HISTORY OF PHYSICAL ASTRONOMY.

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to him the data he had just obtained, under the full assurance of thatillustrious mathematician’s competency to investigate the intricacies ofthe moon’s perturbations. As Poisson and Lubbock had already shewnthat the errors of mean longitude could not be traced to the direct actionof the sun, or the disturbing influence of the earth’s figure, the questionwas narrowed to the investigation of the effects produced by the action ofthe planets, which had been hitherto supposed to he almost insensible.Hansen, guided by his profound knowledge of the theory of perturbation,undertook a rigorous scrutiny of all the inequalities of long period whichappeared likely to afford an explanation of the errors of the tables. liecalculated many equations of this nature that were found to be insensible;hut he finally discovered two depending on the action of Venus , themagnitudes of which were totally unexpected. One of these arises fromthe fact that sixteen times the mean motion of the earth, plus the meanmotion of the moon, minus eighteen times the mean motion of Venus , isa very small quantity. Indeed, when its numerical value is computed, itis found to amount only to about th of the moon’s mean motion.Now the terms of the perturbing function which have this quantity for amultiplier of the time under the symbols sine or cosine, although of thethird order with respect to the eccentricities and inclinations, will acquireby double integration its square as a divisor, and will on this accountform an equation of considerable magnitude in the expression for thelongitude. Hansen found the maximum value of this equation to amountto 27".4, and its period to 273 years. The other equation depends onthis—that the difference between thirteen times the mean motion of theearth, and eight times the mean motion of Venus , is a very small fraction.Its maximum value amounts to 23".2, and its period to 239 years. Theinequality represented by this equation is manifestly analogous to the longinequality in the earth’s epoch discovered by Mr. Airy. Indeed, it is notdifficult to see that the latter inequality will occasion a variation in themean value of the disturbing action of the sun, and will thereby give riseto an inequality of a similar nature in the moon’s longitude.

These two inequalities discovered by M. Hansen, when applied to themoon’s computed longitude, completely account for the errors in the tables,which had so long perplexed the astronomers and mathematicians ofEurope . The lunar theory may, therefore, now be considered as divestedof all serious embarrassment; and in its present state it undoubtedlyconstitutes one of the noblest monuments of intellectual research whichthe annals of science offer to our contemplation. Erom the age ofHipparchus down to the present day, the complicated movements of themoon have formed the subject of anxious enquiry. One by one have hernumerous inequalities been detected, and their laws ascertained, until theastronomer is finally enabled to predict her place with all the accuracycalled for by the most refined appliances of modern observation. Perhapsno other part of astronomy exhibits so many unequivocal triumphs of thetheory of gravitation as the researches connected with the moon’s motion.The coincidence between the deductions of the geometer and the resultsof actual observation is truly astonishing, when one considers the intricacyof the subject. Newton furnished incontestable evidence of the truth ofhis principles when he calculated the motion of the moon’s nodes to within■j^th part of the actual motion. In the present day, however, the theoriesof Plana and Damoiseau assign the motions of the apsides and nodes to