III. On scalar and clinant algebraical coordinate geometry, introducing a new and more general theory of analytical geometry, including the received as a particular case, and explaining ‘imaginary points,’ ‘intersections,’ and ‘lines.’

Scalar Plane Geometry .— With O as a centre describe a circle with a radius equal to the unit of length. Let OA, OB be any two 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 =u . 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 =u . OA +v . 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 =x . OI +y . OJ, and any relation be assigned between the values ofx andy , such asy =fx orϕ (x, y ) = 0 , then the possible positions of P are limited to those in which for any scalar value ofx there exists a corresponding scalar value ofy . The ensemble of all such positions of P constitutes the ‘locus ’ of thetwo equations, viz. the ‘concrete equation’ OP =x . Ol +y .OJ, and the ‘abstract equation’y =f. x. The peculiarity of the present theory consists in the recognition of thesetwo 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.