Apr 292011

A Cartesian coordinate system is named after René Descartes. But Descartes did not first invent the Cartesian coordinate system nor did he first establish analytic geometry on the ground of the Cartesian coordinate system. Still it deserves its name, because the method of the Cartesian coordinate system is similar to that of Cartesian philosophy.

1 : Who first invented the Cartesian coordinate system?

A Cartesian coordinate system is a mathematical system that identifies the position of each point by its distance from a set of perpendicular lines that intersect at the origin of the system. The following figures show examples of Cartesian coordinate systems for 2D (left) and 3D (right)

Illustration of the Cartesian coordinate systems for 2D (left) and 3D (right). Source: K. Bolino and Jorge Stolfi

The adjective Cartesian is attributed to René Descartes, whose family name was Cartesius in Latin. Thanks to this name Descartes is usually thought to be the first inventor of the Cartesian coordinate system and even a plausible legend flourished that Descartes, lying on the bed, thought of the coordinate system so as to track the movements of a fly on the wall.

But Descartes did not use the term coordinate nor did he explain the idea equivalent to it explicitly in his writings. Of course it is true that together with Pierre de Fermat he is the founder of modern analytic geometry, which is often called Cartesian geometry. Although geometry developed highly in Ancient Greece, geometrical laws they discovered was scarcely analyzed algebraically before Descartes. Descartes used letters from the beginning of the alphabet, such as a, b, c…, for constants and those near the end of the alphabet, such as x, y, z…, for variables, and expressed a line as an equation composed of these alphabets. This practice continues today.

Let me introduce an example quoted from The geometry of Descartes in 1637.

The geometry of Descartes. Source: La Géométrie (author) René Descartes (page) 321

Suppose the length of AG, KL, NL, AB, BC is respectively a, b, c, x, y, of which a, b, c are constants and x, y are variables that can vary according to a rotary movement of the ruler GL around G. The following function of x and y indicates that the locus of the point C is a quadratic curve.


If you regard AK and AG as x-axis and y-axis that intersect at the origin A, this equation is the function of the locus of the point C identified by a Cartesian coordinate system. But Descartes himself regarded x and y as just a variable length and he did not locate the point by reference to any Cartesian coordinate system. It was Gottfried Wilhelm Leibniz’s article in 1692 that first treated a line as a function (functiones) specified with reference to coordinate (coordinata) with x-axis (absciss) and y-axis (ordinata) [1].

Some might imagine that Descartes presupposed the coordinate system to set up the equation without explaining it explicitly in his writings. Even if it should be the case, it would be doubtful whether Descartes was the first inventor of the Cartesian coordinate system, because similar systems had been used before the age of Descartes.

For example, the system of latitude and longitude that Eratosthenes used in his world map created in the year -193 is an orthogonal coordinate system. As the Earth is a sphere, the identification system of latitude and longitude itself is a non-Cartesian orthogonal coordinate system. Eratosthenes, however, recognized that the Earth is round and he created a 2D map, projecting the sphere on the plane surface. The following is Eratosthenes’ map of the known world and the cylindrical projection made his map a Cartesian coordinate system with latitude and longitude.

The modern reconstruction of Eratosthenes’ map of the known world. Source: World map according to Eratosthenes

Descartes did not first invented the Cartesian coordinate system nor did he first establish analytic geometry on the ground of the Cartesian coordinate system. Should we stop calling the coordinate system Cartesian? No. It deserves its name, because we can consider the Cartesian coordinate system to be an application of Cartesian philosophy to mathematics. In order to recognize this, let’s look back upon the philosophical method of Descartes.

2 : How did Descartes establish the foundation for knowledge?

Descartes proposed four precepts to reach the truth in Discourse on the Method.

The first was never to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgement than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt.

The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution.

The third, to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence.

And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted. [2]

The order of the first and the second should be reverse. Our research starts from uncertainty. It is not until we encounter uncertainties that we divide them into elements in order to reduce uncertainty and spread certainty step by step to the whole. The process of methodological skepticism followed the second precept, the discovery of “I think, therefore I am” was the result of the first, the proof of the existence of God and the foundation of the scientific knowledge on it resulted from the third and its verification put the fourth into practice.

Let’s see this process in detail. What Descartes thought the most certain was the ego as res cogitans. The existence of the self-conscious ego is evident to itself in spite of and because of the radical skepticism and Descartes chosen it as the starting point of his philosophy and science. But, if what he found was that the existence of Descartes’ consciousness was evident to Descartes’ consciousness, it was too trivial to be called the truth. So, starting from the ego as res cogitans, he tried to prove the existence of God and thereby establish the foundation of scientific knowledge.

According to Descartes, if the existence of the finite ego is evident, the existence of the infinite, namely God, must be even more evident.

