Cartesian coordinate systemDirection angles
Left: a Cartesian coordinate system. Right: direction angles of in Cartesian coordinates.
In mathematics, the Cartesian system or Cartesian coordinate system is a three-dimensional representation of space, in which a point P is referenced with respect to three axes, the x-axis (perpendicular to the y-axis), y-axis (perpendicular to the x-axis), and z-axis (perpendicular to the xy-plane). [1] The system is named after the French mathematician and philosopher René Descartes, who developed the system.

The idea of this system was developed in 1637 by mathematician and philosopher René Descartes in his Discourse on Method, specifically in an appendix titled La Géométrie. The conception of the system was independently developed by French mathematician Pierre de Fermat, although Fermat did not publish the discovery. [2]

In part two of his Discourse on Method, Descartes introduces the new idea of specifying the position of a “point” or object on a surface, using two intersecting axes as measuring guides. In La Géométrie, he further explores the above-mentioned concepts. [3] Descartes' publication was the first to propose the idea of uniting algebra and geometry into a single subject “algebraic geometry” or analytic geometry, which means reducing geometry to a form of arithmetic and algebra and translating geometric shapes into algebraic equations.

The Cartesian coordinate system is often a starting point in the derivation of the science of thermodynamics, particulary in the 1850s derivations of German physicist Rudolf Clausius, in relation to a force acting on an infinitesimally small body, at a give point “‚óŹ” P, causing it to move through an small displacement ds, thus producing an amount of work dW. [4] This type of analytic geometric analysis was, in a way, a sort of mathematical quantification of productive results, i.e. functional work output obtained in the cyclical contraction and expansion of a body of steam (used to push a piston up and down, to which a rotative crank arm is attached), derived or inherent in the various "laws of Boerhaave", one in particular being that:

“Ever body, whether solid or fluid, is augmented in all its dimensions by any increase of its sensible heat.”

In other words, that "heat" causes bodies to expand or contract, through the action of a force, was a general law of the universe established as far back as 1724 by Dutch physician and chemist Herman Boerhaave. [6] The forceful movement of an infinitesimally small body through space is what is called work. The Cartesian system facilitates this type of analysis.

In particular, these types of calculations are facilitated via a vector analysis. For a vector in a plane, it is convenient to measure direction in terms of the angle, measured counterclockwise, from the positive axis to the vector. In three dimensional space, it is more convenient to measure direction in terms of the angles between the vector v and the three unit vectors i, j, and k, as shown (adjacent). The angles α, β, and γ are called the “direction angles” of v, and the quantities cos α, cos β, and cos γ, are called the “direction cosines” of v. [5]

1. (a) Daintith, John. (2005). Oxford Dictionary of Science. Oxford University Press.
(b) Maxwell, James C. (1878). “Tait’s ‘Thermodynamics’ (I)”, (pgs. 257-59). Nature, Jan. 31.
2. "analytic geometry". Encyclopædia Britannica (Encyclopædia Britannica Online). (2008).
3. Descartes, R. La Géométrie, Livre Premier: Des problèmes qu'on peut construire sans y employer que des cercles et des lignes droites (Book one: Problems whose construction requires only circles and straight lines).
4. Clausius, Rudolf. (1879). The Mechanical Theory of Heat, (Section: “Mathematical Introduction”, pgs. 1-20). London: Macmillan & Co. (2nd. Ed.). 5. Larson, Roland, Hostetler, Robert P., and Edwards, Bruce H. (1990). Calculus – with Analytic Geometry, (section: “Direction cosines”, pg. 730). Lexington, Mass.: D.C. Heath and Co.
6. Boerhaave, Herman. (1724). Institutiones et Experimenta Chemiae, (an unauthorized publication of his chemical lectures at the University of Leyden). Paris.

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