Part 6

86. _Cause of Precession._--We have seen that the earth is flattened at the poles: in other words, the earth has the form of a sphere, with a protuberant ring around its equator. This equatorial ring is inclined to the plane of the ecliptic at an angle of about 23-1/2°. In Fig. 100 this ring is represented as detached from the enclosed sphere. _S_ represents the sun, and _Sc_ the ecliptic. As the point _A_ of the ring is nearer the sun than the point _B_ is, the sun's pull upon _A_ is greater than upon _B_: hence the sun tends to pull the ring over into the plane of the ecliptic; but the rotation of the earth tends to keep the ring in the same plane. The struggle between these two tendencies causes the earth, to which the ring is attached, to wabble like a spinning-top, whose rotation tends to keep it erect, while gravity tends to pull it over. The handle of the top has a gyratory motion, which causes it to describe a curve. The axis of the heavens corresponds to the handle of the top.

II. THE MOON.

Distance, Size, and Motions.

87. _The Distance of the Moon._--The moon is the nearest of the heavenly bodies. Its distance from the centre of the earth is only about sixty times the radius of the earth, or, in round numbers, two hundred and forty thousand miles.

The ordinary method of finding the distance of one of the nearer heavenly bodies is first to ascertain its horizontal parallax. This enables us to form a right-angled triangle, the lengths of whose sides are easily computed, and the length of whose hypothenuse is the distance of the body from the centre of the earth.

Horizontal parallax has already been defined (32) as the displacement of a heavenly body when on the horizon, caused by its being seen from the surface, instead of the centre, of the earth. This displacement is due to the fact that the body is seen in a different direction from the surface of the earth from that in which it would be seen from the centre. Horizontal parallax might be defined as the difference in the directions in which a body on the horizon would be seen from the surface and from the centre of the earth. Thus, in Fig. 101, _C_ is the centre of the earth, _A_ a point on the surface, and _B_ a body on the horizon of _A_. _AB_ is the direction in which the body would be seen from _A_, and _CB_ the direction in which it would be seen from _C_. The difference of these directions, or the angle _ABC_, is the parallax of the body.

The triangle _BAC_ is right-angled at _A_; the side _AC_ is the radius of the earth, and the hypothenuse is the distance of the body from the centre of the earth. When the parallax _ABC_ is known, the length of _CB_ can easily by found by trigonometrical computation.

We have seen (32) that the parallax of a heavenly body grows less and less as the body passes from the horizon towards the zenith. The parallax of a body and its altitude are, however, so related, that, when we know the parallax at any altitude, we can readily compute the horizontal parallax.

The usual method of finding the parallax of one of the nearer heavenly bodies is first to find its parallax when on the meridian, as seen from two places on the earth which differ considerably in latitude: then to calculate what would be the parallax of the body as seen from one of these places and the centre of the earth: and then finally to calculate what would be the parallax were the body on the horizon.

Thus, we should ascertain the parallax of the body _B_ (Fig. 102) as seen from _A_ and _D_, or the angle _ABD_. We should then calculate its parallax as seen from _A_ and _C_, or the angle _ABC_. Finally we should calculate what its parallax would be were the body on the horizon, or the angle _AB'C_.

The simplest method of finding the parallax of a body _B_ (Fig. 102) as seen from the two points _A_ and _D_ is to compare its direction at each point with that of the same fixed star near the body. The star is so distant, that it will be seen in the same direction from both points: hence, if the direction of the body differs from that of the star 2° as seen from one point, and 2° 6' as seen from the other point, the two lines _AB_ and _DB_ must differ in direction by 6'; in other words, the angle _ABD_ would be 6'.

The method just described is the usual method of finding the parallax of the moon.

88. _The Apparent Size of the Moon._--The _apparent size_ of a body is the visual angle subtended by it; that is, the angle formed by two lines drawn from the eye to two opposite points on the outline of the body. The apparent size of a body depends upon both its _magnitude_ and its _distance_.

The apparent size, or _angular diameter_, of the moon is about thirty-one minutes. This is ascertained by means of the wire micrometer already described (19). The instrument is adjusted so that its longitudinal wire shall pass through the centre of the moon, and its transverse wires shall be tangent to the limbs of the moon on each side, at the point where they are cut by the longitudinal wire. The micrometer screw is then turned till the wires are brought together. The number of turns of the screw needed to accomplish this will indicate the arc between the wires, or the angular diameter of the moon.

In order to be certain that the longitudinal wire shall pass through the centre of the moon, it is best to take the moon when its disc is in the form of a crescent, and to place the longitudinal wire against the points, or _cusps_, of the crescent, as shown in Fig. 103.

89. _The Real Size of the Moon._--The real diameter of the moon is a little over one-fourth of that of the earth, or a little more than two thousand miles. The comparative sizes of the earth and moon are shown in Fig. 104.

