THE MOON’S MOTIONS.

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may employ a simple method available with quantitiesthat are nearly equal. Thus, suppose two bodies placedat distances represented by 100 and 101 respectivelyfrom a certain centre of force, then the attractions in thetwo bodies are inversely proportional to the squares of100andl01,orareinthe ratio 10,201 to 10,000; but thisratio is appreciably the same as the ratio of 102 to 100.Therefore in this case, and in all such cases * wherethe distance from one body exceeds the distance fromthe other by a relatively minute quantity, we canobtain the relative forces by representing them as lineshaving a relative difference twice as great.

Now let us apply this principle to the moon andearth. Suppose E (fig. 17, Plate V.) to be theearth, M the moon, and that the lines E s, M s'(appreciably parallel) are directed towards the distantsun. We may suppose the globe S to represent thesun, and we may regard S s' and S s as the pro-longations of M s' and E s, if we recognize the factthat the gap at ss', ss', would, on the scale of ourfigure be some ten yards across. Now suppose thatthe sun’s attraction on a unit of the moon’s mass t is

* The student should test this assertion by a few calculations.Thus he can take the numbers 45,681 and 45,682, and show thatthe ratio of the squares of these numbers is approximately repre-sented by the ratio 45,681 to 45,683 ; and therefore the inverseratio of the squares by the ratio 45,683to 45,681. We may equallywell take 45,680 to 45,682 for the ratio of the squares, and 45,682to 45,680 for that of the inverse squares.

+ Throughout the explanation it must be carefully borne innund that when the attraction on the moon or earth is spoken of,