1 History of Calculus

The history of Calculus can be separated into three key periods: the anticipation period, the developmental period, and the rigorization period.

The anticipation period was a time of basic discovery, where the first fundamentals were established that would later lead to the revolutionary processes developed by Sir Isaac Newton and Gottfried Leibniz during the developmental period [11]. Dating back to 1650 B.C. the anticipation period begins with an Ancient Egyptian papyrus scroll. Considered to be one of the oldest mathematical texts in the world, the scroll was first purchased by Henry Rhind in 1858. Now known as the Rhind Papyrus, it is said to contain some of the first fractions and number theories in recorded history.

One of the first transcriptions of the Rhind Papyrus, published by Professor A. Eisenlohr in 1877  [39], contained

This was one of the first signs of a civilization having a knowledge of rational numbers. If we look beyond any ancient text we can observe mathematical advancements in Ancient Egypt through the large rectangular prism structures we know as pyramids. The longevity of the pyramids gives us great insight into the Egyptians knowledge of volumes and engineering.

More small discoveries can be traced through the year following the Ancient Egyptians. In 1300 A.D. the first graph of a function can be found in the works of Nicole Oresme, who believed that the total area found under this curve was also the total distance covered by the object represented by that function [10].

In 1630 A.D. Bonaventura Cavalieri, an Italian mathematics professor at the University of Bologna, developed and later published his discoveries in analytic geometry and indivisibles, titled A Certain Method for the Development of a New Geometry of Continuous indivisibles. Caveliera also introduced the use of logarithms as a method of computation which was published in his work A General Directory of Uranometry. His work would later be seen as a 17th century version of the integration we use today [27].

It could be debated that the next major contribution could be introduced in the developmental stage as his discoveries were vital to the methods and theorems produced in the developmental era of calculus. In the early 17th century Pierre De Fermat, the creator of the number theory later named the Fermat numbers, introduced his own methods for determining the area under a curve for a power function which led the way for the development of integration. Fermat also published a work titled Method for Determining Maxima and Minima and Tangents for Curved Lines which introduced a new method to the 17th century, recognized as one of the first methods of differentiation [33].

We now come to what most consider the true development stage of present day calculus. In calculus classes we learn that, to mathematicians, Sir Isaac Newton and Gottfried Wilhelm Leibniz are considered the fathers of modern calculus. Having made similar discoveries in the same time frame this title, claimed by both, has long been the topic of a heated debate. Although their discoveries and concepts are roughly the same, the methods they developed for solving problems differ in process from each other.

We begin with Sir Isaac Newton. Born in England on December 25, 1642, Newton’s approach to calculations were very geometric and entirely for personal development. This conclusion is supported by many observations made on his many works which contain a wide array of notations, lacking the consistency and keys used in published works for the intent of universal comprehension [11].

Newton developed his work Method of Fluxions and Infinite Series where calculations were thought of in terms of motion where time was constantly moving. As modern calculus uses variables, Newton called them fluent which again relates to his emphasis on movement in his concept of derivatives. Newton’s incorporation of physics led to his concept that a derivative is taken with respect to time and the derivative of is considered a moment of which leads to a change in velocity of over small periods of time [24]. Newton also believed in solving for 0 and considered small intervals of time to be insignificant.

Gottfried Wilhelm von Leibniz, born in Leipzig, Germany on July 1, 1646, approached his calculations in terms of sums and differences. Leibniz also introduced notations for integration, which we will see later on in the course. Leibniz concluded that in order to find the tangent line to a curve one only needed to find the ratio for the change in rise, , over the change in run, . This ratio was noted as as Leibniz believed that a tangent line between two points could be calculated in minute increments of and denoted and , respectively, giving us a triangle made of sides , , and the slope of tangent line [24]. This would later lead to the discovery of Leibniz’s version of the power rule that he denoted as .

By studying Newton’s and Leibniz’s methods for differentiation, it would be immediately clear that their methods are quite different. But although they differed greatly in approach, both mathematicians would come to the same conclusions. It should be noted that Leibniz’ version of differentiation is far closer to modern differentiation that of Newton’s. Leibniz also created formal rules for integration that we use today. We will be introduced to many of these in our coming section on integration.

These differing methods caused a major dispute between the two men as Newton believed Leibniz had stolen his ideas to be published as Leibniz’s own works [24]. Leibniz and Newton were later individually credited for their independent discoveries, unfortunately far after Leibniz’s death.

Despite any rifts between mathematicians or conflicting discoveries the importance of calculus can be seen throughout our day to day lives. Calculus can be found in economics, physics, medicine, engineering, and math modeling. Calculus can be broken into two parts: Differential calculus and Integral calculus. Both will be discussed in depth later on [5].

For now differential calculus can be described as the branch of mathematics involved with derivatives with the purpose to observe the behavior of variables and rate at which variables change.  We can begin with a given position of an object and whose first derivative gives you the velocity, or rate of change, of that object at that given position. Examples of this type of differentiation can be seen in studying the spread of disease in order to calculate the rate at which the disease is affecting a population. We can also see this in chemical reactions, marginal costs and revenues to name a few.

The purpose of integration is to find the anti-derivative of a function. Real world situations involving anti-derivatives are the same examples as given before but used in reverse. Instead of finding the rate of change of a quantity we are given a rate of change and by finding the anti-derivative of said rate we are able to find the original value at a given time. You may start with a rate at which an infection spreads for a sample population and through integration you are able to find the population of infected people at a given time within a margin of error. The one obvious downfall of integration is unless you know the position or population value at a given time your calculations using the anti-derivative may have a larger margin of error than if you were to have a given value with which you will be able to complete the anti-derivative, by finding the value for a constant in the anti-derivative function.

To conclude, it took thousands of years to develop the meticulous calculations we are introduced to in calculus I through III and which we apply, every day, to help develop new technologies, study populations, research diseases, run small and large economies, and so much more. We will now discuss more in depth the fundamentals that make up the study of calculus.