Philip Beeley wrote: > Dear Pat, > > It is not without justification that Wallis is credited with the development > of this concept. In chapter LXVI (Of negative squares, and their imaginary > roots in algebra) of his Treatise of Algebra (1685) he sets out (and > illustrates) something akin to the modern concept of the number line: > > "As for instance: Supposing a man to have advanced or moved forward, (from A > to B,) 5 yards; and then to retreat (from B to C) 2 yards: If it be asked, how > much he had advanced (upon the whole march) when at C? Or how many yards he is > now forwarder than when he was at A? I find (by subducting 2 from 5,) that he > is advanced 3 yards. (Because +5 -2 = +3.) > > D A C B > |...|...|...|---|---|---|---|---| > > But if, having advanced 5 yards to B, he thence retreat 8 yards to D; and it > be then asked, How much is he advanced when at D, or how much forwarder than > when he was at A: I say -3 yards. (Because +5 -8 = -3.) That is to say, he is > advanced 3 yards less than nothing." > > As you see, his concept certainly includes negative values.
I think this understates the case because Wallis was dealing not just with the "number line" but the "complex plane". In fact, the passage you cite was preliminary to his discussion of imaginaries, where he conceives the imaginary unit as the ratio of a line to an equal but perpendicular line. Hamilton (1847 paper "On Quaternions" in Proceedings of the Royal Irish Academy) credits this idea to "Mr. Warren and Mr. Peacock" but does not mention Wallis, but both Hamilton and Tait build on the concept in an attempt to base quaternions on a similar notion. They fail in this because four dimensions are required, and, of course, there are only three. In the paper mentioned Hamilton toys with using time as the fourth.
Regards, Bob Robert Eldon Taylor philologos at mindspring dot com