It is said that geometry became analytic with the publication of René Descartes' (1596–1650) masterpiece in 1637: Discourse on the Method. In an appendix to the Discourse, titled La Gèomètrie, he included applications of algebra to geometry, giving a push to the use of geometry to solve some algebraic problems.
Almost simultaneously, Pierre de Fermat (1601–1665), a contemporary citizen, was making research on special curves and their geometrical solutions. Some authors claim that the discovery of the basis of a coordinate system to plot curves and find solutions to algebraic equation should be attributed to Fermat and not Descartes because Fermat had more geometrical insight than Descartes. However, even before both of them, some rudimentary forms of solving geometrical problems are attributed to being used by Apollonius in Greece almost two thousand years earlier.
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Simultaneity in discoveries are not so rare in science. Another coincidental discovery in math is the invention of the modern calculus by Isaac Newton (1642–1727) and Gottfried Wilhelm von Leibniz (1646–1716), the former in England, and Leibniz in Germany.
E. T. Bell1, a historian of mathematics, writes the following about the Fermat—Descartes controversy:
There is no doubt that he preceded Descartes. But as his work of about 1629 was not communicated to others until 1636, and was published posthumously only in 1679, it could not be possibly have influenced Descartes in his own invention, and Fermat never hinted that it had.
More explicitly, James Newman2, expressing his thought about to whom should be attributed the 'modern' use of the coordinate system states:
Fermat may have preceded Descartes in stating problems of maxima and minima; but Descartes went far past Fermat in the use of symbols, in "arithmetizing" analytic geometry., in extending it to equations of higher degree. The fixing of a point's position in the plane by assigning two numbers, co-ordinates, giving its distance from to lines perpendicular to each other, was entirely Descartes' invention.
In order to develop a useful geometric coordinate system to solve mathematical problems related to geometry and physics two important steps are needed: the recognition of the zero as a number, and the introduction of the negative numbers. Keep in mind that mathematics was invented to solve real life problems and none of the mentioned concepts come from everyday reality. A clear example of that is the Roman "mathematics" where they had no symbol nor numeral for the number zero, much less for the negative numbers.
The number zero has no zero-history, on the contrary, its uses, manipulations and rejections has been traced as far as the beginning of civilization itself. Some form of acceptance can be found in Greece, India, Babylon. On the other side, the Greek mathematics had no negative numbers, thus, in the cases where they used geometry to solve some problems, the concepts ordinates and abscissas were applied a posteriori and not a priori when working with a problem.
With the acceptance of the negative numbers the story is similar: some mathematicians argued against the existence of numbers "below zero". Other considered that subtraction from zero was "nonsense".
It may sound strange, but Descartes never used "the Cartesian coordinates" in his treatise of 1637 nor ever in his life; he wrote about "coordinates" in the sense of distances to describe the locus of a curve, and even more, he didn't used negative distances. Citing Bell about this:
In details, Descartes' presentation differs from that now current. Thus, he used only an x-axis and did not refers to a y-axis. For each value of x he computed the corresponding y from the equation, thus getting the coordinates x and y. The use of two axes obviously is not a necessity but a convenience. In our terminology, he used the equivalent of both rectangular and oblique axes.
In Descartes' work every thing was measured with positive distances. Later, Bell writes:
He considered equations only in the first quadrant, as it was thence that he translated the geometry into algebra. ... As analytic geometry evolved and negative numbers were fearlessly used, the restriction was removed.
Respect to the study of the correlation between geometry and algebra, there were some predecessors in that field. Quoting Newman again, from his Commentary on Descartes and Analytical Geometry:
The study of curves by means of their equations, defined as the "essence" of analytic geometry, was known to the Greeks ... Menaechmus, the tutor of Alexander the Great, is reputed to have made this discovery. Among Descartes' other predecessors were the French theologian Nicole Oresme, whose system of "latitudes and longitudes" roughly foreshadowed "the use of co-ordinates in the graphical representation of arbitrary functions", and François Viète, the sixteenth–century counselor to the King of France, whose improvement in notation substantially facilitated the development of algebra.
The earliest use of negative coordinates is attributed to Isaac Newton (1642–1727) in a collection of figures and graphs of polynomials of the third degree of his book Enumeratio linearum tertii ordinis, or Enumeration of Curves of Third Degree.
In this publication Newton used perpendicular axes and included both positive and negative numbers. In fact, in some of the figures he used the capital letter X to label the horizontal axis, the capital letter Y for the vertical axis, and even the capital letter O to label the point of intersection of both axes. In some sketches he didn't included none of the labels.
The systematization of the present day coordinate system using perpendicular axes meeting at a common "origin" is the final result of the human endeavor through more that two millennia to give significance to real mathematical problems that give zero or negative results.
The Cartesian coordinate system is widely used in mathematics to represent 2-dimensional and 3-dimensional curves and solids. But beyond this there is also 4-dimensional systems and spaces. For more about 4-dimensioned "worlds", see: Quaternions and transcomplex numbers, and Can there be different fourth dimensions?
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