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The Folium of Descartes


Calculus of the
     Area of the Loop


Descartes' Tangent
from On the Nature of Curved Lines in
La Géométrie

Find the line tangent to a given curve

Tangent to a curve

by constructing a circle tangent to the curve at point  P.

Tangent to a curve

The radius of the circle is perpendicular to the tangent of both the circle and the curve at point  P.

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The Folium of Descartes and his Equiangular Spiral . . . . 

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Descartes is equally famous among mathematicians,
philosophers and theologians.

One of the best known quotes throughout the world is his

"Cogito ergo sum"
( I think, therefore I am.).

The Logarithmic Spiral or the Equiangular Spiral

Construction of spirals

The investigation of spirals is known to date from the ancient Greeks.  The Spiral of Archimedes is the quintessential example.  Descartes discovered the Logarithmic Spiral, also know as the Equiangular Spiral in 1638 while studying dynamics.  Its special feature is that the curve cuts all radii vectors at a constant angle.  Any radius drawn from the center  O  to any point of tangency  P  on the spiral will form the same angle between the radius and the tangent line.  Thus, this curve features a property of self-reproduction. 

Jacob Bernoulli ( 1654 - 1705 ) was so fascinated by the Equiangular Spiral that he requested it be carved on his tombstone with the phrase "Eadem mutata resurgo"  ("I shall arise the same, though changed").
This spiral is related to Fibonacci numbers, the Golden Ratio, and the recursion method widely used in computer science.

Biographical Sketch

René Descartes ( 1596 - 1650 ):  Gentleman, Soldier, Philosopher, and Mathematician

Modern mathematics began with two great advances, analytic geometry and the calculus.  The man who finally crystallized the method of wedding algebra to geometrical proof was René Descartes.  Moreover, his persistent rational skepticism, his questioning of how one could ever know truth, has led to what we generally today call "the scientific method,"  i.e., controlled experiments based on the application of rigid mathematical reasoning.

His life spanned one of the greatest intellectual periods in the history of all civilization. Map of Descartes, the cityTo mention only a few of the giants, Fermat and Pascal were his contemporaries in mathematics.  Shakespeare died when Descartes was twenty;  Descartes outlived Galileo by eight years;  and Newton was eight when Descartes died.  Descartes was twelve when Milton was born and Harvey outlived Descartes by seven years.  Father Mersenne, the famous amateur of science and mathematics, was Descartes' older chum, schoolmate, and life-long friend.  Cardinal Richelieu was his supporter.

The concept of analytic geometry came to Descartes in a dream on November 10, 1619;  thus, this day marks the official birthday of modern mathematics.  Its formal debut to his contemporaries came on June 8, 1637, with the publication of La Géométrie as an appendix to his now famous Discours de la Méthode.  Descartes was then forty-one years old.

As a soldier, he joined armies and survived fierce battles.  As a gentleman and traveler, he visited most of the major sites of late Renaissance Europe.  As a teacher, he enjoyed the companionship of royalty.  He died in Stockholm of complications - some say poisoning, others say lung disorders - acquired while delivering 5 AM instruction to Queen Christina of Sweden, the daughter of Gustavus Adolphus.  He is now buried in the Pantheon in Paris.

May we encourage you to read further.  Such an interesting, varied, and accomplished life deserves depth of investigation. 

The small town where Descartes was born has been renamed in his honor.

A Brief Listing of classical references that should be in most university libraries.

Bell, E. T.,  Men of Mathematics, Chapter 3 in various editions and publishers.  Bell's essay entitled Descartes: Gentleman, Soldier, and Mathematician is a wonderfully written description of a "particularly full and interesting life."

Boyer, Carl B.  A History of Mathematics, various editions and publishers.

Descartes supported a cosmic scheme based on vortices.  He seemingly tried to avoid the raging controversy between Galileo and the papacy in Italy.

Descartes mistaken view of vortices
Click on the stamp to see an enlarged image.
Issued by France in 1996 to commemorate the 400th anniversary of Descartes' birth.

Notice the Folium of Descartes on the stamp issued by Albania in 1966.  This stamp apparently features the misspelling of the Latin form of Descartes which should be  "Cartesius."


Issued by France in 1937 in commemoration of the third centenary of the publication of
Discours de la Méthode.

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