The distance and apparent size of the moon being known, her real diameter is found by means of a triangle formed as shown in Fig. 105. _C_ represents the centre of the moon, _CB_ the distance of the moon from the earth, and _CA_ the radius of the moon. _BAC_ is a triangle, right-angled at _A_. The angle _ABC_ is half the apparent diameter of the moon. With the angles _A_ and _B_, and the side _CB_ known, it is easy to find the length of _AC_ by trigonometrical computation. Twice _AC_ will be the diameter of the moon.

The volume of the moon is about one-fiftieth of that of the earth.

90. _Apparent Size of the Moon on the Horizon and in the Zenith._.--The moon is nearly four thousand miles farther from the observer when she is on the horizon than when she is in the zenith. This is evident from Fig. 106. _C_ is the centre of the earth, _M_ the moon on the horizon, _M'_ the moon in the zenith, and _O_ the point of observation. _OM_ is the distance of the moon when she is on the horizon, and _OM'_ the distance of the moon from the observer when she is in the zenith. _CM_ is equal to _CM'_, and _OM_ is about the length of _CM_; but _OM'_ is about four thousand miles shorter than _CM'_: hence _OM'_ is about four thousand miles shorter than _OM_.

Notwithstanding the moon is much nearer when at the zenith than at the horizon, it seems to us much larger at the horizon.

This is a pure illusion, as we become convinced when we measure the disc with accurate instruments, so as to make the result independent of our ordinary way of judging. When the moon is near the horizon, it seems placed beyond all the objects on the surface of the earth in that direction, and therefore farther off than at the zenith, where no intervening objects enable us to judge of its distance. In any case, an object which keeps the same apparent magnitude seems to us, through the instinctive habits of the eye, the larger in proportion as we judge it to be more distant.

91. _The Apparent Size of the Moon increased by Irradiation._--In the case of the moon, the word _apparent_ means much more than it does in the case of other celestial bodies. Indeed, its brightness causes our eyes to play us false. As is well known, the crescent of the new moon seems part of a much larger sphere than that which it has been said, time out of mind, to "hold in its arms." The bright portion of the moon as seen with our measuring instruments, as well as when seen with the naked eye, covers a larger space in the field of the telescope than it would if it were not so bright. This effect of _irradiation_, as it is called, must be allowed for in exact measurements of the diameter of the moon.

92. _Apparent Size of the Moon in Different Parts of her Orbit._--Owing to the eccentricity of the moon's orbit, her distance from the earth varies somewhat from time to time. This variation causes a corresponding variation in her apparent size, which is illustrated in Fig. 107.

93. _The Mass of the Moon._--The moon is considerably less dense than the earth, its mass being only about one-eightieth of that of the earth; that is, while it would take only about fifty moons to make the bulk of the earth, it would take about eighty to make the mass of the earth.

One method of finding the mass of the moon is to compare her effect in producing the _tides_ with that of the sun. We first calculate what would be the moon's effect in producing the tides, were she as far off as the sun. We then form the following proportion: as the sun's effect in producing the tides is to the moon's effect at the same distance, so is the mass of the sun to the mass of the moon.

The method of finding the mass of the sun will be given farther on.

94. _The Orbital Motion of the Moon._--If we watch the moon from night to night, we see that she moves eastward quite rapidly among the stars. When the new moon is first visible, it appears near the horizon in the west, just after sunset. A week later the moon will be on the meridian at the same hour, and about a week later still on the eastern horizon. The moon completes the circuit of the heavens in a period of about thirty days, moving eastward at the rate of about twelve degrees a day. This eastward motion of the moon is due to the fact that she is revolving around the earth from west to east.

95. _The Aspects of the Moon._--As the moon revolves around the earth, she comes into different positions with reference to the earth and sun. These different positions of the moon are called the _aspects_ of the moon. The four chief aspects of the moon are shown in Fig. 108. When the moon is at _M_, she appears in the opposite part of the heavens to the sun, and is said to be in _opposition_; when at _M'_ and at _M'''_, she appears ninety degrees away from the sun, and is said to be in _quadrature_; when at _M''_, she appears in the same part of the heavens as the sun, and is said to be in _conjunction_.

96. _The Sidereal and Synodical Periods of the Moon._--The _sidereal period_ of the moon is the time it takes her to pass around from a star to that star again, or the time it takes her to _make a complete revolution around the earth_. This is a period of about twenty-seven days and a third. It is sometimes called the _sidereal month_.

The _synodical period_ of the moon is the time that it takes the moon to _pass from one aspect around to the same aspect again_. This is a period of about twenty-nine days and a half, and it is sometimes called the _synodical month_.

