Math Forum Discussions

Robert (Bob) Eldon Taylor

Posts: 101
Registered: 12/3/04

Re: Number Line
Posted: Jan 5, 2008 12:22 PM

Plain Text

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