Dinostratus' chief contribution to mathematics was his solution to the problem of squaring the circle. To solve this problem, Dinostratus made use of the trisectrix of Hippias, for which he proved a special property (Dinostratus' theorem) that allowed him the squaring of the circle. Due to his work the trisectrix later became known as the quadratrix of Dinostratus as well. Although Dinostratus solved the problem of squaring the circle, he did not do so using ruler and compass alone, and so it was clear to the Greeks that his solution violated the foundational principles of their mathematics. Over two thousand years later it would be proved impossible to square a circle using straight edge and compass alone.
^ abBoyer (1991). "The age of Plato and Aristotle". A History of Mathematics. pp. 96–97. Dinostratus, brother of Menaechmus, was also a mathematician, and where one of the brothers "solved" the duplication of the cube, the other "solved" the squaring of the circle. The quadrature because a simple matter once a striking property of the end point Q of the trisectrix of Hippias had been noted, apparently by Dinostratus. If the equation of the trisectrix (Fig. 6.4) is πrsin θ = 2aθ, where a is the side of the square ABCD associated with the curve, [...] hence, Dinostratus' theorem is established - that is, AC/AB = AB/DQ. [...] Inasmuch as Dinostratus showed that the trisectrix of Hippias serves to square the circle, the curve more commonly came to be known as the quadratrix. It was, of course, always clear to the Greeks that the use of the curve in the trisection and quadrature problems violated the rules of the game - that circles and straight lines only were permitted. The "solution" of Hippias and Dinostratus, as their authors realized, were sophistic; hence, the search for further solutions, canonical or illegitimate, continued with the result that several new curves were discovered by Greek geometers.