This is a historical question that needs some background to make sense. Let me start with the longer version of the question:

When did negative numbers, algebra and coordinate plane come together?

Here are some useful facts:

  1. For a long time, even after recognition of negative numbers, there were mathematicians who actively tried to clean algebra from negative numbers (at least, up until late eighteen century).

  2. For a long time, the use of letters in algebra was confined to positive quantities. Simply speaking $-a$ stood for a negative number by default (again, at least, up until late eighteen century).

  3. And, this is the most surprising fact, and the reason that I ask this question. George Peacock (1791-1858), one of the pioneers of modern symbolic algebra, the person who gave an abstract treatment of negatives, and the person who gave a treatment of algebra in which letters could admit negatives as input, when came to a geometric interpretation of imaginary numbers, treats coordinates in a way that only the first quarter is used (1830).

Let me finish this long post with a very concrete question somehow summarizing all my historical points: When you simply write $x+y=1$ as the equation of the line passing through $(0,1)$ and $(1,0)$, you work with the "standard" coordinate plane, and you know that $x$ and $y$ admit certain negative numbers as input. Historically, when did such an understanding come into play?

I hope the question makes sense at least for those who are interested in history of mathematics.

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    $\begingroup$ This doesn't quite answer the question as posed, but I think they are called Cartesian coordinates for a reason (there's a passage in the treatise on this, too). $\endgroup$ – Christian Remling Jul 27 '14 at 23:36
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    $\begingroup$ @ChristianRemling Just to remind you that Pell's equation is called as such simply because Euler confused Brouncker with Pell! :) $\endgroup$ – Amir Asghari Jul 28 '14 at 22:06
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    $\begingroup$ As I wrote, Descartes (with some pride) reports on this the Treatise, so this is not a case of a mistaken identity. $\endgroup$ – Christian Remling Jul 29 '14 at 21:33
  • $\begingroup$ @ChristianRemling sure, I've just mentioned Pell (or better to say Brouncker) story as a reminder of the often complexity of "for a reason" part of your comment. Though, Descartes famously regarded negatives as "false". Thus, it would be safe to claim that Cartesian coordinates "as we know it" is not that same as Descartes himself knew it $\endgroup$ – Amir Asghari Jul 29 '14 at 22:12

One landmark could be the 1693 paper "An Instance of the Excellence of the Modern ALGEBRA, in the Resolution of the Problem of finding the Foci of Optick Glasses universally"(alt. link), where Edmond Halley (of comet fame) introduced the rules of signs in optics:

This Dioptrick Problem is that of finding the Focus of any sort of Lens, exposed either to Converging, Diverging or parallel Rays of Light, proceeding from, or tending to a given Point in the Axis of the Lens, be the ratio of Refraction what it will (..., p.961)

if $d$ be so small, as that $2r\rho$ exceed $dr+d\rho$, then will it be $-f$, or the focus will be Negative, which shows that the Beams after both Refractions still proceed Diverging. (..., p.963)

If the Lens be double Concave, the focus of converging Beams is negative, where it was affirmative in the case of diverging Beams on a double Convex (..., p.964)

A second use is to find what Convexity or Concavity is required, to make a vastly distant Object be represented at a given focus (...) in Glass $\smash[b]{\frac{rf}{2r-f}}=\rho$, whence if $f$ be greater than $2r$, $\rho$ becomes Negative, and $\frac{rf}{f-2r}$ is the Radius of the Concave sought. (..., p.967)

So here clearly $\rho$ can have "negative numbers as input", as you say.

According to Shapiro (1990, p.151) "An inherent limitation of Barrow (1669) and his contemporaries is that all line segments must be positive... Edmond Halley in his landmark paper (1693) overcame this drawback and derived a single equation that yielded the image point for all varieties of thick lenses."

According to Delambre (1816, p.X), reading this paper was what converted Lagrange to Mathematics, "and revealed to him his true destiny", no less.

Edit: From some more authorities' writings:

  • Struik (A Source Book in Mathematics, 1200-1800, p.168) says: "Newton's Enumeratio linearum tertii ordinis was first published together with his Opticks in London in 1704, but it was written much earlier, perhaps in or after 1676 (...) he freely used (a novelty at the time) positive and negative values of the coordinates."

  • Michel Serfati (Landmark Writings in Western Mathematics 1640-1940, p.18) says: "From 1655, the De sectionibus conicis of John Wallis used the Cartesian method for expressing algebraically the ancient geometric definitions and properties of the conics of Apollonius, systematically interpreting $x$ and $y$ as having any sign."

