Thymaridas

Thymaridas of Paros (Greek: Θυμαρίδας; c. 400 – c. 350 BCE) was an ancient Greek mathematician and Pythagorean noted for his work on prime numbers and simultaneous linear equations.

Life and work

Although little is known about the life of Thymaridas, it is believed that he was a rich man who fell into poverty. It is said that Thestor of Poseidonia traveled to Paros in order to help Thymaridas with the money that was collected for him.

Iamblichus states that Thymaridas called prime numbers "rectilinear", since they can only be represented on a one-dimensional line. Non-prime numbers, on the other hand, can be represented on a two-dimensional plane as a rectangle with sides that, when multiplied, produce the non-prime number in question. He further called the number one a "limiting quantity".

Iamblichus in his comments to Introductio arithmetica states that Thymaridas also worked with simultaneous linear equations.[1] In particular, he created the then famous rule that was known as the "bloom of Thymaridas" or as the "flower of Thymaridas", which states that:[2]

If the sum of n quantities be given, and also the sum of every pair containing a particular quantity, then this particular quantity is equal to 1/(n + 2) [this is a typo in Flegg's book – the denominator should be n − 2 to match the math below] of the difference between the sums of these pairs and the first given sum.

or using modern notation, the solution of the following system of n linear equations in n unknowns:[1]

\( {\displaystyle {\begin{aligned}x+x_{1}+x_{2}+\cdots +x_{n-1}&=s,\\x+x_{1}&=m_{1},\\x+x_{2}&=m_{2},\\&~~\vdots \\x+x_{n-1}&=m_{n-1}\end{aligned}}} \)

is given by

\( {\displaystyle x={\frac {(m_{1}+m_{2}+\cdots +m_{n-1})-s}{n-2}}.} \)

Iamblichus goes on to describe how some systems of linear equations that are not in this form can be placed into this form.[1]
References

Heath, Thomas Little (1981). A History of Greek Mathematics. Dover publications. ISBN 0-486-24073-8.
Flegg, Graham (1983). Numbers: Their History and Meaning. Dover publications. ISBN 0-486-42165-1.

Citations and footnotes

Heath (1981). "The ('Bloom') of Thymaridas". A History of Greek Mathematics. pp. 94–96. "Thymaridas of Paros, an ancient Pythagorean already mentioned (p. 69), was the author of a rule for solving a certain set of n simultaneous simple equations connecting n unknown quantities. The rule was evidently well known, for it was called by the special name [...] the 'flower' or 'bloom' of Thymaridas. [...] The rule is very obscurely worded , but it states in effect that, if we have the following n equations connecting n unknown quantities x, x1, x2 ... xn−1, namely [...] Iamblichus, our informant on this subject, goes on to show that other types of equations can be reduced to this, so that the rule does not 'leave us in the lurch' in those cases either."
Flegg (1983). "Unknown Numbers". Numbers: Their History and Meaning. pp. 205. "Thymaridas (fourth century) is said to have had this rule for solving a particular set of n linear equations in n unknowns:
If the sum of n quantities be given, and also the sum of every pair containing a particular quantity, then this particular quantity is equal to 1/(n + 2) of the difference between the sums of these pairs and the first given sum."

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Ancient Greek and Hellenistic mathematics (Euclidean geometry)
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