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Cartesian equation: xy = cx3 + dx2 + ex + f

This curve was investigated by Newton, who dubbed it the "trident". It was also investigated by Descartes, giving it the other name of the "Parabola of Descartes" even although it is not a parabola.

The curve occurs in Newton's study of cubics. It is appears in his classification of cubic curves Curves by Sir Isaac Newton in Lexicon Technicum by John Harris published in 1710.

Newton was the first to undertake such a systematic study of cubic equations and he classified them into 72 different cases. However, his classification was ultimately incomplete, as he missed six cases. The trident is the 66th species in his classification and Newton provides an illustration quite similar to the ASCII art above.

Some properties of his trident are stated by Newton in the Lexicon. He states that the curve "has four infinite legs" and that the y-axis is "an asymptote to two tending towards contrary parts ... And this figure is that parabola by which D. Cartes constructed equations of six dimensions".

Newton's classification of cubics was criticised by Euler because it lacked general principle, which was likely the cause of the six missing types. Plucker later gave a more detailed classification with 219 types.

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