Clebsch described the plane representations of various rational surfaces, he was especially interested in that of the general cubic surface.
Other results on cubic surfaces were proved by Clebsch which included: there exists a covariant of order nine which intersects the cubic surface in exactly 27 lines; and every smooth cubic surface can be represented in the plane using four plane cubic surfaces through six points and vice-versa.
Starting from the construction of a cubic surface given by a straight line, three groups of three points on a line, and six other points, Le Paige was led to the construction of a cubic surface given by a line, three points on a line and twelve other points.
This is to avoid mistaken 'theorems', based on fallible intuitions, of which many instances have occurred in the history of the subject (for example, in mathematical analysis).
Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classificationtheorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.
However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers.
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