25 Sep 2015 Principal’s Prize For Maths – Read The Winning Essay
Congratulations to Nadia on winning the Principal’s Prize for Maths. You can read her winning essay below.
“Leibniz and Newton discovered calculus at about the same time, independently of each other. Give a historical account of their discoveries. Of the two, who do you think has the strongest claim to the discovery and why? Compare their methods and discuss which notation is more used now.”
Like any theory, calculus was not plucked out of thin air and then premiered in the late 17th century, despite it being a pivotal time for the field, but in fact is part of a long chain of discoveries, with its earliest accounts existing as far back as Ancient Greece. Calculus, split into differential calculus, a method of working out instantaneous rates of change and integral calculus, which allows you to work out areas of a given function, is however predominantly credited to two men- Gottfried Wilhelm Leibniz and Isaac Newton.
Leipzig Germany, July 1st 1646, a child prodigy was born⁽¹⁾. Leibniz, at the ripe age of twelve, still sought to exceed expectations; he taught himself Latin and Greek, became proficient in the standard text-books of mathematics, philosophy, theology and law, as well as studying books authored by his father, Friedrich Leibniz, a professor of moral philosophy, who died when Leibniz was only six, in extension to his academic studies⁽¹⁾. It only took two more years for him to become a student of the University of Leipzig, reading law, after which, he received a doctorate at Altdorf, the university town of Nürenberg⁽¹⁾. Despite being offered positions in academia, he chose to enter the legal, diplomatic profession which led him to cross paths with princes, electors, archbishop and dukes⁽²⁾.
His diplomatic work provided countless opportunities to travel and meet up with scholars from a multitude of fields. In 1672, Leibniz was sent to Paris, to convince the king, Louis XIV and the French government to severe any impending attacks on German soil⁽¹⁾. During his stay, Leibniz, continued to pursue his other interests, in particular maths and physics, as he studied under Christiaan Huygens, a prominent Dutch mathematician and philosopher in Europe⁽³⁾. Huygens steered Leibniz into work around summing series, whereby using the relationship between sequences of sums and differences he discovered that he can calculate the summation of finite and infinite series⁽³⁾. This idea contributed to the creation of calculus, as Leibniz compared the summation of series, to area, epitomising it to be a sum of infinitesimal differences⁽³⁾.
November 11, 1675, Leibniz found the area under a graph with the equation y=f(x), using the idea of a summation of infinitely thin rectangles, to which we know as integration⁽⁴⁾. Alongside his discovery, he invented a whole new set of notation, allowing all arguments to be written in the form of symbols and formulas. The elongated s, represented the first letter of the Latin word summa, meaning summation and the d, in dy and dx meaning differential, an infinitesimal distance, is the first letter of the Latin translation differentia⁽⁵⁾. The concept of the ‘characteristic triangle’, was fundamental in the creation of calculus; the infinitesimal two sides of the triangle denoted small increments of change, between the horizontal and vertical coordinates of two infinitesimally neighbouring points along a curve and it is their ratio, that determined the tangent line to the curve at that point⁽³⁾. Leibniz saw that similar to the summing of sequences and differences, that the integral and differential are inverse functions of each other⁽³⁾. The differential triangle led to the derivation of the chain, power, product and quotient rule⁽⁵⁾. Leibniz, despite his breakthroughs, delayed the publication of his work on differential calculus until 1684 and 1686 marked the release date of his work on integral calculus⁽²⁾.
Sir Isaac Newton, born in 1642, is considered one of the best scientific intellects of all time, but unlike his counterpart, Leibniz, he didn’t display any prodigious behaviour during his childhood. Newton had an erratic upbringing; the death of his father when he was only three months old, his mother abandoning him for her newly wed at three and when she reunited with Newton at twelve, she removed him from school so that he can take a permanent role, tending to their farm⁽⁶⁾. Fortunately, this plan didn’t pan out and it took Newton’s uncle, a Cambridge graduate, to see his innate abilities and convinced his mother to enrol him at Cambridge. ⁽⁶⁾
Isaac Barrow, another great player in the invention of calculus was a maths professor at Cambridge and took Newton under his wing, steering him into unanswered mathematical problems, including calculus⁽⁷⁾. 1666, the bubonic plague savaged thousands to their agonising deaths, Cambridge as a result sent many students home in account of the outbreak⁽⁶⁾. Newton was among those students⁽⁶⁾. During the study of his degree, Newton, on the surface was a seemingly average student, graduating without honours or distinction, but in that eighteen-month break after being sent home, Newton had somewhat of a revelation, in fact fair to say he had a multitude of epiphanies⁽⁶⁾. In this time he came up with his theories of motion and gravitation, the components of white light and of course calculus⁽⁶⁾. Newton’s theory of calculus, was built in the pursuit to solve difficult problems in physics, especially with his discovery and quantification of gravity, he sought to find a method to describe the rate of change of objects at any given moment in time, hence creating a whole new branch of mathematics⁽⁸⁾.
Newton named his calculus the method of fluxions, with fluxion being the term for the derivative of a continuous function. In his notation, he took the dependent variable x, to be the fluent (coordinates describing the position of object in motion) and it velocity to be the fluxion⁽⁵⁾. He notated the fluxion in respect to the time t, as and the differential of x, which signified the change in velocity of x in an infinitesimal increment of time⁽⁵⁾. Newton mainly used geometric proof to explain his theory, one example is his proof of the product rule, using a rectangle with length and width of A and B considered as the flux and in a given time, there is increase of a and b⁽⁹⁾. The area can be split into three rectangles and a square and the difference between the outer and inner rectangles formed the derivation of the moment (the differential) of the product AB, otherwise more universally known as the product rule⁽⁹⁾. Similar to Leibniz, Newton saw the link between the method of fluxions (differentiation) and the method of fluents (integration) being inverse operations⁽¹⁰⁾. Newton also calculated the area bounded by a curve and its x-axis; by breaking it into infinitesimally thin vertical columns, which helped him solve problems like calculating the distance travelled by an accelerating object, in a given time, using his method of fluents on a velocity-time graph⁽¹⁰⁾.
