Menelaus

Menelaus of Alexandria (ca. 70-140 AD) was a Roman empire mathematician prolific in geometry. He was born in Rome but spent a great deal of his life in Alexandria, Egypt, then a Roman colony by conquest.

His best known work in Euclidean geometry is known as the Theorem of Menelaus, regarding the relationship of a triangle with three collinear points on a line that cuts the triangle. He also worked on spherical geometry, defining a spherical triangle as that region bounded by three great circles. Spherical geometry is different from Euclidean geometry in one important way. In Euclidean geometry, two triangles are similar but not congruent if their interior angles are all known (the "AAA" condition). In spherical geometry two triangles meeting AAA are congruent, which means their areas are the same. (In Euclidean geometry, if two triangles meet AAA their sidelengths and thus their area can scale.)

Not much is known about Menelaus. The mathematician Pappus of Alexandria (c. 290-350) and the philosopher Proclus (412-485) both mention him in their writings. The writer Plutarch (46-120) mentions a conversation he had with Lucius in Rome. Ptolemy (90-168) mentions two astronomical observations made by Menelaus in Rome in 98 AD: the occultation of the stars Spica and Beta Scorpii by the moon, a few days apart. Ptolemy used these observations to show the precession of the equinoxes in his famous book on astronomy, The Almagest.

Menelaus' surviving works include the following:

• Sphaerica, in Arabic translation. Three books on spherical geometry.
• On the Calculation of the Chords in a Circle (Six books. Not sure if they have survived, but historians are aware that they used to exist.)
• Elements of Geometry (Three books - again, not sure if any survive.)
• On the Knowledge of the Weights and Distributions of Different Bodies

The eponymous Theorem of Menelaus deals with a triangle ABC cut on two sides by a line at points D and E, and also cutting the extension of the third side at point F. The theorem states that if a product of ratios of line segments on each side is equal to -1, then the three points are collinear. Since collinearity is a wonderful thing in the world of geometry, Menelaus' Theorem is often employed to show that three points fall on the same line, or that a ratio of side lengths equals -1 if three points are in a same line.

Incidentally, the negative number might sound absurd, since all distances are usually thought of as positive. However, in analytic geometry great use of directed distances between points is made, and usually distances are positive if on one side of a point and negative on another. (Think of the parametric equation of a line P = P₁ + k*(P₂ - P₁), where P₁, P₂, and P are points (with x,y coordinates) and k is the scalar that is negative if the point P is on the opposite side of the line centered at P₁ from P₂, and positive if it is on the same side.) The use of directed distance, where a distance can take positive or negative values, is useful when determining if a point is inside or outside an object, such as a triangle or a circle.

Everything2 Writeups: Articles on (topic)

1. eipi10, Theorem of Menelaus, Aug., 2002
2. eipi10, Ceva's Theorem, Aug, 2002
3. IWhoSawTheFace, Triangle and Circle Geometry, (Nov, 2011 - not finished yet)
4. IWhoSawTheFace, Giovanni Ceva, (Nov, 2011 - not finished yet)
5. IWhoSawTheFace, Cevian, (Nov, 2011 - not finished yet)
6. IWhoSawTheFace, Heron, (unfilled nodeshell)
7. cbustapeck, Pappus of Alexandria, Oct, 2001
8. legbagede, Library of Alexandria, June, 2000
9. cbustapeck, Euclid, 2001
10. avjewe, Euclid's Elements, June, 2001

References: Useful books and references on geometry

1. H.S.M. Coxeter, Introduction to Geometry, 2nd Ed., (c) 1969
   
2. Dan Pedoe, Geometry: A Comprehensive Course
3. J.L. Heilbron, Geometry Civilized, ©2000
4. David Wells, Ed., The Penguin Dictionary of Curious and Interesting Geometry, ©1991
5. Melvin Hausner, A Vector Space Approach to Geometry, ©1965
6. Daniel Zwillinger, Ed., The CRC Standard Mathematical Tables and Formulae, 30th Ed, ©1996
   Ch. 4, Geometry,
   esp. § 4.5.1, “Triangles,” p. 271
   

Internet References

1. Wikipedia, "Menelaus of Alexandria"
2. Weisstein, Eric W. "Menelaus' Theorem" From MathWorld--A Wolfram Web Resource.
3. Biographies of Mathematicians, Menelaus of Alexandria, School of Mathematics and Statistics, University of St. Andrews, Scotland. A wonderful resource for mathematical history.
4. Alexander Bogomolny, "The Menelaus Theorem," Cut the Knot
5. Alexander Bogomolny, "Pole and Polar with Respect to a Triangle," from his Cut the Knot web site. Wonderful use of Java applet to let you move around an interior point P and see how the perspector affects things such as lines of concurrency.