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obtained, as shown in fig. 102, by drawing the quadrant H a and A; I parallel to E a.* And therefore (remembering
* This is a very important general proposition, continually involved when we are considering the apparent changes in the presentation of a globe, orbit, or ring;—as the changes in the Earth's presentation towards the Sun (on which the seasons depend), or the changes in the apparent figure of Saturn's rings, or of the orbits of his satellites, or, again, the changes (which belong more particularly to the subject of this treatise),
in the presentation of the Sun's latitude parallels—the paths apparently followed by the spots as seen from the Earth. The following simple geometrical proof of the property is worth noticing, and is, I believe, new:—
Let E E' bo a circle seen edgewise from a very distant station, so as to appear as a straight line, and suppose that the circle rotates bodily about the axis <t a' through its centre o; it is required to determine the aspect of the circle after any definite amount of rotation. Enclose the circle in a sphere about o as centre, and let poi'.n polar axis of the circlo at the beginning of the motion, rotate with it. Then, from the distant station, P and P" will seem to move along right lines p Q and p' Q' parallel to A o A', though in reality the line pop' will be revolving conically and uniformly about a a. And it is clear that if a always what the rotation corresponds to) it follows that the time occupied by the orbit in opening out from an apparent line to an oval having a half-axis E&, will bear to the time occupied by the orbit in obtaining its full opening the same proportion that the angle IE a bears to the right angle M E a. But we require this proportion to be such that k E shall bear to II E the ratio 5 to 18; and it follows, therefore, that the angle Is. a must be one of as nearly as possible 16^- degrees.*
circle be described on p Q as diameter (only half the circle is shown in the figure), then, as p really revolves uniformly round a circle of this size, but seen edgewise, we can determine the apparent position of p after rotating through any given angle, by simply taking p N L equal to this angle, and drawing a perpendicular lf onpa. At this moment, then, p o p' represents the apparent position taken up by P O P'; and clearly e o 1! at right angles to p op' is the greater axis of the ellipse now presented by the moving circle. The minor axis bob' lies of course on p p'. Now in order to determine the length of b o h\ conceive the circle to rotate on e o ef, till b and A'appear to coincide with 0. Then plainly p has moved to m on the edge of the disc presented by the sphere (o p m straight), and it is obvious that the amount of rotation about e o e' necessary to effect this change is measured by the arc m k, (p k being drawn square to o m), so that o b must be the projection of a radius inclined to the eye as o k is inclined to o m. Therefore o b must be equal to p k,—that is to L p (for L p and p k are obviously equal, since the square of each is equal to the rectangle dp, p P).
Hence we have a very simple construction for determining the position of the ellipse e b t' b' for any amount of rotation round a o ct. This construction in full (starting from nothing given, save K o K', the position of a a', and the amount of rotation) runs thus:—
Describe the circle K p E', draw Pop' square to E E', and p N square to a a'. Describe the arc P l equal to the given rotation-angle, round N as centre, and draw L p square to N P. Then p o p' is the position of the lesser axis of the apparent ellipse now formed by the circle originally seen as the line Eoi ; i.p is the length of the half-axes o b and o V', which we can now measure off along pop'; and of course the major axis is simply the diameter tod square topp'.
Also, since the greatest amount of opening is obviously obtained by drawing E n, E' B' parallel to A o A', and since b o' B' is obviously equal to P N Q, the statement made in the text is shown to be just
* The angle must have a sine equal to 0 277777, and the sine of 161° is 0 277734.
Hence the required time is obtained by reducing half the before-mentioned period, 173-3 days, in the proportion of HH to 90; that is, as nearly as possible, 15i days, which must be doubled, because we have to consider the range on either side of the epoch corresponding to the presentations I., v., IX., in fig. 98. Thus, so far as this rough process is concerned, the eclipse-season lasts 31 days, or thereabouts. The real mean is somewhat greater, for the Moon's diameter is more than one-fourth of the Earth's. But, as we have only had in view the general principles on which the recurrence and duration of our eclipse-seasons depend, exact accuracy has not (thus far) been necessary. For our present purpose we shall take thirty-three days as about the average, and consider one or two consequences of this relation.
