@ChristianRemling sure, I've just mentioned Pell (or better to say Brouncker) story as a reminder of the often complexity of "for a reason" part of your comment. Though, Descartes famously regarded negatives as "false". Thus, it would be safe to claim that Cartesian coordinates "as we know it" is not that same as Descartes himself knew it

@FrancoisZiegler thanks a lot indeed. Do you know any English translation of his work?

@ChristianRemling Just to remind you that Pell's equation is called as such simply because Euler confused Brouncker with Pell! :)

John, could you help me to find Huygens' writing on folium. I searched but I failed to find it. I am very curious since it is for a while that I am working on the development of negatives in the context of algebra for some educational reasons, and if your interpretation would be correct I need to revise the main part of the historical support of my thought.

I was worried that the distinction between input and output does not make sense.Happy it did, and thanks for clarifying your answer.

I was aware of that drawing, indeed through your work you have cited. My point is not just the acceptance of negative coordinates, rather it is when negative coordinates where algebraically used. Do you consider Huygens' drawing as an example of such a use?

Euler's Elements of Algebra is full of such use of letters. For example, he admits negative solution of equations, say x=-13, but after working out an equation of the form ax+by=c, in which a, b, and c represents positive numbers, writes: Now, if b is negative, and the equation has to form ax-by=c...!consider that how he bypasses the use of negatives as input. Basically, when he writes "if b is negative" he means "let the coefficient of y be negative".

Dear Francois, your answer as usual is enlightening, but unfortunately this time, is not answer the question! Here are some reasons: (1) there is a big difference between a signed letter and a letter that admits signed numbers. (2) The so-called rules of signs is quite an aspect of "arithmetical algebra" in which letters stand for positive numbers. (3)ρ, in the formula you mentioned, is an example in which a letter can have a negative number as "output" rather than "input"... (to be continued)

Here is a nice paper, "Was Cantor surprised?", showing the earlier works of Cantor on some related ideas: maa.org/sites/default/files/pdf/pubs/AMM-March11_Cantor.pdf

@smyrlis that's very true.