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Scene: Various mathematicians of the seventeenth century in conversation, seated or standing, as the last decade of the century begins:
JOHN WALLIS (aged English mathematician, Savilian Professor of Geometry at Oxford); ANTOINE ARNAULD (aged French theologian and mathematician); GOTTFRIED LEIBNIZ (German diplomat, mathematician and universal sage, at the peak of his philosophical powers).
Seated frontstage: LEONHARD EULER (Swiss mathematician in his late twenties, representing the early 18th century looking back). He gives introduction and commentary for the benefit of the audience, but eventually finds himself drawn into the conversation .
EULER [to audience ]
Look! These are some of the truly great mathematicians of the seventeenth century  there is John Wallis from England: he was a Professor at Oxford. And that Frenchman is Antoine Arnauld, the great antagonist of the Jesuits  he was more a theologian than a mathematician. And that is the renowned Gottfried Leibniz himself! I have always wished I could have met him and discussed with him some of the mathematical problems that have baffled me, but he died when I was just a boy, in the year 1716, I think.
We, in the eighteenth century, are beginning to make some progress, I think, with solving the legacy of mathematical problems we inherited from them. Except for these incorrigible negative numbers! They are still giving us grave difficulties. Not that we any longer doubt their existence or usefulness  it's their strange behaviour that perplexes us.
[Wallis begins to wave his stick excitedly ]
Come! Let us watch them struggling to come to terms with these newly admitted numbers.
WALLIS^{2} [in pompous, pedantic manner; illustrates his argument by pacing up and down upon an imaginary line, and gesturing back and forth with his walkingstick ].
It is impossible (everyone agrees) that any quantity can be negative ; since it is not possible that any magnitude (or geometric length) can be less than nothing, or any number fewer than none.
Yet ... that supposition (of negative quantities) is not either unuseful or absurd, when rightly understood. And though, as to the bare algebraic notation, it imports a quantity less than nothing, yet, when it comes to a physical application, it denotes as real a quantity as if the sign were "plus" [gestures + in the air with his stick ], but to be interpreted in a contrary sense.
As for instance: supposing a man to have advanced or moved forward (from A to B) 5 yards [begins pacing ]; and then to retreat (from B to C) 2 yards. Suppose it be asked, how much he had advanced (upon the whole march) when at C, or how many yards he is now forwarder than he was at A. I find (by subducting 2 from 5) that he is advanced 3 yards. (Because plus 5 minus 2 equals plus 3).
But if, having advanced 5 yards to B, he thence retreat 8 yards to D; and if it be then asked, how much he is advanced when at D, or how much forwarder than when he was at A: I say minus 3 yards! (Because plus 5 minus 8 equals minus 3). That is to say, he is advanced 3 yards less than nothing!
Which, in propriety of speech, cannot be (since there cannot be less than nothing). And therefore as to the line AB forward , the case is impossible.
But if (contrary to the supposition) the line from A be continued backwards , we shall find D, 3 yards behind A (which was presumed to be before it).
And thus to say, "he is advanced minus 3 yards", is but what we should say (in ordinary form of speech): "he is retreated 3 yards", or "he wants 3 yards of being so forward as he was at A".
Which doth not only answer negatively the question asked: that he is not (as was supposed) advanced at all; but tells, moreover, that he is so far from being advanced (as was supposed) that he is retreated 3 yards; or that he is at D, more backward by 3 yards than he was at A.
And consequently minus 3 doth as truly design the point D, as plus 3 designed the point C. Not forward, as was supposed, but backward from A.
So that plus 3 signifies 3 yards forward; and minus 3 signifies 3 yards backward: but still in the same straight line. And each designs (at least in the same infinite line) one single point: and but one!
EULER [to audience ]
I think Dr Wallis can rest assured that he has made his point! It is when we settle down to learn to live with this new breed of numbers, that the real trouble begins.
ARNAULD^{3}
Thankyou, Dr Wallis, for your most entertaining and exhaustive exposition  and your exhausting perambulations; I am sure I speak for us all in affirming that I cannot hope to find anywhere so eloquent an advocate for these poor illused quantities!
However  if you will bear with me, Sir I must confess to feeling very uneasy about their behaviour with respect to multiplication. If we are to follow your illustrious example and take negative numbers as lying in the opposite direction to positive numbers, it seems clear that any negative number, being less than nothing, must be less than any positive number.
But then I simply cannot see how to reconcile this with a basic principle of multiplication of two factors, which is that the ratio of unity to the first factor is equal to the ratio of the second factor to the whole product. This is true whether we are talking of whole numbers or fractions. Thus:
"three times four is twelve", yields "one is to three as four is to twelve";

or "onethird times onequarter is onetwelfth" yields "one is to onethird as onequarter is to onetwelfth".

Now, when multiplying two negatives, I meet a contradiction :
minus four times minus five is supposed to be plus twenty;

so that we must have plus one is to minus four as minus five is to plus twenty.

I find myself quite unable to accept this! For plus one is bigger than minus four, and minus five is less than plus twenty, so we have that bigger is to smaller as smaller is to bigger! But this is ridiculous, when we know that in all such proportions, if the first term is bigger than the second, then the third term must be bigger than the fourth.
EULER [to audience ]
Monsieur Arnauld's dilemma is perhaps made clearer by drawing the seemingly intolerable conclusion that plus one is to minus one as minus one is to plus one; two numbers in reciprocal proportion yet not equal!

LEIBNIZ^{4}
Your difficulty here appears to me to originate in unfounded expectations of the properties of these mixed ratios. There is a metaphysical principle that every property has its proper domain of application. We cannot expect these mixed ratios to inherit the wellknown properties of ordinary ratios of positive quantities, and to behave legitimately, for they are not truly ratios at all, as we know them. However, for my part, I believe there is nothing hindering us from formally calculating with them, just as we calculate with the square root of negative unity. We may even reach some useful conclusions, in spite of the lack of rigour surrounding these pseudoratios, and hence I think we may regard them as tolerable and useful. At least no harm can come of it!
Gavin Hitchcock , "Alien Encounters  Act 1: Confusion and Paradox," Convergence (May 2011)