Previous "Tales from Watnall Hall" have looked at RAF Watnall's pioneering use of radar that helped us win the air battles of WW2 but this time we look at an even older radar story from almost 200 years ago. It concerns the work of maths prodigy George Green, owner of Green's Windmill in Sneinton, which he first published in 1828. Green's unique mathematical formulas were used by American radar boffins in WW2 to develop the gun-aiming radar responsible for shooting down Nazi V1 flying bombs over southern England.
Green taught himself advanced mathematics "in hours stolen from my sleep" when he was not grinding corn at his windmill. He had received only about one year of formal schooling as a child, between the ages of 8 and 9 in 1801 at Robert Goodacre's Academy in Upper Parliament Street, Nottingham. It's still a mystery¹ how he progressed so much on his own but he was an early member of Nottingham's scholarly Bromley House Subscription library. The library opened in 1816 and still exists today on Angel Row over 200 years later.
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| Bromley House library where Green published his first and most important work. |
In 1828, aged 35, he finished his first and most important work "An essay on the application of mathematical analysis to the theories of electricity and magnetism" and, almost apologetically, published it. It was sold on a subscription basis by Bromley House library to just 51 people, most of whom were friends who probably could not understand it.
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| Green's Mill with Wollaton Hall and the architectural delights of Nottingham behind⁶. |
Sadly he got ill, returned to Nottingham and died there in 1841 with the significance of his work largely unrecognised. It was rediscovered and re-published in 1871 and then used by 20th century quantum physics pioneers. Albert Einstein on a visit to Nottingham in 1930 said Green was years ahead of his time³.
In England, in summer 1944, it was also used in Operation Diver to shoot down increasing numbers of V1 "Doodlebug" flying bombs... "an average of almost 120 flying bombs a day were launched in the first week of July. After the first two weeks of bombing, some 1,769 people had been killed and in the Strand the Air Ministry itself was hit and 198 people killed. On July 1 a flying bomb crashed in Chelsea, killing 124; four days later the total death roll was 2,500." Aircraft and manually-aimed anti-aircraft guns found it almost impossible to hit the V1's but the same guns fitted with the SCR-584 radar-aiming technology eventually brought down up to 82% of them as they flew over the south coast of England².
Schwinger published a tribute to George Green and the influence he had on his work entitled "The Greening of Quantum Field Theory: George and I" in 1993. He concluded his explanation of the usefulness of Green's work in today's quantum physics by saying... "Very fascinating, indeed. So ends our rapid journey through 200 years. What, finally, shall we say about George Green? Why, that he is, in a manner of speaking, alive, well, and living among us."
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| The damage caused by a single V1 flying bomb on the Cleverly estate, Shepherd's Bush in London. |
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| June 1930 - Einstein at Nottingham University paid tribute to his idol George Green. He wanted to visit two places in the area, Isaac Newton's house and Green's mill. |
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| A page from Green's essay |
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| Green's memorial slab in Westminster Abbey next to those of Newton, Kelvin, Faraday and Maxwell. |
Sources
Green's portrait is sourced from Cambridge University although there is speculation that it may not be authentic. It's the best we have though. Here he is with other famous Cambridge mathematicians.. https://www.maths.cam.ac.uk/undergrad/admissions/files/history.pdfhttps://en.wikipedia.org/wiki/George_Green_(mathematician)
https://www.britannica.com/technology/radar/Advances-during-World-War-II
https://arxiv.org/abs/physics/0606153
https://www.infoage.org/history-ia/world-war-ii-radar/camp-evans-developed-radar-key-at-anzio-italy/
https://mathshistory.st-andrews.ac.uk/Biographies/Green/
https://en.wikipedia.org/wiki/Green%27s_Mill,_Sneinton
http://www.fiddlersgreen.net/models/aircraft/V1.html
Einstein's visit and Schwinger's tribute https://youtu.be/vK9NQ6e6rng?t=589
https://aperiodical.com/2012/02/george-and-julian/
Explainng Green's functions in "simple" terms and why they are so useful... https://www.youtube.com/watch?v=ism2SfZgFJg&ab_channel=Mathemaniac
Mary Cannel's Guardian obituary covers her championing of Green https://mathshistory.st-andrews.ac.uk/Obituaries/Cannell_Guardian/
Notes
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| Green's Mill in 1870 |
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| Green's Mill c.1900 by Thomas Hammond |
2 - Operation Diver https://en.wikipedia.org/wiki/Operation_Diver
Operation Diver was the British code name for the V-1 flying bomb campaign launched by the German Luftwaffe in 1944 against London and other parts of Britain. Diver was the code name for the V-1, against which the defence consisted of anti-aircraft guns, barrage balloons and fighter aircraft. Anti-aircraft guns proved the most effective form of defence in the later stages of the campaign, with the aid of radar-based technology and the proximity fuse. Anti-aircraft gunners found that such small, fast-moving targets were difficult to hit. At first, it took an average of 2,500 shells to bring down a V-1. The average altitude of the V-1, between 2,000–3,000 ft (610–910 m) was in a narrow band above the optimum engagement height range for light 40 mm Bofors guns. The rate of traverse of the standard British QF 3.7 inch mobile gun was too slow for the heights at which V-1s flew and static gun installations with faster traverses had to be built at great cost. The development of centimetric (roughly 30 GHz frequency) gun laying radars based on the cavity magnetron and the development of the proximity fuze helped to neutralise the advantages of speed and size which the V-1 possessed. In 1944 Bell Labs started delivery of an anti-aircraft predictor fire-control system based around an analogue computer, which supplanted the previous electro-mechanical Kerrison Predictor) just in time for use in the campaign.
