Calculus: Understanding Its Concepts and Methods

Heron of Alexandria (Hero) (10-75)    Historical Sketch

Historians generally refer to this impressive historical figure as Heron of Alexandria, for from his writings it seems clear that he taught mathematics and related topics at the Museum in Alexandria. It seems to be agreed that he lived from about 10 A.D. to about 75 A.D. He is often referred to as Hero rather than Heron, just as Platon is also known as Plato.

His lecture notes and textbooks that have survived reveal that he taught mathematics and physics, in addition to mechanics, measurement, astronomy, surveying, plane and spatial geometry, number theory (see Problems 1 and 2 below), approximation theory (Problems 3 and 4 below), and numerous other topics.

Heron is best known for his formula---which is not a formula for the birds---to compute the area A of any triangle, which is stated: If a, b, and c are the lengths of the sides of a triangle, then the area of the triangle is

   hero1.png (1785 bytes)

where s is the semiperimeter of the triangle---that is,

    semiperimeter.png (1365 bytes)

This is called Hero's formula in most books. You probably used it for the first time in high school geometry. He proved this in Book I of his treatise Metrica, where he also developed formulas for areas of regular polygons of four sides, five sides, etc., through twelve sides, surfaces of cones, cylinders, prisms, pyramids, spheres, and so forth.

  1. You may already know this, but if not, show that if a and b are positive numbers with a > b, then a + b, a - b, and 2ab form the lengths of the sides of a right triangle. This was known to, but is not due to, Heron.
  2. The converse of the statement in Problem 1 is also true. For the right triangle with sides of length 5, 12, and 13, find numbers a and b for which a+ b = 13, a - b = 5, and 2ab = 12.
  3. Heron approximated the square root of p + q by p + (q/(2p)). Try this for some specific numbers p and q to see what you think of this.
  4. Use Heron's formula of Problem 3 to find rational number approximations to the square root of each of the following. Choose p so that p is as close to the radicand as possible.

a. 11
b. 48
c. 77
d. 150

Square your answers as a measure of how good your approximations are.

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Calculus: Understanding Its Concepts and Methods