This manuscript page from 1665 shows a 23-year old Isaac Newton calculating the area under a hyperbola ( the curve drawn on the top left of the page).
He calculates no less than 55 decimal places, meticulously adding values from each term of an infinite series. The series emerges naturally when the space under this curve is cut up into an infinite number of thin rectangular strips, and their areas are added up. Because Newton does not have a mechanical computer, his entire thought process (known by the archaic term quadrature) is completely visible on paper.
One can imagine sitting on the shoulders of Sir Isaac Newton as he invented the symbolic machinery needed to describe his system of the universe. In the next 3 months we will try to experience exactly that, by studying Newton’s original masterworks, including the Method of Fluxions and his magnum opus the Principia.
What is there in thee, Man - that can be known? —
Dark fluxion, all unfixable by thought,
A phantom dim of past and future wrought
Newton’s own name for the full-blown architecture of calculus was the fluxions, a word that would feature almost a hundred years later in the poem above by Samuel T. Coleridge.
Calculus is a language of movement and change, and underneath its facade lies the vast scaffold of infinite sums such as the one created by Newton. However, the historical origin of these ideas lies thousands of years ago in ancient Greece.
The first infinite series were discovered in ancient Greece with Zeno’s paradox and Archimedes’ calculation of the area under a parabola. This proto-science lay largely dormant for centuries, with some important breakthroughs made by Nicole Oresme in 14th century France and his contemporary Madhava in India, for purposes of astronomy.
From Clockwork To Chaos
When Kepler began constructing his theory of orbital motion, and his Platonic vision of a universe in harmony - areas of shapes were indeed thought of as a sum of infinite lines. Volumes were similarly imagined as a collection of infinite discs. Naturally then, the summation of infinite series was always one of the most important and time-consuming tasks of any Renaissance mathematicus.
Here is a page from Bonaventura Cavalieri's Geometria indivisibilibus continuorum nova quadam ratione promota (Geometry, developed by a new method through the indivisibles of the continua, 1635):
After Isaac Newton described the laws of gravity ( not before having infinite series and fluxions firmly in his grasp ) the prevailing view of the universe was described famously by Pierre Simone, Marquis de Laplace in these famous words:
We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.
However, Laplace was to be proved wrong…..the problems of celestial mechanics did not yield so easily to the laws of Newton. Neither were the orbits of the moon, the planets and their gravitational effects on each other so predictable as to be some kind of clockwork.
The solution to the motion of a two-body system, by Newton and his eighteenth century mathematical successors, is one of the triumphs of Newtonian mechanics.
In our own solar system, which has many more than two bodies, things are much more complicated. The planets follow orbits that are almost, but not exactly, ellipses, the discrepancy being due to the fact that each planet has its own gravitational field, which influences – or perturbs – the motion of all the others. Consequently, the planets’ orbits are not exactly periodic: they return to a slightly different position, and their time of revolution about the sun varies slightly, from year to year.
They needed a better mathematics to describe the perturbed mess that was our solar system. Passing through the deft hands of mathematicians like Barrow, Wallis, Gregory, Newton, Leibniz, Gauss, Euler, Laplace, d’Alembert, Clairaut & Lagrange….infinite series became one of the formidable weapons of mathematical physics.
After burying the all-knowing demon god of Laplace, these new methods yielded a completely new vision of the cosmos - the radical theme of which was a beautiful chaos. The damning shock of this new vision came from Henri Poincaré, who explained it thus:
….imagine a small asteroid, moving back and forth between two larger bodies - call them planets A and B. Given the right conditions, it is possible for the asteroid to alternate between the two planets, spending some of its time revolving around A, and some revolving around B, like a bee flitting back and forth between two flowers. If we track which planet the asteroid goes around at each revolution, we will get a sequence of A’s and B’s which can look statistically like a sequence of random coin tosses.
The workshop will take the participants through this short journey from clockwork to chaos in the most interesting way possible. Throughout this period, we will be looking at original sources and manuscripts from history wherever accessible.
Who can Join? Absolutely anyone. This course is for self-taught hobbyists, not for experts; and it is designed to require absolutely no prior knowledge of history, science or mathematics. Here is some feedback (on Twitter) from attendees of the most recent workshops. Apart from the existing members ( 66 from 13 countries ) we hope to enroll at least 75-80 new participants worldwide in this round.
How does it work? The workshop will be conducted via the online ZetaTrek mailing list, which has been active for almost 3 years now. Participants will be guided through interactive modules according to a syllabus (always being updated). The duration of the workshop is roughly 3 months, extending from 19 July-19 October, 2014. [ Note: This workshop has been extended by a few months to accommodate the large syllabus. ]
The expected commitment is roughly 2-5 hours per week, depending on your enthusiasm.
This workshop is also the final 3 months (or, second semester) of The Age of Re:discovery online workshop. Our first semester was wide-ranging and diverse in ideas, cultures and images. In this second semester, we would like us to keep a sharp focus on Newton’s corpus of ideas and keep miscellany on the sidelines.
For existing members of the Zetatrek expedition, this phase intends to bring a familiarity with the historical development of calculus, geometry and more importantly infinite series. We could then study Euler’s work on the zeta function ( because that too is an infinite series ) and thus by the end of October 2014, we can finally segue back into the original goal of Zetatrek - the Riemann Hypothesis.
Registrations: This is an independent platform without any institutional funding. Participants are expected to contribute a fee of $250 (approx. Rs. 15000) for the entire duration. All new participants will get access to our 3 year archives and a lifetime membership of the Zetatrek expedition. All future workshops will thus be free.
You can pay using our online ticketing facility DoAttend, or Paypal ( the linked Gmail ID is “fadebox” ). Please contact me at the address above for any further queries or assistance.
Scholarships & Gifts: Since the fee may be too much for some people, we always create some free scholarships. This time, for every 10 people who register with the fee, we will offer one free scholarship. So if 200 people register, 20 others will be awarded a free seat. The details of applying will be announced later. You are also welcome to sponsor a scholarship for a friend, or send one as a gift.
Convenor: Rohit Gupta (38, M) is an autodidact interested in the history of science and mathematics. In particular, interdisciplinary interactions such as between astronomy and geometry; or colonial science and its Oriental reception. Some of the previous workshops are listed here, along with a recent interview. His older projects have been featured at Wired and the BBC. Gupta also writes the blog Compasswallah, and tweets as @fadesingh. A complete CV is available on request.
- The Zetatrekker’s Guide To The Galaxy, online documentation of the project is constantly updated, by Rohit Gupta & Ajinkya Kulkarni
- The secret writings of Isaac Newton, by Sarah Dry…author of The Newton Papers.
- A video lecture on the historical development of infinite series by N.J. Wildberger. The whole series is worth watching.