TTTE MOON’S MOTIONS.

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1I 4 , and lastly with maximum velocity again at M 4 .It will be clear, then, that near the points TOj, to 2 , m 3 ,and to 4 , the moon moves with mean velocity; the arcsm 4 m 4 and m 2 m t are traversed with a velocity exceedingthe mean ; and the arcs m 1 m 2 and m s w 4 with a velocityfalling short of the mean. Thus at or near m 4 the moonceases to gain, and therefore the amount by which sheis in advance of her mean place has attained its max-imum * when the moon is at or near mj. Similarly theamount by which the moon is behind her mean place hasattained its maximum when the moon is at or near m 2 .

* It is singular how frequently the very simple principles onwhich the attainment of a maximum, mean, or minimum valuedepend are misunderstood; and how commonly the mistake ismade of supposing that a maximum or minimum value is attainedwhen the increasing or diminishing cause is most effective. Itis precisely when an increasing cause is most effective that the rateof increase is greatest, and therefore the maximum value is thenclearly not attained. And so of a minimum value ; there canclearly he no minimum while the decreasing cause is still effect-ive. It is when a cause neither tends to increase nor diminish,—that is, when it has a mean value,—that the maximum or minimumof effect is attained. Thus in spring the sun’s daily elevation isincreasing more rapidly than at any other time, and in autumn thedaily elevation is diminishing most rapidly ; but it is not at theseseasons that the sun attains his maximum or minimum degree ofelevation ; this happens in the summer and winter, when his dailyelevation changes least. In an illustrative case such as this there canhe no mistake ; yet very often, when less familiar instances are dealtwith, the mistake to which I have referred is made. Thus in Mr.Lockyer’s Elementary Lessons in Astronomy, we have the seasonswhen the equation of time is zero described as those when thereal sun’s motion is the same as the mean sun’s ; the fact really beingthat it is precisely at these seasons that the real sun’s motion attainseither its maximum or minimum value.