"Our" Notation from Their Quarrel: The Leibniz-Newton Controversy in Calculus Texts

by Shelley Costa, Cornell University

In Philosophers At War, Rupert Hall details the historical controversy between Gott-fried Wilhelm von Leibniz and Isaac Newton over the development of the infinitesimal calculus. The controversy itself had achieved fame before Hall's account was published in 1980. Hall attested to this renown in his preface, where he wrote that he was telling "the story of the bitter quarrel between two of the greatest men in the history of thought, the most notorious of all priority disputes." Given that their quarrel achieved such renown, how have Newton and Leibniz, the famous creators of calculus, have been introduced in textbooks to beginning calculus students, both now and in the past?

As I detail below, 20th-century American texts depict the men in specific ways, and their styles reveal how each author views the nature of mathematics. Most important, in none of these 20th-century texts is a sense of conflict or controversy evident. Rather, the authors describe Newton and Leibniz simply as co-con-tributors to the great assembly of knowledge that makes up the calculus. Simlarly, late 17th- and early 18th-century texts, though published in Europe contemporaneously with the dispute itself, fail to mention outright the fact that Newton and Leibniz were embroiled in a controversy over priority. Instead, they couch the debate in terms of method. Thus, these early texts define, in essence, a new debate: one of notation and method, and not of priority.

First, let me briefly summarize the Newton-Leibniz priority dispute (see Hall 1980). Newton developed his infinitesimal calculus between 1664 and 1666 when he was temporarily con-fined to his estate in Woolsthorpe, quarantined from an outbreak of Bubonic plague in England. However, Newton did not publish his mathematics. Years later Leibniz published his own infinitesimal methods in a recently established scholarly journal circulated around Europe, the Acta Eruditorum. Leibniz's methods appeared in the Acta in two different articles, one in 1682 and the other in 1684. Thus, while Newton's techniques were developed first, Leibniz was the first to publish. The Swiss mathematicians Jakub and Johann Bernoulli interpreted Leibniz's articles and extended his methods; Johann then traveled to Paris and, beginning in 1692, disseminated the Bernoullis' work to eager French intellectuals. Thus the first version of the calculus to reach large numbers of people was Leibnizian. Meanwhile, Newton began to publish his own methods, most explicitly in Wallis's Works and in his own Opticks.

Leibniz originally developed his calculus in order to find methods by which discrete infinitesimal quantities could be summed up to calculate the area of a larger whole. This approach reflected his philosophical view that individually imperceptible metaphysical entities were the basis of existence and that humans experience the world as the sum of these entities. Newton, on the other hand, had been occupied with problems of gravitation and planetary motion, and his mathematical methods attempted to understand motion and force in terms of infinitesimal changes with respect to time. Because of these distinct approaches, most modern commentators consider Leibniz as the inventor of integral calculus, while Newton is typically seen as the inventor of differential calculus. This is very much a simplification, however, as both Leibniz and Newton developed versions of both integration and differentiation.

As Rupert Hall's chronology (1980) reveals, Newton and Leibniz were in communi-cation--through Henry Oldenberg--before Leibniz published his calculus, and that proved crucial in the later debate. In 1710 Leibniz was accused of plagiarism; during the following year he unsuccessfully petitioned the Royal Society for redress. In 1713, Newton in turn was accused of plagiarism. Leibniz died only three years later, at the height of the dispute. The controversy continued for some years. Newton died eleven years after Leibniz, in 1727.

Some introductory textbooks still com-memorate Leibniz and Newton. They typically credit Leibniz with integration and Newton with differentiation, though as mentioned above this is a simplification. The two men also live on through certain mathematical symbols. The modern integration symbol, for example, originated with Leibniz, as did the dy/dx notation for differentiation. The "characteristic triangle," a standard geometric approach to conceptuali-zing differentiation graphically as a function of y versus x, also came from Leibniz. Newton's own notation was a dot directly above the variable (read as "x-dot" or "y-dot"), and survives in many physics classrooms.

It makes sense for modern texts to present history in terms of its method and notation. I have found, however, that most introductory calculus books--when they mention Newton and Leibniz at all--discuss these men in terms of method and notation without giving any hint of their priority dispute. Why?

20th-Century Calculus Texts

• Shanks and Gambill, Calculus of the Elementary Functions (1969: Holt, Rinehart and Winston, Inc.)

