Hypatia is estimated to have been born in Alexandria around 370 AD [4]. Interestingly, there is more known about her death than her life [5]. She is the first woman mathematician who we have evidence of [4]. Her commentaries on older works helped them survive and be better understood for years after.

Her story begins with her father, Theon of Alexandria, who was a famous mathematician and philosopher. He was the director of the library of Alexandria during its peak and raised his daughter, Hypatia, in an environment filled with education and thought [4]. They had a strong bond and shared passions [6]. Though her father desired for her to become as well-rounded as possible, he focused mostly on mathematics and astronomy [4]. Hypatia studied astronomy, astrology, religion, and mathematics, and kept a strict physical routine to maintain a healthy body [6]. She took the various teachings of her father to a new level by studying philosophy and in about 400 AD she became head of the Platonist school at Alexandria [4]. She was viewed highly for her way of teaching and respectable morals [9].

As a teacher, she focused primarily on mathematics and the philosophy of Neoplatonism, based on the ideas of its founders, Plotinus and Iamblichus. This particular ancient Greco-Roman philosophy reflected upon centuries of intellectual culture and merged the scientific and moral theories of Aristotle, Plato, and the Stoics with literature, myth, and religious practice. The Neoplatonists were very idealistic and laid their foundation on the meaning of what is now phrased as “mind over matter.” The stark newness they brought to the realm of philosophy, and perhaps the reason they are separated from the Platonists by the prefix “neo” was their investigation of the beginning of the universe [2]. Hypatia taught these ideas with more scientific background than the former Neoplatonists [4].

Many students traveled from around the world to learn from her and some wrote correspondences from afar [6]. One of her most famous pupils was Synesius of Cyrene, who went on to become the Bishop of Ptolemy. His letters provide greater insight into the life Hypatia lived. He gives her credit for creating two important astronomical devices, the astrolabe and the planesphere. He also refers to her creation of instruments to distill water, measure the level of water, and determine specific gravity of certain liquids [9]. Synesius shows great reverence for Hypatia in these letters, addressing her as “the Philosopher” [10]. Hypatia was indeed well respected in a male-dominated field [1].

Her works are not completely known, but there is evidence that she wrote commentaries and treatises on several important mathematical works [4]. Though many were lost, some remain and it is through these editions that ancient Greek mathematical thought was preserved and more easily studied [1]. The first of these is her commentary on Ptolemy’s Book III of the Almagest, written with her father Theon. In it’s introduction reads “the edition having been prepared by the philosopher, my daughter Hypatia” [1]. The Almagest was the top resource for astronomical study in the world from its writing until 16^th century Copernicus. Another important commentary which she used to teach her students was that on Diophantus’ Arithmetica. Below, [10] explores the general structure of a problem found in Arithmetica in Book I:6.

(a) General problem

Example: To divide a given number into two numbers such that a given fraction of the first number exceeds a given fraction of the second number by a given number.

(b) Necessary condition

Example: It is necessary, that the later given number be smaller than the number obtained when the greater given fraction is taken of the first given number.

(c) Specific values for the given number(s)

Example: Let it be required to divide 100 into two numbers such that the difference of one fourth of the first number and one sixth of the second number be 20 units.

(d) Detailed solution in paragraph form followed by careful check that the found quantities indeed satisfy the given conditions

Example: Let one sixth of the second number be 1 arithmos, then this number will be 6 arithmoi. From then on, one forth of the first number will be 1 7 Not all problems have necessary condition. Some necessary conditions prevent having negative solutions since Diophantus avoided these as being absurd numbers, such as the condition in I:6; others are genuinely necessary for the proper solution of the equation, such as the condition in V:11. 4 arithmos plus 20 units; so the second number will be 4 arithmoi plus 80 units. Moreover, we want the numbers added together to produce 100 units. Now, these two numbers added together form 10 arithmoi plus 80 units, which equal to 100 units. Subtract like from like: it remains that 10 arithmoi are equal to 20 units, and the arithmos becomes 2 units. Returning to our conditions, it was proposed that one sixth of the second number be 1 arithmos, which is 2 units, then the second number will be 12 units. On the other hand, since one fourth of the first number is 1 arithmos plus 20 units, which is equal to 22 units, then the first number will be 88 units. From then on, it is established that one fourth of the first number exceeds one sixth of the second number by 20 units and that the sum of the required numbers is the given number.

