Did the coordinate system used by Eratosthenes influence the development of the Cartesian coordinate system?

Question

The first coordinate system was used by Eratosthenes to locate things using 2 points. Did this influence the development of the Cartesian coordinate system in any way?

Answer 1

It seems like it should have, but not really. Woodward in Medieval Mappaemundi tries to draw the connection thus:

"In the words of a modern historian of mathematics: "The development of graphical representation forged a link between the intuitive concepts of continuously varying quantities arising from physical phenomena and the geometry of the Greeks. The connection between the graphing concept and cartography is seen in Oresme's use of the terms "longitude" and "latitude" for the independent and dependent variables plotted on a graph. Once these ideas were associated with algebraic symbolism, the seventeenth-century mathematicians Rene Descartes and Pierre de Fermat were able to formulate analytical geometry in the form familiar today."

It is true that Oresme drew bar graphs of distances and velocities in the 14th century and called the axes longitude and latitude. But there is no evidence that Fermat and Descartes, or their immediate successors, were aware of his work and plenty of evidence that they came to coordinates from a very different tradition, unrelated to cartography. They did not even draw two axes, only one, the second one was introduced later by van Schooten. Even Boyer, who is more open to it, writes in The Invention of Analytic Geometry:

"Because Descartes carefully avoided any reference to his predecessors, one cannot say with assurance that he was familiar with the work of Oresme. It seems quite probable that he was. Yet the differences between Descartes' system, called Cartesian geometry, and the graphical representation of the latitude of forms are so great as to make questionable any decisive influence."

Instead, the main source of inspiration was Apollonius's Conica (c. 200 BC), where he anticipates the use of coordinate equations ("symptoms") to study conic sections, see When do we see for the first time the use of the Cartesian coordinates? The names "abscissa" and "ordinate", instead of longitudes and latitudes, come from translations of Apollonius's terms, see Pierce, Abscissas and Ordinates. The association of geometry to algebraic symbolism, on which Descartes and Fermat did rely, was done by Vieta in Supplementum Geometriae (1593), where he systematically assigns letters to segments and areas and solves the resulting equations, see Viète's Relevance and his Connection to Euler. But again, so far as we can tell, Vieta was not aware of Oresme or motivated by cartography.

Was there some vague influence through some general cultural osmosis, as Woodward suggests? Maybe. Descartes and Fermat probably saw maps even if they did not read Oresme. But why wasn't there more definitive influence? A speculation in the accepted answer to Was there some prior idea that inspired both Fermat & Descartes to invent coordinates? on Math SE suggests a possible answer. Coordinates in cartography and geometry were used for two very different purposes, at least in the early stages. In cartography they were used to identify locations of individual points. In geometry, since Apollonius, the idea was to study curves by relating them to coordinate equations. Ancient and medieval cartographers did not write equations for curves on a map, and mathematicians had little interest in associating points to pairs of numbers as such. So the two fields were of little help to each other.

Answer 2

Just an extended remark. There is nothing special about Eratosphenes here. Coordinates were used in astronomy since its very beginning, to describe stars positions. At least three of them were used since antiquity: ecliptic, equatorial and horizontal. Astronomers routinely did analytic geometry in these coordinates; it is called spherical trigonometry. Whether this influenced Descartes, it is hard to say; one cannot penetrate Descartes mind.

Another remark is that probably neither Descartes nor Fermat could read Erathosphenes, since all his writings were already lost by that time. They could learn about coordinate systems from Ptolemy or later astronomers and geographers.