# Greek Anthology Book XIV: Metrodorus: 124

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## Arithmetical Epigram of Metrodorus

*The sun, the moon, and the planets of the revolving zodiac spun such a nativity for thee;**for a sixth part of thy life to remain an orphan with thy dear mother,**for an eighth part to perform forced labour for thy enemies.**For a third part the gods shall grant thee home-coming, and likewise a wife and a late-born son by her.**Then thy son and wife shall perish by the spears of the Scythians,**and then having shed tears for them thou shalt reach the end of thy life in twenty-seven years.*

## Solution

Let $n$ be the length in years of the subject's life.

- $\dfrac n 6$ is spent as an orphan

- $\dfrac n 8$ is spent as a slave

- $\dfrac n 3$ is spent at home with a wife and son

- $27$ years remain.

We have:

\(\ds n\) | \(=\) | \(\ds \dfrac n 6 + \dfrac n 8 + \dfrac n 3 + 27\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds 24 n\) | \(=\) | \(\ds 4 n + 3 n + 8 n + 24 \times 27\) | multiplying through by $24 = \lcm \set {6, 8, 3}$ and simplifying | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds \paren {24 - 4 - 3 - 8} n\) | \(=\) | \(\ds 24 \times 27\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds 9 n\) | \(=\) | \(\ds 24 \times 27\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds n\) | \(=\) | \(\ds \dfrac {24 \times 27} 9\) | |||||||||||

\(\ds \) | \(=\) | \(\ds 72\) |

So the subject lived $72$ years, of which:

- $12$ years were spent as an orphan

- $9$ years were spent as a slave

- $24$ years were spent as a family man at home

and $27$ years remained before his death.

$\blacksquare$

## Source of Name

This entry was named for Metrodorus.

## Sources

- 1918: W.R. Paton:
*The Greek Anthology Book XIV*... (previous) ... (next): Metrodorus' Arithmetical Epigrams: $124$