When did coordinate plane "as we know it" come into play?

Question

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.

Answer 1

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."

Answer 2

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.

Answer 3

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.