Greek Anthology Book XIV: Metrodorus: 143/Historical Note

From ProofWiki
Jump to navigation Jump to search

Historical Note on Metrodorus' Arithmetical Epigram no. $143$

The father perished in the shoals of the Syrtis, and this, the eldest of the brothers, came back from that voyage with five talents.
To me he gave twice two-thirds of his share,
on our mother he bestowed two-eighths of my share,
nor did he sin against divine justice.


In W.R. Paton's $1918$ translation of The Greek Anthology Book XIV, he gives the answer as:

the elder brother had $1 \frac 5 7$ talents
the narrator had $2 \frac 2 7$ talents
the mother had $1$ talent.


The discrepancy between this and the shares reported in the calculation appears to be due to a misinterpretation of the wording.


If we replace:

on our mother he bestowed two-eighths of my share

with:

on our mother he bestowed two-eighths of our combined share

then we arrive at the solution given by The Greek Anthology Book XIV.


Proof

Let $a$ talents be the elder brother's share.

Let $b$ talents be the narrator's share.

Let $c$ talents be the mother's share.


We have:

\(\ds a + b + c\) \(=\) \(\ds 5\)
\(\ds b\) \(=\) \(\ds 2 \times \dfrac 2 3 a\)
\(\ds \) \(=\) \(\ds \dfrac {4 a} 3\)
\(\ds c\) \(=\) \(\ds \dfrac 2 8 {a + b}\)
\(\ds \leadsto \ \ \) \(\ds c\) \(=\) \(\ds \dfrac 1 4 \paren {a + \dfrac {4 a} 3}\)
\(\ds \) \(=\) \(\ds \dfrac {7 a} {12}\)
\(\ds \leadsto \ \ \) \(\ds a + \dfrac {4 a} 3 + \dfrac {7 a} {12}\) \(=\) \(\ds 5\)
\(\ds \leadsto \ \ \) \(\ds 12 a + 16 a + 7 a\) \(=\) \(\ds 12 \times 5\)
\(\ds \leadsto \ \ \) \(\ds 35 a\) \(=\) \(\ds 60\)
\(\ds \leadsto \ \ \) \(\ds a\) \(=\) \(\ds \dfrac {12} 7\)
\(\ds \) \(=\) \(\ds 1 \dfrac 5 7\)
\(\ds \leadsto \ \ \) \(\ds b\) \(=\) \(\ds \dfrac 4 3 \times \dfrac {12} 7\)
\(\ds \) \(=\) \(\ds \dfrac {16} 7\)
\(\ds \) \(=\) \(\ds 2 \dfrac 2 7\)
\(\ds \leadsto \ \ \) \(\ds c\) \(=\) \(\ds \dfrac 1 4 \times \paren {\dfrac {16} 7 + \dfrac {12} 7}\)
\(\ds \) \(=\) \(\ds \dfrac 1 4 \times \dfrac {28} 7\)
\(\ds \) \(=\) \(\ds 1\)


So:

the elder brother takes $1 \frac 5 7$ talents
the narrator takes $2 \frac 2 7$ talents
the mother takes $1$ talent.

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Greek_Anthology_Book_XIV:_Metrodorus:_143/Historical_Note&oldid=399807"