Part 13

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 ..

Weight of the Earth and Precession.

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.

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.

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.