How Archimedes used infinitesimals
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The ancient Greek mathematician, physicist, and engineer Archimedes of Syracuse was the first mathematician to make explicit use of infinitesimals. His work with infinitesimals is found in the celebrated Archimedes Palimpsest. The palimpsest embodies Archimedes' account of his "mechanical method", so called because it relies on the concepts of torque exerted on a lever and of center of gravity. Both of those concepts were first introduced by Archimedes.
Ironically, Archimedes disbelieved in the existence of infinitesimals, and therefore said explicitly that his arguments fall short of being finished mathematical proofs.
The proof of the first proposition in the palimpsest appears below.
The first proposition in the palimpsest
The curve in this figure is a parabola.
The points A and B are on the curve. The line AC is parallel to the axis of the parabola. The line BC is tangent to the parabola. The first proposition states:
The area of the triangle ABC is exactly three times the area bounded by the parabola and the secant line AB.
Proof: Let D be the midpoint of AC. The point D is the fulcrum of a lever, which is the line JB. The points J and B are equidistant from the fulcrum. As Archimedes had shown, the center of gravity of the interior of the triangle is at a point I on the "lever" so located that DI:DB = 1:3. Therefore, it suffices to show that if the whole weight of the interior of the triangle rests at I, and the whole weight of the section of the parabola at J, the lever is in equilibrium. If the whole weight of the triangle rests at I, it exerts the same torque on the lever as if the infinitely small weight of every cross-section EH parallel to the axis of the parabola rests at the point G where it intersects the lever. Therefore, it suffices to show that if the weight of that cross-section rests at G and the weight of the cross-section EF of the section of the parabola rests at J, then the lever is in equilibrium. In other words, it suffices to show that EF:GD = EH:JD. That is equivalent to EF:DG = EH:DB. And that is equivalent to EF:EH = AE:AB. But that is just the equation of the parabola. Q.E.D..
Other propositions in the palimpsest
A series of other propositions of geometry are proved in the palimpsest by similar arguments. Some of them have the location of a center of gravity as the conclusion. One of those states that the center of gravity of the interior of a hemisphere is located 5/8 of the way from the pole to the center of the sphere.