Propositiones ad Acuendos Juvenes/Problems/24 - De Campo Triangulo/Historical Note
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Historical Note on Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $24$: De Campo Triangulo
When the answer is given in perches, it is clear that square perches is really meant.
Note that the acre as given is defined as $144$ square perches rather than the modern definition as $160$ of them.
As David Singmaster points out, the answer is in fact numerically incorrect: $22 \frac 1 2$ acre is in fact $126$ square perches.
However, using Heron's Formula:
- $\AA = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$
where:
| \(\ds s\) | \(=\) | \(\ds \dfrac {30 + 30 + 18} 2\) | \(\ds = 39\) | |||||||||||
| \(\ds s - a\) | \(=\) | \(\ds 39 - 30\) | \(\ds = 9\) | |||||||||||
| \(\ds s - b\) | \(=\) | \(\ds 39 - 30\) | \(\ds = 9\) | |||||||||||
| \(\ds s - b\) | \(=\) | \(\ds 39 - 18\) | \(\ds = 21\) |
we obtain:
| \(\ds \AA\) | \(=\) | \(\ds \sqrt {39 \times 9 \times 9 \times 21}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {66 \, 339}\) | ||||||||||||
| \(\ds \) | \(\approx\) | \(\ds 257.56\) |
At $144$ square perches to the acre, this equals $1$ acre and $113.56$ square perches.
Sources
- 1992: John Hadley/2 and David Singmaster: Problems to Sharpen the Young (Math. Gazette Vol. 76, no. 475: pp. 102 – 126) www.jstor.org/stable/3620384