the mechanical principle of the equality of action and reaction, it happens that while the mean motion of one planet is increasing, that of the other is diminishing, and vice versa.  We have supposed that the orbits of both planets are situate in the same plane.  In reality, however, they are inclined to each other, and this circumstance will produce an effect exactly analogous to that depending on the eccentricities of the orbits.  It is plain that the more nearly the mean motions of the two planets approach a relation of commensurability, the smaller will be the displacement of every third conjunction, and consequently the longer will be the duration, and the greater the ultimate accumulation, of the inequality.—­Translator.

[35] The utility of observations of the transits of the inferior planets for determining the solar parallax, was first pointed out by James Gregory (Optica Promota, 1663).—­Translator.

[36] Mayer, from the principles of gravitation (Theoria Lunae, 1767), computed the value of the solar parallax to be 7".8.  He remarked that the error of this determination did not amount to one twentieth of the whole, whence it followed that the true value of the parallax could not exceed 8".2.  Laplace, by an analogous process, determined the parallax to be 8".45.  Encke, by a profound discussion of the observations of the transits of Venus in 1761 and 1769, found the value of the same element to be 8".5776.—­Translator.

[37] The theoretical researches of Laplace formed the basis of Burckhardt’s Lunar Tables, which are chiefly employed in computing the places of the moon for the Nautical Almanac and other Ephemerides.  These tables were defaced by an empiric equation, suggested for the purpose of representing an inequality of long period which seemed to affect the mean longitude of the moon.  No satisfactory explanation of the origin of this inequality could be discovered by any geometer, although it formed the subject of much toilsome investigation throughout the present century, until at length M. Hansen found it to arise from a combination of two inequalities due to the disturbing action of Venus.  The period of one of these inequalities is 273 years, and that of the other is 239 years.  The maximum value of the former is 27".4, and that of the latter is 23".2.—­Translator.

[38] This law is necessarily included in the law already enunciated by the author relative to the mean longitudes.  The following is the most usual mode of expressing these curious relations:  1st, the mean motion of the first satellite, plus twice the mean motion of the third, minus three times the mean motion of the second, is rigorously equal to zero; 2d, the mean longitude of the first satellite, plus twice the mean longitude of the third, minus three times the mean longitude of the second, is equal to 180 deg..  It is plain that if we only consider the mean longitude here to refer to a given epoch, the combination