luminous object shining upon it; and let the line A B be drawn joining A and B, and passing through the sphere at D; and let BC be a tangent to the sphere. It is evident that C will be a point in the boundary of the illuminated part of the sphere, for no point farther from B than C is, can receive a ray of light from B, as some part of the opaque sphere will be between them; and every point between C and D must receive such rays, for there is nothing to interpose and prevent them from arriving there.

Now, all tangents drawn to a sphere from the same point are equal to each other; at every point therefore in the boundary of the illuminated part the value of C B is the same; A C, the radius of the spherical body, is equal at all points, and AB is always the same line: the value therefore of the angle DAC is the same in the case of every point in this boundary, for all the triangles, formed as ABC is, have all their sides equal, and consequently their angles equal also. Every point then in this boundary is at the same angular distance from the point B, or from Ď, and therefore in the circumference of a circle whose pole is D, and consequently whose plane is perpendicular to the line A B. This circle differs almost imperceptibly, in the case of the sun and any body on which he shines, from a great circle of the sphere; for, as the angle A CB is a right angle, being that made by a tangent with the radius which it meets, the angles C A B and C BA together are equal to a right angle; and as the sun's distance is exceedingly great in proportion to the radius of any heavenly body on which he shines, the angle CBA is very small indeed, or the angle C A B very nearly equal to a right angle *.

In drawing this conclusion, however, we have treated the light as proceeding from a single luminous point, B; and the result will require to be a little modified, as light does actually proceed from every part of a very large body, namely

[In the case of the moon, the mean value of the

distance A B is very nearly equal to the distance of the earth from the sun (for the greatest distance exceeds that quantity by the moon's distance from the earth, and the least distance falls short of it in the same degree), or to about 23,500 times the earth's radius, (p. 60); the moon's radius on 3 the other hand is only about ths of the earth's radius (p. 61). CA therefore is to A B, or, which is very nearly the same, C B, in the proportion only of IT to 23500, or of 3 to 2585 00 and the

3

the sun. If the sun were of the same size as the moon, the extreme, or tangential rays, would be parallel; if smaller, they would continually diverge from each other; if larger, they would converge to a point. These results are obvious in themselves; or they will immediately appear by the inspection of fig. 18, where, if S represents Fig. 18.

the sun, and A, B C, three bodies, the first equal to the sun in size, the second larger, the third smaller, the extremities of the shaded figures beyond them will evidently represent the course of the extreme rays, and the figures themselves the shadows cast by the bodies, which, in the two former cases, would be prolonged to an infinite distance; in the latter, they would terminate as in the figure. The third case represents that of the moon, which is smaller than the sun: the boundary of the illuminated part therefore will be determined by rays, not diverging from the illuminating body S, but converging to a point on the other side of the moon. But still, as they all meet in a point, the boundary of the illuminated part will be a circle, in the same manner as before; and as the difference of the diameters of the sun and moon is small in comparison with their distance, the rays will converge very slowly, and the extremity of the shadow, the point to which the extreme rays converge, will be very distant from the moon; and the boundary therefore, in this case also, will differ very little from a great circle. The whole part illuminated therefore rather exceeds half the sphere, angle ABC, which is very nearly equal to 57°.29578. CB' is consequently only

AC

but so little, that we may say generally that half of any spherical heavenly body is illuminated by the sun, and that the boundary of the illuminated part is a great circle of the sphere, whose plane is perpendicular to the line joining the centre of the body with the centre of the sun.

Again, if we suppose B, in fig. 17, instead of being the position of an illuminated body, to be the position of an observer looking at the spherical body A, he will evidently see all between D and C, and nothing beyond C. C therefore, in this case, will represent a point in the boundary of the part of the sphere visible to the observer; and by the same reasoning as before, this boundary must be circular, and its plane perpendicular to AB: and further, it also may, if the distance A B is sufficient, be considered as a great circle of the sphere. In the case of the moon and earth, this distance

is too considerable to be entirely neglected*. Still, even in this case, the supposition that the boundary is a great circle is so nearly correct, that we may use it to explain the manner in which the magnitude of the visible part of the object varies; and in the others to which we shall hereafter apply it, it differs quite imperceptibly from the truth.

