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)

Mathematicians

(timeline)

Anaxagoras Anthemius Archytas Aristaeus the Elder Aristarchus Apollonius Archimedes Autolycus Bion Bryson Callippus Carpus Chrysippus Cleomedes Conon Ctesibius Democritus Dicaearchus Diocles Diophantus Dinostratus Dionysodorus Domninus Eratosthenes Eudemus Euclid Eudoxus Eutocius Geminus Heliodorus Heron Hipparchus Hippasus Hippias Hippocrates Hypatia Hypsicles Isidore of Miletus Leon Marinus Menaechmus Menelaus Metrodorus Nicomachus Nicomedes Nicoteles Oenopides Pappus Perseus Philolaus Philon Philonides Porphyry Posidonius Proclus Ptolemy Pythagoras Serenus Simplicius Sosigenes Sporus Thales Theaetetus Theano Theodorus Theodosius Theon of Alexandria Theon of Smyrna Thymaridas Xenocrates Zeno of Elea Zeno of Sidon Zenodorus

Treatises

Almagest Archimedes Palimpsest Arithmetica Conics (Apollonius) Catoptrics Data (Euclid) Elements (Euclid) Measurement of a Circle On Conoids and Spheroids On the Sizes and Distances (Aristarchus) On Sizes and Distances (Hipparchus) On the Moving Sphere (Autolycus) Euclid's Optics On Spirals On the Sphere and Cylinder Ostomachion Planisphaerium Sphaerics The Quadrature of the Parabola The Sand Reckoner

Problems

Angle trisection Doubling the cube Squaring the circle Problem of Apollonius

Concepts/definitions

Circles of Apollonius

Apollonian circles Apollonian gasket Circumscribed circle Commensurability Diophantine equation Doctrine of proportionality Golden ratio Greek numerals Incircle and excircles of a triangle Method of exhaustion Parallel postulate Platonic solid Lune of Hippocrates Quadratrix of Hippias Regular polygon Straightedge and compass construction Triangle center

Results

In Elements

Angle bisector theorem Exterior angle theorem Euclidean algorithm Euclid's theorem Geometric mean theorem Greek geometric algebra Hinge theorem Inscribed angle theorem Intercept theorem Pons asinorum Pythagorean theorem Thales's theorem Theorem of the gnomon

Apollonius

Apollonius's theorem

Other

Aristarchus's inequality Crossbar theorem Heron's formula Irrational numbers Menelaus's theorem Pappus's area theorem Problem II.8 of Arithmetica Ptolemy's inequality Ptolemy's table of chords Ptolemy's theorem Spiral of Theodorus

Centers

Cyrene Library of Alexandria Platonic Academy

Other

Ancient Greek astronomy Greek numerals Latin translations of the 12th century Neusis construction

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