Greek Anthology Book XIV: Metrodorus: 121

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Arithmetical Epigram of Metrodorus

From Cadiz to the city of the seven hills
the sixth of the road is to the banks of Baetis, loud with the lowing of herds,
and hence a fifth to the Phocian soil of Pylades -- the land is Vaccaean, its name derived from the abundance of cows.
Thence to the precipitous Pyrenees is one-eighth and the twelfth part of one-tenth.
Between the Pyrenees and the lofty Alps lies one-fourth of the road.
Now begins Italy and straight after one-twelfth appears the amber of the Po.
O blessed am I who have accomplished two thousand and five hundred stades journeying thence!
For the Palace on the Tarpeian rock is my journey's object.


Solution

The city of the seven hills, and the location of the Palace on the Tarpeian rock, is of course Rome.

Let $n$ be the number of stades between Cadiz and Rome.

$\dfrac n 6$ is the distance from Cadiz to the Baetis
$\dfrac n 5$ is the distance from the Baetis to Pylades
$\dfrac n 8 + \dfrac n {120}$ is the distance from Pylades to the Pyrenees
$\dfrac n 4$ is the distance from the Pyrenees to the Alps
$\dfrac n {12}$ is the distance from the Alps to the Po
$2500$ is the distance from the Po to Rome itself.


So we have:

\(\ds n\) \(=\) \(\ds \dfrac n 6 + \dfrac n 5 + \dfrac n 8 + \dfrac n {120} + \dfrac n 4 + \dfrac n {12} + 2500\)
\(\ds \leadsto \ \ \) \(\ds 120 n\) \(=\) \(\ds 20 n + 24 n + 15 n + n + 30 n + 10 n + 120 \times 2500\) multiplying through by $120 = \lcm \set {6, 5, 8, 4, 12, 120}$ and simplifying
\(\ds \leadsto \ \ \) \(\ds \paren {120 - 20 - 24 - 15 - 1 - 30 - 10} n\) \(=\) \(\ds 120 \times 2500\)
\(\ds \leadsto \ \ \) \(\ds 20 n\) \(=\) \(\ds 120 \times 2500\)
\(\ds \leadsto \ \ \) \(\ds n\) \(=\) \(\ds \frac {120 \times 2500} {20}\)
\(\ds \) \(=\) \(\ds 15 \, 000\)


So it is $15 \, 000$ from Cadiz and Rome, of which:

$2500$ stades is the distance from Cadiz to the Baetis
$3000$ stades is the distance from the Baetis to Pylades
$2000$ stades is the distance from Pylades to the Pyrenees
$3750$ stades is the distance from the Pyrenees to the Alps
$1250$ stades is the distance from the Alps to the Po

and as we know:

$2500$ is the distance from the Po to Rome itself.

$\blacksquare$


Source of Name

This entry was named for Metrodorus.


Sources

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