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The Parabola of Fermat:

 A Member of the Spiral Family of Plane Curves. . . . 

The Parabola of Fermat
a = 1, 2, or 3.

Replay the animation
Polar Equations

For n = 2,

The Spiral Family of Curves is analogous to the Cartesian

This parabolic curve was investigated by Fermat as early as 1636.  There are plus and minus values of  r  for any positive angle.  Thus the equation for the single spiral may have the form . . .

This spiral may be easily sketched on polar graph paper.

MATHEMATICA®Code for a Parabolic Spiral
The spiral curves are easily entered and modified on a graphing calculator.
The spiral on the tomb of Jakob (James) Bernoulli. 

Eadem mutata resurgo. 
I shall arise the same though changed.


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.) investigated spirals centuries before Fermat, Descartes and Bernoulli, but he 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 slight 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 Animations 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
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.
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.