The History of Mathematics: A Brief Course

1. HELLENISTIC GEOMETRY 321

FIGURE 2. Heron's proof of his direct method of computing the

area of a triangle.

gave as an example a triangle whose sides were 7, 8, and 9 units. His prescription

was: Add 9 and 8 and 7, getting 24. Take half of this, getting 12. Subtract 7

units from this, leaving 5. Then subtract 8 from 12, leaving 4. Finally, subtract 9,

leaving 3. Multiply 12 by 5, getting 60. Multiply this by 4, getting 240. Multiply

this by 3, getting 720. Take the square root of this, and that will be the area of the

triangle. He went on to explain that since 720 is not a square, it will be necessary

to approximate, starting from the nearest square number, 729.

This result seems anomalous in Greek geometry, since Heron is talking about

multiplying an area by an area. That is probably why he emphasizes that his results

are numerical rather than geometric. An examination of his proof of the formula

shows that he need not have multiplied two areas together. He must have made a

deliberate choice to express himself this way. His proof is based on Fig. 2, in which

one superfluous line has been omitted to streamline it. In the following proof, some

rewording has been done to accommodate this minor modification of the figure.

The lines AB and AH are perpendicular respectively to BT and HT. The proof

follows easily once it is shown that the quadrilateral ΑΒÇÃ is cyclic, that is, can

be inscribed in a circle. In fact, if Ó denotes the semiperimeter, then

Ó^2 : Ó · (Ó - ΒÃ) :: Ó : (Ó - ΒÃ) :: (Ó - ÁÃ) : ÊΕ :: (Ó - ΑÃ) ÃΕ.ÊΕÃΕ

= (Ó - ΑÃ) • (Ó - AB) : ΕÇ^2.

Here ÊΕ • ÔΕ = EH^2 because EH is the altitude to the base of the right triangle

HKY.

Heron could have stated the result in Euclidean language if he had wanted to.

If he were to regard each term in the proportion

Ó^2 : Ó • (Ó - ΒÃ) :: (Ó - ΑÃ) • (Ó - ΑΒ) : ΕÇ^2

as an area and take the sides of squares equal to them, he would have four squares

in proportion, of sides Ó, á, â, EH, where a is the mean proportional between Ó

and Ó — ΒÃ and â the mean proportional between Ó - ΑÃ and Ó - AB. It would

need to be proved that if four squares are in proportion, then their sides are also

in proportion; however, that fact follows immediately from the Eudoxan theory of