Fig. 86.
Fig. 87.
Fig. 88.
Fig. 86 shows the tide that would be produced by the moon alone; Fig. 87, the tide produced by the combined action of the sun and moon; and Fig. 88, by the sun and moon acting at right angles to each other.
The tide is affected by the distance of the moon from the earth, being highest near the time when the moon is in perigee, and lowest near the time when she is in apogee. When the moon is in perigee, at or near the time of a new or full moon, unusually high tides occur.
76. Diurnal Inequality of Tides.—The height of the tide at a given place is influenced by the declination of the moon. When the moon has no declination, the highest tides should occur along the equator, and the heights should diminish from thence toward the north and south; but the two daily tides at any place should have the same height. When the moon has north declination, as shown in Fig. 89, the highest tides on the side of the earth next the moon will be at places having a corresponding north latitude, as at B, and on the opposite side at those which have an equal south latitude. Of the two daily tides at any place, that which occurs when the moon is nearest the zenith should be the greatest: hence, when the moon's declination is north, the height of the tide at a place in north latitude should be greater when the moon is above the horizon than when she is below it. At the same time, places south of the equator have the highest tides when the moon is below the horizon, and the least when she is above it. This is called the diurnal inequality, because its cycle is one day; but it varies greatly in amount at different places.
Fig. 89.
77. Height of Tides.—At small islands in mid-ocean the tides never rise to a great height, sometimes even less than one foot; and the average height of the tides for the islands of the Atlantic and Pacific Oceans is only three feet and a half. Upon approaching an extensive coast where the water is shallow, the height of the tide is increased; so that, while in mid-ocean the average height does not exceed three feet and a half, the average in the neighborhood of continents is not less than four or five feet.
78. The Day.—By the term day we sometimes denote the period of sunshine as contrasted with that of the absence of sunshine, which we call night, and sometimes the period of the earth's rotation on its axis. It is with the latter signification that the term is used in this section. As the earth rotates on its axis, it carries the meridian of a place with it; so that, during each complete rotation of the earth, the portion of the meridian which passes overhead from pole to pole sweeps past every star in the heavens from west to east. The interval between two successive passages of this portion of the meridian across the same star is the exact period of the complete rotation of the earth. This period is called a sidereal day. The sidereal day may also be defined as the interval between two successive passages of the same star across the meridian; the passage of the meridian across the star, and the passage or transit of the star across the meridian, being the same thing looked at from a different point of view. The interval between two successive passages of the meridian across the sun, or of the sun across the meridian, is called a solar day.
79. Length of the Solar Day.—The solar day is a little longer than the sidereal day. This is owing to the sun's eastward motion among the stars. We have already seen that the sun's apparent position among the stars is continually shifting towards the east at a rate which causes it to make a complete circuit of the heavens in a year, or three hundred and sixty-five days. This is at the rate of about one degree a day: hence, were the sun and a star on the meridian together to-day, when the meridian again came around to the star, the sun would appear about one degree to the eastward: hence the meridian must be carried about one degree farther in order to come up to the sun. The solar day must therefore be about four minutes longer than the sidereal day.
Fig. 90.
Fig. 91.
The fact that the earth must make more than a complete rotation is also evident from Figs. 90 and 91. In Fig. 90, ba represents the plane of the meridian, and the small arrows indicate the direction the earth is rotating on its axis, and revolving in its orbit. When the earth is at 1, the sun is on the meridian at a. When the earth has moved to 2, it has made a complete rotation, as is shown by the fact that the plane of the meridian is parallel with its position at 1; but it is evident that the meridian has not yet come up with the sun. In Fig. 91, OA represents the plane of the meridian, and OS the direction of the sun. The small arrows indicate the direction of the rotation and revolution of the earth. In passing from the first position to the second the earth makes a complete rotation, but the meridian is not brought up to the sun.
