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I'm interested in learning more about the Bloom of Thymaridas, a description of which can be found here.

Obviously the mathematics behind the identity is not particularly deep from a modern standpoint. However, does anyone have any information on why it was considered a notable arithmetical method during its time? Were there some common scenarios in which you would be given those particular sums and needed to solve for x?

Thanks!

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    ... (continuing my comment) as for why it was later considered notable, my guess is that medieval scholars were simply enormously pleased to have any mathematics at all from antiquity. Europe had gone through centuries in which (roughly) no mathematics was done at all, and to find things like this must have astonished them. (The general practice of solving systems of equations is of course enormously useful. But I do not think later scholars put ancient results on specific problems like this to any particular "use".) Again, this is just my personal speculation.2012-01-10

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Here are two papers about the problem, written in Greek-

"The Mathematician Thymaridas the Parian and the Thymaridian Bloom"

"Thymaridas the Parian and his contribution to Theoretical Arithmetics"

Two problems are proposed by Iamblichus that are approached using the Thymaridian Bloom. Specifically one is stated as follows (presumably directly translated from the original text).

Find four integers such that:

$1)$ The sum of the first two is twice the sum of the third and the fourth.
$2)$ The sum of the first and third is thrice the sum the second and fourth.
$3)$ The sum of the first and fourth is four times the sum of the second and third.
$4)$ The sum of all four is five times the sum of the second and third.

No mention is made on "real-life" problems to which the Thymaridian bloom was applied by Thymarides' contemporaries.