339. Thus it appears, that from the vernal equinox to the autumnal, the North Pole is enlightened; and the Equator and all its parallels appear Semi-ellipses as seen from the Sun, more or less curved as the time is nearer to or farther from the Summer Solstice; and bending downwards or towards the South Pole; the reverse of which happens from the autumnal Equinox to the vernal. A little consideration will be sufficient to convince the reader, that the Earth’s Axis inclines towards the Sun at the Summer Solstice; from the Sun at the Winter Solstice; and sidewise to the Sun at the Equinoxes; but towards the right hand, as seen from the Sun at the vernal Equinox; and towards the left hand at the autumnal. From the Winter to the Summer Solstice, the Earth’s Axis inclines more or less to the right hand, as seen from the Sun; and the contrary from the Summer to the Winter Solstice.
How these positions affect solar Eclipses.
340. The different positions of the Earth’s Axis, as seen from the Sun at different times of the year, affect solar Eclipses greatly with regard to particular places; yea so far as would make central Eclipses which fall at one time of the year invisible if they fell at another, even though the Moon should always change in the Nodes and at the same hour of the day: of which indefinitely various affections, we shall only give Examples for the times of the Equinoxes and Solstices.
Fig. IV.
In the same Diagram, let FG be part of the Ecliptic, and IK ik ik ik part of the Moon’s Orbit; both seen edgewise, and therefore projected into right lines; and let the intersections N, O, D, E be one and the same Node at the above times, when the Earth has the forementioned different positions; and let the spaces included by the Circles P, p, p, p be the Penumbra at these times, as its center is passing over the center of the Earth’s Disc. At the Winter Solstice, when the Earth’s Axis has the position NNS, the center of the Penumbra P touches the Tropic of Capricorn t in N at the middle of the general Eclipse; but no part of the Penumbra touches the Tropic of Cancer T. At the Summer Solstice, when the Earth’s Axis has the position NDS (iDk being then part of the Moon’s Orbit whose Node is at D) the Penumbra p has its center on the Tropic of Cancer T at the middle of the general Eclipse, and then no part of it touches the Tropic of Capricorn t. At the autumnal Equinox the Earth’s Axis has the position NOS (iOk being then part of the Moon’s Orbit) and the Penumbra equally includes part of both Tropics T and t at the middle of the general Eclipse: at the vernal Equinox it does the same, because the Earth’s Axis has the position NES: But, in the former of these two last cases, the Penumbra enters the Earth at A, north of the Tropic of Cancer T, and leaves it at m, south of the Tropic of Capricorn t; having gone over the Earth obliquely southward, as its center described the line AOm: whereas in the latter case the Penumbra touches the Earth at n, south of the Equator ÆQ, and describing the line nEq (similar to the former line AOm in open space) goes obliquely northward over the Earth, and leaves it at q, north of the Equator.
In all these circumstances, the Moon has been supposed to change at noon in her descending Node: had she changed in her ascending Node, the Phenomena would have been as various the contrary way, with respect to the Penumbra’s going northward or southward over the Earth. But because the Moon changes at all hours, as often in one Node as the other, and at all distances from them both at different times as it happens, the variety of the Phases of Eclipses are almost innumerable, even at the same places, considering also how variously the same places are situated on the enlightened Disc of the Earth, with respect to the Penumbra’s motion, at the different hours that Eclipses happen.
How much of the Penumbra falls on the Earth at different distances from the Nodes.
341. When the Moon changes 17 degrees short of her descending Node, the Penumbra P 18 just touches the northern part of the Earth’s Disc, near the North Pole N; and, as seen from that place the Moon appears to touch the Sun, but hides no part of him from sight. Had the Change been as far short of the ascending Node, the Penumbra would have touched the southern part of the Disc near the South Pole S. When the Moon changes 12 degrees short of the descending Node, more than a third part of the Penumbra P 12 falls on the northern parts of the Earth at the middle of the general Eclipse: had she changed as far past the same Node, as much of the other side of the Penumbra about P would have fallen on the southern part of the Earth; all the rest in the expansum, or open space. When the Moon changes 6 degrees from the Node, almost the whole Penumbra P6 falls on the Earth at the middle of the general Eclipse. And lastly, when the Moon changes in the Node, the Penumbra PN takes the longest course possible on the Earth’s Disc; its center falling on the middle thereof, at the middle of the general Eclipse. The farther the Moon changes from either Node within 17 degrees of it, the shorter is the Penumbra’s continuance on the Earth, because it goes over a less portion of the Disc, as is evident by the Figure.
The Earth’s diurnal motion lengthens the duration of solar Eclipses, which fall without the polar Circles.
342. The nearer that the Penumbra’s center is to the Equator at the middle of the general Eclipse, the longer is the duration of the Eclipse at all those places where it is central; because, the nearer that any place is to the Equator, the greater is the Circle it describes by the Earth’s motion on its Axis: and so, the place moving quicker keeps longer in the Penumbra whose motion is the same way with that of the place, tho’ faster as has been already mentioned § [337]. Thus, (see the Earth at D and the Penumbra at 12) whilst the point b in the polar Circle abcd is carried from b to c by the Earth’s diurnal motion, the point d on the Tropick of Cancer T is carried a much greater length from d to D: and therefore, if the Penumbra’s center goes one time over c and another time over D, the Penumbra will be longer in passing over the moving place d than it was in passing over the moving place b. Consequently, central Eclipses about the Poles are of the shortest duration; and about the Equator of the longest.