Given arbitrary triangle ABC with base AB, within there can be inscribed a parabola which passes through points A, B, tangent to line BC at point B. Archimedes 1st Proposition of the Palimpsest says that the area of the inscribed parabolic segment cut from the parabola by line ABequals one third the area of the triangle.

Archimedes proof of the 1st Proposition leaves the reader somewhat puzzled: First, the condition  which guarantees a balance of the lever, is specified in advance. Perhaps, it would be more logical to produce this during the derivation. It is shown that the condition can be obtained by a series of simple manipulations.

Second, the validity of the proof of lever balance depends upon a property of parabolas which is stated as a proportion (PT). In his works, Archimedes gives a long derivation of this property, but it is not easy reading. A less complex, short derivation of the proportion property (PT) for parabolic curves is presented.

Palimpsest, Archimedes mechanical method, Archimedes 1st Proposition of the Palimpsest, proportion form of the parabolic equation.