by the breadth of the ring when the Sun is centrally eclipsed, is affected in a contrary manner when the shadow of the Moon falls near the centre of the Earth's disc; for that point being nearest to the Moon, the Moon appears relatively larger there, and the annulus therefore relatively narrower. It is true this part of the Earth is nearest also to the Sun, but his apparent magnitude is little affected, whereas the Moon's (owing to her relative proximity) is appreciably enlarged.

We may thus sum up the general characteristics and relations of our eclipse-seasons, the note on the preceding paragraph supplying the details on which the results are founded:

:

The most common of all orders of eclipse-seasons are those in which two eclipses take place. Of these one of course is sclar, the other lunar, and most commonly the solar eclipse is central, the lunar one partial, but in a considerable proportion of cases the solar eclipse is partial and the lunar one total. Very seldom does a total lunar eclipse accompany a central solar one, and yet more seldom are both partial. Next in order of frequency to the seasons of two eclipses are the seasons of but one eclipse, always a central solar one. Lastly come the seasons in which there are three eclipses, which are always-in order-a partial solar eclipse, a total lunar eclipse, and again a partial solar eclipse.*

From the preceding note it follows that the average frequency of the several orders of eclipses-omitting the case of two partial eclipses as of such infrequent occurrence—are fairly presented in the following table, in which the letters refer to fig. 104:

Now, with regard to the succession of these eclipse-seasons, it needs only to be noted that three seasons in which there are three eclipses never occur in succession.

We can now easily determine the greatest and least number of eclipses which may occur in any single year. The average interval between successive eclipse-seasons is 173-3 days. Two such intervals amount together to 346-6 days, or fall short of a year by about 19 days. Hence there cannot be three eclipse-seasons in a year. For each eclipse-season lasts on the average 33 days. Now suppose an eclipse-season to begin with the beginning of a year of 366 days. The middle of the season occurs at about midday on January 17; the middle of the next eclipse-season 173-3 days later, or on the evening of July 8; and the middle of the third occurs yet 173.3 days later, or on December 29, early in the forenoon; so that nearly the whole of the remaining half belongs to the following year. Now this is clearly a favourable case for the occurrence of as many eclipses as possible during the year. If all three seasons could be of the class containing three eclipses, there would be eight eclipses in the year, because the second eclipse of the third season would occur in the middle of that season. This, however, can never happen. But there may be two seasons, each containing three eclipses, followed by a season containing two eclipses, only one of which can occur in the fragment of the eclipse-season falling within the same year. In this case there would be seven eclipses in the year. So also there would be seven if in the first season there were three, in the second two, and in the third three, for then the fragment of the third falling within the year, being rather more than onehalf, would comprise two eclipses. So also if the three successive seasons comprise severally two, three, and three eclipses. The same would clearly happen if the year closed with the close of an eclipse-season.

There may then be as many as seven eclipses in a year, in which case at least four eclipses will be solar, and at least three of these partial, while of the lunar eclipses two at least will be total.

As regards the least possible number of eclipses, it is obvious that, as there must be two eclipse-seasons in the year, and at least one eclipse in each, we cannot have less than two eclipses in the course of a year. In this case each eclipse is solar and central.

As regards intermediate cases, we need make no special inquiry. Many combinations are possible. The most common case is that in which there are four eclipses-two solar and two lunar. Further, it may be noticed that, whatever the number of eclipses, from two to seven inclusive, there must always be two solar eclipses at least in each year.

And now we may turn from the particular mode of considering eclipses, which we have thus far followed. There is another by which we might have arrived at similar results almost as readily. We might, instead of viewing the Earth and Moon in imagination from the Sun, have traced the course of the Sun and Moon around the heavens. Both methods of dealing with eclipses are employed by astronomers, the method used in the preceding pages corresponding to what is termed the method of projection, the other to the method of direct calculation from the celestial ordinates of the Sun and Moon. For our present purpose, however, one method is all that need be considered.*

And now, before closing this essay, I will consider in the usual manner the nature of the Moon's shadow-cone in solar eclipses, and of the Earth's shadow-cone in lunar eclipses.