This work on algebraic methods veers from the typical focus of Greek mathematics on topics relating to geometry and number theory. For this reason, it demonstrates Hypatia’s versatility and rank in the field [9]. Her comments likely brought this material to a level where more students could grasp it.

Apollonius also studied in Alexandria and developed the important work, Conics, which Hypatia also wrote and taught about. This was again heavy material that she not only was able to understand but well enough to teach it. The concepts presented in this work influenced Ptolemy’s study of planetary orbits as well as the development of analytical geometry by Descartes and Fermat [5]. It introduced the ideas of hyperbolas, parabolas, and ellipses [6]. The potential of the effect Hypatia’s comments and teachings on this work had in future mathematics is astounding. Hypatia also may have written on Archimedes’ Measurement of the Circle. Arabic manuscripts found in the 20^th century have the careful explanation that matches the style of Hypatia’s teachings [3]. Proposition 3 in Measurement of the Circle that gives a bound for pi, the ratio of the circumference to the diameter of a circle.

It could be argued that mathematical achievements are just as important as their accessibility to others. Hypatia took these works and probably many more important mathematical ideas and presented them in ways that were more easy to digest. She created activities and scenarios that demonstrated the material [6]. It is difficult to know for certain the impact she had on future mathematical achievements. However, one can speculate that if it were not for her efforts the timeline of mathematical history probably would have taken a large hit. For one example, imagine if Diophantus’ ideas had not been so effectively translated and studied leading up to the time of Fermat. Would he have made the same strides that he did in number theory? How would it have affected the topic of calculus? She is linked directly by [6] to the later work of Descartes, Newton, and Leibniz. Anyone who has studied the connections and building nature of mathematical history can understand the amazing role of this woman despite any solid proof. One could even argue that every mathematician who read one of her editions of mathematical works was impacted in some way by her orate skill. 

Unfortunately, the to the religious climate of Alexandria during Hypatia’s life was not ideal. There was a divide between the Christians, led by Bishop Cyril, and the Neoplatonists, led by the prefect Orestes. Cyril was conservative, dogmatic, and power-hungry. He organized a political reprisal against Orestes and created a riot. Hypatia, an easy target due to her fame, outspokenness, and ties to Orestes, was accused of being a heretic [3]. She was taken from her chariot, brutally tortured, and burned. Her story becomes one of a scientific martyr. At a time when most women were occupied with domestic duties, Hypatia is remembered as one of the last great thinkers and teachers of ancient Alexandria [1]. 

References

[1]   B. Bibel, “Meet Hypatia, the ancient mathematician who helped preserve seminal texts,” July 2018, https://massivesci.com/articles/hypatia-math-science-heroes/.

[2]   C. Brown, Christianity & western thought, Apollos, 1 (1990) 34.

[3]   J. Lienhard, Hypatia’s mathematics, The Engines of Our Ingenuity, 215 (1997), https://uh.edu/engines/epi215.htm.

[4]   J. O’Connor & E. Robertson, “Hypatia of Alexandria,” April 1999, https://mathshistory.st-andrews.ac.uk/Biographies/Hypatia/. 

[5]   L. Osen, Women in Mathematics, Cambridge, MA, MIT Press (1974) 21—32. 

[6]   L. Riddle, “Hypatia,” January 2017, https://www.agnesscott.edu/lriddle/women/hypatia.htm. 

[7]   M. Betrand, “Archimedes and Pi,” May 2014, http://nonagon.org/ExLibris/archimedes-pi.

[8]   M. Mercer, “Hypatia,” http://www.math.wichita.edu/history/Women/hypatia.html.

[9]   S. Greenwald & E. Mendez, “Hypatia, the First Known Woman Mathematician,” https://mathsci2.appstate.edu/~sjg/ncctm/activities/hypatia/hypatia.htm. 

[10]   T. Davis, “Forty Two Problems of First Degree from Diophantus’ Arithmetica,” December 2010, http://www.math.wichita.edu/~pparker/classes/Davis_Tinka_FL2010.pdf.

