section_path: History/Origins/Diophantus
section_name: Diophantus
paragraph_text: Diophantus
  
  Very little is known about Diophantus of Alexandria; he probably lived in the third century AD, that is, about five hundred years after Euclid. Six out of the thirteen books of Diophantus's Arithmetica survive in the original Greek and four more survive in an Arabic translation. The Arithmetica is a collection of worked-out problems where the task is invariably to find rational solutions to a system of polynomial equations, usually of the form f (x, y) = z 2 {\displaystyle f(x,y)=z^{2}} or f (x, y, z) = w 2 {\displaystyle f(x,y,z)=w^{2}}. Thus, nowadays, we speak of Diophantine equations when we speak of polynomial equations to which rational or integer solutions must be found.
  
  One may say that Diophantus was studying rational points, that is, points whose coordinates are rational—on curves and algebraic varieties; however, unlike the Greeks of the Classical period, who did what we would now call basic algebra in geometrical terms, Diophantus did what we would now call basic algebraic geometry in purely algebraic terms. In modern language, what Diophantus did was to find rational parametrizations of varieties; that is, given an equation of the form (say) f (x 1, x 2, x 3) = 0 {\displaystyle f(x_{1},x_{2},x_{3})=0}, his aim was to find (in essence) three rational functions g 1, g 2, g 3 {\displaystyle g_{1},g_{2},g_{3}} such that, for all values of r {\displaystyle r} and s {\displaystyle s}, setting x i = g i (r, s) {\displaystyle x_{i}=g_{i}(r,s)} for i = 1, 2, 3 {\displaystyle i=1,2,3} gives a solution to f (x 1, x 2, x 3) = 0. {\displaystyle f(x_{1},x_{2},x_{3})=0.}
  
  Diophantus also studied the equations of some non-rational curves, for which no rational parametrisation is possible. He managed to find some rational points on these curves (elliptic curves, as it happens, in what seems to be their first known occurrence) by means of what amounts to a tangent construction: translated into coordinate geometry(which did not exist in Diophantus's time), his method would be visualised as drawing a tangent to a curve at a known rational point, and then finding the other point of intersection of the tangent with the curve; that other point is a new rational point. (Diophantus also resorted to what could be called a special case of a secant construction.)
  
  While Diophantus was concerned largely with rational solutions, he assumed some results on integer numbers, in particular that every integer is the sum of four squares (though he never stated as much explicitly).