal to nothing, and the two values of t, corresponding to this substitution, are also equal to each other, the edge of the Moon's disc becomes a tangent to the pure shadow, and no eclipse, properly speaking, takes place, but only a simple appulse. If it were required to find the time at which the disc of the Moon was just wholly immersed in the shadow, it would be obtained, making d=p + p'(D + D'). Another interesting epoch is the middle of the eclipse, which evidently takes place when the two values of t are equal to each other, and the radicle vanishes. Then the distance of the centres of the Moon and the Earth's shadow is equal to cos a, and As the middle of the eclipse takes place when the centres are at the least possible distance from each other, that moment is consequently the time of the greatest phase, the magnitude of which is therefore easily found; for if to the distance of the centres, Icosa, the apparent semidiameter of the Moon be added, the sum will be the distance of the Moon's exterior limb from the centre of the shadow; and from this sum the semidiameter of the shadow being subtracted, the remainder will be the part of the Moon's diameter which is not eclipsed; and which is therefore equal to (D+D') + cos a— (p + p'). When the value of this quantity is positive, subtracting it from the diameter of the Moon gives the part eclipsed equal to (D'-D) + p + p' —l cos a. Astronomers conceive the diameter of the Moon to be divided into 12 equal parts, which they call digits; the number of which may readily be found by a simple proportion, by saying, as the whole diameter : to the part eclipsed: 12: to the number of digits answering to this part. When the value of the above quantity is equal to nothing, then it shows that the eclipse is just total; and when it becomes negative, it expresses the distance which the exterior limb of the Moon is immersed within the Earth's shadow. The total duration of an eclipse may also be easily found by subtracting the time answering to the beginning from that corresponding to the end; and which will give the 2 sin a duration= {d — l2 cos2 a)* ; n in which expression the values of d, corresponding to these epochs, must be substituted; that is, d= (D' —D) + p + p'. The business of calculating lunar eclipses is therefore reduced to finding the given quantities for the exact time of the opposition, by means of solar and lunar tables, and substituting them in the preceding formulæ our limits, however, prevent us from giving an example of this process; and we shall therefore supply its place with the following easy geometrical construction, which such of our readers as are not accustomed to calculations of this kind will most likely prefer. From what has been shown in the former part of this article, the semidiameter of the Earth's shadow, at the time of the eclipse, may be easily determined; and with this as a radius, and C as a centre, describe the circle OQR (fig. 11) to represent it, Let ECW be a part of the ecliptic, and through C draw CN or CS, according as the Moon's latitude, at the time of the eclipse, is north or south, perpendicular to ECW. Make the angle NCD equal to that which the orbit of the Moon forms with the ecliptic; the mean value of which has already been stated, in a previous article, to be about 5°.15, or nearly 5° 10. Bisect this angle with the line CF in which the opposition of the Sun and Moon, as given by the tables, takes place. From a convenient scale of equal parts, representing minutes of a degree, take the Moon's latitude at the true time of opposition, and set it off from C to G, on the line CF. Through the point G, and at right angles to CD, draw the line HKGLI, to represent the path of the Moon's centre. Then L is the place of the Moon's centre at the middle of the eclipse, G the point of that centre at the time of full Moon, as given by the tables; and K the point occupied by her centre at the instant of her ecliptic opposition; also, I the place of her centre at the beginning of the eclipse, and H at the end of it. Then, with the Moon's semidiameter as a radius, and the centres I, L, and H, describe circles which will represent the Moon at the commencement, middle, and termination of the eclipse. From the length of the line IH, measured on the same scale, the du ration of the eclipse may be determined, by saying, as the Moon's horary motion from the Sun (that is, the excess of her horary motion above that of the Sun) the length of the line IH:: 60': the duration required. Finally, by measuring the distance OP on the same scale, and making the proportion above pointed out, the digits eclipsed will readily be obtained. [To be concluded next Month.] The Naturalist's Diary. The wood-path is carpeted over with leaves, The Goddess of Plenty has bound up her sheaves, With dissonant guns, hills and vallies resound, GLOOMY as this month usually is, yet there are some intervals of clear and pleasant weather: the mornings are, occasionally, sharp, but the hoarfrost is soon dissipated by the Sun, and a fine open day follows. A few soft days succeed Of pleasing mildness; but the varying storm The trees are now stripped of their foliage. The separation of the leaves from their branches is termed the fall; and, in North America, the season in which this takes place is universally known by that name. The falling of leaves is not always in consequence of the injuries of autumnal frosts, for some trees have their appropriate period of defoliation, seemingly independent of external causes. The lime (tilia europaea) commonly loses its leaves before any frost happens; the ash seems, on the contrary, to wait for that event; and at whatever period the first rather sharp frost takes place, all its leaves fall at once. The fall of the leaf can be considered only as a sloughing or casting off diseased or worn-out parts,' whether the injury to their constitution may arise from external causes or from an exhaustion of their vital powers. Hence a separation takes place, either in the foot-stalk, or more usually at its base, and the dying part quits the vigorous one, which is promoted by the weight of the leaf itself, or by the action of autumnal winds upon its expanded form. Sometimes, as in the hornbeam, the beech, and some oaks, the swelling of the buds for the ensuing season is necessary to accomplish the total separation of the old stalks from the insertions. How fall'n the glories of these fading scenes! WHITE, Leaves undergo very considerable changes before they fall; ceasing to grow for a long time previous to their decay, they become gradually more rigid and less juicy, often parting with their pubescence, and always changing their healthy green colour to more or less of a yellow, sometimes a reddish hue. One of the first trees that becomes naked is the walnut the mulberry, horse-chesnut, sycamore, lime, and ash, follow. The elm preserves its verdure for some time longer the beech and ash are the latest deciduous forest trees in dropping their leaves. All lopped trees, while their heads are young, carry their leaves a long while. Apple-trees and peaches re ; |