Leibniz and the Discovery of Calculus The introduction of calculus to the world in the seventeenth century is often associated with Isaac Newton, however on the main continent of Europe calculus would be near-simultaneously discovered by Gottfried Wilhelm Leibniz. With such a strong affinity to knowledge and learning from a young age, Leibniz s contributions in mathematics would lay the cornerstone for calculus as we know it today. Leibniz was born in Leipzig, Germany on July 1st, 1648 to a pious and well educated family. His father was notable professor and philosopher. His father was a notable professor at the local university, but would end up dying when Leibniz was still a young child. Significantly, Leibniz was full access to a large library left behind by his father and would take excellent advantage of this. Perhaps due to his family s background, Leibniz initially used this library to study historical and religious works. Although Leibniz would get a more standard education at the University of Leipzig where he would study philosophy and humanism, his ability to self teach himself afforded to him by his early life circumstances would prove far more impactful. Ultimately, Leibniz would travel to Paris, the greatest learning hub in all of Europe at the time, 1 where he would meet Christiaan Huygens. Recognizing his brilliance, Huygens would guide Leibniz into studies of mathematics and physics. It is at this point where Leibniz would begin to make true breakthroughs and form the basis of calculus. In 1672, Huygens presented Leibniz with a problem regarding triangular numbers. These are the numbers that can form a triangular arrangement of dots, the first of which are 1, 3, 6, 10, 1 Totally History contributors. (2012). Gottfried Leibniz. Totally History. Retrieved from http://totallyhistory.com/gottfried-leibniz/
2 and 15. Huygens asked Leibniz to find the total sum of the reciprocal of all triangular numbers. Knowing the equation to find a triangular number is i (i + 1 )/2, Leibniz knew the reciprocal would be 2 /(i(i + 1 )). He split this reciprocal into the form 2/i 2/(i + 1 ) and started writing down the first few terms of the sum: ( 2 2 /2) + (2 /2-2/3) + (2/3-2/4) + + (2/i - 2/(i+1)). He noticed that the second part of each term would be cancelled out by the first part of the next term, and thus simplified the summation to 2 2 /(i + 1 ). Given that i is a very large number, it was obviously apparent to Leibniz that the solution to this problem was 2. The way the entire summation collapsed to leave only the difference between the first and final term fascinated Leibniz. He began to use this problem as an analogy for finding the area of something as the summation of infinitesimal differences. Leibniz then started looking at a curve of a function as a series of right triangles whose legs are the infinitesimal difference between the vertical and horizontal difference between a point on the curve and a point infinitesimally close to it, making the slope of the hypotenuse equal to the slope of the tangent line at the point. Slowly, this would lead to the discovery of what we refer to today as the fundamental rule of calculus. Particularly recall the second part of the fundamental theorem of calculus which states that b a f (x)dx = F (b) F (a). It can be seen how Leibniz went from the initial problem of the triangular numbers and, through his brilliance and learning ability, arrived at this breakthrough in mathematics and the dawn of calculus. With this invention of calculus, Leibniz wanted to also develop a clear and efficient notation that was logical and easy to understand. Initially the symbol he used for integration was 2 Laubenbacher, Reinhard. (1998). Leibniz's Fundamental Theorem of Calculus. NMSU. Retrieved from https://www.math.nmsu.edu/~history/book/leibniz.pdf
3 o mn ω which referred to summing all of the ω. It was not long, however, until he switched to using the symbol which is used today. Years later he would actually end up switching to a different symbol that was similar to the modern day symbol with the exception that it was not curved at the bottom, but would ultimately revert back to today s symbol. A lot of the inconsistencies in the notation Leibniz used in his work is said to be because Leibniz was greatly concerned with the clarity of his work and would adjust his notation based on his audience's opinion. Strongly believing in the importance that the symbolism be in agreement by mathematicians, Leibniz consulted with his colleague Johann Bernoulli. Bernoulli favored the symbol I denoting integration, but when asked of his