ON MOTION. 16 



could be distinguished. Supposing a ship to move at the rate of three miles 

 in an hour, and a person on board to walk or to be drawn towards the 

 stern at the same rate, he would be relatively in motion with respect to the 

 ship, yet we might very properly consider him as absolutely at rest : but 

 he would, on a more extended view, be at rest only in relation to the 

 earth's surface ; for he would still be revolving round the axis of the earth, 

 and with the earth round the sun, and with the sun and the whole solar 

 system, he would be slowly moving among the starry worlds which surround 

 them. Now with respect to any effects within the ship, all the subsequent 

 relations are of no consequence, and the change of his rectilinear distance 

 from the various parts of the ship is all that needs to be considered in deter- 

 mining those effects. In the same manner, if the ship appear, by compari- 

 son with the water only, to be moving through it with the velocity of 

 three miles an hour, and the water be moving at the same time in a con- 

 trary direction at the same rate in consequence of a tide or current, the 

 ship will be at rest with respect to the shore ; but the mutual actions of the 

 ship and the water will be the same as if the water were actually at rest 

 and the ship in motion. 



It is not sufficient to observe the increase or decrease of distance of a 

 moving point from another single point only ; we must compare its succes- 

 sive situations with many other points surrounding it ; and for this 

 purpose these points must be at rest among themselves, in order to be 

 considered as belonging to a quiescent space or surface ; which may be 

 denned as a space or surface of which all the points remain always at equal 

 distances from each other without any external influence. In this sense 

 we must call the deck of the ship a quiescent surface, whether the ship be 

 at anchor or under sail ; but we must not consider a surface revolving 

 round a centre as a quiescent surface, for it will appear hereafter that no 

 such motion can exist without the influence of a centripetal force ; which 

 renders it improper for determining the affections of a moving body. 



When a point is in motion with respect to a quiescent space, it is often 

 simply denominated a moving point, and the right line joining any two 

 of its places immediately contiguous to each other is called its direction. 

 If it remains continually in one right line drawn in the quiescent space, 

 that line is always the line of its direction ; if it describes several right lines, 

 each line is the line of its direction as long as it continues in it ; but if its 

 path becomes curved, we can no longer consider it as perfectly coinciding at 

 any time with a right line, and we must recur to the letter of the defini- 

 tion, by supposing a right line to be drawn through two successive points 

 in which it is found, and then if these points be conceived to approach each 

 other without limit we shall have the line of its direction. Now such a line 

 is called in geometry a tangent : for it meets the curve but does not cut it, 

 provided that the curvature be continued. (Plate I. ,Fig. 1 3.) 



Having formed an accurate ^idea) of the nature of motion, and of the im- 

 port of the terms employed in speaking of its properties, we may proceed 

 to consider the mechanical laws to which it is subjected, and which are de- 

 rivable from the essence of tKe~aennitions that have been premised. The 

 first is, that a moving point never quits the line of its direction without a 



