Introduction
Sir Isaac Newton is credited with initiating and developing mathematical theories of ancient geometry into fully developed branch of mathematics we have today. He combined different ideas of early geometers into two unifying themes of the derivation and the integral. The transmission of mathematical knowledge from one culture to the other is a matter of research and debate we don’t intend to go into details. The route of some mathematical ideas can be traced from the ancient Indian astronomers, the exemplary work of Persian geometers and the entire classical Greek world, to mention just a few. He turned these ideas into explicit connections and exploited them as tools for mathematical discoveries. He used integration method to show connections between mathematical formulas and later turned to calculus into great problem solving tool we have today.
Development of mathematical theory
Newton’s special talent in mathematics started at an early age. By the age of 22, his passion in mathematics had grown even bigger when he started investigating into physics and geometry that later transpired into calculus. He devised various methods of dealing with mathematical problems based on ideas from past geometers like the Greek, Indians, Persians and even Arabs. His first mathematical discoveries were differential and integral calculus and gravitational law. He introduced his own theory of infinitely small numbers (infinitesimals) which was the best solution to finding the areas of derivatives or slopes. He also invented a mathematical symbolic language subsequently creating a new mathematical subject known us calculus which was part of his scientific inventions of motions and magnitudes (Sandnes & Rasmussen, 2005).
By 1666, Newton had developed a new form of mathematics known as the calculus subsequently developing into derivation and integration methods. Though afraid of criticism, his methods were never publicized and shifted his focus into tangent problems since he believed calculus was a metaphysical explanation of change he planned to look into in future. His central focus was to formalize the inverse properties between the integral and the differential; this was the first calculus system where he created new rhetoric and descriptive terms. Primarily due to his hard work and ability to synthesize the insights around him into a universal algorithmic process enabled him to form a new mathematical platform that influenced the direction of the modern mathematics (Sandnes & Rasmussen, 2005).
Through 18 century, calculus enlarged into mathematical application which is used today in natural and physical sciences. This involved a major shift away from working just with calculus towards working with integration and derivation (Chapter 3, 2005, p.11). Newton’s many mathematical discoveries established transition between the ancient calculus and the modern mathematics. He’s incredible aptitude was recognized by many geometers all over the world. At an early age he quickly learned about the current theories and by 1664 he had already advanced the binomial theorem formula which included fractions and negative exponents and extended this by applying algebra of finite quantities in an analysis of infinite series. His approach demonstrated how he viewed infinite series as alternative forms (Smith, 2007).
Fluxions Theory
Differential calculus known to Newton as “Theory of Fluxions” was developed by early 1666. The formula was applied calculating the areas of indefinitely small triangles whose areas were considered a function of x and y he called fluents. He argued that when infinitesimal increase in abscissa, a new formula will be created where x=x+O (letter O not a digit 0). He then progressed to calculating the area using the binomial theorem by removing all quantities with letter O and formed algebraic expression of the area. His formula created a solution of an area under curve by considering a momentary increase at a central point at this effect; a formula he called fundamental calculus. In his development admitted to the errors that evolved in mathematics and referred to them as continual flowing motion. He considered variable magnitudes to be generated by motion and not by infinitesimal elements (Smith, 2007: Whiteside, 1981).
Newton not only explicitly recognized a connection between derivation and tangents but also gave calculus a more rigorous approach when he complied the Methodus fluxionum et Serierum infinitarum into definite terms in 1671. This new method shaped and defined how fluxional calculus could be approached leading to a greater understanding of these phenomena in a more general and mathematical setting. This method used methodological tool in calculating fluxional calculus in explaining the physical world other than the existing instantaneous motion. His revised calculus became continuity like the continual flowing calculus. The Methodus Fluxionum defined the quantity generated (fluent). For instance, if x and y are fluents, then x and y and the velocity of this fluents was termed as fluxion. His inventions continued to develop gradually and were termed in 1676 as “De Quadratura Curvarum”. This method defined the present day derivative as the ultimate ratios of change which defined the ratio between evanescent increments. He explained the ultimate ratio by appealing to motion-ratio as the increments vanish into nothingness (Smith, 2007).
In Newton’s days, a variable was termed as a fluent that flows with time and its derivative or rate of change changes with time, later termed as “fluxion”. It should be noted that his solid foundation in mathematics helped influence the direction of derivation which would have otherwise been ignored. His recognition in magnitudes brought major changes to the current mathematics as he had extensive literature and pursued his own line of analysis. His discovery of binomial theorem, a tool that employed infinitesimals in finding areas under curves and slopes of curves brought major development to calculus (Smith, 2007).
Conclusion
Despite the attempts of many geometers to find solutions to mathematical problems, it was Sir Isaac Newton who finally provided the transition between ancient and modern mathematics. He clarified the notion of derivation and integration and resolved many foundational difficulties, allowing analysis to develop further into calculus-now used by many. Although calculus received numerous criticisms at the time of invention, Newton succeeded in putting calculus on a firm foundation. He synthesized concepts and results of calculus with the deductive methods of ancient geometers thereby stimulating full development of mathematics we have today. The inter-relationship between the ancient and modern calculus can be seen in the invention of computers which has provided simpler forms of solving calculus problem which were considered impossible.
References
Chapter 3. (2005). Analysis: Calculating Areas and Volumes. Opaque, 97, 1-15.
Sandnes, A., & Rasmussen, M. (2005). How integral calculus has developed. Web.
Smith, E. D. (2007). History of Modern Mathematics. New York: Cosimo books Whiteside, D. (1981). The Mathematical Papers of Isaac Newton. Cambridge: Cambridge University Press.