or motions of the moment as to "do work," but can only produce deflection or transference of an energy to whose potential they do not contribute. I regard them as quasi-perpendicular to all physical forces: i.e. their action upon any physical atom is, in the above sense, always as if at right angles to its path, so that of course they cannot increase or diminish its kinetic or potential energy, except indirectly by deflecting it into new positions and conditions, where, however, all transfers of energy will still be made under purely physical laws. [Here, and in what follows, we use the term "physical force" only in the sense of a push or pull exerted at a material point by attraction, repulsion, or vis inertiæ; and analogously, the term "spiritual force": so that a group of agencies and phenomena, like electricity or intelligence, would not be called "a force."]
Even if the concept of quasi-perpendicularity, which perhaps has been already suggested by Maxwell and others, should rest upon nothing deeper than a metaphor or convention, yet it is the more presumably natural and helpful here because of its known value in Pure Algebra. Thus, the perfect symmetry between the relations of i (=) to +1 and to -1; or of 1 to +i and -i; i.e. that mutual independence of the two abstract units 1 and i which would seem so analogous to the independence of physical and spiritual forces, — has led to the use of the "complex plane," with resulting methods of great power and beauty. On this plane every value of A+Bi is located at a point having longitude A and latitude B, as in the annexed scheme; so that i is treated as quasi-perpendicular to 1.
| -1+i | i | 1+i |
| -1 | 0 | 1 |
| -1-i | -i | 1-i |
In mathematics, this notion of quasi-perpendicularity, though as I think often implied, goes unnamed. It is very flexible: a translation and a rotation would be quasi-perpendicular, however their respective directions were related, and so would be the several parameters of a curve when they were regarded as a new system of variables. Let us hold the concept in a correspondingly free way. Let it simply accentuate for us the fact that