Today, Newton and Leibniz are both credited with the invention of calculus independently of each other, but that was certainly not the case at the time, as the timing of their publications led to their embittered controversy, which spanned their lifetimes. It is thought to believe that Newton made his discoveries around 1666 before Leibniz’s timeframe of 1672-1676⁽⁵⁾; however like with most of Newton’s work, he kept them under wraps, until in 1687 he published his findings in The Philosophiae Naturalis Principia Mathematica⁽⁸⁾. In it contained his laws of motion, gravitation and other theories, in which he continuously used calculus in his calculations, to support his new theories. However, since Leibniz was the first to publish his work and in essence introduce the world to this new mathematical tool, he was given sole credit to the invention of calculus. This did not sit well with Newton and so took it upon himself to make slanderous allegations against Leibniz, accusing him of plagiarism. It didn’t take long for Newton to construct his case, as there was a lot of circumstantial evidence against Leibniz. His correspondence with Newton, the sharing of mutual colleagues, as well as being under the suspicion of reading Newton’s manuscripts⁽¹¹⁾, in which he was accused of drawing from the fundamental ideas of calculus and took as his own, but with his added notation⁽⁵⁾. The case was taken to be reviewed by the Royal society⁽⁵⁾, but with the appointed committee members compromising of mostly Englishmen, as well as members like Halley, Jones, De Moivre and Machin, who were friends of Newton and him being president of the society, there was little room for objectivity⁽¹¹⁾. And so in 1711, Leibniz’s guilty verdict served as no surprise⁽⁵⁾.
In fairness, Leibniz’s and Newton’s theories of calculus shared many resemblances, hence the claims of plagiarism seeming somewhat probable, but after Leibniz was exonerated, years after his death, there was a continental divide between Britain and the rest of Europe, mathematically. While the rest of Europe continued with Leibniz’s theories of calculus, Britain, with its pretentious need to uphold national pride, continued with the native Newton’s fluxional methods⁽⁵⁾. I, like the millions of schoolchildren across the country are taught calculus using Leibniz’s theories and notation, as it was and still is the most efficient and applicable method, yet throughout the late 18th century and early 1800s, Britain discordance with Leibniz’s methods let them to languish mathematically, while the rest of Europe flourished⁽⁵⁾. Acting clandestinely will never allow mathematics, or any other field for that matter to advance, as ideas should be shared holistically and respected. Newton and Leibniz will forever be marked as some of the cleverest minds that ever existed but their feud, led to a trail of deceit, treachery and foul-play, which blinded them of the true triumph of their discoveries and impeded the development of mathematics.
⁽¹⁾J.J O’Connor and E.F Robertson. 1998. Gottfried Wilhelm von Leibniz, viewed on 14 July 2015, http://www-history.mcs.st-andrews.ac.uk/Biographies/Leibniz.html
⁽²⁾W. W. Rouse Ball. 1908. Gottfried Wilhelm Leibnitz (1646 – 1716) From `A Short Account of the History of Mathematics’ viewed on 14 July 2015, http://www.maths.tcd.ie/pub/HistMath/People/Leibniz/RouseBall/RB_Leibnitz.html
⁽³⁾2002. Leibniz’s Fundamental Theorem of Calculus, viewed on 14 July 2015, https://www.math.nmsu.edu/~history/book/leibniz.pdf
⁽⁴⁾Luke Mastin. 2010. 17th century mathematics – Leibniz, viewed on 14 July, http://www.storyofmathematics.com/17th_leibniz.html
⁽⁵⁾2010. Newton and Leibniz: the Calculus Controversy, viewed on 14 July 2015, Available at: http://ugrad-conf.fitchburgstate.edu/other/Sample-Math-Poster.pdf
⁽⁶⁾Chris Pinaire. 2015. Isaac Newton, viewed on 14 July 2015 http://www.math.wichita.edu/history/men/newton.html
⁽⁷⁾iWonder. 2015. Isaac Newton: The man who discovered gravity, viewed on 14 July 2015. http://www.bbc.co.uk/timelines/zwwgcdm
⁽⁸⁾Jason Garver 2013. How and why did Newton Develop Such Complicated Mathematics? Viewed on 14 July 2015, http://www.fromquarkstoquasars.com/how-and-why-did-newton-develop-such-a-complicated-math/
⁽⁹⁾Science and technology past. 2001. Newton – The Method of Fluxions., viewed on 14 July 2015, http://clivemabey.me.uk/SciTech/gravity/fluxions.php
⁽¹⁰⁾Luke Mastin. 2010. 17th century mathematics – Newton, viewed on 14 July 2015, http://www.storyofmathematics.com/17th_newton.html
⁽¹¹⁾Dorothy V. Schrader. 2012. The Newton-Leibniz controversy concerning the discovery of the calculus. Viewed on 14 July 2015, http://www.personal.psu.edu/ecb5/Courses/M475W/WeeklyReadings/Week03/01-TheNewton-LeibnizControversyDorothySchrader.PDF