Isaac Newton – Pioneer in Differential Equations

All elementary schoolchildren are familiar with Mathematics as a subject of numbers and operations on numbers. As our education ventured forward, our mathematical abilities grew. In High School, we are no longer confined to doing mathematical operations on numbers as we learned to add, subtract, multiply and divide variables in a field called Algebra. Later on, our mathematical world expanded once again as we learned to manipulate something called functions. As our mathematical abilities expanded so has our ability to represent real world objects with mathematics. As children we learned that two apples plus three apples equals five apples. In high school, we learned that the only number apples that gives us a total of five apples if we have two apples is three. Mathematics becomes not only a tool but a language as our abilities grew. A language that we can translate to and from English. Consider for example the second law of motion. In English, we can express it as the following:

“A force applied upon a subject will result in accelerating the object at a rate proportional to its mass”

We may express this mathematically as F = MA. While this completely satisfies our English text, our mathematics is limited to constant forces, and constant acceleration. Instead, we can rewrite our second law as the following:

(Wylie & Barrett, 1092)

In this case, we represent Newton’s second law not only as an equation but as a differential equation. The equation represented our acceleration as the derivative of motion – a function. Depending on the initial conditions, this equation may be satisfied by constants or by time varying functions. The last part is important. Its beauty is that it can accurately give us the acceleration of our object over time when the forces against it are time varying. Indeed, differential equations has been an important tool for science and engineering for the past 300 years. All of this thanks to the pioneering work done by one of the most brilliant scientists and mathematicians who ever lived – Sir Isaac Newton.

Isaac Newton was born on Christmas, 1642 at the town of Wolstrop in Britain. He was born into a wealthy family: his father had been the Lord of the Manor of Wolstrop. At age 12, he was sent to a public school at Grantham but was taken from his studies a few years later to learn to take over the family business. His parents soon discovered that young Newton had no predisposition towards business and was still devotedly studying his books. He was then allowed to continue his education at Grantham. In 1600, the 18 year old Newton was admitted to Trinity College at Cambridge University. It was here in Cambridge that he would develop into one of the most brilliant mathematicians and minds the world has ever known (de Fontanelle).

As shown by our second law example, the importance of differential equations is in modeling the time varying behavior of systems. These systems can be anywhere from electric circuits, to weighted springs, to the heat of a radiator to the skin of a drum. The key in differential equations is to understand continuously varying quantities.

Rene Descartes and Pierre de Fermat had attempted to solve these problems before Newton. Their solutions were cumbersome for general use and sacrificed elegance and simplicity. Moreover, they only attempted to provide a solution for specific problems, not the general case (Christianson, 36).

While a student at Cambridge under Isaac Barrow, Newton’s interest in mathematics was spurred. It was here that he started studying the previous analytical work done by Descartes. He used the Cartesian coordinate system to understand the conic sections as well as the equations that governed them. Before 1664 drew to a close, Newton was able to measure the slope of a curve at any given point – a problem called the “problem of tangents”. This method of finding the tangent at any given point is essentially the process of getting the derivative – differentiation (Iliffe, 25).

Between 1663 and 1665, Isaac Newton embroiled himself in studying the work of mathematician John Wallis. Wallis offered a way of measuring the areas under sections of a curve of the form x = yⁿ. His method relied on dividing the area under the curve into small sections – a forerunner of integration. By the winter of 1664, Newton was able to build upon Wallis’ work. He was able to refine Wallis’ technique to work with curves of fractional powers (Iliffe, 25).

By the start of 1665, Newton was beginning to see that the process of differentiation and integration were but inverse operations of one another – the fundamental theorem of calculus. At this point, Newton was treating curves as traces carved out by traveling points. He was calling his work the method of  fluxions – referring to how the curve was flowing from one point to another (Iliffe, 26).

Newton’s fluxions brought him immense power and he realized it. Using his fluxions, he can accurately predict the variations in acceleration of a falling body. He used his fluxions to compute the arc of a revolving planet. The method of fluxions enabled him to measure how a rolling ball inches to a stop. Newton realized that his fluxions allowed him to calculate the smallest, most infinitesimal variations of a body’s motion. When he presented his work to Humphrey Babington in 1665, a Cambridge scholar, he carried out one division to 55 decimal places. However, Newton was curiously silent about his work. He did not publicize his discovery to the mathematical world until much later (Christianson, 37).

In 1668, the German mathematician Nicholas Mercator published his book titled Logarithmotechnia. In the book, he outlined his method of solving the area under a hyperbola via an infinite series. This was the first time that such a method of solution had been widely published. Isaac Darrow, recognized these as similar to the work done by one of his students – Isaac Newton. Moreover, his students’ work was not confined to the hyperbola but extended into all kinds of curves, even solids and surfaces of solids (de Fontenelle).

Newton wasn’t worried upon seeing Mercator’s work. He was convinced that Mercator had discovered his secret (referring to fluxions) and that if not, it would have been discovered by others before he was of the proper age to publish it. It was clear that Newton, aged 27, was quite humble about his achievements. It was only upon pressure from Darrow that Isaac Newton gave out his manuscript in 1669 to William Brouncker and John Collins. The Manuscript titled “A method which I had formerly discovered” discusses his invention and use of his method of fluxions (de Fontenelle).

