The word “calculus” is related to the words “calculation” and “calculate,” but the root of all of these words is the Latin root meaning “pebble” [29]. It also can mean “stone formed in a body” [7], for instance “calculi”, which are small stones formed in the gallbladder, kidney, or other organs [29]. In the field of mathematics, calculus has two main components: differential calculus and integral calculus. Though the typical progression in modern day schools is to teach differentiation before integration, they were actually established in the opposite order, first from attempting to calculate area of irregular shapes, then out of a need to describe continuous change [7]. Differential calculus is most commonly used to optimize scenarios. Some financial applications are minimizing cost and maximizing profit. Integral calculus is often associated with the area under a curve on the plane, but it has many real-world applications as well such as statistical predictions. Rigorous calculus connected algebra and geometry topics in a way that had not been done before [7] and formally described infinitesimals for the first time in history [29]. This paper highlights the major contributions in the study of calculus from the ancient world to the start of real analysis in the 19th century.
The foundational ideas necessary to the development of calculus began in ancient Greece with Pythagoras, Euclid, Eudoxus, and Archimedes. Pythagoras lived from about 569 BC to about 475 BC who left behind no writings due to the secrecy of the religious-mathematical society he led. His beliefs were the following: reality is mathematical, philosophy can purify one’s soul, the soul can rise to be with the divine, certain symbols are mystical, and all brothers in his society should swear to loyalty and secrecy. Though the Babylonians used what is called Pythagoras’ Theorem thousands of years before him, Pythagoras is most likely the first to prove it [25].
Unlike Pythagoras, who left nothing behind, Euclid left behind one of the most famous ancient mathematical texts called Elements [16]. This ancient codex is made of 13 books and contains definitions, postulates, and axioms, several of which are still used today. The first six books describe plane geometry, such as properties of parallelograms and circles. Books seven through nine deal with number theory, containing algorithms and geometrical progressions. Euclid wrote the tenth book to merge Theaetetus’ work on irrational numbers with new definitions from Eudoxus. The last three books are about three-dimensional geometry, including proportions relating the diameters of circles and spheres with their areas, respectively. The method Euclid used to prove this and many other theorems in Elements was that of Eudoxus, called the “method of exhaustion” [13]. This was the closest concept to modern day limits at the time. It started with applying rectilinear area to circular area, was later used to explain the geometry of curves [16], and eventually recognized as an early form of integration [9].
Perhaps the most important Greek contributor to the topic of calculus was Archimedes, who likely studied under students of Euclid. Though he invented machines of war that helped defend Syracuse from the Romans, he was primarily interested in pure mathematics [9]. His work with geometric series implies his understanding of the concept of limits [5]. He also systematically found the area under a curve using the sum of infinite rectangles. One groundbreaking discovery of Archimedes that is named after him and still used in modern day analysis is the Archimedean Property, which states that for every element there exists an element such that Mathematics owes the concepts of infinite summations of areas and vigorous proof to these classic Greeks.
Building off of the logic of the Greeks, European mathematicians of the early 17th century were able to pave more of the road towards calculus. The group most influential in this period and who eventually helped establish the French Academy in 1666, were Marin Mersenne, René Descartes, Pierre de Fermat, Blaise Pascal, and Christiaan Huygens. Mersenne acted as the hub of the group, receiving, copying, and distributing new results to the entire group. Descartes essentially founded analytic geometry, forming monumental connections between geometry and algebra through use of what is now known as the Cartesian coordinate system [28]. Fermat, a lawyer and politician of the time, considered mathematics as a hobby. He is most known for “Fermat’s Last Theorem” [28]. His largest contributions to modern day calculus was his work involving areas bounded by curves done through a summation process [3] and his use of the tangent line parallel to the x-axis in his investigation of maxima and minima [8]. Fermat discussed properties of probability with Huygens, but steered more towards number theory, creating a method of infinite descent to prove that every prime of the form could be written as the sum of two squares [23]. Fermat also corresponded closely with Pascal to develop what is known as the calculus of probabilities [30], though Huygens was the first to publish a work on the subject [12]. Huygens focus on proof influenced the later contributions of Leibniz [8]. Pascal’s work with cycloids resemble methods of modern integral calculus and his work with the Archimedean spiral helped lay the groundwork for Newton’s infinitesimal calculus [28].
Bonaventura Cavalieri and Gilles Roberval were two other important contributors during the same time period as the group of Frenchmen above. They made steps toward integration, each in a different way from the other. Cavalieri viewed the area as the sum of lines (see image below [2]).
Roberval viewed the area between a line and a curve as the sum of infinitely narrow rectangles. He proved, more rigorously, an approximate value of Cavalieri’s integral. [8].
