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primary planets are greater in proportion as the cubes of the numbers representing the mean distances of their satellites are greater, and also in proportion as the squares of their periodic times are less.
NOTE 136, p. 55. As hardly anything appears more impossible than that man should have been able to weigh the sun as it were in scales and the earth in a balance, the method of doing so may have some interest. The attraction of the sun is to the attraction of the earth as the quantity of matter in the sun to the quantity of matter in the earth; and, as the force of this reciprocal attraction is measured by its effects, the space the earth would fall through in a second by the sun's attraction is to the space which the sun would fall through by the earth's attraction as the mass of the sun to the mass of the earth. Hence, as many times as the fall of the earth to the sun in a second exceeds the fall of the sun to the earth in the same time, so many times does the mass of the sun exceed the mass of the earth. Thus the weight of the sun will be known if the length of these two spaces can be found in miles or parts of a mile. Nothing can be easier. A heavy body falls through 16.0697 feet in a second at the surface of the earth by the earth's attraction; and, as the force of gravity is inversely as the square of the distance, it is clear that 16.0697 feet are to the space a body would fall through at the distance of the sun by the earth's attraction, as the square of the distance of the sun from the earth to the square of the distance of the centre of the earth from its surface; that is, as the square of 95,000,000 miles to the square of 4000 miles. And thus, by a simple question in the rule of three, the space which the sun would fall through in a second by the attraction of the earth may be found in parts of a mile. The space the earth would fall through in a second, by the attraction of the sun, must now be found in miles also. Suppose mn, fig. 4, to be the arc which the earth describes round the sun in C, in a second of time, by the joint action of the sun and the centrifugal force. By the centrifugal force alone the earth would move from m to T in a second, and by the sun's attraction alone it would fall through Tn in the same time. Hence the length of T n, in miles, is the space the earth would fall through in a second by the sun's attraction. Now, as the earth's orbit is very nearly a circle, if 360 degrees be divided by the number of seconds in a sidereal year of 365 days, it will give m n, the arc which the earth moves through in a second, and then the tables will give the length of the line CT in numbers corresponding to that angle; but, as the radius Cn is assumed to be unity in the tables, if 1 be subtracted from the number representing CT, the length of Tn will be obtained; and, when multiplied by 95,000,000, to reduce it to miles, the space which the earth falls through, by the sun's attraction, will be obtained in miles. By this simple process it is found that, if the sun were placed in one scale of a balance, it would require 354,936 earths to form a counterpoise.
NOTE 137, p. 59. The sum of the greatest and least distances SP, SA, fig. 12, is equal to PA, the major axis; and their difference is equal to twice the excentricity CS. The longitude op SP of the planet, when in the point P, at its least distance from the sun, is the longitude of the perihelion. The greatest height of the planet above the plane of the ecliptic E N en, is equal to the inclination of the orbit PN An to that plane. The longitude of the planet, when in the plane of the ecliptic, can only be the longitude of one of the points N or n; and, when one of these points is known, the other is given, being 180° distant from it. Lastly,
the time included between two consecutive passages of the planet through the same node N or n. is its periodic time, allowance being made for the recess of the node in the interval.
NOTE 138, p. 60. Suppose that it were required to find the position of a point in space, as of a planet, and that one observation places it in n,
fig. 34, another observation places it in n', another in n", and so on; all the points n, n', n", n'", &c., being very near to one another. The true place of the planet P will not differ much from any of these positions. It is evident, from this view of the subject, that Pn, Pn', Pn", &c., are the errors of observation. The true position of the planet P is found by this property, that the squares of the numbers representing the lines Pn, P n', &c., when added together, is the least possible. Each line Pn, Pn', &c., being the whole error in the place of the planet, is made up of the errors of all the elements; and, when compared with the errors obtained from theory, it affords the means of finding each. The principle of least squares is of very general application; its demonstration cannot find a place here; but the reader is referred to Biot's Astronomy, vol. ii. p. 203.