Mercator’s book was not the only case wherein someone published Newton’s work before him. One of the most bitter mathematical conflicts arose because of Newton’s failure to publish. In 1684 Gottfried Leibniz published the rules of differentiation and integration. He had been using his calculus since 1675, almost a decade after Newton started using his fluxions. He also used a notation that was more elegant than Newton. Swiss mathematician Nicolas Fatio de Duillier suggested to Leibniz that his calculus was inferior and younger than Newton’s own calculus. Thus started a bitter exchanges between the two mathematicians (Iliffe, 121). As early as 1728,  it had been the consensus by mathematicians that it was Isaac Newton discovered calculus first and that Leibniz discovered it independently of Newton. Leibniz however had been the first to make his work public (de Fontenelle). It wasn’t until Newton’s second book, Optics in 1704, did Newton made public his fluxional calculus. The book contained an appendix on the method of fluxions (Ball, 353).

Calculus is but one thing which has been attribute to Sir Isaac Newton. He is also widely known for his contributions in understanding the field of mechanics. His major work, Philosophiæ Naturalis Principia Mathematica or simply Principia laid the foundation for modern mechanics and gravitation. It is regarded as one of the most important books ever written and its methods and laws are still being used by scientists, engineers and students more than 300 years later.

It is in the Principia that Newton outlined his famous three laws of motion including the second law of motion. Newton stumbled onto the second law after his investigations into the motion of the planets. He reasoned that it was the gravitational pull of the sun which prevents the planets from going off into outer space (Christianson, 84).

Incidentally, Newton already had a complete understanding of calculus through his fluxions even before Principia was written. However, few traces of calculus is present in the Principia with the majority of the mathematics in Principia being framed using geometrical analysis. The question therefore is raised of how much did Newton’s knowledge of calculus aid him in writing the Principia. This has been a matter of debate since the 17^th century when mathematical historian Bernard de Fontenelle asserted that the whole of Principia is about differential calculus. Additionally, it is known that Newton and his students used calculus in the analysis of forces and motion. It is therefore concluded that Newton did use calculus in writing the Principia.

It is however still a mystery why Newton decided to keep the knowledge of fluxional calculus within his inner spheres (Guicciardini, 303-344).  Some authors have speculated that this was because Newton celebrated discovery itself. Understanding and mathematics were joys he was content with and that recognition was only secondary to this (Christianson, 37). Aside from his work on optics, each of his works were only published due to the urgings of his colleagues. For example, it was Edmund Halley who paid for the publication of the Principia. There have also been multiple instances wherein he agreed to publish papers on the condition that he was not named as the author (Ball, 355). While this may be indicative of someone who is bashful and uncertain of his work, we can certainly say that it is conduct unbecoming of someone who has been described by his contemporaries as having made the greatest production of the human mind (attributed to Lagrange)  (Ball, 355).

Together with calculus, the second law is an indispensable tool for the field of dynamics.  The two are used to understand nearly every system where a force is acting upon it. That system can be as simple as a rock on a spring or as complex as the solar system. We can even say that the two Newtonian contributions serve as the language for the entire field of classical dynamics.

It is true that the second law may be used to analyze motion without having to use calculus. Indeed, it may even simpler to use the F = MA form of the second law. Simpler as it may be, it is not as elegant as the differential form of the second law. Using the differential form, we can quickly translate the effects of forces as displacement or velocity, not just acceleration. Moreover, we can use the differential form to make a parametric valuation of the second law wherein both acceleration and force are time varying. Thus with the loss in simplicity, we gain flexibility in analyzing dynamic systems and the forces acting on them. This is especially more important since there are probably more systems that have time varying forces than constant forces.

Students studying differential equations have Isaac Newton to blame for making their lives much more difficult. Getting through ordinary differential equations, partial differential equations and other forms of advanced calculus are one of the hardest points in any science, engineering or mathematics students life. These students therefore must thank Isaac Newton for the sleepless nights and the difficult exams. These same students would use their newfound knowledge in analyzing our world and building upon it. Therefore, as a society we can thank Isaac Newton and his creations -calculus and the second law- for the space program, modern automobiles, satellite communications and for any other piece of technology that is based upon non uniform forces and motion.

Bibliography

Ball, W.W. Rouse. A short account of the history of mathematics. 1^st. New York: Dover Publications, 1893.

Christianson, Gale. Isaac Newton and the Scientific Revolution. 1st. New York: Oxford University Press, 1996.

Crowell, Benjamin. Newtonian Physics. 2.3. Morrisville, NC: Lulu, 2008.

Guicciardini , Niccolo. “Did Newton use his calculus in the Principia? .” Centaurus 40(1998): 303-344.

Iliffe, Rob. Newton: A Very Short Introduction. 1^st. Oxford: Oxford University Press, 2007.

de Fontenelle, Bernard Le Bovier . The Life of Sir Isaac Newton. 1st. London: 1728.

Wylie, Ray, and Louis Barrett. Advanced Engineering Mathematics. 6th ed. New York: McGraw Hill, 1995.