## Should Gottfried Wilhelm von Leibniz be considered the founder of calculus

## Should Gottfried Wilhelm von Leibniz be considered the founder of calculus

MATHEMATICS AND COMPUTER SCIENCE

# Should Gottfried Wilhelm von Leibniz be considered the founder of calculus?

**Viewpoint:** Yes, Leibniz should be considered the founder of calculus because he was the first to publish his work in the field, his notation is essentially that of modern calculus, and his version is that which was most widely disseminated.

**Viewpoint:** No, Leibniz should not be considered the founder of calculus because the branch of mathematics we now know by that name was first expressed by Isaac Newton, and only later was it developed by Leibniz.

Deciding who deserves to be called the founder of calculus is not altogether easy. Some fiercely contend that the English scientist Isaac Newton (1643-1727) should be considered the founder, while others are convinced the recognition should belong to Gottfried Wilhelm von Leibniz (1646-1716), the German philosopher, diplomat, and mathematician. As is true with many scientific discoveries, the publication process often cements the scientist's place in history. But is "Who published first?" always the fairest way to credit someone with an innovative idea? Isn't it possible that the first person to be published is not necessarily the innovator of the idea? After all, being published not only has to do with submitting work that has merit, but it also with academic connections and public reputation. What happens when it is unclear who is the mentor and who is the student? Often colleagues inspire each other, and should credit for a discovery be given, at least in part, to both?

Besides, when and how an idea is discovered is not always clear. Especially in mathematics, concepts often develop over time, like snapshots taken of the interior of a house. As time goes by, snapshots will become outdated; changes over time naturally occur. Scientific discovery is often the same way. It evolves; it changes. So, how then, without accurate and completely honest disclosure, can one ever hope to credit an idea to a single person?

When making up your mind on who is the true founder of calculus, there are a variety of issues to consider. First, bear in mind the nature of the student/professor relationship. How does that relationship affect the scientific discovery?

The timing and complexity of the discovery are also important. Is the discovery entirely new or is it the expansion of a previously held belief? Part of the recognition process also involves whether the discovery can be proved or whether it is merely an hypothesis. In mathematics, a problem solved has tremendous merit, but hypotheses also have value. Indeed, much controversy surrounds the application of both Leibniz's and Newton's work.

Finally, the importance of political alliances and how they affect public opinion cannot be underestimated in this debate. The glory of discovery tends to expand over time beyond the actual discoverer. Nations, for a variety of political reasons, take great pleasure in recognizing the achievements of their native sons and daughters. Unfortunately, sometimes this can complicate the issue, making the truth even harder to find.

In any event, don't be surprised if you find it hard to choose a side. People have been arguing over this issue for a long time, and we may never be certain who "found" calculus.

—LEE ANN PARADISE

## Viewpoint: Yes, Leibniz should be considered the founder of calculus because he was the first to publish his work in the field, his notation is essentially that of modern calculus, and his version is that which was most widely disseminated.

Gottfried Wilhelm von Leibniz (1646-1716), the German philosopher, diplomat, and mathematician, is considered by many to be the founder of calculus. Although other men may lay claim to the same title, most notably the English scientist Isaac Newton (1643-1727), several arguments favor Leibniz. Not only was Leibniz the first to publish an account of this new branch of mathematics, his notation and conceptual development of calculus became the acknowledged and accepted form throughout Europe and eventually the world. Without Leibniz, modern calculus would have a much different look.

### Leibniz's Early Life

Leibniz was born in Leipzig (now part of Germany) and died in Hanover (also now part of Germany). Although he was born and died in the German states, Leibniz was a true world traveler who learned from and consulted with the greatest scholars of Europe. Leibniz was awarded a bachelor's and a master's degree in philosophy from the University of Leipzig, as well as a bachelor's degree in law. Later he was awarded a doctorate in law from the University of Altdorf in Nuremberg.

Leibniz was an incredibly versatile scholar. His lifelong goal was to unite all of humankind's knowledge into one all-encompassing philosophy. To this end he spent much of his energy organizing and encouraging scientific and other scholarly societies. He also believed that he could develop a sort of mathematical system to quantify human reasoning. For these reasons, along with more concrete accomplishments such as his development of calculus, Leibniz is known as one of history's most important philosophers and scientists.