Technological advances - By mid-August 1944, the threat was all but overcome by the expedited arrival of two enormously effective electronic aids for anti-aircraft guns, the first developed by the Radiation Laboratory at the Massachusetts Institute of Technology (MIT Rad Lab) radar-based automatic gun-laying (using, among others, the SCR-584 radar) and the proximity fuze. Both of these had been requested by Anti-Aircraft Command and arrived in numbers, starting in June 1944, just as the guns reached their free-firing positions on the coast. Seventeen percent of all flying bombs entering the coastal gun belt were destroyed by guns in the first week on the coast. This rose to 60 per cent by 23 August and 74 per cent in the last week of the month, when on one day 82 per cent were shot down. The rate increased from one V-1 for every 2,500 shells fired to one for every hundred.
Barrage balloons - Barrage balloons were also deployed against the missiles but the leading edges of the V-1's wings were equipped with balloon cable cutters and fewer than 300 V-1s are known to have been destroyed by hitting cables.
Aircraft - Part of the area which the Divers had to cover was given over for fighter operations. Most fighter aircraft were too slow to catch a V-1 except in a dive and even when intercepted, the V-1 was difficult to bring down. Machine-gun bullets had little effect on the sheet steel structure and 20 mm cannon shells were explosive projectiles; detonating the warhead could destroy the fighter as well. The V-1 was nearly immune to conventional air-combat techniques because of its design, which dispensed with a pilot and piston engine with a cooling system. One hit on the pilot or oxygen system can damage or shoot down a conventional aeroplane but there is no pilot in a cruise missile. The Argus pulse jet of the V-1 could be shot full of holes and still provide sufficient thrust for flight. The only vulnerable point of the engine was the valve array at the front. The only other vulnerable points on the V-1 were the bomb detonators and the line from the fuel tank; three very small targets inside the fuselage. An explosive shell from a fighter's cannon or anti-aircraft gun hitting the warhead was most effective.
3 - Einstein visited Nottingham University in June 6th 1930. He gave a lecture and the blackboard he wrote on has been kept as a tribute. Supposedly the lecture itself was a bit of a disaster as Einstein only spoke German. To help the situation the Uni had invited 2 types of student, the physics students and the German language students who it was hoped could translate or at least understand. Unfortunately Einstein's subject matter baffled both sets of students!
This video is of Einstein at the Uni. https://www.youtube.com/watch?v=161UNSza_qk&t=24s&ab_channel=MichaelMerrifield The man Einstein is speaking to is Henry Brose, the head of Physics at Nottingham University at the time and also fluent in German. Einstein gave a lecture that day, and the blackboard with his writing is still on display in the Physics department tea room.
In this video Professor Martyn Poliarkoff (who often gets mistaken for Einstein https://www.youtube.com/watch?v=qYZZkI-N3M4&ab_channel=nottinghamscience) takes us on a tour of the Uni to find the Einstein blackboard and tells some amusing tales of the day in cluding Einstein borrowing £5 (a week's wages for a lecturer back then) and forgetting to pay it back. His wife had to write a letter of apology explaing that her husband does not deal with money!... https://www.youtube.com/watch?v=KIwpGEvmgvs&ab_channel=PeriodicVideos
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| Schwinger in 1965 in his lab with his pen just after winning the Nobel Prize for Physics |
In Schwinger's own words, why Green was so influential in his microwave radar work... "I spent the War years helping to develop microwave radar. In the earlier hands of the British, that activity, famous for its role in winning the Battle of Britain, had begun with electromagnetic radio waves of high frequency, to be followed by very high frequency, which led to very high frequency, indeed. Through those years in Cambridge (Massachusetts, that is), I gave a series of lectures on microwave propagation. A small percentage of them is preserved in a slim volume entitled Discontinuities in Waveguides. The word propagation will have alerted you to the presence of George Green. Indeed, on pages 10 and 18 of an introduction there are applications of two different forms of Green's identity. Then, on the first page of Chapter 1, there is Green's function, symbolized by G. In the subsequent 138 pages the references to Green in name or symbol are more than 200 in number. As the War in Europe was winding down, the experts in high power microwaves began to think of those electric fields as potential electron accelerators. I took a hand in that and devised the microtron which relies on the properties of relativistic energy. I have never seen one, but I have been told that it works. More important and more familiar is the synchrotron. Here I was mainly interested in the properties of the radiation emitted by an accelerated relativistic electron. I used the four dimensionally invariant proper time formulation of action. It included the electromagnetic self-action of the charge, which is to say that it employed a four-dimensionally covariant Green's function. I was only interested in the resistive part, describing the flow of energy from the mechanical system into radiation, but I could not help noticing that the mechanical mass had an invariant electromagnetic mass added to it, thereby producing the physical mass of an electron. I had always been told that such a union was not possible. The simple lesson? To arrive at covariant results, use a covariant formulation, and maintain covariance throughout. "
George and Julian By Peter Rowlett. Posted February 13, 2012 in Travels in a Mathematical World...