And I must not imagine that I do not apprehend the infinite by a true idea, but only by the negation of the finite, in the same way that I comprehend repose and darkness by the negation of motion and light: since, on the contrary, I clearly perceive that there is more reality in the infinite substance than in the finite, and therefore that in some way I possess the perception of the infinite before that of the finite, that is, the perception of God before that of myself.[3]

This is an ontological argument for the existence of God and the proof is not correct, but let’s put it aside here. Descartes doubted even the mathematical truth, supposing a cunning demon that had conceived him. If, however, the existence of God is proved, we have no reason to suppose such a demon and can believe the sincerity of God.

For, in the first place, I discover that it is impossible for him ever to deceive me, for in all fraud and deceit there is a certain imperfection: and although it may seem that the ability to deceive is a mark of subtlety or power, yet the will testifies without doubt of malice and weakness; and such, accordingly, cannot be found in God.

In the next place, I am conscious that I possess a certain faculty of judging [or discerning truth from error], which I doubtless received from God, along with whatever else is mine; and since it is impossible that he should will to deceive me, it is likewise certain that he has not given me a faculty that will ever lead me into error, provided I use it aright. [4]

In this way, he started from the ego as a thinking thing and reached the truth of the world through the medium of the Almighty God.

3 : Why does the Cartesian coordinate system deserve its name?

We have seen how Descartes established the foundation for scientific knowledge in general. The Cartesian coordinate system applies a similar method to geometry, which explains why the Cartesian coordinate system is Cartesian.

The starting point of Cartesian philosophy, the ego, corresponds to the origin of the Cartesian coordinate system. This correspondence is not arbitrary, because the origin of the origin is the ego. That is to say the origin of the visual 3D space was originally located at the ego and, even if it is not physically at the position of the ego, one understands it, imagining transferring it from the position of one’s ego to the standpoint of alter ego.

Human beings have made a conceptual distinction between the top and the bottom, the front and the rear, the right and the left before Descartes or Eratosthenes. This indicates that we have located qualitatively, if not quantitatively, the position in 3D space around us in reference to a naïve Cartesian coordinate system where the ego forms the origin, the gravity direction the top/bottom axis, the direction of sight perpendicular to it the front/rear axis and the direction perpendicular to both the right/left axis.

As the visual image we have is based on a perspectively distorted projection of 3D space onto 2D retina, it does not correctly represent the object. Although the Almighty God could perceive the entire object immediately and correctly, it is impossible for human beings with limited recognition ability. All we can do is approach the objective truth, transferring the origin and thus changing the view.

I will explain it by means of a simple example. There are black and red coordinate axes in the graph below and the coordinates of the triangle ABC and its center of gravity (mass center) G differ in relation to the axes.

The choice of the axes

Suppose vectors of the points whose origin is the black O are a, b, c and g. Since O, A, B are on the same line, the equation below is true.

mathbf{b} =kmathbf{a}

Since G is the center of gravity, the equation below is also true.

mathbf{g} =frac{mathbf{a}+mathbf{b}+mathbf{c}}{3}

If vectors of the points whose origin is the red O are a, b, c and g, the former equation is false, but the latter is true. If your eyes are located in the black O, you might have an illusion that A and B are identical, which you will find false, when you see them from the viewpoint of the red O. Although the coordinates of A, B, C and G vary in relation to an origin O, the relation between the triangle and its mass center is independent of it. Still, as you must set the origin and axes anywhere to prove it, the origin and axes are methodologically necessary.

If the ego as the limited is the origin, God as the unlimited is the coordinate axes that have unlimited extension and the world that the ego can recognize by means of God is the space that the origin can specify by means of axes. The unlimitedness of axes enables us to link the limited origin to any point in the unlimited space. This unlimitedness, however, is not that of God. To approach the complete unlimitedness of God, the ego must transcend its singularity as the origin and reach the universality of the truth.

Descartes pursued the universality beyond the singularity of starting points not only in philosophy but also in geometry. The quadratic equation is based on the assumption that AK is the x-axis and AG is the y-axis in the figure in La Géométrie, but the locus of the point C is a quadratic function no matter where the origin and axes are set up. Descartes’ assertion that curves should not be classified according to tools for drawing them but according to their order is still valid today.

If Descartes had applied his method for founding his philosophy to geometry, he might have been the founder of the Cartesian coordinate system in reality as well as in name. In reality, however, he was not aware of the Cartesian coordinate system and it was Leibniz who succeeded to Cartesian geometry as well as Cartesian philosophy and founded Cartesian geometry on the Cartesian coordinate system.

4 : References

  1. De linea ex lineis numero infinitis ordinatim ductis inter se concurrentibus formata, easque omnes tangente, ac de novo in ea re Analysis infinitorum usu (author) Gottfried Wilhelm Leibniz (journal) Acta Eruditorum, vol.11 (date) 1692 (page) 168-171 (medium) Mathematische Schriften, Bd. 5 (editor) Carl Immanuel Gerhardt (page) 266-269
  2. Discours de la méthode, Deuxième partie (author) René Descartes (page) 141-142
  3. Meditationes de prima philosophia, Meditatio III (author) René Descartes (page) 24
  4. Meditationes de prima philosophia, Meditatio IV (author) René Descartes (page) 2-3
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