Now in fig. 19, let E represent the situation of the earth, S that of the sun, M the centre of any spherical body reflecting the sun's light to the earth; the body itself being represented by the circle A Bab, which, however, is excessively magnified to admit of drawing the necessary lines within it. Draw the lines MS, ÉS, and E M, and produce EM to N. The angle ME S, subtended at E by the places of M and the sun, or their apparent angular distance from each other, is called the elongation of M, and the angle N MS is called the

exterior angle of elongation. Draw A Ma perpendicular to MS; then, by what we have already shown, A and a will each be points in the extreme boundary of the eniightened part of the body, or Aba will represent the enlightened part. In the same manner, if B Mb be drawn perpendicular to E M, B and b will be points in the extreme boundary of the part of the body which is turned towards E, or Bab will represent that part: and ab consequently will represent the part actually visible at E by reflected light. Draw Eac through a to meet the line B b, and the points a and c would appear to the observer at E, in the same direction. If he could see the whole part B a b, which is turned towards him, he would see it

3

1

57.29578

220

AC

AB

=

11x60220, a quantity so small that the angle and the sine may be considered to correspond, and consequently the angle ABC = .260435, or very little more than a quarter, of a degree, and the angle C A B falls short of a right angle only by that quantity. The same result may be obtained yet more simply thus: the angle CAB falls short of a right angle by the angle ABC; but A B C is the angle subtended at the earth by the radius of the moon, or it is half the apparent diameter of the moon, which we have seen, p. 68, to vary from 29° 22" to 33′ 31": the mean value of the half will therefore be something more than 154-1

=

that angle is equal to the radius, or half the face of the moon is visible*. The elongation of the moon, together with the angle M S E, always are equal to this exterior angle; and consequently, in this case, the elongation = 90°° MSE, or very nearly 90°; for, as M E is about sixty times the earth's radius, (p. 60) and ES nearly 24,000 times the same quantity, the angle M SE must necessarily be very small. It is obvious that the observations which we have already detailed correspond with these results; and the uniform manner in which they do so, shows that the figure of the moon may, without material inaccuracy, be considered as spherical.

It is perhaps necessary to observe, that when we speak of the moon as reflecting light from the sun, we do not mean that she reflects it so as to present an image of the sun on one point of her surface, like a mirror; but that the light gets broken and diffused from part to part of her surface, and finally sent forward to us in such a manner as to render the whole surface of the enlightened part visible; just as light is diffused over bodies on the earth, which, if perfectly smooth, would only form an image of the sun, but do actually show, by reflected and broken light, the whole of their own surface, its form and colour.

It is evident, from the law of variation which we have deduced, that almost immediately after the moon and sun cease to be in the same line, there is some portion of the illuminated part of the moon turned towards the earth. It is however very small at first: the versed sine increases very slowly while the angle continues small, and it is therefore some time before its magnitude becomes considerable. During this period also the moon is apparently near the sun, and consequently in a very light part of

If therefore we can observe the time when exactly half the moon is visible, or the inner line of light quite straight, we shall know that the exterior angle of elongation is 900. We can observe the actual elongation at that time, and hence we may know the value of MS E, the other angle of the triangle. Hence the proportion of the sides may be ascertained, or of the sun's and moon's distances from the earth. We cannot make the necessary observation of the exact time of half moon with accuracy enough to make the result of any value, as we have now better means of ascertaining it; but this method is worth notice, as being the first ever employed for the purpose, having been adopted by Aristarchus, about 280 B.C.

It follows hence that, as N MS MES+

MSE always, in the case of the moon, NMS=

MES, very nearly. The visible part therefore will vary very nearly as the versed sine of M ES,

or of the elongation itself, in this case,

the heavens; and it is therefore a good while before she really becomes visible, for she cannot be seen till the light which she reflects is sufficient to be distinguished from that which the sun spreads generally over the region of the atmosphere through which the rays which proceed from her must pass. The length of time therefore during which she is not actually seen, furnishes no exception to the correctness of our results.

