Exploring Archimedes' Method for Calculating Parabolic Areas

Archimedes of Syracuse is renowned as one of the most significant scientists and mathematicians from ancient times. He was a versatile thinker whose contributions spanned various fields, including mathematics, physics, astronomy, and engineering. Additionally, Archimedes excelled as an inventor and designer of weaponry.

His many notable achievements include:

Laying the groundwork for mechanics and hydrostatics, introducing key concepts like the laws of levers, the center of gravity, and the famous Archimedes' principle, which describes the buoyant force on bodies in fluids.
Establishing formulas for calculating the volume and surface area of spheres and other geometric figures.
Pioneering the use of physics to tackle mathematical problems, a reversal of the usual approach. An example of this can be found in the article linked below.
Foreseeing methods akin to integral calculus, which would not be fully realized until two millennia later.

As mathematician Steven Strogatz remarked, "[t]o say [Archimedes] was ahead of his time would be putting it mildly."

Inventing various war machines, including the use of parabolic mirrors to ignite enemy ships attacking Syracuse, though the authenticity of this account has long been debated.

In this article, I will clarify, based on Simmons' work, how Archimedes determined the area of a parabolic segment (refer to Figure 6 below). His proof is documented in "The Quadrature of the Parabola," written in the 3rd century BC, long before modern calculus emerged through the efforts of mathematicians like Barrow, Descartes, Fermat, Pascal, Wallis, Cavalieri, Gregory, Newton, and Leibniz.

Archimedes' Construction

Figures 6 and 7 illustrate Archimedes' construction for calculating the area S bounded by chord AB. He constructs three triangles: ?ABC, ?ADC, and ?CEB, by following these steps:

Identifying point C where the tangent is parallel to the chord AB.
Selecting point D where the tangent is parallel to AC.
Choosing point E according to the same principle, along with chord BC.
Repeating this process for areas that do not yet contain inscribed triangles, continuing indefinitely (this process is known as the method of exhaustion). Further details will be shared in the conclusion of this article.

Archimedes' objective was to prove that the area S within the parabolic segment is equal to 4/3 of the area of triangle ?ABC.

This can be demonstrated by first establishing that

and then applying the aforementioned method of exhaustion to triangles ?ACD and ?BCE, among others.

Proving the Area Relation

In this section, I will demonstrate how to establish the area relation, following Simmons' methodology. Subsequently, I will prove that the area of the parabolic segment equals 4/3 of the area of triangle ?ABC using a straightforward application of the method of exhaustion.

The equation for a parabola can be expressed as (with an appropriate choice of axes):

Next, we define points A, B, and C, illustrated in Figure 7:

From Figure 7, we derive the following relation:

According to this relation, the vertical line through point C bisects chord AB at point P. It suffices to show that

This will automatically validate the area relation (for more details, see the linked resource). From Figure 7, we observe that the vertical line through E bisects segment BC at point G and segment BP at point H. If we can prove that:

we will establish the following:

Now, since:

we directly derive the necessary relation. The final step to confirm this area relation is to verify that:

or, equivalently, as illustrated in Figure 7:

To validate this, we will apply analytical geometry to Figure 7. Archimedes' original proof relied solely on geometry, as analytical geometry was not developed until the 17th century by René Descartes.

Through straightforward algebraic manipulation, we can derive:

Final Step: Utilizing the Method of Exhaustion

Now, applying the method of exhaustion and repeating the previously used steps with smaller triangles (specifically ?ACD and ?BCE), we arrive at a geometric series whose sum yields the result we sought.

Thank you for reading! Your constructive feedback is always appreciated!

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