TRANSLATION. This word is used in mechanics, as distinguished from ROTATION, in the following manner :—A body has motion of translation when all its points move in parallel straight lines ; when, in fact, all its points have the same motion. If all have not the same motion, there is either simple rotation, that ia, about one permanent axis ; or rotation about a varying axis; or else a compound of trans lation and rotation.
The point which is called the centre of gravity of a system, and which is of no small importance in the theory of equilibrium, has yet more in that of motion. The motion of any free system is com pounded of the translation of its centre of gravity, and the rotation about an axis (whether always in one direction or not) passing through its centre of gravity. Now whatever the forces may be by which such a system is either set in motion, or acted on while in motion, the translation of its centre of gravity may always be made a distinct problem from the rotation about its centre of gravity, by the following simple rules : 1. The centre of gravity moves just as it would do if the whole system were there collected, and all the forces were there applied.
2. The rotation of a system about its centre of gravity is no other than what it would be if that centre were made a fixed point, and all the forces applied in their proper places.
Suppose, for instance, a bar A ts, whose centre of gravity is at o, is sent spinning into void space by a certain blow in the direction n communicated at D. Let there be another similar bar• ub, whose centre of gravity e moves on a fixed pivot without friction, and which, being parallel to A V, is struck at the same instant with a similar blow in the direction d e. To find the position of the bar at the end of any given time, say three seconds, is a twofold problem, as follows :— First suppose all the mass of the bar concentrated at c, and let the blow be struck, with the same force and direction, at the point C. This point c will then describe a certain parabola c m n; say that in three seconds it is at n. Next turn to the bar which moves on a fixed pivot, and let f g be its position in three seconds. Draw F a, a position of the bar parallel to f y, et being its centre of gravity, and F a will be the real position of the bar at the end of the given time, three seconds ; and similarly for any other given time.
Thus much of translation, mechanically considered : we now speak of the wider use which the word has, or might have, in geometry ; at any rate we have the thing to consider, and perhaps transference might be preferable to translation, as applied to the motion of a figure from one part of space to another. The conception of the possibility of figures differing only in position, and composed of perfectly equal and similar parts of space, similarly bounded, is one which is demanded of the beginner in geometry. Euclid requires this when he speaks of equal figures ; and his test of equality, namely, the possibility of creating a perfect coincidence, requires the notion of one figure being transferred in any requisite manner, whether by what is called in mechanics trans lation, or by rotation, or both. It must be a sort of copy, or facsimile,
of one part of space which is thus moved into and made to occupy another : for it is impossible to imagine space removed, or any part of space made to change place. And this copy, or whatever it may be, must have rigidity, that it may not change form by the way : it must be rigid in our thoughts, at least. We are thus required to imagine space endowed with some of the essential qualities of matter, before we can prove the fourth proposition of Euclid's first book : there must be the consistence of matter without its impenetrability, but whether it require force and time to change place, or not, is of no consequence. Even a plane figure must bo a sort of rigid consistence with two sides to it, for it is necessary to imagine it turned round, so as to present a different face to the spectator. In the fifth proposition of the first book, the very first step is the application of the fourth proposition to prove the equality of two triangles. Now the fourth proposition requires one triangle to be placed upon the other, which cannot be done in the figure of the fifth, unless one of the triangles be turned round, so as to show the other front to the spectator. If Euclid meant, by giving the triangle two handles, to make it easier to turn, he has been unfortunate, for the proposition has acquired the name of the ass's bridge, probably as being that which stops a dull reader. The following proof is as correct as that of Euclid, and it is not much to say that those who do not understand it will not understand the one he gave.
Let A B c be an isosceles triangle, having A B = A c. Let it be turned round (for illustration, the dotted lines show the tracks of the three points, and two intermediate positions are shown) into the position D E F. Then in the two triangles A B C, D E F, we have A B=D E, for D E i8 A C (= A B by hypothesis) removed. Also A C m D F, for a similar reason. And the angle n a a = the angle E D F, the second being only the removal of the first. Hence we have AB=DE, AC=DF, and La A C= LED F, and now by the fourth proposition it follows that LA BC=LD EY. But ZDEF is only another position of LA on; whence Zs n c=L A c is, which was to be shown. If preferred, the triangle A BC might be turned round upon itself, and the reasoning of the fourth proposition applied at once.
It is not to be supposed that Euclid did not see the preceding : but he iA a writer who very rarely goes out of the most obvious path without some cogent reason connected with his system. The proof given above would not serve to demonstrate the equality of the ex ternal angles without the previous introduction of the properties of adjacent angles ; and it happens that the knowledge of the equality of the external angles is immediately wanted.