The Controversy over the Invention of Calculus between Newton and Leibniz - Coursework Example

Analysis and discuss the controversy over the invention of calculus between Newton and Leibniz The invention of calculus is a controversy that remained throughout the lives of Isaac Newton and Gottfried Leibniz until the end of their lives, and the debate has continued to present day (Boyer, 1959:27). The major question is who between Newton and Leibniz invented calculus? The neutral arguments hold that both Newton and Leibniz invented calculus independently, differently and on their own, despite the consistent claims they made against each other, that one of the parties stole the works and ideas of the other. Calculus did exist in the ancient times, but it was not actually advanced and clarified to the level of practical application in solving real life problems. Thus, in the 17th century, the works of Newton and Leibniz came not as entirely new inventions, but basically building on the existing medieval calculus, to generate the modern calculus concepts as they are known to present day (Hall, 1980:12). The Egyptian scientists, the Greek Mathematicians and even some Middle East academic scholars had applied basic calculus concepts in the calculation and determination of certain basic calculus measurements such as the area of spaces and the volume of objects. One of the most prominent mathematicians of the ancient times was the Archimedes of Syracuse, who developed basic formulas of calculating the area of spaces and surfaces, as well as measuring the volume and quantities of liquids contained within certain objects and volumetric cylinders (Simmons, 2007:62). In this respect, while calculus has been attributed to be the invention of the 17th century scientists, Newton and Leibniz, the credit goes back to the scientists before them. Nevertheless, while trying to define who exactly invented calculus between Newton and Leibniz, it is necessary to view the two individual scholars as the co-contributors to the development of infinitesimal calculus (Meli, 1993:36). Analysis The rise of the controversy The discovery of calculus by Isaac Newton and Gottfried Leibniz happened at different time, but definitely putting a finger on the exact time at which each scholar came up with the calculus methods is highly problematic. At the age of 23, that being the year 1664, Isaac Newton claims to have developed the calculus methods, but only documented rather than published them, in one of his annotations (Meli, 1993:17). On the other hand, Gottfried Leibniz claims to have discovered his calculus methodologies in 1674, and then went on to make his first publication of the methodologies in 1682 (Hellman, 1998:26). This controversy therefore becomes an issue of seeking for the justification of the claims made by the two scholars, owing to the fact that none of the scholars claims to have published his calculus methodologies immediately after discovering them. Further, the fact that the documentation related to each time that the scholars claims to have developed the calculus methodologies does no exists also puts the controversy into a more challenging path. The controversy was initially sparked in Leibnizs calculus in 1696, after the publication of Leibnizs calculus in 1696 recognized Isaac Newton’s principle of mathematics that was published by Newton in 1687 in his book Principia although the principle was not put in print until later in part in 1693, and in full in 1704 (Boyer, 1959:14). The controversy over the invention of calculus between Newton and Leibniz does not necessarily focus on who came up with what methods of calculus were developed by each scientist, but centered more on the aspect of priority as to who did it first (Meli, 1993:78). The controversy of time is further crowded by the fact that Isaac Newton discovered his calculus methodologies between 1664 and 1666, only that he did not publish them until later. On the other hand, the calculus techniques developed by Gottfried Leibniz first appeared in the journalist publication of a newly developed journal at the time, Acta Eruditorum, which was circulating throughout Europe, first in 1684, and then later re-appeared in the journal two years later in 1684 (Hellman, 1998:56). In this respect, it can be easily concluded that it is Gottfried Leibniz who published his calculus techniques first, despite the fact that Isaac Newton had discovered his calculus techniques way earlier (Hall, 1980:35). The calculus methods developed by Gottfried Leibniz therefore generated great interest and started circulating throughout different parts of Europe, with further mathematical scholars of the time continuing to develop and extrapolate the calculus methods. Eventually, the Leibnizian calculus became popular and well circulated in Europe by 1692, which is the time that Isaac Newton was starting to publish his calculus methods (Boyer, 1959). Although both Newton and Leibniz are credited with the invention of calculus, they in fact understood and conceived calculus in very different ways. The focus of Leibniz calculus, which eventually became his specialization, was on how the discrete infinitesimal quantities could b applied to measure the size or area of a larger whole (Hellman, 1998:77). This was informed by his perception that an individual component is definitely comprised of the sum of its discrete entities. On the contrary, the focus of Isaac Newton was on the gravitational and planetary motion, such that even the development of his calculus was majorly based on geometry, as opposed to the integration of parts to form the sum-total area of the whole (Meli, 1993). Consequently, most commentators have argued that Isaac Newton developed the differential calculus, while Gottfried Leibniz developed the integral calculus. However, when the two individual contributions to calculus are assessed, it can be seen that each of their different calculus methodologies comprised of both the differential and the integral calculus (Boyer, 1959:64). It is convergence of their distinct inventions into some similar calculus principles and methodologies that comprised of both the differential and the integral calculus, which exacerbates the controversy between Gottfried Leibniz and Isaac Newton regarding the invention of calculus. To the extent that the two scholars would have developed distinct differential calculus and integral calculus principles on their own, then, the controversy would not have accelerated to the modern times. However, the fact that both Gottfried Leibniz and Isaac Newton had a convergence