Shanks and Gambill do not mention Leibniz or Newton in the preface, nor do they mention there a desire to incorporate history into their text. Nevertheless, Newton and Leibniz take a more conspicuous place in this text than they do in most others. Shanks and Gambill have chosen to introduce the first two sections of their book with these portraits and biographies of Newton and Leibniz, including timelines of their lives and work. The Newton page is the first in "Part I: Differential Calculus"; The Leibniz page begins "Part II: Integral Calculus." Shanks and Gambill also have a "Part III: Calculus of Variations," but that section has no portrait or biography to introduce it. In this way Shanks and Gambill symbolically divide the calculus between Newton and Leibniz, implying by omission that no other mathema-ticians are relevant to the subject.

The Shanks and Gambill book draws more attention to our two historical actors than do most other calculus texts by including portraits and devoting a large amount of space to each biography. However, the treatment of historical actors in Shanks and Gambill is definitely not what serious historians would call historical. For example, (p. 11): "Fermat had (in 1629) a method for finding slopes of tangent lines, but it did not involve limits (though it should have)." With this parenthetical clause the authors anachronistically use a 19th-century technique--that of approximating by limits--to find fault with the methods of a man whom they are introducing as an important player in the origins of calculus. The authors encourage in this way two contradictory perspectives. One perspec-tive considers historical actors as having created the methods that currently belong to an accepted collection of techniques. The other judges these same actors according to how accurately they foresaw all the methods currently belonging to that collection. The basis for the contradiction is the dominant role assigned--out of all social context--to the body of technique that forms what we now call calculus.

Pertaining to Leibniz and Newton, Shanks and Gambill write (p. 4), "It is hard for the beginner to understand why calculus was not invented earlier. Indeed, Isaac Barrow (the teacher of Newton) and the great Pierre Fermat were aware of all the pieces of both problems, namely Tangents and Area. But the fact remains that it was left to the genius of Newton and Leibniz to show the way." Shanks's and Gambill's portrayal of Leibniz and Newton as insightful geniuses who guided their stumbling contemporaries is actually quite typical of 20th-century calculus texts (although many texts choose to mention only Newton in this role). It is interesting that Shanks and Gambill use the same information -- that is, evidence that mathematicians before Newton and Leibniz worked with methods quite similar to the calculus -- to lionize Newton and Leibniz, while historian Carl Boyer used the same evidence in an attempt to whittle away at their pedestals: in his History of the Calculus, published in 1949, (p. 187) Boyer wrote that "Few new branches of mathematics are the work of single individuals. The analytic geometry of Descartes and Fermat was the outgrowth of several mathematical trends which converged in the sixteenth and seventeenth centuries. . . . Far less is the development of the calculus to be ascribed to one or two men."

This contrast between Boyer and Shanks/Gambill reveals the importance of a respect for context when interpreting the history of mathematics. No thinker works in a vaccuum. With their statement "it was left to the genius of Newton and Leibniz..." Shanks and Gambill ignore the influence of predominant social and intellectual trends on both men in order to stress their exceptional genius and promote an illusion of independence.

• Bers, Calculus (1969: Holt, Rinehart and Winston, Inc.)

Bers explicitly mentions in his preface (p. vii) a desire to use history to enrich his exposition of calculus. "Like every other subject, calculus is learned best with due regard to its history. References to the history of mathematics are therefore made at various places in the book. . . . Yet, I hope this is a modern calculus book, since it is written by a modern working mathematician. Wherever possible, examples from recent scientific developments are used. But no attempt is made to use artificially modern language; traditional notation and terminology are preserved, with only a few exceptions." Bers's assumption here that most calculus notation is "traditional" as opposed to "modern" is in sharp contrast with the perspectives of many other contemporary authors of introductory calculus books, who emphasize that their mathematical vocabulary is modern--not traditional--and prefer to set it apart from the thought of historical actors. Consider as an example the comments of the authors (Wilcox, Buck, Jacob, and Bailey, Houghton-Mifflin 1971) of a 1971 calculus text (p. 2): "We have borrowed freely from modern mathematical language and definition because these can make Newton's ideas clearer than he himself could have made them." (These authors, by the way, also mention Newton only as "Father of the Calculus.")

To return to Bers's text, his way of incorporating history is primarily through biographical margin notes of Leibniz and Newton. At first glance this may seem to be a more marginal, if you will, way to include a historical perspective in a calculus text than large portraits and chronologies à la Shanks and Gambill. However, when Bers mentions Leibniz's differential calculus, he allows the reader to feel something in common with Leibniz by drawing an implicit connection between present-day methods (recall that he has termed these "traditional") and the original ones. Accompanying a figure on p. 143, Bers writes, "This, essentially, was the description given by Leibniz in his first published paper. . . This paper contains a drawing not unlike our Fig. 4.3." In making this connection between "our" present-day math (represented by a figure recognizable and understandable to a modern readership) and Leibniz's paper, Bers's treatment of Leibniz is quite different from Shanks's and Gambill's haughty pronounce-ments. Unfortunately, Bers's treatment is much less representative of 20th-century calculus texts than Shanks and Gambill's.