Yesterday, the @mathshistory Twitter feed tells me, was the anniversary of the birth of Julian Schwinger (1918-1994), one of the great physicists of the 20th century. (Technically I queued this tweet up but there are a lot of days and a lot of mathematicians to remember…)
Schwinger is known to me particularly through his connection to the story of George Green. Green was a Nottingham mathematician who did work on electricity and magnetism (among other things) that, largely unrecognised in his lifetime, was discovered and brought after his death to further attention by William Thomson (later Lord Kelvin). The application of Green’s work in 19th century science was impressive but it found a new legacy in the 20th century.
At the 1993 celebration in Nottingham of the bicentenary of Green’s birth, Schwinger spoke about his use of Green’s work (a talk written up as The Greening of Quantum Field Theory: George and I).
Schwinger’s account is worth reading. He describes his use of Green’s work first on microwave radar during World War II, then in the development of the microtron and synchrotron particle accelerators, and finally to solve a problem on quantum electrodynamics, work which earned him a share, with Sin-Itiro Tomonaga and Richard Feynman, of the 1965 Nobel Prize for Physics.
In the preface to his most famous work, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (1828), Green had written:
Should the present Essay tend in any way to facilitate the application of analysis to one of the most interesting of the physical sciences, the author will deem himself amply repaid for any labour he may have bestowed upon it.
Schwinger’s account helps us to understand how Green not only impacted the physics of his age, but how it continued to have impact beyond anything Green could have imagined."
https://aperiodical.com/2012/02/george-and-julian/
5 - MRI Scanning and George Green - Sir Peter Mansfield with a MRI scanner
Royal Standard House is a grade II listed building converted from Memorial House, originally built in 1921 as a nurses' home and featuring 1, 2 and 3 bedroomed apartments. https://www.standardhill.co.uk/site-history
George Green Makes the First Attempt to Formulate a Mathematical Theory of Electricity and Magnetism (1828) - views 2,761,905updated
George
Green Makes the First Attempt to Formulate a Mathematical Theory
of Electricity
and Magnetism (1828)
Overview
Over the course of the nineteenth century the science of
electricity and magnetism advanced from laboratory curiosity to a fully
developed theory that would provide the basis for several major technologies.
Essential to this development was the development of a mathematical apparatus
to describe the behavior of fields, physical states characterized by a vector
or scalar at every point in space. A critical initial step was provided in 1838
by George
Green (1793-1841), then a self-taught amateur mathematician. A
complete formulation of the behavior of electromagnetic fields was achieved
over the next 35 years.
Background
While the ancient Greeks were familiar with both static
electricity and permanent magnets, the nature of the two phenomena
remained a subject of speculation until the beginning of the nineteenth
century. In 1800 the Italian physicist Count Alessandro
Volta (1745-1827) created the Voltaic "pile," a dependable
source of electric
current, and vast new experimental possibilities arose. In 1820, the Danish
physicist Hans
Christian Oersted (1777-1851) reported that a current carrying wire
had an effect on compass needles placed around it, a report that quickly
brought new investigators into the field. By 1821 Oersted's experiments were
being reproduced and expanded upon by two men who would play a major role in
the development of the new science of electromagnetism—André Marie Ampère
(1775-1836) in France and Michael
Faraday (1791-1867) in England.
It was recognized that the electric and magnetic forces had
some of the same characteristics as the gravitational force, but were somewhat
more complex in character. It was easier, particularly in the case of
magnetism, to think of each charged or magnetic object as setting up a
disturbance in the space around it which would determine the force that would
act on a charged or magnetic object placed at that point. The electric and
magnetic fields each assigned a vector quantity to each point in space. As one went
from any point to neighboring points, the magnitude and direction of the field
would change—the rate of change being determined by the material present.