After completing his education, Leibniz began a life of travel, scholarship, and service to various royal courts. Two trips made by Leibniz, one to France and the other to England, have special significance in relation to his scientific and mathematical work. An extended stay in Paris, beginning in 1672, enabled Leibniz to study with many of the leading scholars of Europe. In particular, his relationship with the Dutch scientist Christiaan Huygens (1629-1695) proved extremely fruitful. Huygens, who produced the first practical pendulum clock, was an outstanding astronomer and mathematician. Huygens helped his young German protégé with his pursuit of a mathematical education. Under Huygens, Leibniz's latent mathematical abilities began to bloom.

### Influence of Leibniz's Visit to England

During a visit to England the following year (1673), Leibniz conversed with many of Britain's leading scientists, and was made a fellow of the Royal Society. A calculating machine of his invention received a lukewarm reception from members of the society and a prolonged animosity arose between Leibniz and Robert Hooke, an important member. Leibniz's correspondence with Henry Oldenburg, the secretary of the society, would prove to be a fateful step in the dispute that later developed with Isaac Newton concerning who deserved priority in the discovery of calculus. It was through Oldenburg that Leibniz received many letters with veiled references to the new mathematical techniques being developed by Newton, Isaac Barrow, James Gregory, and other British mathematicians.

It has never been completely clear to historians exactly how much Leibniz's trip to England influenced his development of calculus. Although he did not meet Newton, Leibniz did have the opportunity to discuss his mathematics with John Pell (1611-1685), and he was exposed to the mathematics of René-François (R. F.) de Sluse (1622-1685) and Isaac Barrow (1630-1677). Since both de Sluse and Barrow had worked on the problem of determining tangents, Leibniz's introduction to their work probably influenced his future conceptions of calculus.

The trip to England also initiated Leibniz's long-running correspondence with John Collins, the librarian of the Royal Society. This new line of correspondence, along with the continuing correspondence with Oldenburg, provided Leibniz with kernels of information concerning the British work in new mathematical processes. In 1676, Leibniz even received two letters from Newton that included sketches of his work with infinite series. Although this correspondence would later be used against Leibniz in the priority dispute, it is very clear that these letters did not contain the detail required to formulate the rules of calculus. Leibniz's discoveries of the next few years were, without doubt, his own.

### Leibniz's Work on Calculus

Back in Paris, Leibniz continued his study of mathematics. His interest in the study of infinitesimals led him to discover a set of new mathematical techniques he called differential calculus. By the end of 1675, Leibniz had developed the basic techniques of his new discovery. Although Leibniz wrote several manuscripts containing these ideas and communicated his discoveries to many of his contemporaries, it was not until 1684 that he published the details of differential calculus in the journal *Acta eruditorum*. Two years later, he published the details of integral calculus in the same journal.

If Isaac Newton developed the method of fluxions (his version of the calculus) in 1665 and 1666, why then argue that Leibniz, who admittedly did not have his inspiration until nearly a decade later, should be considered the true founder of calculus? One reason is simply that Newton delayed publishing his work, and in fact did not clearly explain his discovery to anyone, until long after Leibniz had independently found his own version of calculus and published his results. Whereas Leibniz published his work in the mid-1680s, the first published account from Newton did not appear until 1704, as an appendix to his work on the science of optics. In addition, Leibniz was confident that his new methods constituted a revolutionary change in mathematics, whereas Newton seemed content to apply his techniques to physical problems without paying heed to their significance. It seems that Leibniz had a greater appreciation for the importance of the discovery than had Newton.

Other than being the first to publish, Leibniz's version of calculus was disseminated widely in Europe. Two Swiss brothers, Jacob Bernoulli (1654-1705) and Johann Bernoulli (1667-1748), were especially important in the circulation of Leibniz's ideas. The Bernoullis were not only innovators in the new field of calculus, but they were also enthusiastic and prolific teachers. One of Johann's students, the Marquis de l'Hôpital (1661-1704), published the first differential calculus textbook in 1696. Much of the credit for the spread of Leibnizian calculus in Europe belongs to these extraordinary brothers.

