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sin. a = sin.d
cos. (v + a)
sin. (0 + 0)
tan. I = tan. a Again, to find the declination and right ascension the formulæ are similar, viz. sin. d = sin. w cos. a sin. I + cos. w sin. a.
tan, a sin. w t sin. l cos, w tan. a = Here, in like manner, taking a subsidiary angle, so that tan. På sin. 1 the resulting formulæ are, sin. d = sin, a
cos. • tan. a = tan. /
sin. 0° The angle of position S may be determined by either of the following theorems, viz. sin. w cos. It
sin. w cos. I sin. S =
or sin. S = The preceding formulæ will answer for all positions of the stars, by making the sines, cosines, or tangents, positive or negative, according to the value of the arcs to which they correspond: they are very convenient in application, and, we think, preferable, on the whole, to the rules of Dr. Maskelyne for the same purpose, given in the first volume of Vince's Astronomy.
One of the most remarkable results to which the theory of attraction has led, is that of the oscillation of all the irregularities of the planetary system within certain limits which they never pass. The variation in the obliquity of the ecliptic is an example of this kind; and M. Biot, in common with many other mathematicians, French and English, ascribes the discovery of this fact to M. Laplace, while, in truth, lie has only the merit of affixing the last link to an interesting chain of deduction. Our countryman, Thomas Simpson, has the honour of forming the first; for, in the resolution of some general problems in physical astronomy, in his ' Miscellaneous Tracts,' applying his results to the lunar orbit, he concludes, by showing that the effect of such terms or forces as are proportional to the cosine of the arch 2, is explicable by means of the cosines of that arch and of its multiples, (no less than the effects of the other terms that are proportional to the cosines of the multiples thereof;) a very important point is determined; for, since it appears thereby that no terms enter into the equation of the orbit but what by a regular increase and decrease do after a certain time return again to their former values, it is evident from thence that
the mean motion and the greatest quantities of the several equations undergo no change from gravity.' - Tracts, p. 179.
The reasoning in the preceding quotation evidently applies to all that has been since done, and is, in fact, the source of every subsequent investigation. It was upon analogous principles that Frisi proved, in his third book De Gruvitate Universali Corporum, prop. 45, that the obliquity of the ecliptic can scarcely ever be more than a degree less than it is now, and that not in less than sixty centuries to come. And, more generally still, M. Lagrange, employing the principles of Simpson, completed the discovery of the permanency of the whole system in a state but little different from what obtains at any assumed period of its existence; as well 'as traced the extent of the oscillations in many particular cases. His method has been thus developed ;—. The law of the composition of forces enables us to express every action of the mutual forces of the sun and planets by the sines and cosines of circular arches, which increase with an uniform motion. The nature of the circle shows, that the variation of the sines and cosines are proportional to the cosines and sines of the same arches. The variations of their squares, cubes, or other powers, are proportional to the sines or cosines of the double or triples, or other multiples of the same arches. Therefore, since the infinite serieses which express those actions of forces, and their variations, include only sines and cosines, with their powers and Auxions, it follows that all accumulated forces, and variations of forces, and variations of variations, through intinite orders, are still expressible by repeated sums of sines or cosines, corresponding to arches which are generated by going round and round the circle. These quantities, as every analyst knows, become alternately positive and negative; and therefore, in whatever way they are compounded by addition of them selves, or their multiples, or both, we must always arrive at a period after which they will be repeated with all their intermediate variations:
Such, in brief, was the process, strictly conformable to the prin ciples originally developed by Simpson, from which Lagrange proved, that the eccentricities of the planetary orbits, though variable, will never vanish entirely, nor exceed certain quantities ; tbat the variation in the obliquity of the ecliptic, and every other apparent irregularity in the system, has its period and its limit. Hence, considering what was accomplished in succession by the three eminent geometers here mentioned, justice compels us to lower considerably the praise ascribed by M. Biot and others to Laplace for his discoveries in this departinent of physical astronomy. His merit consists in carrying their principles into the details. Thus, taking 1750 for the origin of any time t reckoned in years, the distance antecedent to that date being reckoned negatively, and the time subsequent to it, positively, calling the retrogradation of the equinoctial point on the fixed ecliptic, and V the obliquity of the equator from the fixed ecliptic, Laplace gives, in his Mécanique Céleste, the following formulæ expressed in the centesimal va tation:
=t. 155."5927 + 3.011019 + 4.25562 sin. (t.155."5927 + 95.°0733)
- 7.035308 cos. t. 99."1227 - 1.7572 sin. t.43."04.46.
= 26.°0812 - 0.° 36766 1.°81876 cos. (t 155."5927 + 95.0733)
+ 0.050827 cos. t. 43:"0446 - 2.°84636 sin. t. 99."1227. If y be the corresponding retrogradation of the equinoxes upon the moveable ecliptic, and the apparent obliquity of the equator from the movable ecliptic, then the theorems for any time whatever, reckoning from the epoch 1750, are,
= t. 155."5927 19.42823 sin. t. 43."0446 + 6.°22038 sin. 2t. 49."5613. v* 26.°0812 1.*03304 sin. t. 99."1227 0.°73532 sin. ?t. 21."5223.
From these theorems, which have not, as yet, we believe, been published in any English work, it follows that, with regard to the obliquity of the equator from the fixed ecliptic, its total change from the time t will be equal to the product of the annual acceleration into the half of t, that is to say, after the time t the obliquity V will become V + 1*. 0."00003037; while, for the annual change of the obliquity with respect to the moveable ecliptic, we have
1."6083_0."2486 sin t.43."0446 +3."2166 sin ' t. 49."5613 which, besides the terms proportional to the time, and to the powers of the time, contains the constant term – 1.“6083, to which there is nothing analogous in the variations of obliquity with regard to the fixed ecliptic.
