Viraj Patel


Introduction

As the one year anniversary of the UK lockdown passed us, I couldn’t help but think of how productive I’d been over the last year, or rather, how unproductive I’d been. I’d probably binged Netflix more times than I’d watched lectures. Sir Isaac Newton, on the other hand, used the university closures of 1665 to his advantage, developing his theory of gravity with the invention of calculus, and discovering the dispersive property of sunlight [1].

At the start of 1665, universities across the UK closed down for about 2 years due to the outbreak of the Bubonic Plague, one of the worst epidemics London has seen, prompting Newton to return to his home in Woolsthorpe. Prior to the closures, he was working on arithmetic series and sequences, but later chose to examine the nature of curves in the late autumn of 1664 [2]. This would lead to calculus, one of the most revolutionary ideas of mathematics, and arguably a physicist’s most important tool.

Calculus and Gravity

In 1664, Newton began working on the expansion of binomial expressions. By 1665, he had developed a way to expand binomial expressions into series, thus being able to represent a function z as an expansion of terms in powers of x. His next step to invert the series; to represent x as a series of powers of z. It was said that Newton had incredible patience for monotonous algebra, but even he gave up after 5-6 terms to find a general rule of the nth coefficient for a given series [3]. The idea of a series expansion of a function proved very useful in the development of calculus.

In the 1630s, several mathematicians, including Cavalieri and Fermat [4], found that the area under the curve y = xn between x = 0 and x = b is

(1)   \begin{equation*} \mathrm{Area}=\frac{b^{n+1}}{n+1} \end{equation*}

Using his works in binomial expansions, Newton applied this formula to find the area under a variety of curves. He then found that, by defining y = sin-1⁡x in terms of an area, he could find a power series for sin-1⁡x, and using his knowledge of inverting series, he was also able to find a power series for sin⁡ x [5].

In May of 1665, he had found the method of drawing tangents by Slusius, in which a collection of tangents at various angles forms the shape of a parabola. Newton was inspired by this work to create his own method of fluxions (commonly referred to as differential calculus nowadays) by November 1665 [2]. A year later, he had started development on the inverse method of fluxions (integral calculus) [6]. In 1669, Newton published his infamous paper, De analysi, that outlined the fundamental theorem of calculus [7]. In this paper, he developed new rules for calculating the area under a given curve .

Calculus led to a new way to think about maths, and a long dispute between Newton and Leibniz over who should be named as the inventor of calculus. After centuries of investigations, both are claimed to have invented calculus, but for different reasons. Leibniz invented it purely for maths, however, Newton invented it to enhance is theory of gravity [8]. It might seem weird, but Newton actually waited about 20 years to publish his famous Principia paper about the universal theory of gravitation. This was because he needed time to develop a strong mathematical formulation, and that was calculus [9].

Colours and Light

Having seen how busy he was inventing calculus, it’s hard to see how Newton could fit anything else into his schedule during the university closures but, as it turns out, he also had time to discover the theory of colours. In 1666, glass triangular prisms were commonly sold at fairs to children to play with light. Most people were fascinated at how a glass prism “converted” light into colours [10], but Newton proved this was due to refraction, a phenomenon discovered over 50 years earlier.

In the second year of university closures (sound familiar?), Newton closed the shutters of his windows but left a single small hole so that a thin beam of sunlight could enter the room. He held a prism to the sunlight and observed a rainbow [11], but the spectrum appeared to have been stretched vertically, as opposed to the colours forming concentric circles like he expected. He first suspected this to be due to an irregularity in the smoothness of the glass. To test this, he placed a second prism upside-down next to the first so that the split beam entered the second prism. He found that a single beam of sunlight exited the second prism exactly the same way the sunlight entered the first. He further developed this experiment to analyse where the different coloured beams would be refracted separately, and found that they all converged to the same place as the exiting beam in the previous experiment [12]. The only way this was possible was if sunlight was made up of waves with different refractility, producing the difference colours of the spectrum.


Figure 1 Newton dispersed sunlight using an upright prism, and then recombined the constituent colours back to the single beam using an upside-down prism to show that white light was made of different colours. Image source: Souvik Das, 2018, Quora

This discovery has affected modern physics massively. Using Snell’s Law, it can be found that the differing refractilities are due to different wavelengths for each colour. When applied to quantum mechanics, it means different electron transitions correspond to different coloured photons being emitted. This has been used to analyse the absorption and emission spectra of various objects ranging from dyes used in cell imaging [13] to black holes [14].

Conclusion

It goes without saying that Newton was one of the greatest minds to have studied physics and maths. He achieved in 2 years, what most people can only dream of achieving in a lifetime, and he was only 22. Much of physics is built on calculus, and Newton’s theory of universal gravitation was left undisputed for centuries, showing that it was a very good model. Moreover, despite Newton not believing in the wave model of light, his work in optics has inspired technologies to analyse the chemical composition of almost any material, with most uses in astrophysics.

If you felt unproductive during this lockdown, then this article might not have helped you feel much better, but it is worth appreciating the incredible contributions this man has made to physics and maths. To sum up this article, I’d like to quote one of my lecture notes taken from a course taught by Dr Kolthammer here at Imperial [15]:

It’s okay if you weren’t as productive over 2020. Newton didn’t have TikTok as a distraction, and rarely has one person seemed to advance science so far in such a brief period.


References

[1] P. Fara, “Years of Wonders 1665-1667,” National Trust, [Online]. Available: https://www.nationaltrust.org.uk/woolsthorpe-manor/features/year-of-wonders#:~:text=Between the summer of 1665,spread rapidly throughout the country.

[2] R. A. Hall, Isaac Newton: Adventurer in Thought, Cambridge University Press, 1996.

[3] W. Dunham, The calculus gallery: Masterpieces from Newton to Lebesgue, Princeton University Press, 2018.

[4] C. B. Boyer, “Fermat’s Integration of X n,” National Mathematics Magazine, vol. 20, no. 1, pp. 29-32, 1945.

[5] V. J. Katz, “Ideas of calculus in Islam and India,” Mathematics Magazine, vol. 68, no. 3, pp. 163-174, 1995.

[6] H. N. Jahnke, A history of analysis, American Mathematical Society, 2003.

[7] F. Cajori, “Historical note on the Newton-Raphson method of approximation,” The American Mathematical Monthly, vol. 18, no. 2, pp. 29-32, 1911.

[8] J. S. Bardi, The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time, Basic Books, 2007.

[9] G. Gamow, Gravity, Courier Corporation, 2002.

[10] A. C. Sparavigna, “The play of colours of prisms,” arXiv preprint arXiv:1207.3504, 2012.

[11] D. Topper, “Newton on the number of colours in the spectrum,” Studies in History and Philosophy of Science Part A, vol. 21, no. 2, pp. 269-279, 1990.

[12] I. Newton and G. Sarton, “Discovery of the dispersion of light and of the nature of color (1672),” Isis, vol. 14, no. 2, pp. 326-341, 1930.

[13] F. Bestvater, E. Spiess, G. Stobrawa, M. Hacker, T. Feurer, T. Porwol, U. Berchner-Pfannschmidt, C. Wotzlaw and H. Acker, “Two-photon fluorescence absorption and emission spectra of dyes relevant for cell imaging,” Journal of microscopy, vol. 208, no. 2, pp. 108-115, 2002.

[14] N. Sanchez, “Absorption and emission spectra of a Schwarzschild black hole,” Physical Review D, vol. 18, no. 4, p. 1030, 1978.

[15] S. Kolthammer, Second Year Atomic Physics: Atom-Light Interaction, Imperial College London, 2021.