of their calculus methodologies into both differential and integral calculus makes the claim that one scholar copied from the other more plausible (Reyes, 2004:159). Nevertheless, in terms of their different methodologies, the two scholars did come up with some different notations that help them get identified with the invention of calculus to present day. The notable calculus methodology that Isaac Newton developed is the geometrical theory of binomial theorem, which represents the powers of the x and y variables in algebraic expansions (Simmons, 2007). Therefore, it is through the work of Isaac Newton that the power function of (x + y)n was derived, and through further expansion and addition of the coefficients a and b, resulted in the development of the Pascal triangle (Reyes, 2004:168). Through the application of the binomial theorem, Isaac Newton was able to expand the application of the power function of (x + y)n through the application of the algebra of finite quantities to analyze an infinite series. The other notable area of Isaac Newton’s calculus discovery is summarized down as the Fluxionary Calculus, which he applied to define the area under a curve, not through the application of the conventional formula of calculating the area under a curve as presented in the modern day calculus, but rather through first calculating the momentary rate of change, which he then extrapolated into the area (Meli, 1993:14). His reasoning was defined by the application of a small triangle whose area he defined in terms of x and y. however, due to his extensive knowledge in physics regarding the geometric, planetary and gravitational motion, Isaac Newton tried very much to avoid the application of the infinitesimal characteristics of derivatives, owing to the fact that he only understood them informally (Boyer, 1959:13). Therefore, he developed his calculations based on the ratios of change, thus eventually introducing the dot concept in the variables x and y (Reyes, 2004:172). Through the application of the dot x and the dot y variables, Isaac Newton was able to define the rate of change generated as a fluxion, represented by a doted letter, and the overall quantity generated was defined as the fluent (Simmons, 2007:67). Therefore, it is through the works of Isaac Newton that the concept of the dot x and dot y, also represented as the land as it is taught in some physics classes, today came into existence. Through this concept of the dot x and the dot y as the fluxions and the outcome generated as the fluent, he produced the revised calculus ratios, which then gave rise to the present day derivative ratio of change (Meli, 1993:21). Therefore, through the works of Isaac Newton in deriving the derivative of the ratio of change, he indicated not only the possibility of viewing the infinite series not only as approximates, but also the possibility of the application of the infinite series as a form of expressing a term (Hall, 1980:33). The invention of calculus attributed to Isaac Newton are traceable in the period 1665-1666, during the plague quarantine in England that saw Newton quarantined for some time, allowing him to define not only his physics and mathematical philosophy, but also generate the calculus derivates that have since been in use to present day. Gottfried Leibniz discoveries on the other hand are not entirely different from those of Isaac Newton, only that Leibniz did not fear to apply the formal infinitesimals in developing and advancing his calculus derivatives (Meli, 1993:44). In this respect, most of the introductory books to calculus have credited Gottfried Leibniz with the invention of integration derivatives, while on the other hand crediting Isaac Newton with the invention of the differentiation derivative (Hall, 1980:21). However, when carefully assessed, it can be seen that Gottfried Leibniz was the brain behind the formal infinitesimals of both the integration and also the differentiation formulas as currently applied in modern day calculus. Therefore, the first invention of the integration symbol ( ∫ ), as currently applied in major calculus integration derivatives is credited to the genius of Gottfried Leibniz (Boyer, 1959:63). In addition, the further refining of the integration derivatives and the eventual generation of the calculus integration formula defined as the dy/dx notation is also credited to the genius of Gottfried Leibniz. Further, the characteristic geometric triangle was also formulated by Gottfried Leibniz, whereby he defined the area under a triangle through the geometric approach, under which the differentiation characteristics of dy/dx notation would be applied to define the area. This resulted in the modern day characteristic definition of the area under a triangle under the geometric approach as a function of y versus x (Simmons, 2007:71). Therefore, while the subject of the controversy over the invention of calculus between Newton and Leibniz initially started as the determination of the priority approach that defines who invented the calculus first, the controversy debate has later been reduced into a controversy of who developed what calculus methods (Reyes, 2004:169). The focus on the different calculus methods and how they were invented has negated the initial focus on the controversy over the invention of calculus between Newton and Leibniz, from when calculus was invented and by whom, to who invented the different methods as they currently operate under the modern day calculus. For example, to illustrate the debate between Newton and Leibniz, most of the calculus texts applied in the modern day do not present the controversy in terms of the history of the two great scholars and their roles in the invention of calculus. Instead, the books have opted to present this controversy through the attribution of the different calculus derivative to the different scholars. Therefore most of the introductory books to calculus are found differentiated in terms of different chapters, for example Part I: Newton’s “Differential Calculus" and Part II: Leibniz’s “Integral Calculus" (Simmons, 2007:62). The categorization of these calculus derivatives into such distinct attributions of character has served to increase the controversy over who invented calculus, since the books have already attributed differential calculus to Newton and integration calculus to Leibniz, yet in the real sense, each of the two scholars individually contributed to both integration and differential calculus derivation. However, the analysis of the books and literature that was developed much earlier, having