"The reason of this difference (says M. Biot) may be traced in the causes which produce the two phænomena. The attraction of the sun and moon, if they acted alone, would produce a constant precession equal to 155." 5927 (centes.) and would not change the obliquity of the equator from the ecliptic, which would then be fixed. But, by the effect of the planetary attraction, the true ecliptic is displaced in the heavens, and carries those two luminaries with it. Their action in consequence varies, and produces a small variation in the obliquity of the equator from the fixed ecliptic. This variation, at first insensible, becomes accelerated proportionably to the time, and the resulting absolute change of obliquity is therefore proportional to the square of the time. But, farther, the attraction of the planets which displaces the true ecliptic, inclines it also towards the fixed ecliptic. This other an -nual variation is at first constant, and its effect is proportional to the time. But the apparent obliquity which we observe is the difference of
the two inclinations of the equator and of the true ecliptic towards the fixed ecliptic. It is, in fact, the excess of the first over the second : it is therefore, the difference of the two preceding results; and it is thence obvious why its expression, which we have developed, should contain the two kinds of variations which characterise them.
Our author gives an interesting account of the subjects of prefession and nutation. But, on comparing his language in the first and second editions of his work, we cannot but notice the singular evidence which they furnish of his progress in national partiality, In his first edition, (speaking of the inferred esistence of these phanomena previously to their discovery by observation,) he says,
. L'existence de ces phénomènes est une suite de la théorie de l'attraction ; ils ont été découverts et calculés par Newton, avant d'être
C'est l'excellent astronome Bradley qui les a le premier reconnus et determinés
par l'observation.' Since that edition was published, however, he seems to have obtained some new light as to these particulars, for his language now is,
• La théorie de l'attraction universelle a fait connaître pourquoi les variations périodiques observées par Bradley dans l'obliquité de l'ecliptique et dans la position des equinoxes, &c. sont en rapport avec la position des næuds de la lune. C'est à d'Alembert que l'on doit cette importante confirmation de la théorie de l'attraction universelle.'
In treating the subject of the motion of the apsides of the sun's apparent orbit, our author presents some particulars worth recording
• According to the observations of Lacaille, the longitude of the perigee, in 1750, was 309.05827 (centes.).
* When the major axis was perpendicular to the line of the equinoxes this longitude would be 300°.
• The difference is 9o.5827, which at the rate of 191."0668 per year, requires a number of years equal to 958270000-1910668, or about
• This phænomenon would therefore take place in the year 1250 ; when the sun's perigee would coincide with the winter solstice, and the apogee with the summer solstice.
• In like manner when the major axis coincided with the line of the equinoxes, the longitude of the perigee was 200°. From that epoch to 1750, it would have advanced 109o.5827. The number of years necessary for this displacement is 10958270000_1910668, or about 5735, which refers this phænomenon to about 4000 years previous to the Christian æra. By a coincidence sufficiently singular it happens that most chronologers refer nearly to this time the first traces of the residence of man upon earth; though it appears by a great number of physical proofs, that the earth itself is much more ancient.'
We shall not stop to expose the folly of this observation, but leave M. Biot to settle the point with his cher et illustre confrère,' Laplace, who, in his • Exposition,' liv. iv. ch. 4, throws a doubt of a contrary kind upon the Mosiac accounts, and eagerly endeavours to adduce proofs of • la nouveauté du monde moral, dont les monu
ne remontent guère, au-delà de trois milles ans. Our author, however, goes on:
• The same phænomenon will occur again when the solar perigee becomes 400°, that is to say, when it has described 100°—9.05827, after the year 1750; and, estimating from the preceding results, we shall see that in order to that there will be required a number of years expressed by 5735-1000=4735, which refers this phænomenon to the year 6485. The solar perigee will then coincide with the vernal equinox, while in the opposite position it coincided with the autumnal equinox. In these two cases the line of the solstices, which is always perpendicular to that of the equinoxes, coincides with the minor axis of the solar ellipse.
M. Biot next proceeds to shew how the position of the apsides affects the relative length of the seasons. Thus, it has been computed that in the year 1800 : * From the vernal equinox to the summer solstice was 92.90588.
From the summer solstice to the autumnal equinox 934.56584. * From the autumnal equinox to the winter solstice 894.69954.
From the winter solstice to the vernal equinox 894.07110. • The spring is, therefore, now shorter than the summer, and the autumn longer than the winter.
So long as the solar perigee remains on the side of the equator, on which it is now, the spring and summer taken together, will be longer than the autumn and winter together. In the present age the difference is about 7 days, as appears from the preceding values. These intervals will become equal about the year 6485, when the perigee will reach the vernal equinox; afterwards it will pass beyond it, and the spring and summer taken together, will become shorter than the autumn and the winter.
. These phænomena could not obtain if the motion of the sun were circular and uniform ; but all the seasons would be equal. The eccentricity of the orbit, therefore, though very small, has a sensible influence on the duration of the seasons; and the displacement of the major axis, though very slow, produces varieties that become perceptible in dif
Book III. on the theory of the moon, contains 21 chapters, and occupies the rest of the second volume. Its subjects are: General phænomena of the lunar notions; theory of the moon's circular motion, (or the first approximation to the true motions); moon's phases; apparent diameter and parallax; theory of the moon's elliptical