Another claim for the priority of Leibniz in the calculus question is that his notation, for the most part, is the notation that was adopted and continues to be in use today. Much of Newton's notation was awkward, and very little remains a part of modern calculus. Leibniz, on the other hand, was the first to use such familiar notation as the integral sign ∫. This symbol represents an elongated *s,* the first letter in the Latin word *summa,* because an integral was understood by Leibniz to be a summation of infinitesimals. Leibniz also invented the differential notation ( *dx, dy,* etc.) favored by most calculus texts. Leibniz's development of the notation of calculus goes hand-in-hand with his life-long dream of creating a sort of "algebra" to describe the whole of human knowledge.

### The Priority Dispute

The priority dispute that developed between Newton and Leibniz, and was continued by their supporters even after the death of the two great mathematicians, is one of the most interesting and controversial episodes in the history of mathematics. One can find historical accounts favorable toward Newton and accounts favorable toward Leibniz. Luckily, there are several historical works (see Further Reading) that present a balanced and unbiased analysis of the dispute.

At first, it seemed that Newton played the part of a good-natured, yet slightly condescending teacher, and Leibniz that of the eager student. Although Leibniz always maintained a high level of confidence in his own abilities, he realized after his visit to England that he had a lot to learn about mathematics. Yet, later Leibniz was to give little credence to Newton's abilities, preferring to believe that Newton's work in infinite series was not the equivalent to his own development of calculus.

The controversy arose when the disputants and their supporters gathered along two lines of thought. Newton and his defenders rightfully claimed that Newton was the first discoverer of calculus. However, Newton's supporters went on to make the erroneous claim that Leibniz had plagiarized Newton's work and therefore had no claim of his own upon the discovery of calculus. Leibniz, on the other hand, was essentially content to be credited as an equal in the priority question, although he continued to believe that his discovery was much broader and more substantial than that of Newton.

The low point of the dispute occurred after Leibniz appealed to the Royal Society to correct what was, in Leibniz's own judgment, unfair attacks by John Keill (1671-1721, a Scottish scientist and member of the Royal Society who supported Newton, on his role in the discovery of calculus. The report of a committee formed within the Royal Society to investigate the question was known as the *Commercium epistolicum*. To Leibniz's chagrin, the report reaffirmed Newton as the "first inventor" of the calculus, and worse, condemned Leibniz for concealing his knowledge of Newton's work. The report stopped just short of calling Leibniz a plagiarist. It later came to light that the author of the *Commercium epistolicum* was none other than Newton himself.

For centuries, the question of who deserved credit for the discovery of calculus depended largely on which side of the English Channel one lived. British mathematicians and historians defended Newton, while the majority of continental mathematicians sided with Leibniz. Today, the two great men are generally accorded equal credit for the discovery, mutually independent of each other. However, Leibniz was the first to give the world calculus through publication, his notation is essentially that of modern calculus, and his version is that which was most widely disseminated. Therefore, if one man was chosen as the founder of calculus, Leibniz could rightfully stake his claim to the title.

—TODD TIMMONS

## Viewpoint: No, Leibniz should not be considered the founder of calculus because the branch of mathematics we now know by that name was first expressed by Isaac Newton, and only later was it developed by Leibniz.

Gottfried Wilhelm von Leibniz (1646-1716) is often considered the founder of calculus. However, many claim this title is not deserved, and for several reasons. First, the history of mathematics is much too complicated for such a simple statement to be true. Leibniz did not create calculus in a scientific vacuum, but rather gathered together the work of many mathematicians over many centuries to derive a "new" way of approaching physical problems through mathematics. Second, if one were to venture an assertion as to the true founder of calculus, the name of the English mathematician Isaac Newton (1643-1727) would certainly be the first to mind. Either of these reasons taken alone forms a solid argument that Leibniz is not the founder of calculus.

