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The standard presentation of "coordinatizing the plane" in 19th century British textbooks on geometry (Salmon, Smith, Besant, and many more) take the plane as being rigorously (at the time) axiomatized by Euclid's axioms, and establishing a bijection between the plane and $\mathbf{R}^{2}$ by picking a point $O$, two distinct lines passing through $O$ meeting at right angles, and a suitable unit length. An arbitrary point of the plane $P$ can then be associated to a pair of real numbers $(a,b) \in \mathbf{R}^{2}$ by projection.

But towards the end of the 19th century, several distinct constructions of $\mathbf{R}$, discovered (mainly) by Continental mathematicians, were already in the air. Extensive work by Cantor on the theory of "point sets", as well as the work by topologists on the dimension of $\mathbf{R}^{n}$ later on made $\mathbf{R}^{2}$ the "standard definition of the plane", and the "synthetic plane" given by axioms fell by the wayside.

The situation then was completely the reverse of the way we do things now: lines, planes, angles, etc. were considered more rigorously established, and more trustworthy, than the real numbers. Thus the real numbers were only considered as "a tool" to study geometric phenomena. Nowadays we take it for granted that $\mathbf{R}^{2}$ is the "Euclidean plane", and in differential geometry and topology, when we talk about "Euclidean space", we really mean $\mathbf{R}^{n}$. The modern way of studying Euclidean space can also be confusing: in manifold theory, for example, $\mathbf{R}^{2}$ can be coordinatized in many different ways by different atlases, which means that we should really picture it as being "featureless" before the imposition of coordinate lines, but people usually talk about $\mathbf{R}^{2}$ as being the coordinate plane, which muddles things a bit. The older counterpart of giving different atlases of $\mathbf{R}^{2}$ can be thought of as the process I described in the first paragraph above.

Unfortunately, the story I proposed in the previous paragraph is only a vague sketch of (the way I see it) how things went; I haven't read proper accounts of why and when mathematicians switched completely to using $\mathbf{R}^{2}$ as the formalization of the "infinitely thin flat plane" of our intuition, and not the object defined by axioms. I would be interested to know more.

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    $\begingroup$ There was another recent question (I don't see it now) where someone was confused by the textbook (Loomis & Sternberg, perhaps) using notations $E^n$ for Euclidean space and $R^n$ for $n$-tuples of real numbers. That guy didn't understand the distinction. $\endgroup$ – Gerald Edgar Sep 28 '17 at 12:07
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    $\begingroup$ @GeraldEdgar I think it's worth noting that almost every modern book on topology/differential geometry refers to $\mathbf{R}^{n}$ with the standard metric as "Euclidean space". $\endgroup$ – Maxis Jaisi Oct 14 '17 at 8:53
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As far as I can tell, "coordinatizing the plane" is a method invented by Rene Descartes and is given as an explicit example in his famous book Discourse on the method of what the method produces when applied to geometry. So I guess that $R^2$ as the plane spread in Europe as fast as the discourse on the method, starting from its publication in 1637. Note that "geometry with coordinates" is often called cartesian geometry, in honor of Rene Descartes.

Let me spell out the four precepts of the method.

The first was never to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgment than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt.

-> we are accepting here the standard rules concerning numbers and their relationship to the line. We see the ruler so clearly, as clearly as $2+2=4$.

The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution.

-> We divide the plane into two lines and decompose all our geometric objects along these lines.

The third, to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence.

-> How lengthy the resulting computations are, we proceed step by step with the uttermost rigor, without omission and without errors. Well, we proceed with method, but I think that we really own that expression to Descartes!

And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.

-> We synthetize everything back to the plane. And it works!

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    $\begingroup$ There are a number of issues involved in attributing $\mathbb R^2$ to Descartes. Descartes (and Fermat) only worked with a single axis, the abscissa. It was Newton and Leibniz who introduced the x-y axes. Also, Descartes' concept of $\mathbb R$ was very different from the modern concept. (ditto Newton and Leibniz.) $\endgroup$ – Nick R May 30 '18 at 18:30
  • $\begingroup$ @NickR The x and y's are introduced by Descartes p383-384 of Adam and Tannery's edition of his treatise geometrie as you can see there: fr.wikisource.org/wiki/… He then proceeds to compute several geometric quantities, namely the distances of points $D$, $F$, $H$ to the point $C$ of coordinates $(x,y)$ assuming the slope (or angles) of the lines are given. $\endgroup$ – coudy Jun 7 '18 at 14:50
  • $\begingroup$ It is my understanding that Descartes chose the ordinate according to the application. This meant that Descartes was usually employing an oblique frame rather than a rectangular frame. Quoting Boyer : "He did not lay down a coordinate frame to locate points as a surveyor or a geographer might do, nor were his coordinates thought of as pairs of numbers." ( I might add that I made the same mistake in my now deleted answer.) $\endgroup$ – Nick R Jun 7 '18 at 16:31

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