The reason why the synodical period is longer than the sidereal period will appear from Fig. 109. _S_ represents the position of the sun, _E_ that of the earth, and the small circle the orbit of the moon around the earth. The arrow in the small circle represents the direction the moon is revolving around the earth, and the arrow in the arc between _E_ and _E'_ indicates the direction of the earth's motion in its orbit. When the moon is at _M_{1}_, she is in conjunction. As the moon revolves around the earth, the earth moves forward in its orbit. When the moon has come round to _m_{1}_, so that _m_{3}m_{1}_ is parallel with _M_{3}M_{1}_, she will have made a complete or _sidereal_ revolution around the earth; but she will not be in conjunction again till she has come round to _M_, so as again to be between the earth and sun. That is to say, the moon must make more than a complete revolution in a synodical period.

The greater length of the synodical period is also evident from Fig. 110. _T_ represents the earth, and _L_ the moon. The arrows indicate the direction in which each is moving. When the earth is at _T_, and the moon at _L_, the latter is in conjunction. When the earth has reached _T'_, and the moon _L'_, the latter has made a sidereal revolution; but she will not be in conjunction again till the earth has reached _T''_, and the moon _L''_.

97. _The Phases of the Moon._--When the new moon appears in the west, it has the form of a _crescent_, with its convex side towards the sun, and its horns towards the east. As the moon advances towards quadrature, the crescent grows thicker and thicker, till it becomes a _half-circle_ at first quarter. When it passes quadrature, it begins to become convex also on the side away from the sun, or _gibbous_ in form. As it approaches opposition, it becomes more and more nearly circular, until at opposition it is a _full_ circle. From full moon to last quarter it is again gibbous, and at last quarter a half-circle. From last quarter to new moon it is again crescent; but the horns of the crescent are now turned towards the west. The successive phases of the moon are shown in Fig. 111.

98. _Cause of the Phases of the Moon._--Take a globe, half of which is colored white and the other half black in such a way that the line which separates the white and black portions shall be a great circle which passes through the poles of the globe, and rotate the globe slowly, so as to bring the white half gradually into view. When the white part first comes into view, the line of separation between it and the black part, which we may call the _terminator_, appears concave, and its projection on a plane perpendicular to the line of vision is a concave line. As more and more of the white portion comes into view, the projection of the terminator becomes less and less concave. When half of the white portion comes into view, the terminator is projected as a straight line. When more than half of the white portion comes into view, the terminator begins to appear as a convex line, and this line becomes more and more convex till the whole of the white half comes into view, when the terminator becomes circular.

The moon is of itself a dark, opaque globe; but the half that is towards the sun is always bright, as shown in Fig. 112. This bright half of the moon corresponds to the white half of the globe in the preceding illustration. As the moon revolves around the earth, different portions of this illumined half are turned towards the earth. At new moon, when the moon is in conjunction, the bright half is turned entirely away from the earth, and the disc of the moon is black and invisible. Between new moon and first quarter, less than half of the illumined side is turned towards the earth, and we see this illumined portion projected as a crescent. At first quarter, just half of the illumined side is turned towards the earth, and we see this half projected as a half-circle. Between first quarter and full, more than half of the illumined side is turned towards the earth, and we see it as gibbous. At full, the whole of the illumined side is turned towards us, and we see it as a full circle. From full to new moon again, the phases occur in the reverse order.

99. _The Form of the Moon's Orbit._--The orbit of the moon around the earth is an ellipse of slight eccentricity. The form of this ellipse is shown in Fig. 113. _C_ is the centre of the ellipse, and _E_ the position of the earth at one of its foci. The eccentricity of the ellipse is only about one-eighteenth. It is impossible for the eye to distinguish such an ellipse from a circle.

100. _The Inclination of the Moon's Orbit._--The plane of the moon's orbit is inclined to the ecliptic by an angle of about five degrees. The two points where the moon's orbit cuts the ecliptic are called her _nodes_. The moon's nodes have a westward motion corresponding to that of the equinoxes, but much more rapid. They complete the circuit of the ecliptic in about nineteen years.

The moon's latitude ranges from 5° north to 5° south; and since, owing to the motion of her nodes, the moon is, during a period of nineteen years, 5° north and 5° south of every part of the ecliptic, her declination will range from 23-1/2° + 5° = 28-1/2° north to 23-1/2° + 5° = 28-1/2° south.

101. _The Meridian Altitude of the Moon._--The _meridian altitude_ of any body is its altitude when on the meridian. In our latitude, the meridian altitude of any point on the equinoctial is forty-nine degrees. The meridian altitude of the summer solstice is 49° + 23-1/2° = 72-1/2°, and that of the winter solstice is 49° - 23-1/2° = 25-1/2°. The greatest meridian altitude of the moon is 72-1/2° + 5° = 77-1/2°, and its least meridian altitude, 25-1/2° - 5° = 20-1/2°.

