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The Hyperbola of Fermat:

 One of the Classic Conic Sections. . . . 

The difference between (x,y) and the two foci remains constant.

Replay the animation
Equations for the Hyperbola:

These equations are in "Cartesian" form.  What is less well-known is that Fermat, not Descartes, might be credited with writing about these curves earlier than his contemporary.   According to E. T. Bell, "...each of them, entirely independently of the other, invented analytic geometry"  and labeled Fermat as "The Prince of Amateurs."

The following are all known as the hyperbola, parabola and spiral of Fermat.

In a letter written to Roberval in 1636, Fermat stated that he had formulated these curves seven years earlier. 

MATHEMATICA®Code for the Hyperbola
Parametric Plot

Polar Plot

The Prince of Amateurs

An Example of Implicit Differentiation from Calculus Applied to a Hyperbola

Note:   Using the slope of the slant asymptotes, not points on the hyperbola, to sketch a hyperbola is far more common and does not require calculus.

But if the slope at any point on the hyperbola is known, a "slope field" may be drawn using a TI-89 or TI-92 Plus.  The calculator screen for the upper branch of a hyperbola might appear as . . . .


Pierre de Fermat (1601? - 1665) is credited with generalizing work on spirals dating from Archimedes.  But he is far more famous for  Fermat's Last Theorem.   Its proof eluded great mathematicians until late in the 20th century when Andrew Wiles patiently and laboriously produced its solution.

Fermat scribbled the following in his 1621 copy of a translation of  Diophantus' Arithmetica:

It is impossible to divide a cube into two cubes, or a fourth power into two fourth powers, or in general, any power greater than the second, into two like powers, and I have a truly marvelous demonstration of it.  But this margin will not contain it.

In modern terms - not Latin - we would write,

Interesting Facts. . . . .

Archimedes (287-212 B.C.) and Apollonius (262 - 190 B. C.) investigated spirals and the conics centuries before Fermat and Descartes, but the "Ancients" did not have the advantage of symbolic algebra or analytic geometry.

Interestingly, both Fermat and his contemporary, René Descartes, were lawyers.    Both were also passionate lovers of number theory.   In 1636 Fermat wrote that 17,296 and 18,416 were  "amicable" numbers.  Descartes replied that he had also found another pair - 9,363,584  and  9,437,056.     As two positive integers are said to be amicable if each is the sum of the proper divisors of the other, their calculations are slightly amazing for pre-calculator or computer mathematics.

Later Fermat made a mistake.  He sought a formula for identifying prime numbers.  He wrote others:

He had calculated for n = 2, 3, and 4.  Later, Euler proved Fermat wrong by finding that when n = 5, Fermat's formula was divisible by 641.   May we suggest you try this on a calculator knowing that these gentlemen were calculating by hand.

Father Mersenne, a Franciscan friar, philosopher, scientist and mathematician asked Fermat if  100,895,598,169   was prime.  Fermat promptly wrote back "no" for its was the product of  112,303  and  898,423!

Other Animated Spirals with MATHEMATICA®Code

Lituus' Spiral


Sinusoidal Spiral



Historical Sketch on Spirals

From the legendary Delian problem in antiquity to modern freeway construction, spirals have attracted great mathematical talent.  Among the more famous are Archimedes, Descartes, Bernoulli, Euler, and Fermat, but there are many more whose work has enormously influenced pure mathematics, science and engineering.

The name spiral, where a curve winds outward from a fixed point,  has been extended to curves where the tracing point moves alternately toward and away from the pole, the so-called sinusoidal type.    We find Cayley's Sextic, Tschirnhausen's Cubic, and Lituus' shepherd's (or a bishop's) crook.  Maclaurin, best known for his work on series, discusses parabolic spirals in Harmonia Mensurarum (1722).  In architecture there is the Ionic capital on a column.  In nature, the spiraled chambered nautilus is associated with the Golden Ratio, which again is associated with the Fibonacci Sequence.

Useful Links and Books
Bell, E. T., Men of Mathematics, Simon and Schuster, 1937, pp. 56 - 72.
Boyer, Carl B., revised by U. C. Merzbach, A History of Mathematics, 2nd ed., John Wiley and Sons, 1991.
Eves, Howard, An Introduction to the History of Mathematics, 6th ed,. The Saunders College Publishing, 1990, pp. 353-354.
FERMAT'S THEOREM,  math HORIZONS, MAA, Winter, 1993, p.  11.
Gray, Alfred,  Modern Differential Geometry of Curves and Surfaces with MATHEMATICA®, 2nd ed., CRC Press, 1998.
Katz, Victor J., A History of Mathematics,  PEARSON - Addison Wesley, 2004.
Lockwood, E. H., A Book of Curves, Cambridge University Press, 1961.
McQuarrie, Donald A., Mathematical Methods for Scientists and Engineers, University Science Books, 2003.
Shikin, Eugene V., Handbook and Atlas of Curves, CRC Press, 1995.
Yates, Robert,  CURVES AND THEIR PROPERTIES, The National Council of Teachers of Mathematics, 1952.

MATHEMATICA® Code and animation contributed by
Gus Gordillo, 2005.