an eclipse. These are, therefore, the lunar ecliptic limits for mean motions.

Now, by means of tables of the mean places and motions of the sun, moon and nodes, it is easy to find the time of mean full moon for any given month, and the mean longitude of the sun and node at that time. If the difference of these longitudes is greater than the greater ecliptic limit for mean motions, there cannot be an eclipse at that full moon; if it is less than the less limit, there must be one. When the difference falls between the limits, further computation is necessary to determine whether there will or will not be an eclipse. In this research it will not be necessary to make computations for all the full moons in the year, for it will at once be seen by the tables at what periods in the year the sun is near to either of the nodes, and it is only at these periods that eclipses can occur.

230. Different kinds of lunar eclipses. When the moon just touches the earth's shadow or passes through the penumbra without entering the shadow, the circumstance is called an appulse. When a part, but not the whole, of the moon enters the shadow, the eclipse is called a partial eclipse; when the moon enters entirely into the shadow, it is called a total eclipse; and when the moon's centre passes through the centre of the shadow, it is called a central eclipse. A central eclipse of the moon seldom, however, if ever, occurs.

It follows from a preceding article (227), that the moon does not in general entirely disappear even in total eclipses.

231. Visibility of a lunar eclipse. As in an eclipse of the moon there is a real loss of light at the moon, the eclipse must be visible, and present the same appearance at all places that have the moon above the horizon, during its continuance.

ECLIPSES OF THE SUN.

232. Length of the moon's shadow. The length of the moon's shadow is about equal to the distance of the moon from the earth; being,, alternately, a little greater and a little less.

Suppose the moon, at new moon, to be at one of her nodes.

Her centre will then be in the plane of the ecliptic, and in the straight line passing through the centres of the sun and earth. Let AB, hg and DG, Fig. 38, be sections of the sun, moon and earth, by a plane passing through their centres S, M, and E. Also, let AC and BC, and AH and BK be tangents common to the sections of the earth and moon, and, therefore, limiting the sections of the shadow and penumbra.

Put a

moon's app. semi-diam. as seen from C,

ang. HCM

R = Ec

earth's radius; and let x, x', &c. be as in Art. 225.

Then, since the parallaxes of bodies, and the apparent semidiameters of the same body, seen at different distances, are inversely as the distances (93 Cor. and 97), we have,

Now, since hCM = AhShSM, or d=d"

d', we have,

By taking the greatest value of ' and least of 8, and the corresponding values of π and л' (100, 95 and 96), the greatest nu

merical value of x'.

π

will be obtained, and will be found to be

less than three tenths of a second. Consequently d is always very nearly equal to 8'. We have therefore (C),

Hence the length of the shadow is greater than the moon's distance from the earth, equal to it, or less, according as è is greater than ', equal to it or less.

Cor. Since (C), MC

ME. 7
we have,

Taking the mean values of 8, d', я, and я', and regarding d as equal to ', we find EC 0.86R. Hence, when the sun and moon are at their mean distances from the earth, the moon's shadow extends a little farther than the nearest part of the earth's surface. By taking the proper values of the quantities, it will be found that the shadow, when longest, extends beyond the earth's centre about three and a half times the earth's radius, and when shortest, does not reach the centre, by about six times the earth's radius.

233. Breadth of the moon's shadow at the earth. The greatest breadth of the moon's shadow at the earth, when it falls perpendicularly on the surface, is about 166 miles.

Now the breadth of the shadow will evidently be greatest when the moon's distance from the earth is least, and the sun's distance, is greatest. Hence, taking the greatest value of 8 and least of d', and the corresponding values of x and x' we find,

π

1° 12′ 13′′. Hence, ab 2o

This gives the angle EaC = 56′ 28′′. Adding d or 8' = 15′ 45′′, to EaC, we have aEc or the arc ac = 1° 12′ 13′′. 24′ 26′′ = 2°.407; and as each degree is 691 miles (71), the breadth of the shadow is 166 miles.

When the moon is at some distance from the node, the shadow falls obliquely on the earth, and its greatest breadth will evidently be increased.

234. Breadth of the moon's penumbra at the earth. The greatest breadth of the moon's penumbra at the earth's surface, when it falls perpendicularly on the surface, exceeds 4800 miles.

Put, d" ang. hLM moon's app. semi-diam. as seen from L.

Then, as hLM

and B),

ShB+hSM d'+d', we have (232. A

=

Or, by substituting the value of ME (93.E), and of (+8"),

From the triangle LEK, we have,

EK: EL:: sin 8" : sin EKd,

The breadth of the penumbra will evidently be greatest when the moon's distance from the earth is greatest and the sun's distance, least. The angle EKd being found with the values of 8, &c., corresponding to these distances, and 8'' being also found (F), the angle KEL or the arc Kc = EKα s", becomes known. Twice Kc gives KH, which is thus found to be 69°.87, or 4833 miles.

235. Different kinds of eclipses of the sun. A partial eclipse is one in which a part, but not the whole of the sun, is obscured. A total eclipse is one in which the sun is entirely obscured. It must occur at all those places on which the moon's shadow falls. A central eclipse is one in which the axis of the moon's shadow or the axis produced, passes through the place at which the eclipse

occurs.

An annular eclipse is one in which a part of the sun's disc is seen as a ring, surrounding the moon. To explain this, let AB, hg and DG, Fig. 39, be sections of the sun, moon and earth, when

the moon's shadow does not extend to the earth. In this case, the tangents AC and BC, which limit the shadow, being produced, cross each other at C and meet the section of the earth at a and b. From e, or any other point between a and b, let tangents to the moon be drawn, as ce and cf. Then it is obvious, that the part of the sun's disc that is without the circle ef, described about the diameter ef, will be visible to an observer at c.

The greatest breadth of the part of the surface in which the eclipse is annular may be found in a similar manner to that of the shadow (233). It is about 200 miles.

As the moon moves in

236. Visibility of an eclipse of the sun. her orbit from m to n, Fig. 38, her penumbra and its axis move. over the earth's surface from west to east, passing in succession over different parts. At all places along the line in which the axis meets the surface, there must be a central eclipse. At all places contiguous to this line, on each side, there must be a total or an annular eclipse. And at places more remote from the central line, but within the limits of the penumbra, there will be a partial eclipse.

As the greatest breadth of the penumbra is less than half the semi-circumference of the earth, it is evident there must be a large part of the earth's enlightened hemisphere in which the eclipse is not visible, even when the extent of the penumbra is greatest.

When the moon is so far from the node at the time of new moon that the axis of the penumbra does not meet the earth, the eclipse cannot be central at any place; and the partial eclipse is only visible in a portion of the northern or southern hemisphere, according as the moon's latitude is north or south.

It follows from the above and a preceding article (231), that the visibility of an eclipse of the sun is of much less extent than that of an eclipse of the moon.

237. General eclipse of the sun. An eclipse of the sun, considered with reference to the whole earth and not to any particular place, is called the general eclipse.

The general eclipse commences at the first contact of the moon's penumbra with the earth, and ends at the last contact. Thus, Fig. 40, the general eclipse begins when the moon is at u, and ends when she is at v.