This is the second of several posts relating to The Modern Major-General's song from G&S's The Pirates of Penzance.
The Modern Major-General asserts that he is "very well acquainted, too, with matters mathematical…”. As evidence, he claims to know "... many cheerful facts about the square of the hypotenuse." This is a reference to the Pythagorean Theorem: The Pythagorean Theorem asserts that for a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2with "^" the exponentiation operator.
(E.g., 3^2 = 3-squared = 3*3 = 9;
2^3 = 2-cubed = 2*2*2 = 8.)
Pythagoras, for whom the famous theorem is named, lived during the 6th century BCE on the island of Samos in the Aegean Sea.
There are hundreds of proofs of this theorem, the sheer number testifying to its fundamental importance in mathematics.
A digression: Pythagorean triples and Fermat’s Last Theorem.
A Pythagorean triple is a set of 3 integers (a,b,c) satisfying the Pythagorean relationship,
a^2 + b^2 = c^2.
E.g.,. (3,4,5): 3^2 + 4^2 = 9 + 16 = 25 = 5^2.
There are an infinite number of such triples.
By contrast, there are no integers (r,s,t) satisfying
r^3 + s^3 = t^3
In fact, there are no integers (x,y,z) satisfying
x^n + y^n = z^n
for any n > 2. This is Fermat’s Last Theorem whose proof by Andrew Wiles in 1994 elicited headlines, and was featured in a PBS Nova episode entitled, “The Proof.”
Pierre Fermat, a 17th-century French mathematician and jurist, jotted in the margin of his copy of the Greek text Arithmetica by Diophantus the following note:A true proof eluded mathematicians for 350 years, until Wiles’s achievement."It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."
Another digression: non-Euclidean geometry – where Pythagoras fails.
The Pythagorean Theorem is true only in Euclidean geometry – the geometry of the plane. In the most familiar non-Euclidean world – the surface of the earth – the theorem is not true.
The surface of a sphere is the most familiar example of a non-Euclidean geometry (it’s the geometry in which we live – the original GeoMetry: “Earth Measure”). Another non-Euclidean geometry, known as hyperbolic geometry, is virtually impossible to visualize, but has been modeled using crochet!Crocheted model of pseudosphere (the hyperbolic equivalent of a cone) by Daina Taimina. Photo courtesy Steve Rowell/The Institute for Figuring.Crocheted model of a hyperbolic plane by Daina Taimina. Photo courtesy Steve Rowell/The Institute for Figuring.
As it turns out, the Pythagorean Theorem is equivalent to Euclid's famous Parallel Postulate. For 2,000 years from the time of Euclid, mathematicians tried to prove the Parallel Postulate, until a few geniuses in the early 19th century constructed consistent geometries without it! - The two main non-Euclidean geometries are Spherical Geometry and Hyperbolic Geometry, both referenced above.
p.s. It was Gilbert's lyric that inspired me, at age 9, to learn the Pythagorean Theorem! - just its statement, not its proof!