Quadrature of the Parabola

Quadrature of the Parabola (Greek: Τετραγωνισμὸς παραβολῆς) is a treatise on geometry, written by Archimedes in the 3rd century BC and addressed to his Alexandrian acquaintance Dositheus.

It is one of the best-known works of Archimedes, in particular for its ingenious use of the method of exhaustion and in the second part of a geometric series.

[1] He then computes the sum of the resulting geometric series, and proves that this is the area of the parabolic segment.

This represents the most sophisticated use of a reductio ad absurdum argument in ancient Greek mathematics, and Archimedes' solution remained unsurpassed until the development of integral calculus in the 17th century, being succeeded by Cavalieri's quadrature formula.

[2] A parabolic segment is the region bounded by a parabola and line.

To find the area of a parabolic segment, Archimedes considers a certain inscribed triangle.

Proposition 1 of the work states that a line from the third vertex drawn parallel to the axis divides the chord into equal segments.

The main theorem claims that the area of the parabolic segment is

Conic sections such as the parabola were already well known in Archimedes' time thanks to Menaechmus a century earlier.

However, before the advent of the differential and integral calculus, there were no easy means to find the area of a conic section.

Archimedes provides the first attested solution to this problem by focusing specifically on the area bounded by a parabola and a chord.

[3] Archimedes gives two proofs of the main theorem: one using abstract mechanics and the other one by pure geometry.

In the first proof, Archimedes considers a lever in equilibrium under the action of gravity, with weighted segments of a parabola and a triangle suspended along the arms of a lever at specific distances from the fulcrum.

[5] Archimedes here deviates from the procedure found in On the Equilibrium of Planes in that he has the centers of gravity at a level below that of the balance.

[6] The second and more famous proof uses pure geometry, particularly the sum of a geometric series.

The main idea of the proof is the dissection of the parabolic segment into infinitely many triangles, as shown in the figure to the right.

In propositions eighteen through twenty-one, Archimedes proves that the area of each green triangle is

Archimedes evaluates the sum using an entirely geometric method,[8] illustrated in the adjacent picture.