80. Inequality in the Length of Solar Days.—The sidereal days are all of the same length; but the solar days differ somewhat in length. This difference is due to the fact that the sun's apparent position moves eastward, or away from the meridian, at a variable rate.
There are three reasons why this rate is variable:—
(1) The sun's eastward motion is due to the revolution of the earth in its orbit. Now, the earth's orbital motion is not uniform, being fastest when the earth is at perihelion, and slowest when the earth is at aphelion: hence, other things being equal, solar days will be longest when the earth is at perihelion, and shortest when the earth is at aphelion.
Fig. 92.
Fig. 93.
(2) The sun's eastward motion is along the ecliptic. Now, from Figs. 92 and 93, it will be seen, that, when the sun is at one of the equinoxes, it will be moving away from the meridian obliquely; and, from Figs. 94 and 95, that, when the sun is at one of the solstices, it will be moving away from the meridian perpendicularly: hence, other things being equal, the sun would move away from the meridian fastest, and the days be longest, when the sun is at the solstices; while it would move away from the meridian slowest, and the days be shortest, when the sun is at the equinoxes. That a body moving along the ecliptic must be moving at a variable angle to the meridian becomes very evident on turning a celestial globe so as to bring each portion of the ecliptic under the meridian in turn.
Fig. 94.
Fig. 95.
(3) The sun, moving along the ecliptic, always moves in a great circle, while the point of the meridian which is to overtake the sun moves in a diurnal circle, which is sometimes a great circle and sometimes a small circle. When the sun is at the equinoxes, the point of the meridian which is to overtake it moves in a great circle. As the sun passes from the equinoxes to the solstices, the point of the meridian which is to overtake it moves on a smaller and smaller circle: hence, as we pass away from the celestial equator, the points of the meridian move slower and slower. Therefore, other things being equal, the meridian will gain upon the sun most rapidly, and the days be shortest, when the sun is at the equinoxes; while it will gain on the sun least rapidly, and the days will be longest, when the sun is at the solstices.
The ordinary or civil day is the mean of all the solar days in a year.
81. Sun Time and Clock Time.—It is noon by the sun when the sun is on the meridian, and by the clock at the middle of the civil day. Now, as the civil days are all of the same length, while solar days are of variable length, it seldom happens that the middles of these two days coincide, or that sun time and clock time agree. The difference between sun time and clock time, or what is often called apparent solar time and mean solar time, is called the equation of time. The sun is said to be slow when it crosses the meridian after noon by the clock, and fast when it crosses the meridian before noon by the clock. Sun time and clock time coincide four times a year; during two intermediate seasons the clock time is ahead, and during two it is behind.
The following are the dates of coincidence and of maximum deviation, which vary but slightly from year to year:—
One of the effects of the equation of time which is frequently misunderstood is, that the interval from sunrise until noon, as given in the almanacs, is not the same as that between noon and sunset. The forenoon could not be longer or shorter than the afternoon, if by "noon" we meant the passage of the sun across the meridian; but the noon of our clocks being sometimes fifteen minutes before or after noon by the sun, the former may be half an hour nearer to sunrise than to sunset, or vice versa.
82. The Year.—The year is the time it takes the earth to revolve around the sun, or, what amounts to the same thing, the time it takes the sun to pass around the ecliptic.
(1) The time it takes the sun to pass from a star around to the same star again is called a sidereal year. This is, of course, the exact time it takes the earth to make a complete revolution around the sun.
Fig. 96.
(2) The time it takes the sun to pass around from the vernal equinox, or the first point of Aries, to the vernal equinox again, is called the tropical year. This is a little shorter than the sidereal year, owing to the precession of the equinoxes. This will be evident from Fig. 96. The circle represents the ecliptic, S the sun, and E the vernal equinox. The sun moves around the ecliptic eastward, as indicated by the long arrow, while the equinox moves slowly westward, as indicated by the short arrow. The sun will therefore meet the equinox before it has quite completed the circuit of the heavens. The exact lengths of these respective years are:—
Since the recurrence of the seasons depends on the tropical year, the latter is the one to be used in forming the calendar and for the purposes of civil life generally. Its true length is eleven minutes and fourteen seconds less than three hundred and sixty-five days and a fourth.