Eclipses of both sorts may be regarded as illustrated together in fig. 105. Here E is the Earth, and the Moon is shown in two

* In the Popular Science Review for July 1868 I have exhibited t line of reasoning by which the results deduced above can be obtained by considering the apparent motions of the Moon and Sun around the celestial vault. By the artifice of regarding these motions as taking place on a sphere which can be viewed from without, and shifted or rotated so as to illustrate the various relations dealt with, the whole subject may be very conveniently discussed. The student of astronomy does well to examine all such questions by as many independent methods as possible; but in the present treatise there is not space for a complete investigation of the theory of eclipses on the second plan.

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places, at м, directly between the Earth and Sun, and at the point opposite м, in the heart of the Earth's shadow-cone. The true geometrical shadows of the Earth and Moon are shown black, the true geometrical penumbræ are shaded. It must be FIG. 105.

m

C

DM

m

understood, however, that the vertical dimensions have had. to be exaggerated; the angle at c ought properly to contain but about half a degree. Such an angle could not be conveniently employed in illustrating our subject.

ce and ce' produced touch the Sun's globe; so also do the boundaries of the Moon's black shadow. The boundaries me and me' touch the Sun's globe after crossing; so also do the boundaries of the Moon's penumbra.

The distance EC is variable, being as great as 870,300 miles when the Earth is in aphelion, and as small as 843,300 miles when the Earth is in perihelion. The Moon's orbit round the Earth has a mean radius of 238,770 miles. Thus the Earth's shadow extends about three and a half times as far from the Earth as the Moon's orbit.

The end of the Moon's shadow is represented on a larger scale (and without the penumbra) in figs. 106 and 107. In fig. 106 the shadow's extreme point c does not reach the Earth; in fig 107 it passes far beyond the Earth. The two

figures represent the extreme possible range of the shadowpoint either way. In fig. 106 there is shown beyond c a shaded anti-cone. From any point within this the Sun will be annularly eclipsed. Thus the section at a a' includes that part of the Earth whence at the moment an annular eclipse of the Sun is visible. On the other hand, the section u u' in fig. 107 includes the part of the Earth whence the Sun is totally eclipsed. It is important to notice that the greatest possible width of u u' is about 173 miles.†

Now in fig. 105 the points m m m' m' may be supposed to lie on the Moon's orbit seen in plano. If we suppose this orbit not to lie in the plane of the paper, but tilted at an angle of 5° (or rather to an angle as much larger than 5° as the shadow-angle at c is increased beyond its true value of half a minute), then by conceiving the whole figure turned about cм, until the Moon's orbit is seen sideways, this orbit, according to the direction in which the tilt existed, would exhibit a shape resembling one of those shown in fig. 98 (only more open on account of the exaggeration of the tilt. And, further, if we could watch from such a standpoint that is (obviously) a standpoint towards which or from which the Earth was travelling during a period of about 346 days, we should see the Moon's orbit passing in

It is obvious that the Sun will seem to be annularly eclipsed from any point within this anti-cone; for lines drawn from any such point to the circle on the Moon in which the shadow-cone begins will form a cone (right or oblique) which, beyond the Moon, will be wholly within the extension of the shadow cone's geometrical surface. Hence a portion of the Sun's globe must lie outside and around this inner cone. This portion will be visible, therefore, from the vertex of this inner cone (the point within c a a') as a ring of light, whose boundaries will be concentric or eccentric according as the inner cone is right or oblique— in other words, according as its vertex lies or not on the axis of the shadow-cone produced.

The extreme limits of central solar eclipses result when, first, the Sun's diameter has its greatest value, 32′ 36′′-4, and the Moon's its least, 29′ 219, in which case a ring of light 1' 37" 2 wide remains; and, secondly, when the Sun's diameter has its least and the Moon's its greatest value, in which case the Moon's disc overlaps the Sun's by 59" 6, and the Sun continues for several minutes totally eclipsed.