The use of negative numbers in most of today's calculations is natural. But how did the use of negative numbers began in physics? What physical quantity required the introduction of negative numbers and why?

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    $\begingroup$ Before use in physics, negative numbers came up in algebra. Some of the renaissance mathematicians, when solving cubic and quartic equations, would talk about "true" and "false" solutions (what we now call positive and negative solutions) of the equation. Even if you are only interested in the true solutions, for some calculations it is useful to consider also false solutions.... [Not an answer to the question, so only a comment] $\endgroup$ Jun 10, 2019 at 12:07
  • $\begingroup$ Good question. Did those Indian or Persian or Chinese writers using negative numbers use them for anything from physics? I guess considering debts as negative assets would not be called "physics". $\endgroup$ Jun 10, 2019 at 12:13
  • $\begingroup$ The field of physics per se doesn't exist without mathematics, as that is the language of physics. Since subtracting y from x is the same as adding -y , I don't think your question has a real meaning. $\endgroup$ Jun 10, 2019 at 12:26
  • $\begingroup$ The opposite, or negative, of moving right is moving left. This was understood at least by Descartes' time, probably earlier. $\endgroup$ Jun 10, 2019 at 14:20
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    $\begingroup$ @CarlWitthoft ... Subtraction was known and used long before any notion that "subtracting $y$ from $x$ is the same as adding $-y$". $\endgroup$ Jun 10, 2019 at 21:32

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Ancient Greeks painstakingly avoided negative numbers, although they could have come handy in astronomical calculations and number theory, among other places. Brahmagupta in Correctly Established Doctrine of Brahma (c. 630 AD) uses the language of "fortunes" and "debts", which suggests the merchant origin of the negative number concept, but that remains a speculation. Bhaskara II casually mentions negative sides when solving triangles in Lîlâvatî (c. 1150):"[The result] is negative, that is to say, in the contrary direction." Even earlier Liu Hui (c. 260 AD) and later medieval Islamic mathematicians accepted the negatives simply to make the algebra work more generally, see Historically, how did people define multiplication for negative numbers?

As for physics, it is not that any problem "required" them. Unlike many other concepts, where physics was an engine driving mathematical advances, when negative numbers finally made their way into physics it was as an afterthought. Medieval natural philosophers, and even Galileo, did not follow Bhaskara. They show considerable ingenuity in avoiding negatives, in the fashion of ancient Greeks, even where they would have been natural, such as in velocities or temperatures. Oresme (c. 1360 AD), the pioneer of coordinates and (bar) graphs, always adds a constant to his velocity and temperature graphs to avoid negative values. Even Fermat and Descartes still followed suit. Galileo in the Dialogues concerning two new sciences (1638) specifically splits projectile parabolas into two semi-parabolas to avoid having the velocity change sign.

The first physical negatives were, apparently, distances and their derivatives. According to Mumford's What’s so Baffling About Negative Numbers?, we should attribute the first application to Wallis and Newton, who worked independently. Wallis in his Treatise on Algebra (1685) speaks of negative distances traveled:

"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 had he Advanced (upon the whole march) when at C? I find... he has Advanced 3 Yards. But if, having Advanced 5 Yards to B, he thence retreat 8 Yards to D; and it then be asked, How much is he Advanced when at D, or how much Forwarder than when he was at A: I say –3 Yards."

Newton allowed negative numbers for velocities, accelerations, etc., in earlier notes, but did not spell it out in print until after Wallis. In the Universal Arithmetick (1707) he writes:

"Quantities are either Affirmative, or greater than nothing; or Negative, or less than nothing. Thus in human affairs, possessions or stock may be called affirmative goods, and debts negative ones. And so in local motion, progression may be called affirmative motion, and regression negative motion; because the first augments, and the other diminishes the length of the way made."

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    $\begingroup$ Excellent answer. Also, I believe Wallis also was the first to use negative numbers in the Cartesian plane. Descartes did not. $\endgroup$ Jun 11, 2019 at 23:27
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    $\begingroup$ I also find particularly relevant this quotation from the same work of Wallis you have cited: «Yet is it not that Supposition (of Negative Quantities) either Unuseful or Absurd when rightly understood. And though, as to the bare Algebraick Notation, it import a Quantity less than nothing: Yet, when it comes to a Physical Application, it denotes as Real a Quantity as if the Sign were +; but to be interpreted in a contrary sense.» $\endgroup$
    – Charo
    Mar 22, 2020 at 19:39
  • $\begingroup$ The link in the answer to a 'Treatise of Algebra' does not link to Wallis's work of 1685, but to Maclaurin's of 1748. The Wallis book can be found online at echo.mpiwg-berlin.mpg.de/MPIWG:GK8U243K . $\endgroup$
    – terry-s
    Jan 26, 2023 at 11:49

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