their dates of publication between 1969 and 1991, a clear cut understanding of the nature of the controversy over the invention of calculus between Newton and Leibniz has been made. The analysis of Isaac Newton’s most renowned book, the Principia, in addition to other of his later publications has gone a long way to show that Isaac Newton ventured into the issue of calculus more than did Gottfried Leibniz (Simmons, 2007:77). However, this fact is not meant to show that the invention of calculus is attributed to Isaac Newton more than it is attributable to Gottfried Leibniz. Instead, this fact illustrates that Isaac Newton went much further to explore the applications of the principles of calculus, while also taking a different path than the one already taken by Gottfried Leibniz (Boyer, 1959:16). The view of Isaac Newton on calculus was based on limit and concrete reality, as opposed to the road that was taken by Gottfried Leibniz, which ventured more into the infinite and abstractedness of the derivative functions. Nevertheless, the literature on the two scholars that was published in the two decades running between 1969 and 1991 have managed to show that despite the two scholars taking two different paths in defining their inventions of calculus, there is a major basis why the controversy over who invented calculus still rage on. These literatures report that unaware that Isaac Newton had already invented and developed his different calculus methodologies, Gottfried Leibniz invented and published his inventions in the field of calculus between 1973 and 1676 in a well circulated European journal of science (Simmons, 2007:65). Thus, through the wide interest that the calculus inventions of Gottfried Leibniz developed throughout Europe, starting with Paris where the journal first published his calculus methodologies, Leibniz discovered that he was up to something big, which was tantamount to a breakthrough in the field of physics and mathematics. However, what he did not know is the fact that Isaac Newton had already ventured into the same field before and developed similar methodologies, only that he had decided to keep his lips tight over his major breakthrough (Reyes, 2004:159). In this respect, the literature published between 1969 and 1991 in the field of calculus has gone further to show that in fact, it was the fact that Isaac Newton had chosen to keep quiet over his calculus inventions rather than publish them immediately he developed them, that became the ultimate source of the whole controversy over the invention of calculus between Newton and Leibniz. The accounts of published calculus indicates that Gottfried Leibniz first published the account of differential calculus in 1684, and then followed the same with the explanation and account of the integral calculus in 1686 (Meli, 1993:45). However, it is only after Isaac Newton’s book was published in 1687 detailing the calculus methodologies under the integral and also the differential calculus that had already been published by Gottfried Leibniz, that it was realized that already the calculus derivatives had been invented before. Therefore, while Gottfried Leibniz accused Isaac Newton of copying his work by publishing the calculus methodologies in 1687, Isaac Newton holds that he was the first inventor of the calculus, only that even after their initial invention and development; he did not publish them in print. Despite the fact that Isaac Newton did not publish his calculus discoveries early, the available evidence has shown that indeed, Isaac Newton discovered fluxional calculus in the period between 1665 and 1666, and recorded them in his writings, only that he did not develop them fully into publishable content (Hellman, 1998:53). This evidence has gone ahead to show that Isaac Newton was the first to establish and document the first calculus method known as the ‘theory of fluxions’, in addition to being the first to state the fundamental theory of calculus (Meli, 1993:45). Additionally the existing and traceable documented evidence from the period 1665-1666 indicates that indeed Isaac Newton was not only the first one to discover both the differential and the integral derivatives, but was also the first to experiment their application in a single work (Hellman, 1998:51). Nevertheless, due to the fact that Gottfried Leibniz was the first one to publish a dissertation on calculus, he was eventually credited with the full credit of having invented calculus over a number of years. However, the credit given to Gottfried Leibniz over the invention of calculus later resulted in his being hurled accusations of plagiarism (Hall, 1980:54). The doubt as to whether Gottfried Leibniz really applied plagiarism in using the ideas of Isaac Newton to develop the calculus derivatives and then publish them as his own have left little doubt in the minds of the proponents of these accusations. This is because, it has been claimed that Isaac Newton used to pass his manuscripts over to several of his colleagues after the discovery of new ideas, some of whom had close contacts with Gottfried Leibniz (Reyes, 2004:177). However, this hypothesis has never been proven to present day. Conclusion The controversy over the invention of calculus between Newton and Leibniz is a debate that has persisted over the past century, and is sure to continue through to the coming decades. However, the fact that Isaac Newton had arrived at his discoveries earlier, between 1665 and 1666 but failed to publish them is indisputable. On the other hand, the fact that Gottfried Leibniz published his discoveries on calculus earlier than Isaac Newton, between 1674 and 1684, before Newton eventually published his inventions in 1687 is also indisputable. Thus, what remains disputable is whether Gottfried Leibniz had copied and developed the calculus ideas from the works of Isaac Newton, and then published them as his own work. References Boyer, C. (1959). The History of the Calculus and Its Conceptual Development. New York: Dover. Hall, R. (1980). Philosophers at War: The Quarrel between Newton and Leibniz. New York: Cambridge University. Press. Hellman, H (1998). Great Feuds in Science. New York: John Wiley and Sons. Meli, D. B. (1993). Equivalence and Priority: Newton vs. Leibniz. Oxford: Oxford Science Publications. Reyes, M. (2004). The Rhetoric in Mathematics: Newton, Leibniz, the Calculus, and the Rhetorical Force of the Infinitesimal. Quarterly Journal of Speech 90: 159–184. Simmons, F. (2007). Calculus Gems: Brief Lives and Memorable Mathematics. Mathematical Association of America. 61-98. Read More

