Leibniz as the Father of Calculus

           Leibniz is recognized today as the Father of Calculus, as he “invented” calculus hundreds of years ago in around 1674.  To put this in perspective, Leibniz was only 28 when he laid the foundations to a field of mathematics that is widely studied today by nearly anyone who attends higher level education.  It is found in his personal journals dating around 1677 that Leibniz turned his early ideas of infinitesimal calculus into a working and operating system.  None of this work was published, however, until 1684.  The bulk of Gottfried Wilhelm von Leibniz’s mathematical discoveries developed in the 10 years between 1682 and 1892 and can be found in the Acta Eruditorum. 

           Rather than thinking in terms of functions as we often do today, the early studies of calculus were thought of in terms of graphs.  Therefore, it is not surprising to learn that Leibniz’s first published discussion of calculus in Acta Eruditorum was concerning maxima and minima of a curve, and hence drawing tangents on curves.  This exposition article that introduced to the world the idea of differential calculus was titled “New Method for the maximum and minimum, and also tangents, …, and a singular type of calculus for them”.  It should be noted that back in 1674, the term singular was a synonym for “remarkable”. 

Leibniz’s first article as seen in Acta Eruditorum’s issue X, published in October 1684.  Note that in the first line, it references TAB XII (see below).

These graphs and diagrams are used in Leibniz’s explanation
 and discussion of calculus.

           He begins his discussion of differential calculus similarly to how it is introduced in modern day high school calculus classes.  He explains the differentiation of powers and radicals. It is on the pages following that we see Leibniz introduce and apply the notation dy/dx that we use today.  As in present day mathematics, variables x and y were used to describe the coordinate plane (for proof that this idea has been around for 300+ years, refer to Leibniz’s diagrams!).  Gottfried explains that x and y can range over sequences of infinitely close values.   Hence, dx and dy denoted the change in values of x and the corresponding change in y, respectively.  Eventually, he presents the idea that dy/dx gives the slope of the tangent at a given value for x. 

           It is important to note the importance that Gottfried Wilhelm von Leibniz placed on keeping good notation.  He placed such an emphasis on upholding clear and consistent notation, that much of the notation that Leibniz used in his personal notebook and published articles are still used in present day.  It’s astonishing to conceptualize that Leibniz was uncovering ideas so new that there did not exist symbols to present his ideas and findings. 

           Leibniz’s next notable article was published in Acta Eruditorum in June of 1686.  This was the first public debut of the integral sign and a proof of the Fundamental Theorem of Calculus.  Leibniz titled his article “On a hidden geometry and an analysis of indivisibles and infinites.”  In this article, he continues his discussion of the principle of continuity, which is assumed in both his article on derivatives and integrals.  In both discussions, he uses the idea of infinitesimals, which are “quantities that are infinitely small, yet not nonzero.”  Despite the success of calculus and the ideas that Leibniz argued, the concept of infinitesimals was highly criticized by mathematicians.  It took over 100 years for calculus to become rigorous, but as soon as it reached maturity in its development, we began to forget the idea of infinitesimals and instead base calculus in terms of limits.  The evolution of calculus is extraordinary in and of itself, but it all began with 28-year-old Leibniz, who founded an entire field called calculus – or was he?  Stay tuned for a calculus controversy that is still debated today. 

References:

Rouse Ball, W. W. (1908) “Gottfried Wilhelm von Leibniz.”  A Short Account of the History of Mathematics’ 4^th edition.  Retrieved from: https://www.maths.tcd.ie/pub/HistMath/People/Leibniz/RouseBall/RB_Leibnitz.html

Swetz, F. J. (2015). “Mathematical Treasure: Leibniz’s Papers on Calculus.”  Retrieved from: https://www.maa.org/book/export/html/641727.