A period of 33 days is a few days more than a lunation. Hence, supposing that when an eclipse-season is beginning, it is either new Moon or full Moon, there will be three eclipses during that season, for the Moon will pass to full or new, and thence to new or full, before the eclipse-season is over. Now of these three eclipses the first and last will be solar or lunar, and the other lunar or solar. Yet we never hear of a lunar eclipse followed by a solar eclipse, and then by another lunar eclipse, in the course of 33 days or thereabouts. We do find instances (as anyone can see by looking through a few successive almanacs) of a solar eclipse followed by a lunar eclipse, and then by another solar one within such an interval; but never of the other succession. The fac 's, the ' Nautical Almanac,' from which all other almanacs take their astronomical facts, pays no attention to a certain order of lunar eclipses, to which, in the case considered, the first and last of a set of three eclipses must necessarily belong. It will easily be seen that if the middle eclipse of a set of three is a solar one, it will be very considerable, the orbit of the Moon being presented as at I., v., or ix., fig. 98. But the two lunar eclipses —one preceding and the other following the solar one—will be very slight affairs, for they will happen when the orbit is barely contracted enough (in aspect as seen from the sun) for an eclipse to occur at all. As a matter of fact, they are of such a nature that though a portion of the Sun is hidden from the Moon the whole of the Sun is not hidden from any part of the Moon's illuminated hemisphere. They correspond to partial eclipses of the Sun; but though a partial eclipse of the Sun is a noteworthy phenomenon to terrestrial observers, and therefore finds a place in the ' Nautical Almanac,' one of these corresponding lunar eclipses (differing altogether from partial lunar eclipses properly so called *) is a very different matter, and can scarcely be recognised at all by the terrestrial observer. Delicate photometric appliances would doubtless show that full sunlight was not shining on parts of the Moon at such a season, but to ordinary observation no trace of the deficiency of light is discernible. No notice is taken, therefore, of these eclipses in the 'Nautical Almanac,' which deals (very properly of course) only with phenomena that can be observed.
It will be seen then that under the circumstances considered there would be three eclipses or one during the eclipse-season, according as full Moon or new Moon occurred at the time when the Moon's orbit was presented as at 1., v., orix., fig 9S.
The case may be illustrated as in fig. 104; only the reader must remember that the just proportions of the orbits and
* In what is properly called a partial hinar eclipse, there is a part of the Moon from which the whole disc of the Sun is concealed (setting aside the refraction of the solar rays by the Earth's atmosphere); but in the eclipses considered, which I have ventured to designate penumlirul lunar eclipses, every part of the moon is illuminated by direct sunlight, though not by the whole solar disc. I believe I may claim to have been the first to calculate a penumbral lunar eclipse. The details are given in the Montltly Notices of Out Royal Astronomical Society for 1867-1868 (vol. xxviii.), the eclipse occurring on the night of September 2 in that year. The theory of eclipses cannot be considered complete without a consideration of these hitherto neglected penumbral eclipses. To lunarians, if such there be, the Sun must appear really eclipsed— though but partially — at such epochs; and in fact, as stated in the text, these eclipses correspond exactly to partial solar eclipses (that is, not to eclipses which, though really total, are partially seen at certain stations, but to solar eclipses which are partial wherever seen).
distances cannot be indicated in a single diagram. If when theMoon's orbit is presented as at I., v., or IX., fig. 98, the Moon is at si, (fig. 104) there will be a central eclipse of the Sun; but when the Moon has passed on to M3, her orbit, as seen from the Sun, will be so far opened out that no part of the J\foon will be concealed from all parts of the Sun. Hence there will only be a penumbral lunar eclipse, of which no notice will be taken in almanacs. And the like must have been the case aa respects the Moon's position when at M3 before the central solar eclipse. On the other hand, if the Moon is at Mj when her orbit is presented as at I., v., or IX., fig. 98, there will be a total eclipse of the Moon. And, further, when the Moon was at M, before this eclipse, and when she is at Sf,
after this eclipse, the Sun must be partially eclipsed, though no part of the Earth will he concealed from all parts of the Sun.
The same holds when the Moon is near to M, or si, in the middle of the 'eclipse-season.'
But when the Moon is at or near M4 or M2, at the time when her orbit is presented edgewise towards the Sun, only two eclipses can take place. Suppose, for instance, she is at M4, then carrying her onwards, we see that she must eclipse the Sun when at M, but that she cannot be herself eclipsed when she gets to M3, for then three quarters of a lunar month will have elapsed since the middle of this eclipse-season, and an eclipse-season cannot last a month and a half under any circumstances. Carrying the Moon backwards from M4, we see that she must have been eclipsed when at M3, but cannot have eclipsed the Sun when at sf,. Hence there occur two