It will be seen that the tropical year is about twenty minutes shorter than the sidereal year.
(3) The time it takes the earth to pass from its perihelion point around to the perihelion point again is called the anomalistic year. This year is about four minutes longer than the sidereal year. This is owing to the fact that the major axis of the earth's orbit is slowly moving around to the east at the rate of about ten seconds a year. This causes the perihelion point P (Fig. 97) to move eastward at that rate, as indicated by the short arrow. The earth E is also moving eastward, as indicated by the long arrow. Hence the earth, on starting at the perihelion, has to make a little more than a complete circuit to reach the perihelion point again.
Fig. 97.
83. The Calendar.—The solar year, or the interval between two successive passages of the same equinox by the sun, is 365 days, 5 hours, 48 minutes, 46 seconds. If, then, we reckon only 365 days to a common or civil year, the sun will come to the equinox 5 hours, 48 minutes, 46 seconds, or nearly a quarter of a day, later each year; so that, if the sun entered Aries on the 20th of March one year, he would enter it on the 21st four years after, on the 22d eight years after, and so on. Thus in a comparatively short time the spring months would come in the winter, and the summer months in the spring.
Among different ancient nations different methods of computing the year were in use. Some reckoned it by the revolution of the moon, some by that of the sun; but none, so far as we know, made proper allowances for deficiencies and excesses. Twelve moons fell short of the true year, thirteen exceeded it; 365 days were not enough, 366 were too many. To prevent the confusion resulting from these errors, Julius Cæsar reformed the calendar by making the year consist of 365 days, 6 hours (which is hence called a Julian year), and made every fourth year consist of 366 days. This method of reckoning is called Old Style.
But as this made the year somewhat too long, and the error in 1582 amounted to ten days, Pope Gregory XIII., in order to bring the vernal equinox back to the 21st of March again, ordered ten days to be struck out of that year, calling the next day after the 4th of October the 15th; and, to prevent similar confusion in the future, he decreed that three leap-years should be omitted in the course of every four hundred years. This way of reckoning time is called New Style. It was immediately adopted by most of the European nations, but was not accepted by the English until the year 1752. The error then amounted to eleven days, which were taken from the month of September by calling the 3d of that month the 14th. The Old Style is still retained by Russia.
According to the Gregorian calendar, every year whose number is divisible by four is a leap-year, except, that, in the case of the years whose numbers are exact hundreds, those only are leap-years which are divisible by four after cutting off the last two figures. Thus the years 1600, 2000, 2400, etc., are leap-years; 1700, 1800, 1900, 2100, 2200, etc., are not. The error will not amount to a day in over three thousand years.
84. The Dominical Letter.—The dominical letter for any year is that which we often see placed against Sunday in the almanacs, and is always one of the first seven in the alphabet. Since a common year consists of 365 days, if this number is divided by seven (the number of days in a week), there will be a remainder of one: hence a year commonly begins one day later in the week than the preceding one did. If a year of 365 days begins on Sunday, the next will begin on Monday; if it begins on Thursday, the next will begin on Friday; and so on. If Sunday falls on the 1st of January, the first letter of the alphabet, or A, is the dominical letter. If Sunday falls on the 7th of January (as it will the next year, unless the first is leap-year), the seventh letter, G, is the dominical letter. If Sunday falls on the 6th of January (as it will the third year, unless the first or second is leap-year), the sixth letter, F, will be the dominical letter. Thus, if there were no leap-years, the dominical letters would regularly follow a retrograde order, G, F, E, D, C, B, A.
