Hypatia Taught Conic Sections and Diophantine Equations

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Philammon declaring his love for Hypatia. Image by H. M. Paget.

Hypatia of Alexandria became famous for being “the first notable woman mathematician,” and for her untimely demise in 415 AD. What were her areas of mathematical expertise?

Hypatia’s Mathematics

Modern researchers have never unearthed Hypatia’s actual writings, but other writers in antiquity attributed several volumes of math to her, emphasizing conic sections in geometry as well as Diophantine equations. Apparently, Hypatia taught and popularized these subjects.

Just what are “conic sections” and “Diophantine equations?”

Conic Sections

In geometry, a conic section is the curve created when one cuts through a cone with a flat plane.

Picture an empty ice cream cone with a point at one end and a circle at the other. That is one “nappe” of a cone in geometry. The complete cone looks like two ice cream cones, attached at the point, and extending infinitely far in opposite directions.

Geometry may use the words “vertex” or “apex” to mean the point of a cone. It often pictures one nappe with a circular base on a flat horizontal plane, with the vertex above the base rather than below it.

This article uses “cone” to mean a “right cone”, where the vertex is directly above the centre of the base. In other cases, a cone might have the vertex somewhere else, so it looks lopsided or tilted.

The Four Conic Sections

The first conic section is the circle. This is the cross-section of a cone as cut to be a plane, perpendicular to the axis of that cone.

Tilting that plane slightly, so it still cuts across one nappe but not the other, creates an ellipse rather than a circle.

If the plane cuts the cone parallel to one “side” of the cone, at the angle of its slope, then the curve is a parabola.

Finally, if the cutting plane goes through the second nappe of the double cone, the curve is a hyperbola.

Modern science applies these geometric shapes to many disciplines, with the simplest examples in rocket science. Most orbits are ellipses. Gravity pulls thrown objects down in a parabolic arc. If a spaceship achieves escape velocity, its path is an hyperbola.

Conic Sections. Image by Magister Mathematicae.

Visual representation of the sections of a cone: Circle, ellipse, parabola, and hyperbola. Image by Magister Mathematicae.

Diophantine Equations

Diophantine equations only permit solutions with integer values:

  • {…-3, -2, -1, 0, 1, 2, 3…}

One famous Diophantine equation is the Pythagorean Theorem. Any right triangle with short sides of length ‘a’ and ‘b’, and the long hypoteneuse side ‘h’, satisfies the equation:

  • h^2 = a^2 + b^2 (where “x^2″ means “x squared”, or “x to the exponent 2″).

Integer solutions for the Pythagorean equation are triples such as (3, 4, 5) and (5, 12, 13).

Another well-known Diophantine equation, very similar to the Pythagorean Theorem except for the exponent, is the heart of Fermat’s Last Theorem, which states there is no integer solution for:

  • a^n + b^n = c^n, for n>2

Diophantine equations are sometimes couched in terms of riddles. “I am twice as old as my son, who is six times the age of his daughter. Next year she will start school at age 6.” That becomes a set of equations:

  • I=2*S
  • S=6*D
  • D+1=6

This set is easy to solve, by beginning with the third equation.

  • D+1=6 implies D=5;
  • Thus S=6*D=6*5=30;
  • Therefore I=2*S=2*30=60.

Pythagorean Equation and Fermat’s Last Theorem: image by Mike DeHaan.

The Mathematical Legacy of Hypatia of Alexandria

Although Hypatia of Alexandria has not received credit for original research or insights, she was considered a fine teacher and writer.

Her areas of specialization, conic sections and Diophantine equations, remain important for learning math, in applied mathematics, and as tools in physics and other disciplines in science.

Resources:

Weisstein, Eric W. Diophantine Equation. Wolfram Math World.  Accessed October 17, 2013.

Weisstein, Eric W. ConeWolfram Math World.Accessed October 17, 2013.

Weisstein, Eric W. Conic Section. Wolfram Math World.Accessed October 17, 2013.

Weisstein, Eric W. Hyperbola.  Wolfram Math World. Accessed October 17, 2013.

Schappacher, Norbert. Diophantus of Alexandria : a Text and its History(2005). Institut de Recherche Mathématique Avancée. Accessed October 17, 2013.

Cosmopolis.com. Hypatia of Alexandria: Mathematician, Astronomer, Philosopher (d. 415 C.E.). Accessed October 17, 2013.

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