Descartes

Descartes, René, Lat. Renatus Cartesius, 1596-1650, French philosopher, mathematician, and scientist, b. La Haye. Descartes' methodology was a major influence in the transition from medieval science and philosophy to the modern era.

Life

Descartes was educated in the Jesuit College at La Flèche and the Univ. of Poitiers, then entered the army of Prince Maurice of Nassau. In 1628 he retired to Holland, where he spent his time in scientific research and philosophic reflection. Even before going to Holland, Descartes had begun his great work, for the essay on algebra and the Compendium musicae probably antedate 1628. But it was with the appearance in 1637 of a group of essays that he first made a name for himself. These writings included the famous Discourse on Method and other essays on optics, meteors, and analytical geometry. In 1649 he was invited by Queen Christina to Sweden, but he was unable to endure the rigors of the northern climate and died not long after arriving in Sweden.

Elements of Cartesian Philosophy

It was with the intention of extending mathematical method to all fields of human knowledge that Descartes developed his methodology, the cardinal aspect of his philosophy. He discards the authoritarian system of the scholastics and begins with universal doubt. But there is one thing that cannot be doubted: doubt itself. This is the kernel expressed in his famous phrase, Cogito, ergo sum [I think, therefore I am].

From the certainty of the existence of a thinking being, Descartes passed to the existence of God, for which he offered one proof based on St. Anselm's ontological proof and another based on the first cause that must have produced the idea of God in the thinker. Having thus arrived at the existence of God, he reaches the reality of the physical world through God, who would not deceive the thinking mind by perceptions that are illusions. Therefore, the external world, which we perceive, must exist. He thus falls back on the acceptance of what we perceive clearly and distinctly as being true, and he studies the material world to perceive connections. He views the physical world as mechanistic and entirely divorced from the mind, the only connection between the two being by intervention of God. This is almost complete dualism.

The development of Descartes' philosophy is in Meditationes de prima philosophia (1641); his Principia philosophiae (1644) is also very important. His influence on philosophy was immense, and was widely felt in law and theology also. Frequently he has been called the father of modern philosophy, but his importance has been challenged in recent years with the demonstration of his great debt to the scholastics. He influenced the rationalists, and Baruch Spinoza also reflects Descartes's doctrines in some degree. The more direct followers of Descartes, the Cartesian philosophers, devoted themselves chiefly to the problem of the relation of body and soul, of matter and mind. From this came the doctrine of occasionalism, developed by Nicolas Malebranche and Arnold Geulincx.

Major Contributions to Science

In science, Descartes discarded tradition and to an extent supported the same method as Francis Bacon, but with emphasis on rationalization and logic rather than upon experiences. In physical theory his doctrines were formulated as a compromise between his devotion to Roman Catholicism and his commitment to the scientific method, which met opposition in the church officials of the day. Mathematics was his greatest interest; building upon the work of others, he originated the Cartesian coordinates and Cartesian curves; he is often said to be the founder of analytical geometry. To algebra he contributed the treatment of negative roots and the convention of exponent notation. He made numerous advances in optics, such as his study of the reflection and refraction of light. He wrote a text on physiology, and he also worked in psychology; he contended that emotion was finally physiological at base and argued that the control of the physical expression of emotion would control the emotions themselves. His chief work on psychology is in his Traité des passions de l'âme (1649).

In geometry, Descartes' theorem, named after René Descartes, establishes a relationship between four kissing, or mutually tangent, circles. The theorem can be used to construct a fourth circle tangent to three given, mutually tangent circles.

History

Geometrical problems involving tangent circles have been pondered for millennia. In ancient Greece of the third century BC, Apollonius of Perga devoted an entire book to the topic. Unfortunately the book, which was called On Tangencies, is not among his surviving works.

René Descartes touched on the problem briefly in 1643, in a letter to Princess Elizabeth of Bohemia. He came up with essentially the same solution as given in equation (1) below, and thus attached his name to the theorem.

Frederick Soddy rediscovered the equation in 1936. The kissing circles in this problem are sometimes known as Soddy circles, perhaps because Soddy chose to publish his version of the theorem in the form of a poem titled The Kiss Precise, which was printed in Nature (June 20, 1936). Soddy also extended the theorem to spheres; Thorold Gossett extended the poem to arbitrary dimensions.

Definition of curvature

Descartes' theorem is most easily stated in terms of the circles' curvature. The curvature of a circle is defined as k = ±1/r, where r is its radius. The larger a circle, the smaller is the magnitude of its curvature, and vice versa.

The plus sign in k = ±1/r applies to a circle that is externally tangent to the other circles, like the three black circles in the image. For an internally tangent circle like the big red circle, that circumscribes the other circles, the minus sign applies.

If a straight line is considered a degenerate circle with curvature k = 0, Descartes' theorem also applies to a line and two circles that are all three mutually tangent, giving the radius of a third circle tangent to the other two circles and the line.

Descartes' theorem

If four mutually tangent circles have curvature k_i (for i = 1,…,4), Descartes' theorem says:

$(1)$

$(k\_1+k\_2+k\_3+k\_4)^2=2,(k\_1^2+k\_2^2+k\_3^2+k\_4^2).$

When trying to find the radius of a fourth circle tangent to three given kissing circles, the equation is best rewritten as:

$(2)$

$k\_4\; =\; k\_1\; +\; k\_2\; +\; k\_3\; pm2\; sqrt\{k\_1\; k\_2\; +\; k\_2\; k\_3\; +\; k\_3\; k\_1\}.$

The ± sign reflects the fact that there are in general two solutions. Other criteria may favor one solution over the other in any given problem.

Special cases

If one of the three circles is replaced by a straight line, then one k_i, say k₃, is zero and drops out of equation (1). Equation (2) then becomes much simpler:

$(3)$

$k\_4=k\_1+k\_2pm2sqrt\{k\_1k\_2\}.$

Descartes' theorem does not apply when two or all three circles are replaced by lines. Nor does the theorem apply when more than one circle is internally tangent, e.g. in the case of three nested circles all touching in one point.

Complex Descartes theorem

In order to determine a circle completely, not only its radius (or curvature), but also its center must be known. The relevant equation is expressed most clearly if the coordinates (x, y) are interpreted as a complex number z = x + iy. The equation then looks similar to Descartes' theorem and is therefore called the complex Descartes theorem.

Given four circles with curvatures k_i and centers z_i (for i = 1…4), the following equality holds in addition to equation (1):

$(4)$

$(k\_1z\_1+k\_2z\_2+k\_3z\_3+k\_4z\_4)^2=2,(k\_1^2z\_1^2+k\_2^2z\_2^2+k\_3^2z\_3^2+k\_4^2z\_4^2).$

Once k₄ has been found using equation (2), one may proceed to calculate z₄ by rewriting equation (4) to a form similar to equation (2). Again, in general there will be two solutions for z₄, corresponding to the two solutions for k₄.

See also

• Ford circles
• Apollonian gasket
• Soddy's hexlet
• Tangent lines to circles

External links

• Interactive applet demonstrating four mutually tangent circles at cut-the-knot
• The Kiss Precise
• XScreenSaver: Screenshots :: An XScreenSaver display hack visualizes Descartes’ theorem, in hack “Apollonian”.
• Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks: Beyond The Descartes Circle Theorem