  • Kline (Mathematical Thought From Ancient to Modern Times, p.319) says: "John Wallis, in De Sectionibus Conicis (1655) (...) was also the first to consciously introduce negative abscissas and ordinates."

  • Boyer (History of Analytic Geometry, p.139) says: "Newton is sometimes given sole credit for the correct use of negative coordinates, but he had been anticipated to some extent by others, notably Wallis and Lahire. (...) the use of negative values of the coordinates were fairly well established before the middle of the [18th] century."

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    $\begingroup$ Imagine if someone nowadays wrote a paper whose title started "An instance of the Excellence of Modern ALGEBRA". You'd immediately dismiss them as a crank. Have we lost something? Or do we just prefer our titles not to sound like Bill and Ted? $\endgroup$ – Tom Leinster Jul 28 '14 at 1:35
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    $\begingroup$ @TomLeinster Someday I hope to write a paper containing a definition of the form: "We say that a topological space is most excellent if ..." $\endgroup$ – Charles Rezk Jul 28 '14 at 2:07
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    $\begingroup$ Meanwhile we do have a notion of "excellent ring". en.wikipedia.org/wiki/Excellent_ring So I guess all is not lost. $\endgroup$ – Todd Trimble Jul 28 '14 at 4:40
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    $\begingroup$ Euler's Elements of Algebra is full of such use of letters. For example, he admits negative solution of equations, say x=-13, but after working out an equation of the form ax+by=c, in which a, b, and c represents positive numbers, writes: Now, if b is negative, and the equation has to form ax-by=c...!consider that how he bypasses the use of negatives as input. Basically, when he writes "if b is negative" he means "let the coefficient of y be negative". $\endgroup$ – Amir Asghari Jul 28 '14 at 5:19
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    $\begingroup$ @AmirAsghari Aha, your input notion is subtler than I thought. I view Halley's whole paper as an exercise in juggling with (Shapiro's words) the "single equation" $$\frac{pdr\rho}{dr+d\rho-pr\rho}=f,$$ where all letters are alternately regarded as input or output. E.g. once it is seen that the focus or radius can be negative as output (when solved for), then it can be negative as input too. But I would agree that Halley is not fully consistent -- perhaps because he is in the process of overturning old habits, which die hard... $\endgroup$ – Francois Ziegler Jul 28 '14 at 10:41

I don't think there is a decisive answer to this question, because some mathematicians accepted negative coordinates long before others did. However, here is another landmark from the 1690s: Huygens' drawing of the folium. Descartes called it the folium (leaf) because he saw only the leaf-shaped loop of the curve in the positive quadrant. Huygens drew the whole curve, plus its asymptote, including the parts in the other three quadrants.

  • $\begingroup$ I was aware of that drawing, indeed through your work you have cited. My point is not just the acceptance of negative coordinates, rather it is when negative coordinates where algebraically used. Do you consider Huygens' drawing as an example of such a use? $\endgroup$ – Amir Asghari Jul 28 '14 at 5:47
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    $\begingroup$ My interpretation of Huygens' drawing is that he is using negative coordinates, and using them algebraically, because he has to multiply negative coordinates in order to plot points outside the positive quadrant. $\endgroup$ – John Stillwell Jul 28 '14 at 5:59
  • $\begingroup$ John, could you help me to find Huygens' writing on folium. I searched but I failed to find it. I am very curious since it is for a while that I am working on the development of negatives in the context of algebra for some educational reasons, and if your interpretation would be correct I need to revise the main part of the historical support of my thought. $\endgroup$ – Amir Asghari Jul 28 '14 at 21:27
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    $\begingroup$ @AmirAsghari: See here and also there. $\endgroup$ – Francois Ziegler Jul 28 '14 at 22:08
  • $\begingroup$ @FrancoisZiegler thanks a lot indeed. Do you know any English translation of his work? $\endgroup$ – Amir Asghari Jul 28 '14 at 22:52

It's an interesting question. Conservatism about negative numbers as such continued indeed into the early decades of the 19th century. But that was mainly a philosophical position. Pedagogic conservatism with a first-quadrant coordinate plane at first sight looks odd: you write the equation of the unit circle and then show the students just a quarter of it? But of course the good teacher sees that circles centered within the first quadrant can be given nice clean graphical representations, not crossing the axes, if you choose the radius well.

I suspect the answer to your question could be found by constructing a timeline of the "analytic geometry" of conics, in texts written in the 18th century. For cost reasons they may well not have many pictures. But I find it hard to believe Euler, author of one such text, had issues with the other three quadrants. The point would be more what the students were supposed to make of the business of conics in general position, and how that was expressed.


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