globes are not, like other planets, impelled with a greater velocity when they approach the sun and thus they are acted upon by two moving forces, one of which produces their proper revolution about Jupiter, the other regulates their motion round the sun." And in another place, (cap. 20,) he attempts to show an effect of this principle upon the inclination of the orbit; though, as might be expected, without any real result. The case which most obviously suggests the notion that the sun exerts a power to disturb the motions of secondary planets about primary ones, might seem to be our own moon; for the great inequalities which had hitherto been discovered, had all, except the first, or elliptical anomaly, a reference to the position of the sun. Nevertheless, I do not know that any one had attempted thus to explain the curiously irregular course of the earth's attendant. To calculate, from the disturbing agency, the amount of the irregularities, was a problem which could not, at any former period, have been dreamt of as likely to be at any time within the verge of human power. Newton both made the step of inferring that there were such forces, and, to a very great extent, calculated the effects of them. The inference is made on mechanical principles, in the sixth Theorem of the third Book of the Principia;-that the moon is attracted by the sun, as the earth is;-that the satellites of Jupiter and Saturn are attracted as the primaries are; in the same manner, and with the same forces. If this were not so, it is shown that these attendant bodies could not accompany the principal ones in the regular manner in which they do. All those bodies at equal distances from the sun would be equally attracted. But the complexity which must occur in tracing the results of this principle will easily be seen. The satellite and the primary, though nearly at the same distance, and in the same direction, from the sun, are not exactly so. Moreover the difference of the distances and of the directions is perpetually changing; and if the motion of the satellite be elliptical, the cycle of change is long and intricate: on this account alone the effects of the sun's action will inevitably follow cycles as long and as perplexed as those of the positions. But on another account they will be still more complicated; for in the continued action of a force, the effect which takes place at first, modifies and alters the effect afterwards. The result at any moment is the sum of the results in preceding instants : and since the terms, in this series of instantaneous effects, follow very complex rules, the sums of such series will be; it might be expected, utterly incapable of being reduced to any manageable degree of simplicity. It certainly does not appear that any one but Newton could make any impression on this problem, or course of problems. No one for sixty years after the publication of the Principia, and, with Newton's methods, no one up to the present day, has added We know anything of any value to his deductions. that he calculated all the principal lunar inequalities; in many of the cases, he has given us his processes; in others, only his results. But who has presented, in his beautiful geometry, or deduced from his simple principles, any of the inequalities which he left untouched? The ponderous instrument of synthesis, so effective in his hands, has never since been grasped by one who could use it for such purposes; and we gaze at it with admiring curiosity, as on some gigantic implement of war, which stands idle among the memorials of ancient days, and makes us wonder what manner of man he was who could wield as a weapon what we can hardly lift as a burden. It is not necessary to point out in detail the sagacity and skill which mark this part of the Principia. The mode in which the author obtains the effect of a disturbing force in producing a motion of the apse of an elliptical orbit (the ninth Section of the first Book), has always been admired for its ingenuity and elegance. The general statement of the nature of the principal inequalities produced by the sun in the motion of a satellite, given in the sixty-sixth Proposition, is, even yet, one of the best explanations of such action; and the calculations of the quantity of the effects in the third Book, for instance, the variation of the moon, the motion of the nodes and its inequalities, the change of inclination of the orbit, are full of beautiful and efficacious artifices. But Newton's inventive faculty was exercised to an extent greater than these published investigations show. In several cases he has suppressed the demonstration of his method, and given us the result only; either from haste, or from mere weariness, which might well overtake one who, while he was struggling with facts and numbers, with difficulties of conception and practice, was aiming also at that geometrical elegance of exposition, which he considered as alone fit for the public eye. Thus, in stating the effect of the eccentricity of the moon's orbit upon the motion of the apogee, he says, "The computations, as too intricate and embarrassed with approximations, I do not choose to introduce." The computations of the theoretical motion of the moon being thus difficult, and its irregularities numerous and complex, we may ask, whether Newton's reasoning was sufficient to establish this part of his theory; namely, that her actual motions arise from her gravitation to the sun. And to this we may reply, that it was sufficient for that purpose, since it showed that, from Newton's hypothesis, inequalities must result, following the laws which the moon's inequalities were known to follow;-since the amount of the inequalities given by the theory agreed nearly with the rules which astronomers had collected from observation; and since, by the very intricacy of the calculation, it was rendered probable, that the first results might be somewhat inaccurate, and thus might give rise to the still remaining differences between the 10 Schol. to Prop. 35, first edit. 12 calculations and the facts. A progression of the apogee; a regression of the nodes; and, besides the elliptical, or first inequality, an inequality, following the law of the evection, or second inequality discovered by Ptolemy; another, following the law of the variation discovered by Tycho;-were pointed out in the first edition of the Principia, as the consequences of the theory. Moreover, the quantities of these inequalities were calculated and compared with observation with the utmost confidence, and the agreement in most instances was striking. The variation agreed with Halley's recent observations within a minute of a degree". The mean motion of the nodes in a year agreed within less than one-hundredth of the whole12. The equation of the motion of the nodes also agreed well. The inclination of the plane of the orbit to the ecliptic, and its changes, according to the different situations of the nodes, likewise agreed". The evection has been already noticed as encumbered with peculiar difficulties; here the accordance was less close. The difference of the daily progress of the apogee in syzygy, and its daily regress in quadratures, is, Newton says, "44 minutes by the tables, 63 by our calculation." He boldly adds, "I suspect this difference to be due to the fault of the tables." In the second edition (1711) he added the calculation of several other inequalities, as the annual equation, also discovered by Tycho; and he compared them with more recent observations, made by Flam 11 1 B. iii. Prop. 29. 12 Prop. 32. 14 Prop. 35. |