NOTE 139, p. 61. The true longitude of Uranus was in advance of the tables previous to 1795, and continued to advance till 1822, after which it diminished rapidly till 1830-1, when the observed and calculated longitudes agreed, but then the planet fell behind the calculated place so rapidly that it was clear the tables could no longer represent its motion.
NOTE 140, p. 65. An axis that, &c. Fig. 20 represents the earth revolving in its orbit about the sun in S, the axis of rotation Pp being everywhere parallel to itself.
NOTE 141, p. 65. Angular velocities that are sensibly uniform. The earth and planets revolve about their axis with an equable motion, which is never either faster or slower. For example, the length of the day is never more nor less than twenty-four hours.
NOTE 142, p. 68. If fig. 1 be the moon, her polar diameter N S is the shortest; and of those in the plane of the equator, Q E q, that which points to the earth is greater than all the others.
NOTE 143, p. 73. Inversely proportional, &c. That is, the total amount of solar radiation becomes less as the minor axis CC', fig. 20, of the earth's orbit becomes greater.
NOTE 144, p. 75. Fig. 35 represents the position of the apparent orbit of the sun as it is at present, the earth being in E. The sun is nearer to the earth in moving through Pop than in moving through op A, but its motion through
Pop is more rapid than its motion through A; and, as the swiftness of the motion and the quantity of heat received vary in the same proportion, a compensation takes place.
NOTE 145, p. 76. In an ellipsoid of revolu
tion, fig. 1, the polar diameter N S, and every diameter in the equator q E Qe, are permanent axes of rotation, but the rotation would be unstable about any other. Were the earth to begin to rotate about C a, the angular distance from a to the equator at q would no longer be ninety degrees, which would be immediately detected by the change it would occasion in the latitudes.
NOTE 146, pp. 50, 80. Let q o Q, and Ee, fig. 11, be the planes of the equator and ecliptic. The angle eoQ, which separates them, called the obliquity of the ecliptic, varies in consequence of the action of the sun and moon upon the protuberant matter at the earth's equator. That action brings the point Q towards e, and tends to make the plane qpQ coincide with the ecliptic Ecope, which causes the equinoctial points and to move slowly backwards on the plane e o E, at the rate of 50"-41 annually. This part of the motion, which depends upon the form of the earth, is called luni-solar precession. Another part, totally independent of the form of the earth, arises from the mutual action of the earth, planets, and sun, which, altering the position of the plane of the ecliptic e op E, causes the equinoctial points op and to advance at the rate of 0" 31 annually; but, as this motion is much less than the former, the equinoctial points recede on the plane of the ecliptic at the rate of 50"-1 annually. This motion is called the precession of the equinoxes.
NOTE 147, p. 81. Let gop Q, e op E, fig. 36, be the planes of the equinoctial or celestial equator
and ecliptic, and p, P, their poles.
have very little effect on the parallelism of the axis of the earth's rotation during its revolution round the sun, as represented in fig. 20. As the stars are fixed, this real motion in the pole of the earth must cause an apparent change in their places.
NOTE 148, p. 83. By means of a transit instrument, which is a telescope mounted so as to revolve only in the plane of the meridian, the instant of the transit or passage of a celestial body across the meridian can be determined. The transits of the principal stars are used to ascertain the time, or, which is the same thing, the amount of the error of clocks. A system of equidistant wires, as represented in the figure, is placed in the
focus of the eye-piece, so that the middle wire is perpendicular and at right angles to the axis of the telescope. It consequently represents a portion of the celestial meridian; and when a star is seen to cross that wire it then crosses the celestial meridian of the place of observation. A clock beating seconds being close at hand, the duty of an observer is to note the exact second and part of a second at which a star crosses each wire successively in consequence of the rotation of the earth. Then the mean of all these observations will give the time at which the star crosses the celestial meridian of the place of observation to the tenth of a second, provided the observations are accurate. Now it happens that the simultaneous impression on the eye and ear is estimated differently by different observers, so that one person will note the transit of a star, for example, as happening the fraction of a second sooner or later than another person; and as that is the case in every observation he makes, it is called his personal equation, that is to say, it is a correction that must be applied to all the observations of the individual, and a curious instance of individuality it is. For instance, M. Otto Struve notes every observation 0"11 too soon, M. Peters 0"-13 too late; M. Struve noted every observation one second later than M. Bessel, and M. Argelander estimated the transit of a star 1"-2 later than M. Bessel. All these gentlemen were or are first-rate observers; and when the personal equation is known it is easy to correct the observations. However, to avoid that inconvenience Mr. Bond has introduced a method in the Observatory at Cambridge in the United States in which touch is combined with sight instead of hearing, which is now used also at Greenwich. The observer at the moment of the observation presses his fingers on a machine which by means of a galvanic battery conveys the impression to where time is measured and marked, so that the observation is at once recorded and the personal equation avoided.