Finally, Bers's text, like Shanks's and Gambill's does not mention the controversy between Newton and Leibniz.

• Gilett, Calculus and Analytic Geometry (2d ed., 1984: Heath and Co) and Stewart, Calculus (2d ed, 1991: Brooks/Cole Publishing Co.)

I will discuss these two texts together because they have a great deal in common. Both of them are currently widely used in American high school and college calculus courses. Neither of them refers to Newton or to Leibniz in their prefaces, and neither includes a capsule biography of one of these men. But the names of Leibniz and Newton are sprinkled throughout each book.

Both of these books mention Newton much more frequently than they do Leibniz, primarily because they include Newton's work in other branches of mathematics and his work in physics, and they do not include Leibniz's work outside infinitesimal mathematics, with the exception of one infinite series formula which Gilett attibutes to Leibniz (p. 559). In addition, Gilett uses the word naive to describe Leibniz's mathematical reputation: he claims that the success of modern nonstandard analysis "is a vindication of Leibniz, whose ideas now seem less naive than formerly supposed" (p. 169). Both authors credit Newton and Leibniz as having recognized the Fundamental Theorem of Calculus; both authors attribute dy/dx notation to Leibniz, and Gilett mentions Newton's dot-notation. Yet, as in our other modern texts, neither author gives us any sense that Newton and Leibniz may have known or communicated with each other, or that there was any sort of controversy surrounding them.

Stewart introduces Newton and Lebniz in his text along with other early modern mathematicians (Fermat, Barrow, and Wallis). Gilett does not explicitly introduce them as historical actors but has a fascinating way of informing his reader of Newton's greatness through a compliment by Leibniz cited at the opening of a chapter (p. 44): "'Taking mathematics from the beginning of the world to the time of Newton, what he has done is much the better half.' --Gottfried Wilhelm von Leibniz (1646-1716) (generally credited with having created calculus independently of Newton)." This citation is the first indication given by Gilett of the role of either man in developing the calculus -- yet Newton is both overtly and subtly favored as the "better" creator of the calculus. More significantly even than the text's championing of Newton, though, is the image of Leibniz as an all-out fan of Newton's, with only a parenthetical note that Leibniz also had a role in creating the calculus. An uninformed reader would never guess that the two men were involved in a bitter dispute.

Conclusions on 20th-century texts

Most importantly, as I mentioned in the introduction, in none of these 20th-century calculus texts is a sense of conflict betweenber Leibniz and Newton present. The biographies of Newton and Leibniz in Shanks's and Gambill's text, for example, do not indicate whether the two men even knew each other. Even Bers, who otherwise seems so sympa-thetic to the history of his discipline, does not reveal any social relationship between the two men -- only that (p. 143) Leibniz developed calculus later than Newton but independently of him. Also, although no sense of actual conflict appears in these texts, Newton is depicted more strongly. As we saw in the more recent Gilett and Stewart texts, Newton is much more heavily cited as a result of the inclusion of his work in other areas, especially physics.

17th- and 18th-Century Mathematical Texts

I must yet give due recognition to the learned Mr. Newton, who has been recognized [in this capacity] by Mr. Leibniz himself: for he has also found something similar to differential calculus...But Mr. Leibniz characteristic [triangle] renders his [calculus] much easier and more expedient (pp. 12-13).

Twelve texts were published between 1703 and 1737, of which eleven are British and one is French. Recall that it was in the first decade of the eighteenth century that Newton began publishing his own calculus methods, and the priority dispute heated up at the end of that first decade. Newton is clearly preferred in the early 18th-century British texts, reflecting hostilities between England and the Continent over method. In 1704 Hayes, for example, lauded "the Great Mr. Newton, whose Immortal Genius will be a lasting Ornament to the English Nation" (p. 2). Two years later Ditton announced his bias in his very title: "An Institution of Fluxions...According to the Scheme Prefix'd by (its first inventor) the Incomparable Sir Isaac Newton." He also mentions the "Incomparable Mr. Leibnitz," however his incomparability, as opposed to Newton's, seems only to have extended so far as having proposed a clever problem in a mathematical journal, not having co-invented the calculus. In seven of the eleven British texts Leibniz is not mentioned at all. Robins, in 1735, claims that infinitesimal meth-ods were "obnoxious to error" when utilized by:

considering all curves ... like parallelo-grams. But Newton saves these methods by basing his principles on motion, and thereby has taught an analysis free from all obscurity and indirectness (pp. 1-2).

Only one of the seventeen texts mentions neither Newton nor Leibniz -- however, that text is also British and refers to Newton as the inventor of calculus without naming him. By contrast, the 1708 French text (by Reyneau) mentions both men and explicitly declares that Leibnizian methods are easier to understand than Newtonian.

Finally, two texts were published in France later in the century, one in 1749 and one in 1768; neither mentions Leibniz. The author of the first has a British name (Walmesley), which probably explains his omission of Leibniz -- but the co-authors of the second text were French and (as evidenced from their prefaces and other comments) were closely associated with the Royal Academy of Sciences at Paris. Not only does this last text omit Leibniz -- it explicitly refers the reader to Newton's work.

Although one might initially attribute the naming patterns in these texts to nationalism, by mid-century even the French were no longer mentioning Leibniz. From this time onward, Newton achieved clear priority in textbook cita-tions. It was extremely uncommon for a French textbook to use Newton's dot-notation, and all of the French texts we have looked at used Leibnizian notation. Yet in both of our French texts that were published after mid-century, Newton was the only inventor mentioned. This is most probably resulted from the growing fascination in 18th-century Enlightenment France with British thought, in general, and Newtonian science, in particular.

In short, during the period between 1695 and 1750 both Leibniz and Newton benefited from authors' choices of whose method was more useful and/or more understandable. But because Leibniz's name dropped out of the discussion by the mid-18th century, textbook authors stopped praising his methods as superior, even if they continued to use his notation.

Finally, even though Newtonian and Leibnizian methods are often contrasted with one another in these early texts, a sense of controversy is not apparent. One might argue that these authors would be wary of using a beginners textbook as a forum for inflammatory remarks. Yet there is at least one instance in these very texts where an author mentions a contemporary debate. Cheyne's 1703 "Methods of Inverse Fluxions" and de Moivre's 1704 "Respone to Cheyne's Treatise" had been in part responses to one another's work. Indeed, Humphry Ditton alluded to the debate between Cheyne and de Moivre in his 1706 preface:

Those that would enquire farther into the Mysteries of the Inverse Method, will do well to consult the writings of Dr. Cheyne & Mr. de Moivre; two Gentlemen, whose names (notwithstanding their warm, though useful disputes) I will venture to set by one another. (pp. 3-4)

Why would disputation between Cheyne and de Moivre be fair game for Ditton to joke about, yet the Leibniz-Newton dispute never be overtly mentioned in any of these prefaces? The actual object of the respective controversies was of quite a different nature. Cheyne and de

Moivre were clashing over method, and in the budding profession of mathematics at this time there were plenty of social mechanisms which could absorb that type of a battle. Yet a controversy over priority -- indeed, over potential plagiarism -- was very different. Our 18th-century authors couched the Newton-Leibniz dispute in terms of method because, on its own, the topic of priority was not gentlemanly to discuss.

Conclusions

The 20th-century texts from which I drew do not mention whether Leibniz and Newton even knew of each others existence, let alone whether they were at the center of a heated debate. For these authors it is enough to note in passing that the techniques that we use today came independently from two different men. By using the term "we," authors invite students and readers to join them, professional mathematicians, in their judgements and procedures, while evaluating how these procedures relate to those of the historical actors whom they introduce. For these textbook authors, the dominant entity is not social context but the aggregate body of technique that is now called "calculus."

Even during the 18th century, we can see evidence of the authorship of the calculus being effaced in favor of an accepted body of techniques. We have seen that the most partisan 18th-century authors couched the priority dispute in terms of method. In addition, after mid-century even French authors ceased to mention Leibniz as an intellectual predeces-sor while continuing to use his notation.

In conclusion, despite Rupert Hall's description of the Newton-Leibniz dispute as "notorious," this survey of the earliest and the most recently published calculus textbooks has revealed an interesting commonality between books of both eras: the tendency to mute, rather than to reinforce, the rivalry between these two men in favor of portraying the calculus as a body of accepted techniques. In The Structure of Scientific Revolutions (1970), Thomas Kuhn claims that introductory textbooks constitute a major vehicle for the perpetuation of "normal science" -- science that follows an accepted paradigm. Judging by the results of this survey, "normal mathematics" as represented by these calculus textbooks is a technique-driven field which interprets its own history primarily as a lineage of method.