The constant e is one of the most widely important numbers in mathematics. It was in use many years before it was formally recognized. It’s current notation, e, comes from Euler in the early 1700s, but logarithms to base e were first seen in the work of Napier in 1618 [7]. Several other mathematicians made connections to this irrational constant in between Euler and Napier, and there exist many applications in different fields of modern mathematics.

John Napier, a Scottish scholar born in 1550, began the story of the number e. Though he did not uncover the number specifically, he did give a table of natural logarithms [7]. The algebra of the time did not permit use of logarithms how we think of them in the modern day. In fact, Napier did not think of them algebraically at all, but rather as individual numbers that were to be calculated [6]. The process he used is called dynamical analogy. He invented a way of transforming multiplication into addition as modern logarithms do by using two lines each with one moving point. He recognized the property:

if and only if

One large issue with Napier’s logarithms were that the logarithm of 1 was not equal to zero [6].

Henry Briggs was drawn to Napier’s work and its potential towards the field of astronomy [5]. He studied and improved upon Napier’s logarithms. During a scheduled meeting in the summer of 1615, Briggs suggested to Napier that new logarithm tables should be made with base 10 with the logarithm of 1 being equal to zero [6]. Napier agreed but his health issues prevented him from recreating the tables, so Briggs did so and gave due credit to Napier [5]. He is known to have given credit to others when appropriate and was viewed by his peers as very honorable. His publications sparked the use of logarithms by scientists [5].

Another friend of Briggs, Edmund Gunter, published seven figure tables of logarithms of sines and tangents in 1620. The English translation of the work is Canon of Triangles: or Tables of Artificial Sines and Tangents, and it is the first publication of logarithms of trigonometric functions. He is also credited with the invention of the words cosine and cotangent. Specifically regarding logarithms, he came up with a physical device that could multiply numbers based on logarithms using a scale and a pair of dividers, an important foundation for the slide rule [3].

Ten years later, William Oughtred used two Gunter rulers, getting rid of the dividers, and invented a circular slide rule. This device is described in Oughtred’s Circles of Proportion, published in 1632. Oughtred treated logarithms slightly different than his contemporaries. He viewed logarithms as numbers and numbers in general as being continuous, rather than discrete. He also placed great emphasis on the understanding of logarithms instead of procedural calculation [8]. This remains an essential idea in math education.

In the decade after Oughtred’s work with logarithms, Gregorius Saint-Vincent worked with the geometric area under a rectangular hyperbola, which may have had an impact on Huygens’ connection between this and the logarithm [7]. Though the focus was still primarily on logarithms with no explicit recognition of the number e this marks a significant connection and resulting movement towards uncovering e.

The first known approximation of the number e comes from Jacob Bernoulli’s exploration of compound interest. He used the binomial theorem to show that the limit of 

as n approaches infinity lies between 2 and 3. This is also the first time in mathematical history that a number was defined by a limit. Though this approximation was done in 1683, Bernoulli did not make the connection between this number and logarithms [7].

The year 1690 marks the first official and proper use of the number e by Gottfried Leibniz in a letter he sent to Huygens. Though he used the notation b, he was the first to recognize the number itself enough to give it a name [7]. This followed a three-year stint of travel where, though it was not his primary goal, he spent time visiting other elite scholars [4].

Perhaps most important in the development in the number e, and in fact how it gained its current notation was the work of Leonhard Euler [7]. In 1731, in correspondence with Goldbach, Euler writes [15]:

Though a translation is not provided, one can see the very first use of the letter e and statements involving calculus with mention of logarithm.

Euler’s notation stuck, but more significantly, his exploration of the properties of this number led to great discoveries in the fields of number theory and complex analysis. In 1748, he published Indroductio in Analysin infinitorum, in which he showed that

as well as an approximation of the number to 18 decimal places (e=2.718281828459045235), likely by calculating the first 20 terms of the above series [7]. He also gives the continued fraction expansion of the expressions (e-1)/2 and e-1and notes a pattern in each [7]. He related the complex exponential e^(it) to the trigonometric functions cos(t) and sin(t) with the definition e^(it)=cos(t)+isin(t). Of course, he is also responsible for the notation of the notation i for the square root of negative 1 in this equation [11]. Taking this equation at the angle pi gives Euler’s Identity, which remains one of the single most important facts in complex analysis today.

