How long has calculus been around

Like most scientific discoveries, the discovery of calculus did not arise out of a vacuum. In fact, many mathematicians and philosophers going back to ancient times made discoveries relating to calculus.

The ancient Greeks made many discoveries that we would today think of as part of calculus — however, mostly integral calculus, which will be discussed in the module Integration. It is common to credit Archimedes with the earliest stirrings of integral calculus. Among the problems he tackled and solved are finding areas under parabolas and inside spirals, and finding the volume of the sphere, spherical segments, and the parabaloid the solid of revolution obtained by revolving a parabola around its axis.

He also showed how to compute the slope of a line tangent to a spiral, the first glimmer of differential calculus. The beauty of calculus as we now know it comes from its simplicity. The Fundamental Theorem of Calculus enables us to solve very difficult problems by applying simple calculational procedures that are justified by the Fundamental Theorem.

Archimedes did not have these, so he had to rely on basic principles and, with a great deal of ingenuity, come up with clever solutions. Archimedes found volumes of the parabaloid and other solids by using a balancing argument in which he compared the moments of different solids. He calculated the area under a parabola not by the usual method of approximating it by a sum of squares, but by using a geometric observation that enabled him to reduce the problem to finding the sum of a geometric series.

This is one of a number of important uses of geometric series that contributed to the development of calculus. The problem that I want to focus on is that of finding the area inside a spiral.

Archimedes employed what has come to be known as the "method of exhaustion. Archimedes attributes this method to Eudoxus of Cnidus — B.

While Eudoxus found the first proof, this result is even older: it was found by Democritus c. The formula for the volume of a pyramid was also discovered in ancient India, and we have record of it in the Chinese book Chiu Chang Suan Ching Nine Chapters on the Mathematical Art that may have been written as early as B.

Even before anyone could prove this formula, finding it required thinking of the pyramid as made up of thin slices, three sets of which could be reconfigured to make a rectangular block.

Later in this article we'll see how this mental three-dimensional geometry led Archimedes to the formula he needed to find the area of the spiral.

The total area lies between. At this point, Archimedes derived a succinct formula for the sum of the first n - 1 squares:. The formula for the sum of squares may not have been new to Archimedes, and there is evidence that it might have been discovered about the same time in India. We do know that it was rediscovered many times. The earliest proofs, including Archimedes's proof, are all geometric.

We can visualize the sum of squares as a pyramid built from cubes Figure 8. In other words, he showed that. The story is that he discovered it when his teacher ordered him to add the integers from 1 through In fact, it is an ancient formula. It can be found, for example, in India in a Jain manuscript from B. It has a very simple geometric proof as shown in Figure 9.

We still have to prove equation 2 , but once we know it is true, we can combine it with the formula for the sum of the first n - 1 integers to get. We split this rectangular pyramid into two pieces Figure These pieces fit around the green pyramid in Figure 8 so that one more inverted pyramid completes the n x n x n - 1 block Figure 12 :.

Figure The n x n x n - 1 block assembled from three sums of squares and one sum of integers. You might think that having seen how useful the sum of squares formula is, Archimedes would then have found the formula for the sum of cubes. He didn't. It took over a thousand years before anyone did. The problem is that sums of squares are easy to see geometrically. Sums of cubes can be visualized, but the object you want to put them together to form is four-dimensional.

In the first two parts of this article, we saw how Archimedes and ibn al-Haytham used formulas for sums of powers to evaluate areas and volumes.

Starting in the 11th century, Arab, Chinese, and Indian mathematicians began to discover techniques that would enable them to find the area under any polynomial. But before anyone could discover such formulas, they had to invent polynomials. Quadratic and cubic polynomials had existed for well over a thousand years, but expressed as areas and volumes. It was not even clear what a fourth power would mean.

Higher-degree polynomials emerged almost simultaneously around the year in the Middle East, India, and China. Histoire Sci. J O Fleckenstein, The line of descent of the infinitesimal calculus in the history of ideas, Arch. Torino 46 1 , 1 - T Guitard, On an episode in the history of the integral calculus, Historia Mathematica 14 2 , - P Kitcher, Fluxions, limits, and infinite littlenesse : A study of Newton's presentation of the calculus, Isis 64 , 33 - A Nikolic, Space and time in the apparatus of infinitesimal calculus, Zb.

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