Secondly, Theon of Smyrna gives the law of formation of the series of 'side-' and 'diameter-' numbers which satisfy the equations _2x²-y²=±1_. The law depends on the proposition proved in Eucl. II. 10 to the effect that _(2x+y)²-2(x+y)²=2x²-y²_, whence it follows that, if _x_, _y_ satisfy either of the above equations, then _2x+y_, _x+y_ is a solution in higher numbers of the other equation. The successive solutions give values for _y/x_, namely 1/1, 3/2, 7/5, 17/12, 41/29, ..., which are successive approximations to the value of √2 (the ratio of the diagonal of a square to its side). The occasion for this method of approximation to √2 (which can be carried as far as we please) was the discovery by the Pythagoreans of the incommensurable or irrational in this particular case. Thirdly, Iamblichus mentions a discovery by Thymaridas, a Pythagorean not later than Plato's time, called the επανθημα {epanthêma} ('bloom') of Thymaridas, and amounting to the solution of any number of simultaneous equations of the following form: _x+x₁+x₂+ ... +x_{n-1} = s_, _x+x₁ = a₁_, _x+x₂ = a₂_, ... _x+x_{n-1} = a_{n-1}_, the solution being _x=((a₁+a₂+ ... +a_{n-1})-s)/(n-2)_. The rule is stated in general terms, but the above representation of its effect shows that it is a piece of pure algebra. The Pythagorean contributions to geometry were even more remarkable. The most famous proposition attributed to Pythagoras himself is of course the theorem of Eucl. I. 47 that the square on the hypotenuse of any right-angled triangle is equal to the sum of the squares on the other two sides. But Proclus also attributes to him, besides the theory of proportionals, the construction of the 'cosmic figures', the five regular solids.