This controversy therefore becomes an issue of seeking for the justification of the claims made by the two scholars, owing to the fact that none of the scholars claims to have published his calculus methodologies immediately after discovering them. Further, the fact that the documentation related to each time that the scholars claims to have developed the calculus methodologies does no exists also puts the controversy into a more challenging path. The controversy was initially sparked in Leibnizs calculus in 1696, after the publication of Leibnizs calculus in 1696 recognized Isaac Newton’s principle of mathematics that was published by Newton in 1687 in his book Principia although the principle was not put in print until later in part in 1693, and in full in 1704 (Boyer, 1959:14).

The controversy over the invention of calculus between Newton and Leibniz does not necessarily focus on who came up with what methods of calculus were developed by each scientist, but centered more on the aspect of priority as to who did it first (Meli, 1993:78). The controversy of time is further crowded by the fact that Isaac Newton discovered his calculus methodologies between 1664 and 1666, only that he did not publish them until later. On the other hand, the calculus techniques developed by Gottfried Leibniz first appeared in the journalist publication of a newly developed journal at the time, Acta Eruditorum, which was circulating throughout Europe, first in 1684, and then later re-appeared in the journal two years later in 1684 (Hellman, 1998:56).

In this respect, it can be easily concluded that it is Gottfried Leibniz who published his calculus techniques first, despite the fact that Isaac Newton had discovered his calculus techniques way earlier (Hall, 1980:35). The calculus methods developed by Gottfried Leibniz therefore generated great interest and started circulating throughout different parts of Europe, with further mathematical scholars of the time continuing to develop and extrapolate the calculus methods. Eventually, the Leibnizian calculus became popular and well circulated in Europe by 1692, which is the time that Isaac Newton was starting to publish his calculus methods (Boyer, 1959).

Although both Newton and Leibniz are credited with the invention of calculus, they in fact understood and conceived calculus in very different ways. The focus of Leibniz calculus, which eventually became his specialization, was on how the discrete infinitesimal quantities could b applied to measure the size or area of a larger whole (Hellman, 1998:77). This was informed by his perception that an individual component is definitely comprised of the sum of its discrete entities. On the contrary, the focus of Isaac Newton was on the gravitational and planetary motion, such that even the development of his calculus was majorly based on geometry, as opposed to the integration of parts to form the sum-total area of the whole (Meli, 1993).

Consequently, most commentators have argued that Isaac Newton developed the differential calculus, while Gottfried Leibniz developed the integral calculus. However, when the two individual contributions to calculus are assessed, it can be seen that each of their different calculus methodologies comprised of both the differential and the integral calculus (Boyer, 1959:64). It is convergence of their distinct inventions into some similar calculus principles and methodologies that comprised of both the differential and the integral calculus, which exacerbates the controversy between Gottfried Leibniz and Isaac Newton regarding the invention of calculus.

To the extent that the two scholars would have developed distinct differential calculus and integral calculus principles on their own, then, the controversy would not have accelerated to the modern times. However, the fact that both Gottfried Leibniz and Isaac Newton had a convergence of their calculus methodologies into both differential and integral calculus makes the claim that one scholar copied from the other more plausible (Reyes, 2004:159).

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