THE MOON’S MOTIONS.

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portion of D to 1; hence, if the velocity of the outerplanet were equal to that of the inner, the period ofthe outer planet would be D. But it is greater, beingDv^D (that is, it is greater in the proportion of N /Dto 1); hence the velocity of the outer planet must beless, in the proportion of 1 to D. Now the sun’senergy causes the direction of the earth’s motion tobe changed through four right angles in the time 1;that of the outer planet being similarly deflected inthe time D^D; and we know that a moving bodyis more easily deflected in exact proportion as itsvelocity is less; so that the outer planet, movingv'D times more slowly, ought to be deflected -/Dtimes more quickly if the sun influenced it as muchas he does the nearer one. Since the outer planet,instead of being deflected "/D times more quickly, isdeflected D-/D times less quickly, the influence ofthe sun on the outer planet must be less than on theearth, v^D xD/fl times,—that is, D x D (or D*) timesless. In other words, the attraction of the sundiminishes inversely as the square of the distance.

Newton had ther efore only to determine whetherthe force continually deflecting the moon from thetangent to her path is equal in amount to the forceof terrestrial gravity reduced in accordance with thislaw of inverse squares, in order to obtain at least afirst test of the correctness of the theory which hadsuggested itself to his mind. Let us consider howthis was to be done; and in order that the accountmay agree as closely as possible with the actual his-E 2