But leap-years have 366 days, which, divided by seven, leaves two remainder: hence the years following leap-years will begin two days later in the week than the leap-years did. To prevent the interruption which would hence occur in the order of the dominical letters, leap-years have two dominical letters, one indicating Sunday till the 29th of February, and the other for the rest of the year.
By Table I. below, the dominical letter for any year (New Style) for four thousand years from the beginning of the Christian Era may be found; and it will be readily seen how the Table could be extended indefinitely by continuing the centuries at the top in the same order.
To find the dominical letter by this table, look for the hundreds of years at the top, and for the years below a hundred, at the left hand.
Thus the letter for 1882 will be opposite the number 82, and in the column having 1800 at the top; that is, it will be A. In the same way, the letters for 1884, which is a leap-year, will be found to be FE.
Having the dominical letter of any year, Table II. shows what days of every month of the year will be Sundays.
To find the Sundays of any month in the year by this table, look in the column, under the dominical letter, opposite the name of the month given at the left.
From the Sundays the date of any other day of the week can be readily found.
Thus, if we wish to know on what day of the week Christmas falls in 1889, we look opposite December, under the letter F (which we have found to be the dominical letter for the year), and find that the 22d of the month is a Sunday; the 25th, or Christmas, will then be Wednesday.
In the same way we may find the day of the week corresponding to any date (New Style) in history. For instance, the 17th of June, 1775, the day of the fight at Bunker Hill, is found to have been a Saturday.
These two tables then serve as a perpetual almanac.
Table I.
| 100 | 200 | 300 | 400 | ||||
| 500 | 600 | 700 | 800 | ||||
| 900 | 1000 | 1100 | 1200 | ||||
| 1300 | 1400 | 1500 | 1600 | ||||
| 1700 | 1800 | 1900 | 2000 | ||||
| 2100 | 2200 | 2300 | 2400 | ||||
| —- | —- | —- | —— | ||||
| C | E | G | BA | ||||
| 1 | 29 | 57 | 85 | B | D | F | G |
| 2 | 30 | 58 | 86 | A | C | E | F |
| 3 | 31 | 59 | 87 | G | B | D | E |
| 4 | 32 | 60 | 88 | FE | AG | CB | DC |
| 5 | 33 | 61 | 89 | D | F | A | B |
| 6 | 34 | 62 | 90 | C | E | G | A |
| 7 | 35 | 63 | 91 | B | D | F | G |
| 8 | 36 | 64 | 92 | AG | CB | ED | FE |
| 9 | 37 | 65 | 93 | F | A | C | D |
| 10 | 38 | 66 | 94 | E | G | B | C |
| 11 | 39 | 67 | 95 | D | F | A | B |
| 12 | 40 | 68 | 96 | CB | ED | GF | AG |
| 13 | 41 | 69 | 97 | A | C | E | F |
| 14 | 42 | 70 | 98 | G | B | D | E |
| 15 | 43 | 71 | 99 | F | A | C | D |
| 16 | 44 | 72 | .. | ED | GF | BA | CB |
| 17 | 45 | 73 | .. | C | E | G | A |
| 18 | 46 | 74 | .. | B | D | F | G |
| 19 | 47 | 75 | .. | A | C | E | F |
| 20 | 48 | 76 | .. | GF | BA | DC | ED |
| 21 | 49 | 77 | .. | E | G | B | C |
| 22 | 50 | 78 | .. | D | F | A | B |
| 23 | 51 | 79 | .. | C | E | G | A |
| 24 | 52 | 80 | .. | BA | DC | FE | GF |
| 25 | 53 | 81 | .. | G | B | D | E |
| 26 | 54 | 82 | .. | F | A | C | D |
| 27 | 55 | 83 | .. | E | G | B | C |
| 28 | 56 | 84 | .. | DC | FE | AG | BA |
Table II.
