The reader will at once see how the successive appearances represented in fig. 98, from I. to IX., would result from this simple rotation of the whole scheme upon a a'.* We see the orbit-ring opening out and becoming level, closing up and becoming inclined with the left extremity uppermost, opening out again and again becoming level, and lastly closing up and becoming inclined, as at first, with the right extremity uppermost. The whole rotation is supposed to take place in the course of a year, because the Earth takes one year in completing her circuit. Further on we shall have to consider a peculiarity which causes all these changes to take place in rather less than a year; but for the present we are not concerned with details of the sort.

Now, it only needs a glance at fig. 98, to see that when the orbit is presented as at I. there will be an eclipse of the Earth if the Moon is on the nearest part of her orbit, and an eclipse of the Moon if the Moon is on the farthest part of her orbit. For it is to be remembered that we are supposed to be stationed at the Sun; so that if м hides any part of E from us (i.e., from the Sun) that part of E must be in shadow; while if E hides any part of м, or the whole of м, from us, that part

right angles to the plane of its actual orbit, in a direction contrary to that of its real motion, and in the same period. In such a case as the illustration of the Earth's seasons, we must, after considering each change of bearing, consider the effect of a complete rotation of the Earth on her own axis; just as in the case dealt with in the text we consider the Moon's revolution in her orbit in addition to the successive changes in the aspect of the orbit. In my Sun Views of the Earth the twelve successive plates correspond to twelve changes in the Earth's general presentation towards the Sun during the course of the twelve months, while the four pictures in each plate correspond to the changes in the course of a day (at intervals, therefore, of six hours), owing to the Earth's rotation. So illustrated, this method of considering the subject of the seasons becomes singularly simple and truthful.

* Of course the arrow on m m', and the globe of the Moon at м, are simply put in each figure at that part of the orbit which is most convenient, and are not supposed to be carried round with the rotation here specially dealt with.

or the whole of M is in shadow. But when the orbit is presented -a quarter of a year later—as at III., there can be no eclipse, wherever the Moon may be on her orbit. A quarter later, when the orbit is presented as at V., the same state of things results as at the beginning; and yet another quarter later, when the orbit is presented as at VII., no eclipse is possible.

The figure is drawn as nearly as possible to scale, and we see that the intermediate presentations of the orbit as at II., III., VI., and VII., are such that there can be no eclipses. We infer, therefore, that eclipses can only happen when the orbit is presented as shown at I., and for some relatively short time before and after that epoch. At such times, whenever the Moon (M) crosses the place of E, on the nearer or farther half of м's orbit, an eclipse must occur. But after that eclipse-season (if I may invent a word) has passed there can occur no eclipses of either sort until nearly half a year has passed and the presentation of the orbit has approached that shown at V. Then, for a while, eclipses are possible. Lastly, after this eclipse-season has passed, another period of nearly half a year passes during which eclipses cannot happen. And so on continually.

All this is perfectly simple and obvious. The recognition of the fact that these eclipse-seasons recur at intervals of about half-a-year tends also importantly to simplify the consideration of the whole matter. Let me note in passing that the term eclipse-season is not ill-chosen, inasmuch as one eclipse at least must needs takes place while the presentation of the orbit is changing through the critical aspects, such as I., V., IX. (fig. 98), and so on.

But let us now enter somewhat more into details. And first let us inquire how much the orbit m m' must be opened out in order that the Moon (M) may pass clear of E, whether on the nearer or farther side, in such sort that there may be no eclipse.

We have hitherto, for convenience, supposed the observer at the Sun's centre. But now we must give him liberty to traverse the whole of the Sun's globe; or rather we may suppose that millions of eyes placed all over the Sun's

surface are viewing the changes pictured in fig. 98. For if any part of the Sun is concealed from the Earth or Moon a solar or lunar eclipse-partial or otherwise-is in progress.

