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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 wellknown 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

For


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 TI89 or TI92 Plus. The calculator screen for the
upper branch of a hyperbola might appear as . . . .

Fame!
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
(287212 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 precalculator 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
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 socalled 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

http://wwwhistory.mcs.stand.ac.uk/history/Curves/Fermats.html 
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. 353354.

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.


