416 



of its unit radii, termed * coordinate axes.' From any point P in 

 the plane AOB draw PM parallel to BO, so as to cut OA, produced 

 either way if necessary, in M. Then there will exist some ' scalars ' 

 ('real' or 'possible quantities') u, v such that OM=w.OA, and 

 MP = v . OB, all lines being considered in respect both to magnitude 

 and direction. Hence OP, which is the 'appense' or 'geometrical 

 sum ' of OM and MP, or =OM + MP, will =w . OA+0 . OB. By 

 varying the values of the ' coordinate scalars ' u, v, P may be made 

 to assume any position whatever on the plane of AOB. The angle 

 AOB may be taken at pleasure, but greater symmetry is secured by 

 choosing OI and OJ as coordinate axes, where IOJ is a right angle 

 described in the right-handed direction. If any number of lines OP, 

 OQ, OR, &c., be thus represented, the lengths of the lines PQ, QR, 

 &c., and the sines and cosines of the angles IOP, POQ, QOR, &c., can 

 be immediately furnished in terms of the unit of length and the 

 coordinate scalars. 



If OP=# . Ol-f-y OJ, and any relation be assigned between the 

 values of x and y, such as y=fx or <j> (#, y)=0, then the possible 

 positions of P are limited to those in which for any scalar value of x 

 there exists a corresponding scalar value of y. The ensemble of all 

 such positions of P constitutes the 'locus' of the two equations, 

 viz. the 'concrete equation' OP=# . Ol+y . OJ, and the 'abstract 

 equation' y=f> x. The peculiarity of the present theory consists in 

 the recognition of these two equations to a curve, of which the 

 ordinary theory only furnishes the latter, and inefficiently replaces 

 the former by some convention respecting the use of the letters, 

 whereby the coordinates themselves are not made a part of the 

 calculation. 



A variation in either of these two equations will occasion a dif- 

 ference either in the form or position of their locus. If the abstract 

 equation be y=ax-\-b, where a and b are any scalars, the concrete 

 equation OP=^p . Ol+y .OJ becomes OP=# . (Ol + a . OJ) + b . OJ, 

 which shows that OP is the appense of a constant line, and a line in 

 a constant direction, and hence its extremity P must lie on a line in 

 that direction drawn through the extremity of the constant line. 

 Also, since the length of OP is VO'+y 2 ) times the length of OI, 

 the locus of the two equations 



OP=# . Ol-f y . OJ and 