### Leibniz's Predecessors

The development of calculus involved the work and ideas of many people over many centuries, and any number of these people might be called the founder of calculus. However, they might be more appropriately labeled the *founders* of calculus, because each one played an important role in the story. The history of calculus begins over 20 centuries ago with the mathematicians of ancient Greece. A method of calculating the areas of various geometric shapes, called the method of exhaustion, was developed by several Greek mathematicians and perfected by Eudoxus of Cnidus (c. 400-350 b.c.). Eudoxus used the method of exhaustion to find the volumes of various solids such as pyramids, prisms, cones, and cylinders, and his work is considered a forerunner of integral calculus.

Archimedes (c. 287-212 b.c.), perhaps the greatest mathematician of antiquity, used the method of exhaustion to find the area of a circle. He did so by first inscribing a series of polygons within a circle and then circumscribing the polygons around the outside of the circle. As the
number of sides of the polygons increased, the average of the areas of the inscribed and circumscribed polygons became progressively closer to the area of the circle. Using this method with a 96-sided polygon, Archimedes was able to approximate the value of *pi* more accurately than anyone had before. As well as circles, Archimedes applied his method to a number of other geometric figures, including spheres, ellipses, cones, and even segments of parabolas. Like Eudoxus, Archimedes also found ways to calculate volumes of various solids. His method for finding the areas and volumes is essentially equivalent to the method of integration in calculus. Archimedes also developed methods for finding the tangents to various curves, an anticipation of differential calculus. Because his technique for finding areas was an early form of integration, and his tangent problems were essentially derivatives, it is easy to consider Archimedes a founder of calculus.

Medieval Europe saw little in the way of scientific or mathematical advances, and calculus was no exception. However, Islamic scholars of the Middle Ages not only kept alive much of the mathematics of Greek antiquity, but also made important advances in certain areas. At least one Islamic mathematician, Sharaf ad-Din at-Tusi (c. 1135-1213), actually used the concepts of calculus in his work. At-Tusi essentially employed the derivative (although not explicitly and not in modern notation) to find the maximum values of certain cubic equations. Although historians disagree over the approach at-Tusi took in developing his mathematics, there is no doubt that some of the underlying concepts of calculus were known and used by certain Islamic mathematicians.

In seventeenth-century Europe, there was a rebirth of ideas that involve what we now call calculus. One of these critical ideas was the assertion that areas could be considered as an infinite sum of lines, or indivisibles. The concept of an indivisible, popularized by the Italian mathematician, (Francesco) Bonaventura Cavalieri (1598-1647), was an important step in the process of "quadrature," or finding the area of a figure. By conceiving of areas as the sum of infinitely small lines, Cavalieri found methods to calculate the area of a variety of geometric figures. This extension of the work of Archimedes provided a fundamental concept behind the method of integration.

Quite possibly the most important contributor to the development of calculus between the time of Archimedes and that of Leibniz and Newton was the French jurist, Pierre de Fermat (1601-1665). Fermat was an amateur mathematician who made many wondrous discoveries while working on mathematics in his spare time. Although far removed both geographically and professionally from the centers of French mathematics, Fermat maintained a long correspondence with many of the most important mathematicians of France. This correspondence inspired an incredibly fruitful episode in the history of mathematics. Fermat is known today as a cofounder of probability theory, in collaboration with Blaise Pascal (1623-1662); a cofounder of analytical geometry along with, but independently of, René Descartes (1596-1650); and one of the first modern mathematicians to show a serious interest in the theory of numbers. He is particularly important to the story of calculus because he worked in both branches, differential calculus and integral calculus. Fermat extended the work of his predecessors by discovering general formulas for finding the areas of various figures. In addition, Fermat used the method we today call differentiation to find the maximum and minimum values of algebraic curves.

Not long after Fermat's work, another important contribution to the development of calculus came from the Scottish mathematician James Gregory (1638-1675). Gregory discovered many of the basic concepts of calculus at about the same time as Newton and well before Leibniz. In his book *Geometriae pars universalis* (The universal part of geometry), published in 1668, Gregory presented the first proof in print of what we now call the Fundamental Theorem of Calculus. His work anticipated both Newton and Leibniz, yet gained very little notoriety in Gregory's lifetime.

Many historians consider Gregory's contemporary, Isaac Newton, to be the founder of calculus. There are certainly several fundamental arguments that favor Newton as the man to whom the credit is due. These arguments will be presented shortly. However, even Newton relied on his predecessors. Not only was Newton obviously influenced by the work of those mathematicians already mentioned, but he was also greatly indebted to his teacher and the man whom he replaced as professor of mathematics at Cambridge University, Isaac Barrow (1630-1677).