opinion on the symbol, meant to be an elongated S, to refer to integration as a sum, had reached an agreement. Through similar conversations with prevalent mathematicians of the time, Leibniz would also settle with the symbol d to signify difference which is used in the notation d y/dx. Most of Leibniz s discoveries in calculus are shared in two of his works. The first, published in 1684, was titled Nova Methodus pro Maximis et Minimis, itemque Tangentibus 4 which translates to, A new method for maxima, minima and tangents In this work Leibniz details differential calculus along with rules such as the product rule, quotient rule, and power rule. Ironically, given Leibniz s strong desire for clarity listed above, no proofs are provided for any of these rules. This led to some confusion even among his colleague Jacob Bernoulli, but this is not to say that Leibniz s work was incorrect. To the contrary, these are the same differentiating 3 Brumbaugh, Zachary. (2000). The Integration Theory of Gottfried Wilhelm Leibniz. Rutgers. Retrieved from http://sites.math.rutgers.edu/~cherlin/history/papers2000/brumbau.html 4 O'Connor, JJ. (1998). Gottfried Wilhelm von Leibniz. University of St. Andrews. Retrieved from http://www-history.mcs.st-andrews.ac.uk/biographies/leibniz.html
rules used in calculus today. Leibniz s second work on calculus, published in 1686, is titled Acta Eruditorum and deals with the integral parts of calculus. This is actually where the symbol made its first appearance. Around the same time as all of this, Isaac Newton is making similar discoveries in calculus which resulted in a great controversy over who gets priority for the discovery. Leibniz, obviously believing himself to be the original discoverer, argued that It is most useful that the true origins of memorable inventions beknown, especially of those that were conceived not by accident but by an effort of meditation. The use of this is not merely that history may give everyone his due and others be spurred by the expectation of similar praise, but also that the art 5 of discovery may be promoted and its method become known through brilliant examples. Meanwhile, Newton argued that Leibniz s work was plagiarism. Newton further argued that even if Leibniz made his discoveries on his own without stealing from his own work, he (Newton) should be credited with the discovery since he claims to have found it first. This claim is accepted now beyond a doubt, but unfortunately for Newton a lot of his work was not initially written and he was very slow to publish any of it, resulting in continental Europe first receiving Leibniz s accounts of calculus. In the end, the Royal Society - who both mathematicians were members of - would conclude that Newton was the first to discover calculus whereas Leibniz 6 was the first to publish calculus. As for the test of time, it is apparent that Leibniz is the victor in this controversy as ultimately his notation is the one that endures prominently today. 5 Bardi, Jason S. The Calculus Wars: The Greatest Mathematical Clash of All Time. New York. Basic Books, 26 April 2007. Print 6 Mastin, Luke. (2010). 17th Century Mathematics - Leibniz. Story of Mathematics. Retrieved from http://www.storyofmathematics.com/17th_leibniz.html
Aided by access to vast amounts of resources at a young age along with a brilliant mind capable of teaching himself anything, Gottfried Wilhelm Leibniz created a lasting legacy in the field of mathematics and calculus. Although the controversy with Isaac Newton followed him to the grave as he died in 1716 in home in Germany, calculus as we know it today would not be the same without the great contributions Leibniz provided to the field of mathematics.
Works Cited Totally History contributors. (2012). Gottfried Leibniz. Totally History. Retrieved from http://totallyhistory.com/gottfried-leibniz/ Laubenbacher, Reinhard. (1998). Leibniz's Fundamental Theorem of Calculus. NMSU. Retrieved from https://www.math.nmsu.edu/~history/book/leibniz.pdf Brumbaugh, Zachary. (2000). The Integration Theory of Gottfried Wilhelm Leibniz. Rutgers. Retrieved from http://sites.math.rutgers.edu/~cherlin/history/papers2000/brumbau.html O'Connor, JJ. (1998). Gottfried Wilhelm von Leibniz. University of St. Andrews. Retrieved from http://www-history.mcs.st-andrews.ac.uk/biographies/leibniz.html Bardi, Jason S. The Calculus Wars: The Greatest Mathematical Clash of All Time. New York. Basic Books, 26 April 2007. Print Mastin, Luke. (2010). 17th Century Mathematics - Leibniz. Story of Mathematics. Retrieved from http://www.storyofmathematics.com/17th_leibniz.html