OF MECHANICS AND GEOMETRY. 271 



the theorems of the one science can be translated into the language of the other, and that the 

 demonstration belonging to figures in which the marks represent straight lines will apply pre- 

 cisely as well to similar figures in which the marks represent forces; for in both cases the 

 representation must be conventional : no inkmark can be a straight line, and no proposition 

 concerning straight lines can be true of the inkmarks which represent them, and thouo-h it 

 requires a greater abstraction of the mind to speak of an inkmark as a force, yet the speaking 

 of it as a straight line is certainly as really conventional, and the proper utility of the figures 

 in both cases is that they assist the mind artificially in drawing deductions from the pure 

 ideas of direction and magnitude. 



Velocity is another instance of a thing physical involving the ideas of direction and magnitude 

 only, and of which therefore it may at once be predicated that the propositions respecting the 

 straight line refer to it mutatis mutandis. 



6. When it is said that every proposition respecting the straight line will have its fellow 

 respecting force, it is of course equally true that each proposition in Mechanics will have its 

 fellow in Geometry, and it will be asked, what proposition in Geometry corresponds to the parallelo- 

 gram, or rather the triangle of forces : to which I reply, that when two 

 lines AB, BC are given in position and magnitude, the straight line 

 joining the points A and C will be as strictly their geometrical 

 resultant, as the force represented by AC will be the resultant of 

 the forces represented by AB, BC : for by speaking of the resultant 

 of two lines we necessarily imply that the two lines are given to 

 determine some third object, and that object must be a straight 

 line, since the resultant of two things of the same kind must be of the same kind with those 

 which produce it, and if there be any line which is to be considered as the resultant of AB BC 

 it must be AC, since this is the only new line whose position and magnitude is in any way 

 whatever determined by the positions and magnitudes of AB and BC. If therefore we mean 

 by the resultant of two straight lines given in position the straight line which is determined 

 in magnitude and position by those straight lines, and this seems the most obvious meaning to 

 give to the term resultant, then AC is the resultant of AB and BC. 



The proposition of finding the resultant of two straight lines given in position may be 

 generalized into that of finding the resultant of any number of straight lines forming an imperfect 

 polygon. For if all the sides of a polygon be given except one, then that one will be the 

 resultant of all the rest, inasmuch as it is the only new line whose position and magnitude 

 becomes determinate in virtue of the other sides being given. It may be said that the extremity 

 of one of the last sides may be joined with one of the angular points, and that thus some 

 other line will be determined, but the obvious answer is, that this will not employ all the data 

 and that the line so determined will be the resultant of all those which are really made use of. 

 In fact, a straight line may be given just as really, though not so directly, by givino- in position 

 all the other sides of a polygon of which this straight line forms the last ; to give those other 

 sides is, I say, precisely the same thing in fact as to give the line itself. 



Conversely, a straight line may be considered as the resultant of any system of straight lines 

 which with it form a polygon ; and also in such polygon any one side may be called the 

 resultant of all the rest; if two be missing, they cannot be replaced; but if one only, then is 

 that missing one just as fixed and determinate as if it were represented as part of the polyo-on. 



In speaking of the direction of lines, it is of course necessary to distinguish between a line 

 AB, and a line BA, the direction of the one being considered 

 exactly the reverse of that of the other. Thus, in the pre- 

 ceding investigation AC is the resultant of AB, BC, not of 

 BA, liC: the resultant of those latter lines would be found by 

 taking AD ])arallel and equal to BC : then BD would be the 

 resultant of BA and BC. 