Finding increasing decimal places of the number e was not nearly as popular as it was for the number pi. Still, some mathematicians who tackled it were Shanks, Glaisher, Boorman, and Adams. These calculations occurred in the mid to late 1800s [7]. Applications and connections using the number seem to be of more importance than the number itself.

 Euler proved with his fraction expansion that the number e is in fact irrational. In 1873, Charles Hermite became the first to prove that the number e is not algebraic and therefore transcendental [7]. A similar proof, which was adapted from one by Herstein’s Topics in Algebra, is given by [12]:

One important example of the number e in the financial world today is continually compounding interest, stemming from Bernoulli’s work. In this case, the number e is the factor by which a bank account will increase if it otherwise would have doubled. This factor also applies in the same way to reproducing populations and, more generally, to any quantity that grows at a rate proportionally to its current value at any given time [10].

 Since the topic of derivative investigates instantaneous rate, it comes as no surprise that the number e also brought new developments to the topic of calculus. Due to nature of its derivative, the number e is preferred as the base in exponentials and logarithms [10]. Not only is the derivative of e^x equal to itself, its integral works the same with addition of a constant.

The number e also finds its place in the study of probability and statistics. For example, rolling a n-sided dice n times approaches 1/e as n gets increasingly large, applied from the Bernoulli trial process [14]. This ratio is found other places in probability such as the topic of derangement of  items. The number e also appears in probability mass functions and density functions of Poisson distributions, exponential distributions, gamma distributions, normal distributions, and chi-squared distributions. A few more probability examples involving the number e are given in [1]:

Example 1. Each of two people is given a shuffled deck of playing cards. Simultaneously they expose their first cards. If these cards do not match (for example, two “four of clubs” would be considered a “match”), they proceed to expose their second cards and so forth through the decks. What is the probability of getting through the decks without a single match? […] the answer is given by the sum,

which is the initial portion of a series for 1/e (based on the Maclaurin series below):

This problem has appeared in the literature under the name “Hat-Check Problem,” wherein the following question is asked: If men have their hats randomly returned, what is the probability that none of the men winds up with his own hat?

Example 2. […] If numbers are randomly selected from the interval what is the expected number of selections necessary until the sum of the chosen numbers first exceeds 1? The answer is e. […]

Example 3. The “Secretary Problem” concerns an employer who is about to interview applicants for a secretarial position. At the end of each interview he must decide whether or not this is the applicant he wishes to hire. Should he pass over an interviewee, this person cannot be hired thereafter. If he gets to the last applicant, this person gets the job by default. The goal is to maximize the probability that the person hired is the one most qualified. His strategy will be to decide upon a number to interview the first k applicants, and then to continue interviewing until an applicant more qualified than each of those first k is found. […] the probability of hiring the most qualified applicant is greatest when k/n is approximately 1/e. Moreover, this number 1/e, is in fact the approximate maximum probability. For example, if there are n=30 applicants, the employer should interview 11 (which is approximately 30/e) and then select the first thereafter who is more qualified than all of the first eleven. The probability of obtaining the most qualified applicant is approximately 1/e.

Example 4. With each purchase, a certain fast-food restaurant chain gives away a coin with a picture of a state capitol on it. The object is to collect the entire set of 50 coins. Question: After 50 purchases, what fraction of the set of 50 coins would one expect to have accumulated? […] this fraction is

which is approximately 1-(1/e).

Example 5. A sequence of numbers is generated randomly from the interval [0,1]. The process is continued as long as the sequence is monotonically increasing or monotonically decreasing. What is the expected length of the monotonic sequence? For example, for the sequence beginning .91, .7896, .20132, .41, the length of the monotonic sequence is three. For the sequence beginning .134, .15, .3546, .75, .895, .276, the length of the monotonic sequence is five. The probability that the length L of the monotonic sequence is greater than k is given by

Example 6. A slight revision of the previous example gives a more pleasing answer. First, we require the sequence to be monotonically increasing and, second, in computing the length of the sequence we include the first number that reverses the increasing direction of the sequence. Hence, the sequence beginning .154, .3245, .58, .432 is assigned a score of four and the sequence beginning .6754, .239 is assigned a score of two. Using an argument similar to that of the [preceding] example, it can be shown that the expected score is that ubiquitous and fascinating number, e.

