# Cartesian coordinate system in Newton's work

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.

• What's surprising about this? Coordinate systems were used since Descartes. – Alexandre Eremenko Jan 13 '16 at 14:36
• Not exactly in this form, I suppose. See e.g. Victor J. Katz's History of Mathematics, 3rd edition, p. 484 "Both [Descartes and Fermat] used as their basic tool a single axis along which one of the unknowns was measured rather than the two axes used today, and neither insisted that the lines measuring the second unknown intersect the single axis at right angles" I'm interested in the evolution of the concept and I'm wondering if this figure belongs actually to the translator, not to Newton. – skol Jan 13 '16 at 16:29
• I wonder if someone can come up with one (or more?) tags to better describe this question... – Danu Jan 13 '16 at 17:43
• @skol, perhaps you should ask about the development of the "cartesian" coordinate system of perpendicular axes instead. – vonbrand Jan 13 '16 at 20:20
• Your link is to the translator's notes, not to the translation itself. Talbot has simply reformulated Newton's argument in modern mathematical language. – fdb Jan 13 '16 at 23:59

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 [...].

The modern representation (as usual), is due to Leonhard Euler into his Introductio in analysin infinitorum, (1748), Tomus Secundus.

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 :

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.