# History of Plato's formula for generating pythagorean triples

I'm interested in the history behind Plato's formula $2m,m^2-1,m^2+1$ for generating pythagorean triples. Was Plato the first mathematician to come up with such a formula?

• Plato was not a mathematician, nor did he come up with this formula or even its verbal equivalent, he was a philosopher and it is simply mentioned in one of his dialogues. The best we can say is that it likely originates with the Pythagoreans, and little more is known due to their practice of attributing everything to Pythagoras (who most likely did none of it), see hsm.stackexchange.com/questions/391/… – Conifold May 8 '16 at 1:58
• @Conifold :that's very interesting,but how come is Plato's name connected to this method,see for example here. – Nicco May 8 '16 at 2:38

## 2 Answers

Plato was not a mathematician and it is unlikely that he invented anything in mathematics. This particular formula is credited to Pythagoras by ancient mathematician and historian of mathematics Proclus; he says this in his commentaries on Euclid. Some special cases come from Babylonian mathematics.

I did not check with Proclus. My source is the article of I. Bashmakova, Diophantus of Alexandria and his "Atithmetic", published as the introduction to the Russian translation of Diophantus (Moscow, Nauka, 1974).

The only conclusion that can be made from Proclus's statement is that some unknown Pythagoreans knew this formula, they were a secret society, and tended to attribute everything to their semi-legendary founder.

• I will not mark this answer for now,as it does not address the question of how Plato's name stuck to this formula,which was my main confusion. – Nicco May 8 '16 at 11:20
• I suppose Plato mentions it in one of his dialogues, as he mentions many other mathematical results. As he lived before Euclid, this is probably the first mentioning in the literature which survived to our day. – Alexandre Eremenko May 8 '16 at 11:27
• @ Alexandre Emerenko :So would it be correct to continue calling the formula by his name(Plato's formula) or should it simply be Pythagoras formula? – Nicco May 8 '16 at 11:37
• @Nicco: I do not know what is "correct". Most of established mathematical terminology does not credit the "first inventors" . Names are just labels, and the main criterion here is that a name is recognizable, not that it gives a fair credit. – Alexandre Eremenko May 8 '16 at 11:48
• There are already many things named after Pythagoras, so it is OK to choose some other name. By the way "Plato solids" is also an established name, though everyone knows that it is not Plato who studied them. It was Theaetetus, and Plato credits him. – Alexandre Eremenko May 8 '16 at 11:52

The names Pythagoras and Plato are associated with methods of generating Pythagorean triples due to their mention in Proclus's commentary on Euclid. However, as far as I have been able to determine, there is no mention of such a method in the extant works of Plato, and, as has been mentioned in the comments, attribution of any mathematical result to Pythagoras or to the early members of his school is problematic.

In his commentary on the proof of the Pythagorean theorem in Book I of Elements, Proclus describes two methods for generating Pythagorean triples, stating that one is attributed to Pythagoras, the other to Plato. Using an anachronistic algebraic formulation, the "method of Pythagoras" yields the triple $$\left(n,\frac{n^2-1}{2},\frac{n^2+1}{2}\right)$$ for any odd integer $$n$$. Examples are $$(3,4,5)$$, $$(5,12,13)$$, $$(7,24,25)$$, $$(9,40,41)$$, and $$(11,60,61)$$. As two elements of each triple differ by $$1$$, these are clearly primitive.

The "method of Plato" yields $$\left(n,\left(\frac{n}{2}\right)^2-1,\left(\frac{n}{2}\right)^2+1\right)$$ for any even integer $$n$$. This is equivalent to the formula you ask about. Examples are $$(4,3,5)$$, $$(6,8,10)$$, $$(8,15,17)$$, $$(10,24,26)$$, and $$(12,35,37)$$. If $$n$$ is twice an odd number, the method yields the same triples as the "method of Pythagoras", but doubled. If $$n$$ is twice an even integer, the triples are primitive as two of their elements are consecutive odd numbers.

The passage in the question starts on page 340 (location 428.6) in Glenn R. Morrow's translation of A Commentary of the First Book of Euclid's Elements. Here it is:

Certain methods have been handed down for finding such triangles, one of them attributed to Plato, the other to Pythagoras. The method of Pythagoras begins with odd numbers, positing a given odd number as being the lesser of the two sides containing the angle, taking its square, subtracting one from it, and positing half of the remainder as the greater of the sides about the right angle; then adding one to this, it gets the remaining side, the one subtending the angle. For example, it takes three, squares it, subtracts one from nine, takes the half of eight, namely, four, then adds one to this and gets five; and thus is found the right-angled triangle with sides of three, four, and five, The Platonic method proceeds from even numbers. It takes a given even number as one of the sides about the right angle, divides it into two and squares the half, then by adding one to the square gets the subtending side, and by subtracting one from the square gets the other side about the right angle. For example, it takes four, halves it and squares the half, namely, two, getting four; then subtracting one it gets three and adding one gets five, and thus it has constructed the same triangle that was reached by the other method. For the square of this number is equal to the square of three and the square of four taken together.

Morrow refers the reader to pages 356–360 of Volume I of Heath's translation of Elements, where Heath presents some speculations on how such methods might have been discovered. Starting on page 360 Heath also discusses at length the early knowledge of the Pythagorean theorem and Pythagorean triples in India that is exhibited in the Śulvasūtras.

Interestingly, in the paragraph preceding the one quoted above, Proclus discusses isosceles and scalene right triangles, echoing Plato's classification in Timaeus. But among scalene right triangles, Plato singles out the $$30^\circ$$-$$60^\circ$$-$$90^\circ$$ triangle, for which, as Proclus notes in the case of the isosceles right triangle, "you cannot find numbers that fit the sides"—so nothing there about Pythagorean triples. Later in the paragraph, Proclus says that there are scalene triangles for which "it is possible to find such numbers", giving the example of "the triangle in the Republic, in which sides of three and four contain the right angle and five subtends it". In a footnote Morrow says the allusion is likely to Rep. 546c. In my reading that passage engages in a complicated numerology involving the numbers 3, 4, and 5, but does not say anything about right triangles. If somewhere in Plato there is a mention Pythagorean triples, that would be interesting, but I have not heard of such a passage.