 

Following the solution of the catenary problem explained in [2], Euler was the first to publish the hyperbolic sine and hyperbolic sine expressions

and

respectively, though he does not use the word hyperbolic or give any special notation for them. Lambert later made connections with these and the circular trigonometric functions [2].

An important application in science is Newton’s Law or Cooling/Heating, given by

The example below, found in [9], shows the graph of the temperature of an object cooling over time. It starts at 120F, cools according to the law equation as it approaches the surrounding temperature of 60F.

The number e indeed has a unique history and immense presence all over mathematics. It was founded through the use of logarithms, recognized and named, then applied and connected to countless mathematical ideas onward.

 

References

 [1]   H. Shultz & B. Leonard, Unexpected occurrences of the number e, Math. Mag. 62 (1989) 269—271.

 [2]   J. Barnett, Enter, stage center: The early drama of the hyperbolic functions, Math. Mag. 77 (2004) 15—30.

[3]   J. O’Connor & E. Robertson, “Edmund Gunter,” February 2017, https://mathshistory.st-andrews.ac.uk/Biographies/Gunter/. 

[4]   J. O’Connor & E. Robertson, “Gottfried Leibniz,” October 1998, https://mathshistory.st-andrews.ac.uk/Biographies/Leibniz/. 

[5]   J. O’Connor & E. Robertson, “Henry Briggs,” July 1999, https://mathshistory.st-andrews.ac.uk/Biographies/Briggs/.

[6]   J. O’Connor & E. Robertson, “John Napier,” April 1998, https://mathshistory.st-andrews.ac.uk/Biographies/Napier/. 

[7]   J. O’Connor & E. Robertson, “The number e,” September 2001, https://mathshistory.st-andrews.ac.uk/HistTopics/e/. 

[8]   J. O’Connor & E. Robertson, “William Oughtred,” February 2017, https://mathshistory.st-andrews.ac.uk/Biographies/Oughtred/. 

[9]   “Newton’s Law of Cooling,” 2011, http://vlab.amrita.edu/index.php?sub=1&brch=194&sim=354&cnt=1.

[10]   P. Spencer, “Simple and Compound Interest,” September 1997, https://www.math.toronto.edu/mathnet/answers/answers_10.html.

[11]    R. Sachs, “Euler’s Formula for Complex Exponentials,” March 2011, http://math.gmu.edu/~rsachs/m116/. 

[12]   R. Schwartz, “Transcendence of e,” https://www.math.brown.edu/~res/M154/.  

[13]   S. Glaz, “The Enigmatic Number e: A History in Verse and Its Uses in the Mathematics Classroom,” May 2010, https://www2.math.uconn.edu/~glaz/My_Articles/TheEnigmaticNumberE.Convergence10.pdf.

[14]   S. Reichert, “e is everywhere,” September 2019, https://www.nature.com/articles/s41567-019-0655-9.

[15]   The Euler Archive, http://eulerarchive.maa.org/.

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 19^th 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 17^th 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.

 

References

[1]   C. Pickover, The Math Book, Sterling Publishing Co., Inc., Toronto (2009).

[2]   D. Struik, A source book in mathematics 1200-1800, Princeton University Press (1990).

[3]   E. Bell, Men of Mathematics, Simon and Schuster, Inc., New York (1937).

[4]   F. Swetz, Mathematical treasure: Leibniz’s papers on calculus – differential calculus, Convergence, The Pennsylvania State University (2005).

[5]   G. Simmons, Calculus with analytic geometry, McGraw-Hill, New York, 1985.

[6]   J. Bardi, The calculus wars: Newton, Leibniz, and the greatest mathematical clash of all time”, Basic Books, New York (2007).

[7]   J. Davidson, “What is Calculus?”, https://www.sscc.edu/home/jdavidso/MathAdvising/AboutCalculus.html.

