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I want to see an exact image of the first use of the Cartesian plane. I guess it came the first time with Descartes.

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  • $\begingroup$ No; in Descartes's Geometry the "coordinates" are not orthogonal. $\endgroup$ – Mauro ALLEGRANZA May 26 '15 at 8:00
  • $\begingroup$ For a more "modern" use you can see Jan De Witt's commentary (1659); see Albert Grootendorst (editor), Jan de Witt’s Elementa Curvarum Linearum Liber Secundus (2010). $\endgroup$ – Mauro ALLEGRANZA May 26 '15 at 8:20
  • $\begingroup$ Astroman what do you mean by "the Cartesian plane"? By definition I would tend to answer "Descartes", but as Mauro pointed out this isn't the plane we use today. $\endgroup$ – VicAche May 26 '15 at 10:02
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    $\begingroup$ In some sense coordinates were implicit in Apollonius' Conics. $\endgroup$ – Jack M May 26 '15 at 10:07
  • $\begingroup$ I mean the plane that we use today, when do we see it for the first time? And why do we call this plane cartesian? $\endgroup$ – copper May 26 '15 at 18:10
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There are several ways to answer this depending on what "Cartesian plane" means. Most literally, Cartesius is the Latinization of Descartes name, so one can look at pictures in Descartes's La Geometrie, which was the first systematic use of coordinates to solve geometric problems. However, coordinate graphs were introduced before Descartes by Oresme, to plot speed against time for example, Oresme's were bar graphs however. Oresme calls the axes longitude and latitude, the independent variable extension, and the dependent one intension.

Descartes got the idea of coordinates from Apollonius of Perga, abscissas and ordinates are Latin translations of Apollonius's terms. In Conica Apollonius uses tangent to a conic section and its diameter as what we now call coordinate axes, in general they are oblique, draws segments parallel to them, abscissas and ordinates, and relates their lengths by what we now call coordinate equations, he calls them symptoms. He even "changes coordinates" by passing to a new diameter, see Pierce's Abscissas and Ordinates. A suggestive figure is on p.176 of Heath's translation, here is another, from an Arabic copy of book V.

However, this was specific to conic sections, Apollonius has no symptoms of this kind for general curves, and his "coordinates" are curve specific, he does not consider even different conics in the same coordinates. So I am not sure if this qualifies as Cartesian. On the other hand, geometrically Cartesian plane is the same as Euclidean plane, only with numerical labels attached, so perhaps Euclid's diagrams already qualify.

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