Brittney Miller Math 5010

To compute the area of circles, mathematicians living in 250 BCE, such as Archimedes, used polygons to approximate the area of circles. While mathematicians of the time did not know how to calculate the area of curved shapes, they could calculate the areas of large polygons by breaking them up into triangles. This was called the “Method of exhaustion” and when calculated with polygons with large numbers of sides, they got closer and closer to approximating the true area of the circle. (Rosenthal). This was essentially taking the limit as the approximation gets closer and closer to the true area. In the book, “The Historical Development of the Calculus”, C.H. Edwards, Jr. explains, “ The so-called method of exhaustion was devised...to provide a rigorous alternative to merely taking a vague and unexplained limit...The Greeks studiously avoided ‘taking the limit’ explicitly, and this virtual ‘horror of the infinite’ is probably responsible for the logical clarity of the method of exhaustion”. Archimedes took this method and created the “ method of compression” (Edwards, 1979). Instead of just dealing with inscribed polygons, he used both inscribed and circumscribed polygons. Then he “compressed” the area of the circle into the areas of the inscribed and circumscribed polygons to get a much closer approximation. The following picture from Edwards’s book illustrates this point:

In the book, “The History of The Calculus and its Conceptual Development”, Carl B Boyer explains that while Aristotle’s understanding on the infinite and the continuous do not align with the current notions in mathematics, his ideas on the indivisible are closely aligned to our modern-day understanding. Aristotle’s approach to understanding continuous magnitude coincided with the nature of motion which led him to Zeno’s paradox.  (Boyer, 2012).

Zeno’s Dichotomy Paradox loosely states that a person attempting to complete a path (such as a walk towards a wall) will never get to the wall. This is because, to get to the wall, they must start by walking half the distance to the wall, then walk half the distance again, and again and again. If the wall was originally 10 or 10⁄₁ feet away, after one iteration, the person would 5 feet, or 10⁄₂  feet away. Following that, the next five iterations would be 2.5 or 10⁄₄, 1.25 or 10⁄₈, 0.625 or 10⁄₁₆, 0.313 or 10⁄₃₂, and then 0.156 or 10⁄₆₄ feet away. After another 5 iterations, the person will be .0048828 feet away from the wall. At this point, the person is pressed up against the wall- but using the logic that the person still has half of a distance to go to the wall, the person will never reach the wall. Carl Boyer explained, “Zeno’s appeal to the infinite was based upon the supposed inconceivability of the notion of completing in a finite time an infinite number of steps.” (Boyer, 2012). Thus, at this point in time, we still did not have a concrete concept of infinity.
The following applet shows how ancient Greeks approximated areas of circles.

By the year 1612, mathematicians such as Johannes Kepler and Bonaventura Cavalieri began to think of shapes and lines as made up of infinite numbers of triangles, points, lines, and surfaces. (Bardi, 2010). This eventually laid the foundation for mathematicians to form a more concrete definition of a limit and infinity which paved the way for new mathematicians to formally invent calculus. René Descartes also paved the way for calculus in what Jason Socrates Bardi, author of “The Calculus Wars'', said was, “perhaps the most major contribution to mathematics since the time of the Greeks, when he invented analytical geometry”. He showed that things in geometric form (such as lines, shapes, and surfaces) can not only be graphed, but also represented algebraically. This tied together geometrical shapes and mathematical equations.

Pierre de Fermat was a French mathematician born in 1607 (Boyer). Fermat studied curves and equations and was eventually able to create a procedure that was equivalent to differentiation. With this procedure, he was able to find maximums, minimums, points of inflection, and more. He also invented formulas that allowed him to take the area under a bounded curve. His process for finding areas under curves are virtually the same as the current process of using summations for finding integrals. He was the first to make significant progress in the calculus world. While many mathematicians and philosophers had contributed to the idea of calculus and indeed laid the foundation for it, it is Leibniz and Newton who typically get the credit for inventing  it.

Isaac Newton, born in 1643, essentially created calculus out of a necessity to be able to describe and use physics. He called it “his method of fluxions and fluents” (Bardi, 2010). While the famous story of him getting hit in the head by a falling apple may not be true, he was indeed fascinated by how an object speeds up on descent. In 1668, Newton was a student at Cambridge University and was finding tangent lines to curves, instantaneous rates of change, curvature, areas, and had other breakthroughs in calculus. The trouble is, as Newton was inventing calculus, he was unable to publish his breakthroughs. This was because of the great fire of London, two years prior, that wiped out most printing shops and drove up the costs associated with the printers. Newton, now twenty-eight years old, was a professor at Cambridge and already had a wide variety of study subjects such as physics, astronomy, mathematics, and theology.

Gottfried Leibniz, born in 1646, was also inventing calculus at this time. He and Newton ended up both working as professors and even shared a few letters between them. However, in the letters, neither mentioned calculus and their invention that could be used to solve infinite series problems. As more details come to light about calculus, tension arose between Leibniz and Newton. Isaac Newton is quoted as saying,“Whether Mr. Leibniz invented it after me, or had it from me, is a question of no consequence, for second inventors have no rights”. Through the use of older letters, journals, publications, and luck, both Leibniz and Newton were eventually able to prove that they both invented calculus independently of one another. Newton was the first to begin inventing calculus while Leibniz was the first to publish. Leibniz created the notation we currently use in calculus. He is credited in using integral calculus for the first time in history to find the area under the graph of a function. He is also credited in establishing the notation "∂" for differential and "∫" as the integral sign, as it represents a summation. (Bardi, 2010). Both inventors created a tool that connects the qualitative to the quantitative. As Boyer says, “The calculus is without doubt the greatest aid we have to discovery and appreciation of physical truth”.

Many years later, Bernhard Riemann was born in 1826. Riemann studied and wrote on a variety of topics, including the study of geometry. He tied together calculus and geometry in a manner similar to the Greeks. However, he extended the geometry by applying calculus to curves, surfaces, and even spaces of dimension. In one of its simpler forms, Riemman’s work can be used to approximate areas under a curve.