Heron's Mathematics Michael Lahanas Ηρων ο Αλεξανδρεύς

Griechische Mathematik: Heron von Alexandria

Heron of Alexandria contributed to mathematics but he had not the mathematical “quality” of Euclid.

Euclid’s definitions of the elementary geometric entities—point, straight line, plane—at the beginning of the Elements have long presented a problem. Their nature is in sharp contrast with the approach taken in the rest of the book, and continued by mathematicians ever since, of refraining from defining the fundamental entities explicitly but limiting themselves to postulating the properties which they enjoy. Why should Euclid be so hopelessly obscure right at the beginning and so smooth just after? The answer is: the definitions are not Euclid’s. Toward the beginning of the second century A.D. Heron of Alexandria found it convenient to introduce definitions of the elementary objects (a sign of decadence!) in his commentary on Euclid’s Elements, which had been written at least 400 years before. All manuscripts of the Elements copied ever since included Heron’s definitions without mention, whence their attribution to Euclid himself. The philological evidence leading to this conclusion is quite convincing.
Sandro Graffi Review of “ La Rivoluzione Dimenticata” (The Forgotten Revolution) Lucio Russo Feltrinelli, Milan, 1996

Heron’s Formula
Let a,b,c be the sides of a triangle, and let A be the area of the triangle. Heron's formula states that A*A = s(s-a)(s-b)(s-c), where s = (a+b+c)/2. The actual origin of this formula is somewhat obscure historically, and it may well have been known for centuries prior to Heron. For example, some people think it was known to Archimedes. However, the first definite reference we have to this formula is Heron's. His proof of this result is extremely circuitious, and it seems clear that it must have been found by an entirely different thought process, and then "dressed up" in the usual synthetic form that the classical Greeks preferred for their presentations.

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For a proof see Dr. McCrory Foundation of Geometry lecture: