At the close of Book IV many geometries give as an exercise, and some give as a regular proposition, the celebrated problem that bears the name of Heron of Alexandria, namely, to compute the area of a triangle in terms of its sides. The result is the important formula
Area = √(s(s - a)(s - b)(s - c)),
where a, b, and c are the sides, and s is the semiperimeter ½(a + b + c). As a practical application the class may be able to find a triangular piece of land, as here shown, and to measure the sides. If the piece is clear, the result may be checked by measuring the altitude and applying the formula a = ½bh.
It may be stated to the class that Heron's formula is only a special case of the more general one developed about 640 A.D., by a famous Hindu mathematician, Brahmagupta. This formula gives the area of an inscribed quadrilateral as √((s - a)(s - b)(s - c)(s - d)), where a, b, c, and d are the sides and s is the semiperimeter. If d = 0, the quadrilateral becomes a triangle and we have Heron's formula.[82]
At the close of Book IV, also, the geometric equivalents of the algebraic formulas for (a + b)2, (a - b)2, and (a + b)(a - b) are given. The class may like to know that Euclid had no algebra and was compelled to prove such relations as these by geometry, while we do it now much more easily by algebraic multiplication.
CHAPTER XVIII
THE LEADING PROPOSITIONS OF BOOK V
