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Is there any actual historical example in which a Cartesian plane with all four quadrants has been used, but with all axes marked with positive numbers? [Please see Sawyer's paper below for a "made-up" example]

I am writing the result of a research the point of which was to teach children negative numbers in an algebraic context. There were many interesting similarities between their conceptions and difficulties and the so-called historical ones. For the one in the question (suggested by one of the children) I have claimed that "It has an interesting historical counterpart". Has it? Or, I have to delete my claim?

Added: In this nice paper of Warwick Sawyer, The Importance of the Unbelievable, he writes"without negative numbers the equation of a line would depend upon which quarter it was in." The question is whether this possibility has ever been practiced in history or Sawyer just made it up for educational purposes?

PS. Following Todd's comments, I changed the wording of the question. I hope it is now clearer.

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  • $\begingroup$ Where you wrote "natural numbers", did you mean "negative numbers"? I'm afraid I don't understand the question. When discussing cartesian coordinates, I will typically mark points along the axes with a few positive and negative numbers, depending on the application. $\endgroup$
    – Todd Trimble
    Commented Feb 13, 2015 at 23:02
  • $\begingroup$ @ToddTrimble No, I exactly meant natural numbers (including zero): 0,1,2,.... Indeed, I guessed the problem that might arise for a modern mind, that is why I added the Sawyer's paper. $\endgroup$
    – Amir Asghari
    Commented Feb 13, 2015 at 23:10
  • $\begingroup$ I'm still not sure what you're asking; it sounds as if you're asking more whether there are historical examples where the positive direction moves left along the $x$-axis or down along the $y$-axis. What do natural numbers have to do with it? (My friend Jim Dolan has been observed drawing so that the positive direction along the $y$-axis is pointing down.) Incidentally: there is hsm.stackexchange.com where you might have more success with getting answers. $\endgroup$
    – Todd Trimble
    Commented Feb 13, 2015 at 23:31
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    $\begingroup$ @ToddTrimble You are right. It is better to change "natural" to "positive". Yet, my question means all axes be positive. That is why I am not sure whether Jim's example works or not. Perhaps, I've used "natural" since we usually mark the axes with units. And, thanks for introducing the other site. I'll try it. $\endgroup$
    – Amir Asghari
    Commented Feb 13, 2015 at 23:45
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    $\begingroup$ If you want to use both sites, please leave a note at each that you cross-posted at the other, and in each case leave a link to the question at the other site. I'm not too fussed about deletion. $\endgroup$
    – Todd Trimble
    Commented Feb 13, 2015 at 23:53

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A very detailed historical account of the debates about clarifying the concept of negative numbers can be found in the book http://link.springer.com/book/10.1007%2F0-387-28273-4 (Conflicts between Generalization, Rigor, and Intuition, by Gert Schubring). It is stated in the book (p. 82) that "Reyneau (1736) was probably the first to explicitly introduce the four quadrants in the coordinate system of the plane, a novelty that appears self-evident to us today" and that (p. 289) in De Prony's lectures (1795) Cours d'Analyse appliquée à la mécanique "for the first time in France the four quadrants of a coordinate system are explicitly assigned to the respective positive and negative values of the x– and y–axes in the plane".

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this should be seen rather as a comment:

Coordinate geometry seems to have been a singular discovery of Rene Descartes, which he attributed to dream; it is therefore likely, that his choice of axis orientations has been copied ever since.

Now for the specific choice of the axis directions the writing direction seems to have played the essential role; the sequence of natural numbers is written from left to right in Europe and thus it would a natural choice (at least in Europe) for the direction of the positive x-axis.

For the choice of the y-axis direction the writing direction (top to bottom) apparently wasn't the key motivation; I suspect, that it was chosen due to perceiving geometric images in texts as embedded "paintings" and, due to to the fact, that in perspectivic drawings of ground floors, the more distant points are "above" the closer ones.

There are however deviations from that "natural" order:
The ciphers of natural numbers run somehow (from the European perspective) in the "wrong" direction, namely from right to left in order of significance, probably due to the Arabic writing direction.
The y-axis is often oriented from top to bottom in device coordinates of computer graphics or in sequence of the rows of tables or matrices.

Edit:
There is an actual example of a coordinate system in use with 4 axes, directed up, to the right, down and, to the left; all coordinates being positive and used in nautics for communicating directions and relative positions.

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  • $\begingroup$ Dear Manfred. Could you please have a look at the revised version of the question? I think my original wording was misleading. $\endgroup$
    – Amir Asghari
    Commented Feb 14, 2015 at 0:09
  • $\begingroup$ It just came to my mind, that nautics uses such coordinates. $\endgroup$
    – Manfred Weis
    Commented Feb 14, 2015 at 0:50
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    $\begingroup$ The current issue of the Notices of the AMS has a remark that Newton modified Descartes's original orientation. ams.org/notices/201502/rnoti-p147.pdf $\endgroup$
    – Timothy Chow
    Commented Feb 14, 2015 at 2:10

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