Barrow conceived of several ideas fundamental to the development of calculus. One of these was the concept of the differential triangle. In the differential triangle, the hypotenuse of a systematically shrinking triangle represents the tangent line to a curve. This idea is conceptually equivalent to the notion of a derivative. Barrow also realized the inverse relationship between the method of finding tangents (differentiation) and the method of finding areas (integration). Although he never explicitly stated the Fundamental Theorem of Calculus, this inverse relationship laid the groundwork for Newton's work.

### Isaac Newton: The True Founder of Calculus

So far, we have addressed the other characters in the history of mathematics who would claim at least part of the credit for the development of calculus. If anyone deserves to be called the founder of calculus, however, it is Isaac Newton. Newton was born and raised on a small farm in Lincolnshire, England. Although he was initially expected to manage the family farm, an uncle recognized the latent scientific talent in the young boy and arranged for Newton to attend first grammar school and later Cambridge University. At Cambridge, Newton discovered his interests in mathematics, but because that subject was given very little attention at the university, Newton acquired his own books containing the most advanced mathematics of the time and taught himself.

On leave from Cambridge, closed due to an outbreak of the plague, Newton spent 1665 and 1666 at his mother's farm. There Newton conceived of many of the ideas that would make his the most famous name in science: the laws of motion, the universal law of gravitation, and the formulation of calculus, which he called the method of fluxions. This period of a little over a year is often referred to as the *annus mirabilis*, or the "year of miracles," because of the almost miraculous inspiration that allowed one man to make so many important scientific discoveries in a short period of time.

In his conception of calculus, Newton imagined a point moving along a curve. The velocity of the point was a combination of two components, one horizontal and one vertical. Newton called these velocities *fluxions* and the inverse of the fluxions Newton called *fluents.* Although the names are different than those we use in calculus today, the mathematical concept is equivalent. Newton's fluxions are our derivatives. More importantly, Newton realized the relationship between the process of differentiation and that of integration, giving us the first modern statement of the Fundamental Theorem of Calculus.

Isaac Newton developed his version of calculus many years before Leibniz, and he used his newfound mathematical techniques in analyzing scientific questions. However, for various reasons, Newton did not publish his work on calculus until much later, even after Leibniz's own version of calculus was published in the 1680s. Newton wrote a short tract on fluxions in 1666 and several other works on calculus in the next decade. However, none of Newton's works addressing calculus were published until the next century.

Although Newton's writings about calculus were not published for many decades, his formulation of the methods of calculus was known in England and continental Europe. Thanks to many friends and supporters, Newton's work was circulated and discussed long before it was formally published. Just how much Leibniz knew of Newton's discovery is debatable. Leibniz did receive correspondence from several people regarding Newton's work, including two letters from Newton himself. None of this correspondence stated explicitly Newton's work on fluxions, but rather offered vague accounts of the new method. For this reason, most historians believe Leibniz developed his version of calculus independently of Newton. However, of one thing there is no doubt—Isaac Newton formulated the new branch of mathematics we now call calculus long before his rival Gottfried Leibniz.

### Calculus after Leibniz

Whether we consider Leibniz or Newton (or even Archimedes or Fermat) to be the original founder of calculus, the calculus we know today is certainly very different than that which came from the pen of any of these men. Learning of the new methods of calculus shortly after Leibniz's discovery, two Swiss brothers, Jacob Bernoulli (1654-1705) and Johann Bernoulli (1667-1748), quite possibly outshone their teacher. Jacob Bernoulli was professor of mathematics at Basel, Switzerland, where he did pioneering work in integration techniques and in the solution of differential equations. It appears that the term *integral* was first used in print by Jacob. Johann was professor of mathematics at Groningen in the Netherlands and later succeeded Jacob at Basel after Jacob's death. Johann also made significant contributions to the theory of integration, as well as serving
as the tutor for the French nobleman, the Marquis de l'Hôpital (1661-1704). De l'Hôpital used (some say stole) Johann's work to write the first differential calculus textbook in 1696. Interestingly, the two gifted brothers bickered constantly throughout their lives and what could have been brilliant collaboration degraded into petty jealousy. The Bernoulli brothers, however, did as much to shape modern calculus as anyone else associated with its conception.

