During the growth of calculus, there are some important inventions. The basic problems of calculus that are used nowadays are mostly based on Newton and Leibniz’s inventions in 17th centuries by two of most greatest mathematician, Isaac Newton (1643-1727) and Gottfried Leibniz (1646-1716). The interesting fact is that Leibniz’s invention, which is so called calculus, was claimed by Newton to be the same with his thought about fluxion (Newton’s version of calculus) in earlier period before Leibniz had invented it.
LEIBNIZ and NEWTON
Leibniz invented the calculus in period between 1672 and 1676 and published his theorem at 1684 for the first time. Newton’s discovery about almost the same theorem came before Leibniz, but he never published anything about it until few mentioning on his book Principia Mathematica in 1687. Again, Newton did not publish his full thought about calculus until 1704 after Leibniz’s invention had been presented in L’Hospital’s book. Their theories indeed similar, but they are famous of their own invention in calculus. Leibniz is well known with his differential and integral calculus. Meanwhile, Newton is famous with his fluxion and binomial theorem to find the coefficient of the various powers of x.
Nowadays calculus about differential and integral were born from the concept given by Leibniz, who was firstly curious about philosophy. Before introducing the term calculus differential in the late of 17th century, Leibniz himself used the term methodus tangentium directa. Meanwhile, Calculus integral itself was called as methodus tengentium inversa. Hence, the most influential actor behind his successful on his mathematics career was Christian Huygens from whom he mostly got encourage to study mathematics during his stay in Paris from 1672 to 1676.
The notations for integral () and differential () were given through a note on 26 October 1675. Not long after that, Leibniz settled in Hanover and became more focus on calculus. Therefore, in 1682 he announced the existence of his calculus and soon started to publish them on 1684. This paper opened the modern period of calculus. Many translations on some languages are followed by the successful of his theorem. It considered as a fast development of Leibniz calculus.
Leibniz publication contains several rules on differentiation. It needs to keep in mind that Leibniz used the term differentials, not derivatives. He gave some rules that absorbed in school calculus nowadays, such as
Moreover, Leibniz also stated the condition for maximum or minimum and for a point of inflection. These conditions have been used nowadays and introduced in school to solve problems like finding extreme or stationary point(s) of a curve, whether it is maximum, minimum or inflection point.
In the relation with Leibniz’s integral, he had published a paper, Supplementum geometriae dimensoriae … similiterque multiplex construction lineae ex data tangentium conditione, where he explained the inverse relation between differentials and integrals by means of a figure. This statement is very important in the field of calculus and is applied mostly in either mathematics or physic.
In summary, Leibniz had contributed many of his thought for the improvement of standard and basic calculus, which we are using nowadays. His contribution is not only on the understanding of concept and invention of some formula and condition, but also name of the terms.
According to the history, Newton was first interested in the growing of mathematics. However, he then moved his concentration to physic. Newton started to work on mathematics in 1664 under Barrow at Cambridge. One of his earliest sources on calculus was Latin edition by F. van Schooten of the Geometrie of Descartes which derived him to the used of instead of (derived from Leibniz) which we are using now.
In terms of calculus differentials and integrals, Newton had discovered similar things with Leibniz, but wrote them with different terms and ways. In his book, Principia Mathematica, newton introduced moments, which is the same concept as differentials by Leibniz. Still in the same very famous book, Newton firstly explained some rules of fluxions (Newton’s name for calculus), in a section dealing with the motion of bodies that move against resistance. He worked with problems resulted from the integration differential equations with his notations (he used and instead of and ).
However, there is still one thing in calculus that invented by Newton invented and definitely not ignorable, the binomial theorem which now being known as binomial newton. The main concept of binomial theorem is to determine the coefficient of the various powers of and/or (depends on the main function). In a sense, binomial theorem is used to find the coefficients of each variable combination in expanding of a function, such as . How to get the coefficient is by employing formula, so called combination formula. In some cases, expanding function using binomial theorem is useful.
In summary, beside undeniable fact that Newton had contributed nice thoughts on calculus, people do not really use his notations. However, his so called Binomial Newton remains important and useful for mathematicians, physicians and engineers.
- Besides the fact that Leibniz published his calculus earlier then Newton did, conflict about which terms and notations were more appropriate to be used was solved in the early years of 19th century when leading mathematicians in English-speaking countries began to adopt Leibniz terms and notations, mainly through the impact of the work of Laplace. However, in minor case, Newton’s notations are still used in present day.
- During Leibniz and Newton’s growing invention on calculus, they did send each other letters in 1676 through the help of Henry Oldenburg, the secretary of the Royal Society, which were published on the correspondence of Isaac Newton. However, Newton never replied to Leibniz after his second letter.
- Both of them also had similar difficulty on their publication about differentiating the meaning of either differentials and differences or Newton’s moments (in calculus) and real moments. Moreover, they both were never very consistent in their explanations about differentials and moments.
Struik, D.J. A Source Book in Mathematics, 1200-1800. Massachusetts Institute of technology. Cambridge: Harvard University Press, 1969.
minor taking from Katz, Victor J. History of Mathematics. third edition. Pearson New International Edition. printed in the USA: Pearson Education Limited, 2014.