In the english translation of Newton's work "Enumeratio linearum tertii ordinis" by C.R.M. Talbot, we can see in a figure the depiction of a Cartesian coordinate system pretty much as we know it today: two perpendicular axes labeled X, Y intersected at the origin O. Is it possible that this figure corresponds to some of the original Newton's drawings? I cannot find it in the latin edition.
The standard way we represent Cartesian coordinates with orthogonal axis is not present into Descartes' La Géométrie nor into Newton's works.
An early occurrence of the "basic" definitions are in :
§2. We are now to consider quantities as represented by lines. [ page 301 ] So that, if $AP$ represent $x$, and the prependicular $PM$ represent the corresponding values of $y$, then there will be as many points ($M$,) the extremities of these perpendiculars or ordinates [...].
See page 3 for an "elucidation" of function and page 4:
Sit igitur $x$ quantitas variabilís, quae per rectam indefinitam $RS$ repraesentetur, atque manifestum est omnes valores determinatos ipsius $x$, qui quidem sint reales, per portiones in recta $RS$ abscindendas repraesentari posse. [...] Vocantur аutem haec intervalla $АР$, ABSCISSAE. Atque ideo Abscissae exhibent variabilis $x$ valores determinatos.
[page 7] Portiones autem Axis $АР$, quibus determinati ipsius $x$ valores indicantur, vocari solent $ABSCISSAE$.
Et perpendiculares $PM$, ex terminis Abscissarum $M$ ad lineam curvam pertingentes, nomen $APPLICATARUM$ obtinuerunt.
Vocantur autem hoc casu Applicatae normales seu orthogonales, quin cum Axe angulum rectum constituunt.
In general, we can see :
- Carl Boyer, History of analytic geometry (1956 - also Dover reprint).
Philippe de la Hire Nouveaux elements de sections coniques published in 1679 seems a good candidate for inventing cartesian coordinates: he was strongly influenced by Descartes and draw conics by referring them to an origin O, a vertical axis called a stem ('tige') with knots 'noeuds' N and horizontal branches 'rameaux' NL. This is probably the source for the popular view that Descartes invented the left handed coordinate system.
Your link is to the translator's notes, not to the translation itself. Talbot has simply reformulated Newton's argument in modern mathematical language.