Although the basic techniques of calculus were discovered by the early eighteenth century, serious questions arose as to its logical foundation. Few of the individuals discussed up to this point, from Archimedes to the Bernoulli brothers, were overly concerned with setting calculus on a sound logical foundation. This problem was pointed out quite effectively by the Irish scientist (and bishop), George Berkeley (1685-1753). In his 1734 work, *The Analyst: or, a Discourse Addressed to an Infidel Mathematician,* Berkeley attacked many of the illogical assumptions upon which calculus was based. The most glaring of these gaps in logic was the inconsistency with which infinitely small numbers were treated. Berkeley called these numbers "the ghosts of departed quantities." Until it was placed on a solid foundation, we cannot maintain that the process of discovering calculus was complete.

The man who finally succeeded in placing calculus on a firm logical foundation was the French mathematician Baron Augustin-Louis Cauchy (1789-1857). In his book, *Cours d'analyse de l'École Royale Polytechnique* (Courses on analysis from the École Royale Polytechnique), published in 1821, Cauchy set a rigorous course for calculus for the first time. Although not the final word on rigor in calculus—the German mathematician Georg Friedrich Bernhard Riemann (1826-1866) later provided the modern and rigorous definition of a definite integral-—Cauchy's work provided the framework within which modern calculus is understood. Without the work of Cauchy and others, calculus would not be what it is today.

Who should receive credit for a new discovery? If that discovery is made independently without the influences of other sources, obviously the first person to make the discovery deserves credit. Unfortunately, that seldom happens, especially in mathematics. The discovery of calculus was a centuries-long process that culminated with the work of modern mathematicians. This makes it very problematic to ask simply the question, "What single person deserves credit?" Nevertheless, as we have also seen, if one is pressed to answer such a question that answer would surely be Isaac Newton, because Newton was the first person to assimilate all the work that came before into a single, rational entity.

—TODD TIMMONS

## Further Reading

Aiton, E. J. *Leibniz: A Biography.* Boston:Hilger, 1985.

Baron, Margaret E. *The Origins of the Infinitesimal Calculus.* New York: Dover Publications, 1969.

Boyer, Carl B. *The History of the Calculus and its Conceptual Development.* New York: Dover Publications, 1949.

Edwards, C. H. *The Historical Development of the Calculus.* New York: Springer-Verlag, 1979.

Guicciardini, Niccoló. *The Development of Newtonian Calculus in Britain 1700-1800.* Cambridge: Cambridge University Press, 1989.

Hall, Rupert. *Philosophers at War: The Quarrel Between Newton and Leibniz.* New York: Cambridge University Press, 1980.

Hofman, Joseph E. *Leibniz in Paris, 1672-1676: His Growth to Mathematical Maturity.* London: Cambridge University Press, 1974.

Westfall, Richard. *Never at Rest.* Cambridge:Cambridge University Press, 1980.

## KEY TERMS

### DIFFERENTIAL CALCULUS:

Branch of calculus that deals with finding tangent lines through the use of a technique called differentiation.

### FLUXIONS:

Name that Isaac Newton gave his version of calculus.

### FUNDAMENTAL THEOREM OF CALCULUS:

Theorem that states the inverse relationship between the process of differentiation and the process of integration.

### INFINITESIMAL:

Infinitely small. In calculus, the idea that areas are composed of an infinite number of infinitely small lines and volumes are an infinite number of infinitely small areas, etc.

### INTEGRAL CALCULUS:

Branch of calculus that deals with finding areas through the use of a technique called integration.

### METHOD OF EXHAUSTION:

Method developed by ancient Greek mathematicians in which areas or volumes are calculated by successive geometric approximations.

### QUADRATURE:

Process of finding the area of a geometric figure.

### ROYAL SOCIETY:

Scientific society in Great Britain founded in 1660; one of the oldest in Europe.

### TANGENT:

Line that touches a given line at one point but does not cross the given line.