Propositiones ad Acuendos Juvenes/Problems/22 - De Campo Fastigioso

Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $22$

There is an irregular field,

measuring $100$ perches along each side,

and $50$ perches along each end,

but $60$ perches across the middle.

How many acres does it contain?

The area of an acre used here is $144$ square perches.

The length of the field is $100$ perches.

The length of the edges of the field is $50$ perches.

At the middle it is $60$ perches.

Join the length of the enda with the middle, which is $160$.

Take the $3$rd part of that, which is $53$.

Multiply it by $100$, which is $5300$.

Divide this into $12$ equal parts, giving $37$.

This gives the number of acres in the field.

$\blacksquare$

Alcuin's solution to this is suspect in a number of places.

$(1): \quad$ Fractions are rounded inconsistently:

At one point $\dfrac {160} 3 = 53 \tfrac 1 3$ is rounded to $53$.

At another point $\dfrac {5200} {12} = 441 \tfrac 2 3$ is truncated to $441$.

At yet another point $\dfrac {441} {12} = 36 \tfrac 3 4$ is rounded up to $37$.

However, the actual value of $\dfrac {16 \, 000} {3 \times 12 \times 12} = 37.037 \ldots$ is actually remarkably accurate.

$(2): \quad$ The method of averaging the $50$s and $60$s to get $\dfrac {160} 3$ is dubious.

If the field is a double trapezoid, then the mean width is $55$.

$(3): \quad$ The length given is measured along the side, which may not be the true length.