Calculus

• 1. Introduction

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  In Latin, the word calculus means "pebble". Romans used pebbles as counters on their abacuses, so they called reckoning "placing pebbles", and settled accounts by "calling someone to the pebbles". That humble image of small stones moved across a board sits underneath one of the most powerful tools in all of mathematics. Calculus is the mathematical study of continuous change. John von Neumann called it the first achievement of modern mathematics, and said it is difficult to overestimate its importance. So how did a word for a counting-pebble come to describe velocity, area, planetary orbits, and the pricing of financial options? Why did two men, working separately in the late 17th century, arrive at nearly the same ideas? And why did the very foundations of calculus take roughly 150 years to make rigorous? The answers run from ancient Egypt to the hyperreal numbers of the 1960s.

• 2. Two Branches, One Bridge

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  Differential calculus studies instantaneous rates of change and the slopes of curves. Where elementary geometry gives the slope of a straight line, calculus describes the changing slope of a complicated curve. The central object is the derivative of a function, found through a process called differentiation. In Lagrange's notation, the derivative is marked with a prime, pronounced "f prime" or "f dash". If the input represents time and the output the position of a ball, then the derivative is the velocity of that ball.

  Integral calculus studies the accumulation of quantities and the areas under or between curves. Its definite integral takes a function and returns a number, the algebraic sum of areas between the graph and the x-axis. Consider distance traveled at a steady 50 mph for 3 hours, which yields 150 miles. Plotted against time, that is simply a rectangle, its area equal to the distance. When velocity fluctuates, the area is approximated by stacking many thin rectangles, a construction called a Riemann sum, then taking a limit.

  The fundamental theorem of calculus binds these two branches together. It states that differentiation and integration are inverse operations, relating antiderivatives to definite integrals. Because computing an antiderivative is usually easier than applying the raw definition, the theorem gives a practical route to definite integrals without performing limit processes. This realization, made by both Newton and Leibniz, was key to the proliferation of analytic results once their work became known.

• 3. Ghosts Of Departed Quantities

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  Calculus is usually developed by working with very small quantities. The first method was infinitesimals, objects treated like real numbers but, in some sense, infinitely small. An infinitesimal could be greater than 0 yet less than every term in the sequence 1, one half, one third, and so on, and thus smaller than any positive real number. From this view, calculus is a collection of techniques for manipulating infinitesimals.

  Bishop Berkeley attacked the idea directly. In his 1734 book The Analyst, he famously described infinitesimals as the ghosts of departed quantities. Michel Rolle was another fierce critic. Working out a rigorous foundation occupied mathematicians for much of the century after Newton and Leibniz. Maclaurin tried to prove that using infinitesimals was sound, but a way to avoid mere notions of infinitely small quantities arrived only about 150 years later.

  The escape came through the work of Cauchy and Weierstrass. In the late 19th century, infinitesimals were replaced in academia by the epsilon, delta approach to limits, which captures small-scale behavior using the intrinsic structure of the real number system. Cauchy's Cours d'Analyse held a prototype of an epsilon-delta definition, and Weierstrass formalized the concept of limit. Bernhard Riemann then used these ideas to give a precise definition of the integral, the same Riemann whose name marks the summed rectangles.

• 4. Reviving The Infinite Small

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  Abraham Robinson revived the banished infinitesimal in the 1960s through non-standard analysis. His approach uses machinery from mathematical logic to augment the real number system with infinitesimal and infinite numbers, much as Newton and Leibniz first conceived. The resulting hyperreal numbers permit a Leibniz-like development of the usual rules of calculus.

  Smooth infinitesimal analysis offers a different revival, one that mandates neglecting higher-power infinitesimals during derivations. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as continuous and incapable of being expressed through discrete entities. One consequence is that the law of excluded middle does not hold within it.

  Henri Lebesgue extended the reach of calculus by inventing measure theory, building on earlier developments by Émile Borel, and used it to define integrals of all but the most pathological functions. Laurent Schwartz went further, introducing distributions, which can be used to take the derivative of any function whatsoever. In modern mathematics the foundations of calculus live within real analysis, the field that contains full definitions and proofs of the theorems of calculus.

