The plane named after him has two fixed, perpendicular axes running to both infinities. La Géométrie, 1637, has one.
Every gallery has Descartes for the coordinate plane, and the plane used everywhere in this corpus — two perpendicular number lines, positive and negative in all four directions, any point named by an ordered pair — is real mathematics that changed what a proof could be. It is also not quite what he printed. La Géométrie (1637) is the third appendix to the Discours de la méthode, and in it Descartes and, independently, Pierre de Fermat each worked with a single reference line: one axis, against which the length of a second, related segment (the ordinate) was measured for the curve at hand. The axis was chosen problem by problem, not fixed in advance for the whole plane. CITED
What he actually built was the method underneath the grid: that a geometric problem — a curve, a locus, a construction — could be converted into an algebraic equation and solved there, then read back into geometry. That reduction, not the two-axis picture, is the brick.
The fixed two-perpendicular-axis convention — the one every chapter in this corpus draws on — was assembled over the decade after 1649, when Frans van Schooten translated La Géométrie into Latin and, with his students (van Schooten himself, Johan de Witt, Johannes Hudde, Hendrik van Heuraet), added the commentary and worked examples that regularized Descartes' problem-specific axis into the general grid taught today. Two axes at right angles were not new even then: Hipparchus and Ptolemy had used a latitude–longitude grid centuries earlier for maps, and the Greeks are recorded using perpendicular reference lines — without turning either into algebra. Nicole Oresme, in the fourteenth century, graphed a varying quality against a base line in a construction historians read as an early, unalgebraized ancestor of the same picture. CITED
So the name is a metonym twice over: the plane is named for the man who supplied the method it runs on, formalized by his editors, for a picture the ancients had already drawn without the algebra to make it move. None of that is a knock on the 1637 book. It is the same kind of correction this corpus has made about its own terms before — a name surviving past the thing it first named.
Book XVII, ch02 puts two real cities on exactly this grid — the van Schooten-formalized one, fixed axes and all — and derives the Distance Formula from it the same way Book XVII's own floor chapters derive their claims: stated, then run. The Pythagorean argument behind that formula needs the fixed perpendicular grid to work; it is not available to Descartes' own problem-by-problem single axis. ch-leibniz, next in this gallery, is the same kind of correction about a different piece of notation everyone assumes is older, or exactly as printed, as it actually is.
That Descartes is undeserving of the name, or that La Géométrie is a minor book — neither is true. The claim is narrower: the specific picture — two fixed perpendicular axes, both directions signed, any point on the plane nameable in advance — is van Schooten's edition and its annotators' contribution, built on Descartes' method, not a page you can turn to in the 1637 original. OPEN Whether Fermat's independent, unpublished Ad Locos Planos et Solidos Isagoge (written c. 1636, printed only in 1679) used a fixed grid or the same single-axis convention is not settled by the sources checked for this page and is not claimed either way.
| source | accessed | tier |
|---|---|---|
| Wikipedia, “Cartesian coordinate system,” History section | 2026-09-22 | WEB |
| Rutgers Dept. of Mathematics, “The Key to Geometry: A Pair of Perpendicular Lines” (course history paper on Descartes) | 2026-09-22 | WEB |
| Encyclopædia Britannica, “Cartesian coordinates” | not fetched this pass | OPEN |
These are web sources, checked live on the date given, not the held-and-hashed PDFs this corpus prefers for its Tier-1 citations (see docs/floor-texts.tsv). book7/ch-descartes-verify.py checks that the exact facts above still print on this page; it cannot re-fetch the web, so it is a presence check against this page's own text, not an independent re-verification of the sources. Re-confirming against a primary edition of La Géométrie or van Schooten's Latin text is future work. OPEN