One of the largest controversies in the history of mathematics is that between Sir Isaac Newton and Gottfried Leibniz over the official discovery of calculus [6]. Not only was this a battle for prestige in the field during the Age of Enlightenment, it was also a matter of national pride, Newton being English and Leibniz being German. They remarkably invented calculus independently no more than a decade apart [6]. However, there is evidence that Newton had arrived there first. Isaac Barrow, Newton’s teacher, gave a method of tangents to a curve and clearly understood the inverse relationship between differentiation and integration, but Newton was the officially the first to explicitly state the Fundamental Theorem of Calculus. Newton did not immediately receive recognition for this feat due to publication issues [8]. Leibniz is credited with the first published work on the subject of calculus in October of 1684, the article titled “Nova Methodus pro Maximis et Minimis,” which translates to “New method for the maximum and minimum, and also tangents,…, and a singular type of calculus for them” [4]. Though he also corresponded with Barrow, Leibniz’s approach to calculus was different than Newton’s. While Newton considered variables changing over time in terms of motion, Leibniz focused on infinitely small differences between successive values [8]. Leibniz is quoted saying, “It is unworthy of excellent men, to lose hours like slaves in the labor of calculation…My new calculus…offers truth by a kind of analysis and without any effort of imagination” [1]. Though both Newton and Leibniz were undoubtedly influenced by Barrow, Newton’s findings stemmed from his work with fluxions, concentrating primarily on the relationship between distance and velocity [27]. Leibniz’s discovery came from the ideas of infinitesimal sums, similar to Cavalieri and Huygens. He had a much clearer notation for differential and integral calculus which is still used to this day. Both Newton and Leibniz, however, thought about their results graphically rather than in terms of functions [8].
Jacob and Johann Bernoulli, Swiss brothers, studied Leibniz’ papers and Johann understood them well enough to lecture on Leibniz’ calculus in Geneva and Paris [18]. Met by journal debates from Brook Taylor [11], Johann Bernoulli investigated integration of differential equations. Other than meeting Bernoulli on a level playing field, some of his other contributions include the creation of calculus of finite differences, the introduction of integration by parts, and his famous series known as Taylor’s expansion [11]. Johann Bernoulli ended up teaching de l’Hôpital during his time in Paris, who he continued correspondence with even after returning home to Basel, Switzerland. De l’Hôpital gave little credit to his teacher when he published many of his lessons in his first calculus book in 1696 [18].
Another student of Johann Bernoulli, Leonhard Euler, offered substantial contributions to the study of calculus. Euler had a depth of knowledge in calculus from Bernoulli’s lessons and his Master’s analysis of ideas of Descartes and Newton. He connected Newton and Leibniz’s calculus methods in his work of mathematical analysis. He created beta and gamma functions as well as integrating factors for differential equations. Modern mathematics continues to use Euler’s notation for functions, summation, and finite differences.
A young math-minded Italian named Joseph Lagrange wrote Euler several letters that impressed him enough to propose that Lagrange be elected into the Berlin Academy. He served for a year then helped establish a scientific society in Turin, later called the Royal Academy of Sciences of Turin. Within this society, Lagrange wrote papers about his results on the calculus of variations and the calculus of probabilities. He made applications to the sciences such as field mechanics and the orbits of Jupiter and Saturn [22].
Pierre-Simon Laplace, born only 13 years after Lagrange, was similarly an eager young man recognized for his mathematical potential in France following the French Revolution [5]. His papers made improvements to Lagrange’s methods and stated several new ideas regarding maxima and minima. Like Lagrange, he also applied his findings to the study of the motions of the planets [24]. One of Laplace’s students, Jean Baptiste Joseph Fourier, pioneered the representation of expansions of functions as trigonometrical series, now known as Fourier series [17].
Shortly following Lagrange, Laplace, and Fourier, Johann Carl Friedrich Gauss, despite losing his father, wife, and son within a short amount of time, made substantial steps regarding differential equations, elliptic orbits, and conic sections in a two-volume treatise on the motion of celestial bodies in 1809 [19]. Gauss unveiled many laws and theorems which helped to shift calculus from heuristic methods to sophisticated proof [7].
Augustin Louis Cauchy, born near France in 1789, followed in Gauss’ footsteps by paying careful attention to strict proofs. He was not liked by his contemporaries due to his religious views and determination to connect religion to his scientific work [10]. One of his most recognized feat was the first proof of the convergence of a Taylor series. He worked diligently with limits, derivatives, and integrals [5] during the majority of his life, with exception of a short two-year break due to political unrest in 1830 [10]. Though Cauchy criticized others for not having rigorous enough work with Fourier series, Dirichlet is considered the founder of the theory of Fourier series since he found errors in Cauchy’s work and created the Dirichlet Integral which correctly gives the th partial sum of the Fourier series. Dirichlet also improved Laplace’s developments in proving the stability of the solar system without using series expansion [20]. Cauchy also had a strong influence on the French mathematician Joseph Liouville, who coincidentally worked closely with Dirichlet until his death in 1959. Liouville created what is now called fractional calculus. He also analyzed the criteria needed for integrals of algebraic functions to be algebraic. He studied linear second order differential equations, the properties of eigenvalues, the behavior of eigenfunctions, and the series expansion of arbitrary functions with Charles-François Sturm and they together came up with the Sturm-Liouville theory for solving integral equations. He also contributed to differential geometry and conformal transformations [21].
The last major contributions to the topic of calculus were made by Bernhard Riemann. He studied under Gauss and Dirichlet, replacing Dirichlet’s as the chair of mathematics at Göttingen, then elected to the Berlin Academy of Sciences [15]. Riemann found the area under a curve using rectangles, called a Riemann sum. His definition of an integral, which takes the limit of the Riemann sum as the rectangle widths grow increasingly smaller remains one used by almost all textbooks today [5]. Though on the shy side, Riemann’s ideas catapulted the study of calculus into the topic of real analysis.
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