578 



REPOET — 1893. 



lines should really be made, two such projections would be necessary to 

 determine the position and length of the resultant, and the lines would 

 have to be projected according to the rules of descriptive geometry.' 



Now the foregoing operation on a plane surface really involves the 

 addition of two independent variables, viz., length and direction, being 

 equivalent to taking the actual position in space of two dimensions of 

 each successive point, instead of in space of one dimension, as in the first 

 case. The result, which gives the final position of a point in the plane, 

 also involves two measurements, which may be expressed as the length of 

 a segment, and its direction relatively to a line. If three dimensions of 

 space be used, then the operation involves three independent variables, 

 and the results may be also expressed as such in terms of either three 

 distances from three fixed points, or as one distance and two directions 

 with fixed planes. 



We look in vain for any analogous operation given in books under 

 the heading of graphical multiplication. The above process, although 

 extended so as to meet the various requirements for which it is employed, 

 is the only other means of graphical operation. We may, however, 

 suggest a corresponding process by which a segment A A, having length 

 and direction may have both these magnitudes multiplied ten times, and 



Fig. 5. 



C^-- 



this can be done as shown in fig. 5. Here A A, is the given length, and 

 O A Ai the given angle. The successive segments, A A], Aj Ag, Aj A3, 

 . . . are set off at the given angle to each other, starting fi-om A A, . 



