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NOTE 131, p. 60. The line described, &c. Let E, fig. 33, be the earth, S the

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centre of the sun, and V the planet Venus. The real transit of the planet, seen from E the centre of the earth, would be in the direction A B. A person at W would see it pass over the sun in the line v a, and a person at O would see it move across him in the direction v' a'.

NOTE 132, p. 61. Kepler's law. Suppose it were required to find the distance of Jupiter from the sun. The periodic times of Jupiter and Venus are given by observation, and the mean distance of Venus from the centre of the sun is known in miles or terrestrial radii; therefore, by the rule of three, the square root of the periodic time of Venus is to the square root of the periodic time of Jupiter as the cube root of the mean distance of Venus from the sun, to the cube root of the mean distance of Jupiter from the sun, which is thus obtained in miles or terrestrial radii. The root of a number is that number which, once multiplied by itself, gives it square; twice multiplied by itself, gives its cube, &c. For example, twice 2 are 4, and twice 4 are 8; 2 is therefore the square root of 4, and the cube root of 8. In the same manner 3 times 3 are 9, and 3 times 9 are 27; 3 is therefore the square root of 9, and the cube root of 27.

NOTE 133, p. 63. Inversely, &c. The quantities of matter in any two 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 134, p. 63. 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 Tn, 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 mn, 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 C n is assumed to be unity in the tables, if 1 be subtracted from the number representing C T, 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 135, p. 67. 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 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 P N 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 Norn; 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.

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Fig. 34.

NOTE 136, p. 68. 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, Pn', &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

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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 137, p. 74. 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 138, p. 74. Angular velocities that are sensibly uniform. The earth and planets revolve about their axes 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 139, p. 78. If fig. 1 be the moon, her polar diameter NS 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 140, p. 83. Inversely proportional, &c. That is, the total amount of solar radiation becomes less as the minor axis C C', fig. 20, of the earth's orbit becomes greater.

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NOTE 142, p. 86. In an ellipsoid of revolution, fig. 1, the polar diameter NS, 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 143, pp. 57, 90. Let q Q, and Ee, fig. 11, be the planes of the equator and ecliptic. The angle e Q, 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 q Q coincide with the ecliptic Ee, which causes the equinoctial points V and to move slowly backwards on the plane e 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 E, causes the equinoctial points 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.

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Fig. 86.

NOTE 144, pp. 75, 91. Let q Q, e V E, fig. 36, be the planes of the equinoctial or celestial equator and ecliptic, and p, P, their poles. Then suppose p, the pole of the equator, to revolve with a tremulous or wavy motion in the little ellipse p c d b in about 19 years, both motions being very small, while the point a is carried round in the circle a A B in 25,868 years. The tremulous motion may represent the half-yearly variation, the motion in the ellipse gives an idea of the nutation discovered by Bradley, and the motion in the circle a A B arises from the precession of the equinoxes. The greater axis p d of the small ellipse is 18"5, its minor axis b c is 13"74. These motions are so small, that they 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.

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NOTE 145, p. 94. Let N be the pole, fig. 11, e E the ecliptic, and Q 9 the equator. Then, N n m S being a meridian, and at right angles to the equator, the arc m is less than the arc Y n.

NOTE 146, p. 96. Heliacal rising of Sirius. When the star appears in the morning, in the horizon, a little before the rising of the sun.

NOTE, 147, p. 98. Let PA, 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 perigee P is between the solstice of winter and the equinox of spring. So that the time of the sun's passage through the arc ✅ A is greater than the time he takes to go through the arc PY. 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 Y. In 6468 of the Christian era, the perigee P will coincide with In 1234 A.D. the major axis was perpen

dicular to, and then P was in the winter solstice.

NOTE 148, p. 99. At the solstices, &c. Since the declination of a 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 e from the plane Q.

NOTE 149, p. 99. Zenith distance is the angular distance of a celestial object from the point immediately over the head of an observer.

NOTE 150, p. 100. 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 151, p. 101. 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 152, p. 103. 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 153, p. 105. If S be the earth, fig. 14, d the sun, and C QOD 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 SQ, or d S D, each of which is 90°.

NOTE 154, pp. 105, 107. Declination. If the earth be in C, fig. 11, and if qQ 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 155, p. 108. 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 156, pp. 110, 145. Fig. 37 shows the propagation of waves from two

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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

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