When the moon's meridian altitude is greater than the elevation of the equinoctial, it is said to run _high_, and when less, to run _low_. The full moon runs high when the sun is south of the equinoctial, and low when the sun is north of the equinoctial. This is because the full moon is always in the opposite part of the heavens to the sun.

102. _Wet and Dry Moon._--At the time of new moon, the cusps of the crescent sometimes lie in a line which is nearly perpendicular with the horizon, and sometimes in a line which is nearly parallel with the horizon. In the former case the moon is popularly described as a _wet_ moon, and in the latter case as a _dry_ moon.

The great circle which passes through the centre of the sun and moon will pass through the centre of the crescent, and be perpendicular to the line joining the cusps. Now the ecliptic makes the least angle with the horizon when the vernal equinox is on the eastern horizon and the autumnal equinox is on the western. In our latitude, as we have seen, this angle is 25-1/2°: hence in our latitude, if the moon were at new on the ecliptic when the sun is at the autumnal equinox, as shown at _M_{3}_ (Fig. 114), the great circle passing through the centre of the sun and moon would be the ecliptic, and at New York would be inclined to the horizon at an angle of 25-1/2°. If the moon happened to be 5° south of the ecliptic at this time, as at _M_{4}_, the great circle passing through the centre of the sun and moon would make an angle of only 20-1/2° with the horizon. In either of these cases the line joining the cusps would be nearly perpendicular to the horizon.

If the moon were at new on the ecliptic when the sun is near the vernal equinox, as shown at _M_{1}_ (Fig. 115), the great circle passing through the centres of the sun and moon would make an angle of 72-1/2° with the horizon at New York; and were the moon 5° north of the ecliptic at that time, as shown at _M_{2}_, this great circle would make an angle of 77-1/2° with the horizon. In either of these cases, the line joining the cusps would be nearly parallel with the horizon.

At different times, the line joining the cusps may have every possible inclination to the horizon between the extreme cases shown in Figs. 114 and 115.

103. _Daily Retardation of the Moon's Rising._--The moon rises, on the average, about fifty minutes later each day. This is owing to her eastward motion. As the moon makes a complete revolution around the earth in about twenty-seven days, she moves eastward at the rate of about thirteen degrees a day, or about twelve degrees a day faster than the sun. Were the moon, therefore, on the horizon at any hour to-day, she would be some twelve degrees below the horizon at the same hour to-morrow. Now, as the horizon moves at the rate of one degree in four minutes, it would take it some fifty minutes to come up to the moon so as to bring her upon the horizon. Hence the daily retardation of the moon's rising is about fifty minutes; but it varies considerably in different parts of her orbit.

There are two reasons for this variation in the daily retardation:--

(1) The moon moves at a _varying rate in her orbit_; her speed being greatest at perigee, and least at apogee: hence, other things being equal, the retardation is greatest when the moon is at perigee, and least when she is at apogee.

(2) The moon moves at a _varying angle to the horizon_. The moon moves nearly in the plane of the ecliptic, and of course she passes both equinoxes every lunation. When she is near the autumnal equinox, her path makes the greatest angle with the eastern horizon, and when she is near the vernal equinox, the least angle: hence the moon moves away from the horizon fastest when she is near the autumnal equinox, and slowest when she is near the vernal equinox. This will be evident from Figs. 116 and 117. In each figure, _SN_ represents a portion of the eastern horizon, and _Ec_, _E'c'_, a portion of the ecliptic. _AE_, in Fig. 116, represents the autumnal equinox, and _AEM_ the daily motion of the moon. _VE_, in Fig. 117, represents the vernal equinox, and _VEM'_ the motion of the moon for one day. In the first case this motion would carry the moon away from the horizon the distance _AM_, and in the second case the distance _A'M'_. Now, it is evident that _AM_ is greater than _A'M'_: hence, other things being equal, the greatest retardation of the moon's rising will be when the moon is near the autumnal equinox, and the least retardation when the moon is near the vernal equinox.

The least retardation at New York is twenty-three minutes, and the greatest an hour and seventeen minutes. The greatest and least retardations vary somewhat from month to month; since they depend not only upon the position of the moon in her orbit with reference to the equinoxes, but also upon the latitude of the moon, and upon her nearness to the earth.

The direction of the moon's motion with reference to the ecliptic is shown in Fig. 118, which shows the moon's motion for one day in July, 1876.

104. _The Harvest Moon_--The long and short retardations in the rising of the moon, though they occur every month, are not likely to attract attention unless they occur at the time of full moon. The long retardations for full moon occur when the moon is near the autumnal equinox at full. As the full moon is always opposite to the sun, the sun must in this case be near the vernal equinox: hence the long retardations for full moon occur in the spring, the greatest retardation being in March.