There is a remarkable appearance presented by the moon when the visible part, according to the principles we have established, would be small, which this is the proper season for explaining. At these periods the whole of the moon's disk is frequently seen, part bright, and having its magnitude the same with that which we have explained as the whole visible magnitude of the moon; the rest visible by a pale and delicate light, and appearing, from the ordinary effect of brightness in augmenting the apparent magnitude of objects, somewhat smaller in its dimensions than the brighter part. The appearance, from this circumstance, and also as being oftenest observed in the evening, soon after the moon's first appearance, or after the new moon, when more persons have the opportunity of seeing it than in the early morning preceding the disappearance of the moon at the latter end of the month, has received, in common speech, the odd name of the old moon in the new moon's arms. The French, with more accuracy of expression, have named it, from the pale colour of the greater part of the moon, lumière cendrée, or ashy light. The cause of this light is obvious: the earth, as well as the moon, reflects light, and consequently, the enlightened part of the earth, or so much of it as is turned towards the moon, will reflect light to that body. Some of that light will again be reflected back to the earth, and thus even that part of the moon which receives no light directly from the sun, may, by indirectly receiving it from the earth, become, as we see it, faintly visible. The appearance, being thus occasioned, has received the name of earthshine. The light thus indirectly supplied must necessarily be far inferior in quantity and brightness to that which the directly enlightened part of the moon receives immediately from the sun; and thus the great inequality of brightness in the two visible portions is accounted for. The only apparent difficulty arises

from the circumstance that the appearance in question is only seen when the directly illuminated part is small. In reality however this seeming difficulty confirms the explanation given, for there are two obvious reasons for it. As the directly illuminated part increases, its light becomes greater, and the light diffused over that part of the atmosphere through which the moon shines, greater also a stronger light therefore is required to be distinguishable. But this is not all; the light actually supplied to the moon from the earth diminishes. The earth being a spherical body, and reflecting light, appearances or phases will be presented by the earth to the moon similar to those which we, on the earth, observe in the moon; and all our results will be true for this case, as well as for that already examined. The order indeed will be different. Thus, when the moon is invisible to us, being between the earth and sun, the earth will turn the same part to the sun and moon, and will be visible to the moon with a full face: when we see the full moon, the earth is between the moon and sun, and therefore invisible to the moon. Without entering into any further detail of these appearances, as visible at the moon, it is evident that the general principles on which they were deduced apply equally to this case, and consequently that the part of the earth visible at the moon varies nearly as the versed sine of the earth's exterior angle of elongation there. The result will be less accurate than in the case of the moon, on account of the earth's greater magnitude: but still it will be very nearly so. In fig. 19, if ME be prolonged to F, EMS is the earth's elongation, as estimated at the moon, and FES is the earth's exterior angle of elongation. But NMS = MES+MSE, and consequently NMS and FES together, are equal to MES +FES+MSE, or to two right angles, +M SE, or very nearly to two right angles, as MSE is necessarily very small. The greater therefore N M S, or the moon's exterior angle of elongation, the less is FES, or the earth's

The real diameter of the earth being about times that of the moon, and their distance very considerable with reference to it, the apparent dia. meter of the earth at the moon will be about greater than that of the moon at the earth, or it will be about 115', nearly 2°. The whole apparent magnitude will be greater than that of the moon to to 1, or of

us in the proportion of the square of

121 to 9, or more than 13 to 1.

exterior angle of elongation; and as their versed sines increase and diminish when the angles do so, and the visible parts are in the proportion of the versed sines, the part of the earth visible from the moon diminishes as the visible part of the moon increases, and of course the quantity of light which the earth reflects to the moon diminishes also. The power therefore of distinguishing the moon by this light reflected from the earth, is diminished as the part visible. by light directly reflected from the sun is increased; both because less light is thus transmitted to the moon, and because more is required before it can be distinguished.

We have thus explained the manner in which the proportion of the moon which is visible at different periods of the month varies. This proportion however is not the only thing which we can observe with respect to her. Many singular marks and spots are apparent upon her, from which various and important conclusions may be drawn: but before we proceed to state these observations, and draw from them the inferences to which they lead, it will be worth while to pause, and deduce from the results already obtained some remarkable consequences, which materially tend to the convenience of mankind.

The most obvious practical service of the moon, as far as mankind are concerned, is the supply of light which she affords during the otherwise dark hours while the sun is below the horizon. But she herself is sometimes above, sometimes below the horizon; and as her declination is continually varying, her periods of continuance above the horizon continually vary also. Her light also is different at different periods of her course: sometimes none; sometimes little, and therefore of little practical utility; generally however enough to be of material service to the sailor, the traveller, and even to the husbandman; and this most when her light is the brightest, or at the full moon. It therefore matters little to man whether she is above or below the horizon at night, as long as her own light is little or nothing; but it is of much importance that she should be above the horizon at night when her light is considerable. Now this, by the conditions of her motion, she necessarily is. The quantity of light which she reflects is greatest when her distance from the sun is greatest, or when the