Now, suppose the Earth and Moon, as seen from the Sun's centre when the Moon is passing close by E in fig. 98, to be represented by the discs E and M (fig. 100.) Then M is either on this side of E or beyond E-that is, it is or full.

either new Moon

First, suppose м on this side of E. Then if our observer

leaves the Sun's centre, and goes to the uppermost point of the Sun's surface,* both E and м will seem lower down on the background of the heavens-E by about sixteen minutes of arc

* I use here the familiar expression uppermost and further on, the familiar expression lowermost. It is not always the case, however, that familiar expressions are the most intelligible. I could cite instances from several popular works on astronomy to show that the use of familiar and ordinary expressions may result in the most perplexing and iu reality untrue statements. In the present instance, the term uppermost refers to the relations presented in fig. 100, and the uppermost point on the Sun will be readily understood to signify the point on the Sun corresponding to the point e on the globe E representing the Earth. In any general sense, the term uppermost has no meaning as applied to the celestial bodies, and must be classed in the same category with many expressions (too often met with in scientific treatises) whose very simplicity is misleading.

(half the Sun's diameter as seen from the Earth), м somewhat more as being nearer to the Sun. And a very little consideration will show that м will be thus thrown downwards with respect to E, to a position as M', such that m m' bears to c m the same proportion that the Sun's semi-diameter (as seen at the time from the Earth) bears to the Moon's. On the other hand, if our observer proceeds to the lowermost point of the Sun's surface he will see the Moon projected as far upwards, or to the position м"m". And clearly, by shifting his place to other portions of the Sun's edge as seen from the Earth, he would see the Moon shifted in other directions; the whole region covered by the Moon during these excursions corresponding to the circle m' m", whose diameter bears to the diameter of м the same proportion which the Sun's apparent diameter, added to the Moon's, bears to the Moon's diameter alone.

Now, if any portion of this circle m'm" overlaps the circle E e, there will be a solar eclipse; and this amounts to saying that if half the lesser axis of the oval orbit m m' in fig. 98 be less than the amount corresponding to ce' together with cm' in fig. 100, there will be a solar eclipse when the Moon is passing by E.

And clearly we shall have precisely the same relations when the Moon is close by E on the further part of the orbit. The only difference in the reasoning depends on the circumstance that when our imagined observer goes to the uppermost part of the Sun, the Moon is raised instead of lowered with respect to the Earth, and vice versa. But we still get a circle such as m' m' in fig. 100.

Hence it appears that so long as the orbit m m' in fig 98 has no greater opening than that corresponding to the sum of the diameters of the circles e e' and m' m' in fig. 100, there will be a solar or lunar eclipse when the Moon is passing by E (fig. 98) either on the nearer or further part of her orbit.

But we can readily tell (with sufficient approximation for our present purpose*) how long the orbit m m' (fig. 98)

* I have purposely omitted, so far, all reference to certain circum

takes in changing from the appearance shown at I., V., and IX. to such a degree of opening as enables the Moon to pass by E without an eclipse, and thus we can readily tell what must happen during our eclipse-seasons.'

6

For this purpose, we may now take into consideration a circumstance hitherto left out of sight--the fact, namely, that the Moon's orbit does not move strictly parallel to itself. It would do so were the Sun powerless to disturb the Moon's motions around the Earth. But as a matter of fact the Sun largely influences those motions. Much in the same way that by acting on the protuberant mass of the Earth's equatorial regions he causes the Earth's axis to sway conically round the direction of a perpendicular to the Earth's orbit― making the pole of the heavens travel in a circle around the pole of the ecliptic in a period of more than 25,000 yearsso he causes the line through the Earth at right angles to the Moon's orbit to sway conically round the direction of a perpendicular to the Earth's orbit, making the pole of the Moon's orbit travel in a circle around the pole of the ecliptic in a period of rather more than 18 years, or, more exactly, 6793-391080 days. The motion of the nodes of the orbitthat is, of the points in which the orbit crosses the plane of the Earth's orbit-is precessional, like the corresponding motion of the nodes of the Earth's equator-that is, the nodes advance as it were to meet the Moon. It is easy to see the effects of this motion on the reasoning applied to fig. 98. Suppose m Em' (fig. 101) to represent the line in which the plane stances which affect the details above considered, without affecting the general reasoning. For example, the orbit of the Moon, as seen from the Sun, is not truly an ellipse around the Earth as centre (without referring to perturbations or the like). Regarding the Moon's orbit for the moment as a circle about the Earth as centre, a diameter of this circle, as seen from the Sun, would not appear to be bisected at its real point of bisection-unless it were at right angles to the line of view-for its two halves would not be at exactly equal distances from the supposed observer. This and many other similar points, though allimportant in an analysis of the details of eclipses, may be safely neglected in considering the general aspect of the subject.