NOTE 149, p. 84. Let N be the pole, fig. 11, e E the ecliptic, and Q q the equator. Then, N n m S being a meridian, and at right angles to the equator, the arcom is less than the arc op n.
NOTE 150, p. 85. Heliacal rising of Sirius. When the star appears in the morning, in the horizon, a little before the rising of the sun.
NOTE 151, p. 87. Let P op A, fig. 35, be the apparent orbit or path of the sun, the earth being in E. Its major axis, A P, is at present situate as in the figure, where the solar perigree P is between the solstice of winter and the equinox of spring. So that the time of the sun's passage through the arc op A is greater than the time he takes to go through the arc Pop. The major axis A P coincided with, the line of the equinoxes, 4000 years before the Christian era; at that time P was in the point op. In 6468 of the Christian era the perigee P will coincide with. In 1234 A.D. the major axis was perpendicular to ∞ 1,
and then P was in the winter solstice.
Since the declination of a
NOTE 152, p. 88. At the solstices, &c. celestial object is its angular distance from the equinoctial, the declination of the sun at the solstice is equal to the arc Q e, fig. 11, which measures the obliquity of the ecliptic, or angular distance of the plane open from the plane op ^.
NOTE 153, p. 88. Zenith distance is the angular distance of a celestial object from the point immediately over the head of an observer.
NOTE 154, p. 89.
Reduced to the level of the sea. The force of gravitation decreases as the square of the height above the surface of the earth increases, so that a pendulum vibrates slower on high ground; and, in order to have a standard independent of local circumstances, it is necessary to reduce it to the length that would exactly make 86,400 vibrations in a mean solar day at the level of the sea.
NOTE 155, p. 90. A quadrant of the meridian is a fourth part of a meridian, or an arc of a meridian containing 90°, as N Q, fig. 11.
NOTE 156, p. 93. Moon's southing. The time when the moon is on the meridian of any place, which happens about forty-eight minutes later every day.
NOTE 157, p. 96. The angular velocity of the earth's rotation is at the rate of 180° in twelve hours, which is the time included between the passages of the moon at the upper and under meridian.
NOTE 158, p. 96. If S be the earth, fig. 14, d the sun, and CQOD the orbit of the moon, then C and O are the syzygies. When the moon is new, she is at C, and when full she is at O; and, as both sun and moon are then on the same meridian, it occasions the spring-tides, it being high water at places under C and O, while it is low water at those under Q and D. The neap-tides happen when the moon is in quadrature at Q or D, for then she is distant from the sun by the angle d S Q, or d S D, each of which is 90°.
NOTE 159, p. 97. Declination. If the earth be in C, fig. 11, and if qpQ be the equinoctial, and N m S a meridian, then m Cn is the declination of a body at n. Therefore the cosine of that angle is the cosine
of the declination.
NOTE 160, pp. 99, 131. Fig 37 shows the propagation of waves from
two points C and C', where
stones are supposed to have fallen. Those points in which the waves cross each other are the places where they counteract each other's effects, so that the water is smooth there, while it is agitated in the intermediate spaces.
NOTE 161, p. 100. The centrifugal force may, &c. The centrifugal force acts in a direction at right angles to NS, the axis of rotation, fig. 30. Its effects are equivalent to two forces, one of
which is in the direction bm perpendicular to the earth, and diminishes the force of gravity at m. direction of the tangent m T, which makes the fluid the equator.
surface Qmn of the The other acts in the particles tend towards