| A | B | C | D | E | F | G | |
|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | |
| Jan. 31. | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
| 15 | 16 | 17 | 18 | 19 | 20 | 21 | |
| Oct. 31. | 22 | 23 | 24 | 25 | 26 | 27 | 28 |
| 29 | 30 | 31 | .. | .. | .. | .. | |
| Feb. 28-29. | .. | .. | .. | 1 | 2 | 3 | 4 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 | |
| March 31. | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
| 19 | 20 | 21 | 22 | 23 | 24 | 25 | |
| Nov. 30. | 26 | 27 | 28 | 29 | 30 | 31 | .. |
| .. | .. | .. | .. | .. | .. | 1 | |
| April 30. | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
| July 31 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |
| 23 | 24 | 25 | 26 | 27 | 28 | 29 | |
| 30 | 31 | .. | .. | .. | .. | .. | |
| .. | .. | 1 | 2 | 3 | 4 | 5 | |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
| Aug. 31. | 13 | 14 | 15 | 16 | 17 | 18 | 19 |
| 20 | 21 | 22 | 23 | 24 | 25 | 26 | |
| 27 | 28 | 29 | 30 | 31 | .. | .. | |
| .. | .. | .. | .. | .. | 1 | 2 | |
| Sept. 30. | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
| 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
| Dec. 31. | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| 31 | .. | .. | .. | .. | .. | .. | |
| .. | 1 | 2 | 3 | 4 | 5 | 6 | |
| 7 | 8 | 9 | 10 | 11 | 12 | 13 | |
| May. 31. | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| 21 | 22 | 23 | 24 | 25 | 26 | 27 | |
| 28 | 29 | 30 | 31 | .. | .. | .. | |
| .. | .. | .. | .. | 1 | 2 | 3 | |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| June 30. | 11 | 12 | 13 | 14 | 15 | 16 | 17 |
| 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
| 25 | 26 | 27 | 28 | 29 | 30 | .. |
85. The Weight of the Earth.—There are several methods of ascertaining the weight and mass of the earth. The simplest, and perhaps the most trustworthy method is to compare the pull of the earth upon a ball of lead with that of a known mass of lead upon it. The pull of a known mass of lead upon the ball may be measured by means of a torsion balance. One form of the balance employed for this purpose is shown in Figs. 98 and 99. Two small balls of lead, b and b, are fastened to the ends of a light rod e, which is suspended from the point F by means of the thread FE. Two large balls of lead, W and W, are placed on a turn-table, so that one of them shall be just in front of one of the small balls, and the other just behind the other small ball. The pull of the large balls turns the rod around a little so as to bring the small balls nearer the large ones. The small balls move towards the large ones till they are stopped by the torsion of the thread, which is then equal to the pull of the large balls. The deflection of the rod is carefully measured. The table is then turned into the position indicated by the dotted lines in Fig. 99, so as to reverse the position of the large balls with reference to the small ones. The rod is now deflected in the opposite direction, and the amount of deflection is again carefully measured. The second measurement is made as a check upon the accuracy of the first. The force required to twist the thread as much as it was twisted by the deflection of the rod is ascertained by measurement. This gives the pull of the two large balls upon the two small ones. We next calculate what this pull would be were the balls as far apart as the small balls are from the centre of the earth. We can then form the following proportion: the pull of the large balls upon the small ones is to the pull of the earth upon the small ones as the mass of the large balls is to the mass of the earth, or as the weight of the large balls is to the weight of the earth. Of course, the pull of the earth upon the small balls is the weight of the small balls. In this way it has been ascertained that the mass of the earth is about 5.6 times that of a globe of water of the same size. In other words, the mean density of the earth is about 5.6.
Fig. 99.
Fig. 99.
The weight of the earth in pounds may be found by multiplying the number of cubic feet in it by 62-1/2 (the weight, in pounds, of one cubic foot of water), and this product by 5.6.
Fig. 100.
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.
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.
Fig. 101.
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.
Fig. 102.
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.
Fig. 103.
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.
Fig. 104.
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.