• 5. Before Newton And Leibniz

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  The Egyptian Moscow papyrus, dated around 1820 BC, already contains calculations of volume and area, one goal of integral calculus, though its formulas are bare instructions with no hint of how they were obtained. Babylonians may have discovered the trapezoidal rule while making astronomical observations of Jupiter.

  Eudoxus of Cnidus, who lived around 390 to 337 BC, laid foundations for integral calculus and foreshadowed the limit. He developed the method of exhaustion to prove the volume formulas for the cone and the pyramid. Archimedes, around 287 BC, pushed the method further during the Hellenistic period. Combining it with a concept of indivisibles, a precursor to infinitesimals, he solved problems now treated by integral calculus, and in The Method of Mechanical Theorems he computed the center of gravity of a solid hemisphere.

  Liu Hui independently rediscovered the method of exhaustion in China in the 3rd century AD to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, established what would later be called Cavalieri's principle to find the volume of a sphere. In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen and active around 965 AD, derived a formula for the sum of fourth powers and used it to calculate the volume of a paraboloid.

  Bhaskara II, who lived from about 1114 to 1185, knew some ideas of differential calculus and suggested that the differential coefficient vanishes at an extremum value of a function. In the 14th century, Madhava of Sangamagrama and the Kerala School of Astronomy and Mathematics stated components of calculus. According to Victor J. Katz, they were not able to combine the differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today.

• 6. The Priority Dispute

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  Johannes Kepler's 1615 work Stereometria Doliorum formed the basis of integral calculus, calculating the area of an ellipse by summing the lengths of many radii drawn from a focus. Bonaventura Cavalieri built on Kepler, arguing that volumes and areas should be computed as sums of infinitesimally thin cross-sections. Pierre de Fermat, claiming he borrowed from Diophantus, introduced adequality, equality up to an infinitesimal error term. John Wallis, Isaac Barrow, and James Gregory then combined these threads, the latter two proving part of the fundamental theorem of calculus around 1670.

  Newton derived his results first in 1665-1666, later published in his Method of Fluxions, and he called his calculus the science of fluxions, a term that endured in English schools into the 19th century. He used calculus to attack the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a cycloid, and the problem of planetary motion, all discussed in his Principia Mathematica of 1687. Leibniz, by contrast, published his Nova Methodus pro Maximis et Minimis first, in 1684, and put painstaking effort into his choices of notation.

  The controversy was bitter. Newton originally accused Leibniz of plagiarism, claiming he stole ideas from unpublished notes Newton had shared with a few members of the Royal Society. The dispute divided English-speaking mathematicians from continental Europeans for many years, to the detriment of English mathematics. A careful examination of the papers shows they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. The difficulty in tracing influence is sharpened by what Leibniz may have learned from the work of Isaac Barrow. One of the first and most complete works on both infinitesimal and integral calculus was written in 1748 by Maria Gaetana Agnesi.

• 7. The Basic Instrument

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  Calculus has been called the basic instrument of physical science, and physics makes particular use of it. All concepts in classical mechanics and electromagnetism are related through calculus. Newton's second law of motion states that the derivative of an object's momentum with respect to time equals the net force upon it. Starting from how an object is accelerating, one uses calculus to derive its path. Maxwell's theory of electromagnetism and Einstein's theory of general relativity are both expressed in the language of differential calculus, and differential equations are prominent in quantum mechanics.

  Vector calculus, a branch of multivariable calculus, handles the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space. It is used extensively in the description of electromagnetic fields, gravitational fields, and fluid flow. Spacecraft use a variation of the Euler method to approximate curved courses within zero-gravity environments. Differential equations, which relate unknown functions to their derivatives, appear across engineering, physics, economics, and biology.

  Chemistry uses calculus in determining reaction rates and studying radioactive decay, and one differential equation describes exponential decay. In biology, population dynamics starts with reproduction and death rates to model population changes. In medicine, calculus predicts the optimal branching angle of a blood vessel to maximize flow, tracks how quickly a drug is eliminated from a body, and models how quickly a cancerous tumor grows. In economics, it locates maximal profit through marginal cost and marginal revenue, and the Black-Scholes model of option pricing employs a differential equation. Calculus also resolves ancient puzzles: the Greek philosopher Zeno of Elea posed famous paradoxes of motion and infinite sums, and the limit and the infinite series are the tools that dissolve them.