Section 1.1 Introduction and History of Calculus
Subsection 1.1.1 Introduction to Calculus
Calculus involves two main problems:
- Finding the slope of a curve, or the tangent line to a curve.
- Finding the areas of regions under a curve in the plane, or more generally volumes.
The very first calculus-like ideas were formulated by Archimedes, ancient Greek mathematician (among other things). Archimedes used some of the foundational ideas of calculus to determine areas of curved regions, such as the area of a parabolic segment. However, the methods used by Archimedes were very ad-hoc, requiring subtle arguments for each specific situation.
In the 1600s, developments in mathematics were introduced to the European world. First, algebra, developed in the middle east by Arab mathematicians, were introduced to Europe. Also, Cartesian geometry (also called analytic geometry), named after French philosopher and mathematician Rene Descartes (1596-1650), allowed for geometric curves to be studied with a coordinate system (Cartesian coordinates), rather than the synthetic geometry of previous civilizations such as the ancient Greeks. The combination of algebra and Cartesian geometry led to the modern conceptualization of functions, graphs of functions, and calculus on these functions.
By themselves, algebra and geometry only allow for the study of relationships between static objects. Calculus provides tools to measure and describe how a quantity changes. The two operations for this are the derivative and the integral, which correspond to the solutions to the two main problems, respectively. Calculus, broadly, is the study of change. It is the study of how quantities change, with respect to time and each other.
Calculus was invented in the 17th century by both Isaac Newton (1643-1727, English mathematician, physicist, and astronomer) and by Gottfried Leibniz (German mathematician). Newton's motivation was to describe the motion of moving objects, forming what is now called classical mechanics (or Newtonian mechanics).
Initially, calculus was revered because its techniques were so useful, in that it worked well to describe phenomena in the world and for solving problems, and it produced many results. In 1673, when Leibniz first introduced the term calculus, he called it “a calculus” or “my calculus”, where in this context, “calculus” meant “method of calculation”, i.e. a system of rules and algorithms for performing computations. After calculus gained popularity, it was called “the calculus”, meaning “the method of calculation”. The algebraic framework of calculus allowed for systematic techniques for computing areas. In some sense, the calculus developed in the 17th century was an extension or generalization of the computations done by Archimedes. It provided routine steps for calculating many results, rather than long and subtle arguments for particular results.
Newton and Leibniz miraculously developed calculus independently of each other, at around the same time. Both accused the other of plagiarism, but their approaches were quite different, and it is currently believed that they truly did discover calculus independently. They built on some minor developments from other mathematicians.
Over the centuries since then, calculus has been refined and build on by many future mathematicians. Today, both differentiation and integration are defined in terms of another mathematical operation called a limit. Basically, limits capture the idea of going to infinity.
Subsection 1.1.2 The Usefulness of Calculus
In the 16th century, the idea that nature was controlled by mathematical laws gained popularity. In 1623, Galileo Galilei (1564-1642), Italian astronomer, physicist, and engineer, wrote in his book “The Assayer”, about “natural philosophy” (the study of the physical universe, what today we would refer to as science),
“Philosophy is written in this grand book - I mean the Universe - which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth.” ―Galileo Galilei
In short, nature is deeply mathematical. Even still, Galileo pre-dated the development of calculus.
In 1960, Eugene Wigner, Hungarian physicist, wrote in his article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”,
“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.” ―Eugene Wigner
In particular, the laws of nature are written in the language of calculus. Richard Feynman (1918-1988) described calculus as “the language that God talks” to Herman Wouk, which led to his 2010 book titled “The Language God Talks”.
On the other hand, while calculus led to many developments in math and science, calculus is not used by almost all people in everyday life.
Subsection 1.1.3 Learning Calculus
Calculus is more than just formulas and methods of computations. Calculus is a system of reasoning, and these ideas of calculus are accessible to most people.
Subsection 1.1.4 The People of Calculus
- Archimedes (287-212 BC), ancient Greek mathematician, physicist, engineer, and astronomer.
- Rene Descartes (1596-1650), French philosopher and mathematician.
- Pierre Fermat (1607-1665), French mathematician.
- John Wallis (1616-1703), English mathematician. Introduced the infinity symbol \(\infty\text{.}\)
- Issac Barrow (1630-1677), Scottish mathematician. Teacher to Isaac Newton.
- James Gregory (1638-1675), Scottish mathematician. Published the first proof of the fundamental theorem of calculus, although from a geometric perspective, and only for certain types of functions. Developed power series representation for trigonometric functions.
- Isaac Newton (1643-1727), English mathematician, physicist, and astronomer.
- Gottfried Leibniz (1646-1716), German mathematician. Proved the fundamental theorem of calculus, introduced the modern integral sign \(\int\text{,}\) and introduced the term “function”.
- Michel Rolle (1652-1719).
- Jacob Bernoulli (1655-1705). Helped develop the integral. Differential equations.
- Leonard Euler (1707-1783). Perhaps the greatest mathematician of all-time, with the possible exception of Archimedes. The analysis text \textit{Introductio in analysin infinitorum} (1748).
- Joseph-Louis Lagrange (1736-1813). Introduced the prime \(f'\) notation for derivatives.
- Karl Freidrick Gauss (1777-1855).
- Bernhard Riemann (1826-1866), German mathematician.
