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The invention of Differential and Integral Calculus is usually attributed to Isaac Newton and Gottfried Leibnitz , who arrived at the same conclusions independently in the late 17th Century. Archimedes, the famous Greek Mathematician, was born in Syracuse, Sicily almost 2000 years before Newton and Leibnitz made their revolutionary discoveries.

Archimedes lived in the 3rd century B.C. and is widely credited for arriving at an approximation of the value of Pi, a number any student of engineering will tell you is instrumental in their studies. Pi is the ratio of the circumference of a circle to the diameter, and was approximated by Archimedes to be 3.14. What is interesting is how he arrived at his approximation.

Archimedes took his circle (a) and drew 3 chords of equal length to create a triangle (b) inside the circle. Inside each chord, he drew another triangle by bisecting the arc and drawing lines from that point to the endpoints of the chord (c). Since he was using straight lines, he could determine their lengths using Euclidean geometry, and approximate the circumference of the circle (d). (it is believed that Archimedes studied under the successors of Euclid in Alexandria, Egypt). To improve the accuracy of this approximation, Archimedes drew triangles on the chords as described a total of 19 times.

Archimedes obtained another approximation of the circumference by drawing 3 tangents lines of equal length to create a triangle on the outside of the circle, and by continuing his iterative process of drawing tangents inside of these triangles. A much better approximation was obtained by taking the average of the inner and outer linear approximations. Once he had a good approximation of the circumference of the circle, dividing that value by the diameter of the circle gave him the value of Pi, and the circumference of any other circle, provided he knew the diameter (or radius).

C = 2 π R

Did Archimedes make the connection, and ponder the fact that by continuing his process an infinite number of times, he would have arrived at the exact value of the circumference of the circle? If so, he would have been thinking less in terms of Euclidean Geometry and more in terms of Calculus!

* * *

The first day of class in a college level Calculus course begins with a discussion of the Derivative, the basic element of Differential Calculus. The definition of the Derivative uses a mathematical construct called the Limit. The Limit is a device that gives mathematicians the ability to express things in terms of infinity, a concept human beings are not capable of fully understanding.

Take for example the number one no-no of computer programming, division by zero. The easiest way to cause a computer program to croak is to have it divide by zero. The computer can't comprehend infinity either. Given the expression 1/x (1 divided by x, or 1 over x), as x gets smaller and smaller, 1/x gets larger and larger ( 1/2 is larger than 1/100). As x approaches zero, the expression 1/x approaches infinity. Mathematicians use the Limit to describe this situation:

The Limit, as X approached zero, of one over X, is equal to infinity. This might seem a convoluted way of describing things, but it begins to makes more sense (or less, depending on your point of view) with the definition of the Derivative:

This is the definition of the Derivative of the function F(x), expressed in terms of a Limit.

* * *

Having just blasted through the first week of Calculus 101 in a few paragraphs, we are now a little better prepared to discuss whether Archimedes was close to, or had in fact had invented Calculus. Returning to Archimedes' approximation of Pi, we can state the following: "The Limit, as the number of graphical arc bisections approaches infinity, of Archimedes' circumference approximation method, is equal to the exact circumference of the circle".

This alone is evidence that Archimedes could have been thinking more in terms aligned with modern Calculus than to what is generally credited to himself and men of his age. There is more evidence. The word method is intentionally highlighted in the previous paragraph to call attention to a work of Archimedes that was lost to the world until just recently, "The Method".

Many medieval European religious manuscripts were made from what are called "palimpsests". Palimpsests are books from antiquity which have been erased in order to be used for something else, usually prayer books. Sometime around the year 1200 A.D. a copy of Archimedes' "The Method" was overwritten by Christian Monks, and his work was lost until 1906 when it turned up at The Church of the Holy Sepulchre in Istanbul. The Danish philologist Johan Ludvig Heiberg found the palimpsest and photographed each page, publishing his findings. The manuscript is lost again as an amateur French collector buys the book and store's it in his family's Parisian home.

In 1998 the French family sells the Archimedes Palimpsest at auction for two million dollars. The new owner lends the book to the Walters Art Museum in Baltimore, Maryland, where a team of modern forensic and computer scientists attempt to decipher the lost text. What is emerging is a treatise on Archimedes' philosophies on mathematics, including his understanding of the infinite.

Did Archimedes invent Calculus? If not, how close did he come? Why did it take another 2000 years before mathematics confronted the concept of infinity, with ideas like Integrals, Derivatives, and Limits?

For more on the Archimedes Palimpsest, see the PBS documentary on Nova.

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