ROTA'TION (Lat. rota). There is, perhaps, no elementary idea which has been the subject of so much popular misconception as that of rotation. This is probably due to the vagueness of the definitions commonly given.
All motion that we can observe is relative; for instance, any fixed object on the earth's surface has a certain motion relative to the earth's axis, in consequence of the diurnal rotation; the earth itself has a certain motion relative to the sun, in consequence of its annual revolution; the sun has a certain motion relative to the so-called fixed stars; and it is possible that the whole stellar system may have a motion relative to something in space beyond its boundaries. Now, the motion of an object on the earth's surface differs according to the way it is measured: a passenger sitting in a railway carriage is at rest if his motion relative to the carriage be considered; lie has the same motion as the carriage if it be measured relative to the rails; and if the carriage were running from e. to w. along a parallel of latitude, so as to complete the circuit in 24 hours, he would be at rest relative.to the earth's axis. If, therefore, we wish to talk of absolute motion, it must be measured relative to FIXED points or directions; and in the violation of this obvi ous conditibn lies the error most commonly met with. Thus, to show that the earth rotates about its axis, we may observe its motion relatively to the line joining it with the moon; and we observe that the moon comes to the meridian at intervals of (roughly) 25 hours. Does the earth rotate in 25 hours? We know that it does not, and the error con sists in treating as an absolute rotation, a rotation measured relative to a line—that join ing the earth and moon—which is itself turning. If we take the intervals of the sun's crossing the meridian, we find 24 hours- -a much closer approximation; but still not exact, because our line of reference—that joining the earth and sun—is slowly turn ing. Would we have an absolute measure, we must choose a fixed line, or one so nearly fixed that its motion is absolutely insensible. Such is the line joining any fixed star with
the earth, and the time of the earth's absolute rotation about its axis is 23 56a 4.09'—the interval between culminations of the same fixed star. The difference between absolute and relative rotation in any planet gives rise to the difference between the sidereal and the solar day; and the planet's year contains just one more of the former than of the latter.
Now, suppose for a moment that the earth were to revolve only part as fast as it now does, there world be one sidereal clay in the year, and there would be no solar day at all—in other words, there would be no rotation of the earth with reference to the line joining it with the sun; that is, the earth would turn always the mine side to the sun; yet•it would be absolutely rotating about its axis once in a year. This is the case which we.observe in the moon's motion relative to the earth, and we see at once that the moon must rotate absolutely—that is, with reference to fixed directions in space—in the exact time in which she completes one revolution about the earth. Those who say the moon does not rotate on her axis make precisely the same mistake as those who fancied that the earth is immovable, and that moon, sun, and stars revolve about it every clay. There is a physical cause for this peculiarity in the moon's motion, which leads to very impor tant consequences with reference to the future of the solar system. See TIDES.
Several elementary theorems regarding rotation may now be enunciated; but the proofs, though very simple, will be given merely in outline. Any displacement what ever given to a plane figure in its own plane—as to a sheet of paper lying on a table—is equivalent to a single rotation about a definite axis. Let A, B be any two points of the figure, and let them be dis placed to A', B' respectively. Join AA', BB', and bisect them in a and b by perpendiculars meeting in 0. Then, it is easy to show that (1.) OA' = OA,