The analogous problem in fluid flow had been treated by
Swiss mathematician Daniel
Bernoulli (1700-1782) in a 1789 book on hydrodynamics. In a paper on
fluids in 1752, the prolific Swiss mathematician Leonhard
Euler (1707-1783) showed that the potential function satisfied a very
simple equation involving second partial derivatives, a equation now generally
known as Laplace's equation after the French mathematician Pierre Simon Laplace
(1749-1827).
In March of 1828, George Green, a self-taught English
amateur mathematician, published a work entitled "An Essay on the
Application of Mathematical Analysis to the Theories of Electricity
and Magnetism." In this work Green introduced the notion of potential
functions for the electric and magnetic field and showed how to construct the
function by adding contributions from each charge. This essay included a very
important formula, now known as Green's theorem.
Green's work, distributed to only the 52 individuals who had financially supported the work, might have been lost had it not been for Sir William Thomson (aka Lord Kelvin; 1824-1907), who had it reprinted in a German mathematics journal in 1850. A few years earlier Thomson had noted that the solution to Laplace's equation that took on a defined set of values on the boundary of a region of space would be, of all sufficiently "smooth" functions that satisfied the boundary conditions, that function which minimizes an integral commonly known as the Dirichlet integral after the German Mathematician Lejeune Dirichlet (1805-1859). Green and Thomson's results together make it possible to determine the potential over a region of space, given either its values or that of its derivatives (the field) over the boundary.
Over the course of the nineteenth century, considerable
effort was devoted to elucidating the nature of light. This was motivated by
the discovery of the polarization
of light on reflection by Etienne
Louis Malus (1775-1812) in 1808 and the strange "double
refraction" of light into polarized beams in crystalline materials such
Iceland spar, a form of calcium
carbonate. Researcher in optics at the time considered light to be a
vibration in a medium, the "luminiferous aether" which filled all
space and somehow interacted with matter, but did not slow the motions of
bodies moving through it. The Irish mathematician and astronomer Sir
William Rowan Hamilton (1805-1865) devoted several years of effort to
this problem, not leaving any results of lasting value to optics, but
developing the mathematical techniques later to be successfully applied by
himself and others, including the German Karl Gustav Jacobi (1804-1851), to
problems in mechanics.
The true nature of light became apparent when the English mathematical physicist James Clerk Maxwell (1831-1879) formulated the set of four Maxwell equations describing the behavior of the electric and magnetic fields in space. Applying the mathematical techniques of Green and others to the experimental observations of Ampère and Faraday, Maxwell derived in 1864 a set of four coupled partial differential equations which in empty space could take on the form of wave equations for the components of the electric and magnetic fields. The velocity of the waves was given in the equations in terms of the fundamental force constants of the electric and magnetic force, and turned out to be 300,000 km per second, exactly the measured speed of light in vacuum. There could then be little doubt that light was a form of electromagnetic radiation and that the effects of matter on light, reflection, refraction, and polarization could be calculated from the interactions of the electromagnetic field with the charged particles making up the matter in question.
Impact
It is interesting to note the unconventional educational
backgrounds of some of the pioneers of electromagnetism. Ampère was born into
an upper middle-class family and would most likely have prepared for legal
practice or the church were it not for the excesses of the French
Revolution, which lead to the execution of his father and the loss of the
family fortune. Although he never received a degree, he would be appointed to
numerous important university posts in post-revolutionary France. Faraday was
born into a working-class family and received an education working in an
institution originally founded to provide scientific instruction to the working
class. Green was a baker's son who left school early but was nonetheless able
to educate himself through independent reading. After making his important
contributions Green was admitted to Caius College as a scholar at the age of
40. The science of electromagnetism was thus developed very rapidly during a
time of rapid social and economic change by men who would not have been
considered educated by traditional standards.
It would be difficult to overstate the impact of the development of electromagnetic theory on the conditions of life in the industrialized world. The principle of electromagnetic induction, discovered by Faraday and incorporated into Maxwell's equations, made possible the design of electric generators and motors, which in turn made it possible to separate the production of electrical energy from its use in industrial production. Maxwell's identification of light as an electromagnetic wave led directly to the discovery of radio waves and the revolution in communications and mass culture that followed. One of the more astonishing conclusions of consequences of Maxwell's theory was that the speed of light would have the same velocity regardless of the relative velocities of the source and the observer. This conclusion led Albert Einstein (1879-1955) to recast the principles of mechanics in the special theory of relativity. It also abolished any need for a "luminiferous aether" in physics. Over the next century, understanding the atomic structure of matter and its interaction with radiation have created a world of lasers and optical fiber communication, none of which would be possible without the mathematical techniques developed by Green and his contemporaries.
DONALD R. FRANCESCHETTI
Further Reading
Bell, Eric Temple. The Development of Mathematics. New
York: McGraw-Hill, 1945.
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