[8]   J. O’Connor & E. Robertson, “A history of the calculus,” February 1996, https://mathshistory.st-andrews.ac.uk/HistTopics/The_rise_of_calculus/.

[9]   J. O’Connor & E. Robertson, “Archimedes of Syracuse,” January 1999, https://mathshistory.st-andrews.ac.uk/Biographies/Archimedes/.

[10]   J. O’Connor & E. Robertson, “Augustin Louis Cauchy,” January 1997, https://mathshistory.st-andrews.ac.uk/Biographies/Cauchy/.

[11]   J. O’Connor & E. Robertson, “Brook Taylor,” May 2000, https://mathshistory.st-andrews.ac.uk/Biographies/Taylor/.

[12]   J. O’Connor & E. Robertson, “Christiaan Huygens”, February 1997, https://mathshistory.st-andrews.ac.uk/Biographies/Huygens/.

[13]   J. O’Connor & E. Robertson, “Euclid of Alexandria,” January 1999, https://mathshistory.st-andrews.ac.uk/Biographies/Euclid/.

[14]   J. O’Connor & E. Robertson, “Eudoxus of Cnidus,” April 1999, https://mathshistory.st-andrews.ac.uk/Biographies/Eudoxus/.

[15]   J. O’Connor & E. Robertson, “Georg Friedrich Bernhard Riemann,” September 1998, https://mathshistory.st-andrews.ac.uk/Biographies/Eudoxus/.

[16]   J. O’Connor & E. Robertson, “How do we know about Greek mathematics?,” October 1999, https://mathshistory.st-andrews.ac.uk/HistTopics/Greek_sources_1/.

[17]   J. O’Connor & E. Robertson, “Jean Baptiste Joseph Fourier,” January 1997, https://mathshistory.st-andrews.ac.uk/Biographies/Fourier/.

[18]   J. O’Connor & E. Robertson, “Johann Bernoulli,” September 1998, https://mathshistory.st-andrews.ac.uk/Biographies/Bernoulli_Johann/.

[19]   J. O’Connor & E. Robertson, “Johann Carl Friedrich Gauss,” December 1996, https://mathshistory.st-andrews.ac.uk/Biographies/Gauss/.

[20]   J. O’Connor & E. Robertson, “Johann Peter Gustav Lejeune Dirichlet,” May 2000, https://mathshistory.st-andrews.ac.uk/Biographies/Dirichlet/.

[21]   J. O’Connor & E. Robertson, “Joseph Liouville,” October 1997, https://mathshistory.st-andrews.ac.uk/Biographies/Liouville/.

[22]   J. O’Connor & E. Robertson, “Joseph-Louis Lagrange,” January 1999, https://mathshistory.st-andrews.ac.uk/Biographies/Lagrange/. 

[23]   J. O’Connor & E. Robertson, “Pierre de Fermat,” December 1996, https://mathshistory.st-andrews.ac.uk/Biographies/Fermat/.

[24]   J. O’Connor & E. Robertson, “Pierre-Simon Laplace,” January 1999, https://mathshistory.st-andrews.ac.uk/Biographies/Laplace/.

[25]   J. O’Connor & E. Robertson, “Pythagoras of Samos,” January 1999, https://mathshistory.st-andrews.ac.uk/Biographies/Pythagoras/.

[26]   O. Rodriguez, “Teaching the Fundamental Theorem of Calculus: A Historical Reflection – Newton’s Proof of the FTC,” https://www.maa.org/press/periodicals/convergence/teaching-the-fundamental-theorem-of-calculus-a-historical-reflection-newtons-proof-of-the-ftc.

[27]   P. Kitcher, Fluxions, limits, and infinite littlenesse: a study of Newton’s presentation of the calculus, Isis 64 (1973) 33–49.

[28]   R. Buckmire, “History of Mathematics,” March 2010, https://sites.oxy.edu/ron/math/395/10/ws/21.pdf.

[29]   R. Coolman, “What Is Calculus?,” May 2015, https://www.livescience.com/50777-calculus.html.

[30]   W. Ball, “A Short Account of the History of Mathematics,” September 2010, https://www.maths.tcd.ie/pub/HistMath/People/Pascal/RouseBall/RB_Pascal.html.