11 November 1675

Gottfried Leibniz first demonstrates integral calculus for the first time to find the area under the graph of y = ƒ(x).

On November 11, 1675, German mathematician and polymath Gottfried Wilhelm Leibniz demonstrates integral calculus for the first time to find the area under the graph of y = ƒ(x). Integral calculus is part of infinitesimal calculus, which in addition also comprises differential calculus. In general, infinitesimal calculus is the part of mathematics concerned with finding tangent lines to curves, areas under curves, minima and maxima, and other geometric and analytic problems. Today, Gottfried Wilhelm Leibniz as well as independently Sir Isaac Newton are considered to be the founders of infinitesimal calculus. We already dedicated an article at the SciHi blog to Leibniz and his works. But, Leibniz was kind of a universal polymath. His achievements are so numerous that we will definitely have more articles in the future about his contributions to science.

To determine the area of curved objects or even the volume of a physical body with curved surfaces is a fundamental problem that has occupied generations of mathematicians since antiquity. For approximation, you don’t need modern integral calculus to solve this problem. Even the ancient Greeks had developed a method to determine integrals via the method of exhaustion, which also is the first documented systematic technique capable of computing areas and volumes. The method of exhaustion was described by the ancient Greek astronomer Eudoxus, who tried to determine areas and volumes by breaking them up into an infinite number of shapes for which the area or volume was known. This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle. But also in the far east, the Chinese independently developed similar methods around the 3rd century AD by Liu Hui, who used it to find the area of the circle. Later used in the 5th century Liu Hui‘s method was further developed by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere.

Further improvement took until the rise of the European renaissance, when Italian mathematician Bonaventura Cavalieri in the 17th century developed his method of indivisibles. In this work, an area is considered as constituted by an indefinite number of parallel segments and a volume as constituted by an indefinite number of parallel planar areas. Such elements are called indivisibles respectively of area and volume and provide the building blocks of Cavalieri‘s method. As an application, he computed the areas under the curves y=xn – up to the degree 9 – which is known as Cavalieri‘s quadrature formula. Together with the work of Pierre Fermat, they began to lay the foundations of modern calculus. Further steps were made by English theologian and mathematician Isaac Barrow and Italian physicist and mathematician Evangelista Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the fundamental theorem of calculus, which links the concept of the derivative of a function with the concept of the integral. English mathematician John Wallis generalized Cavalieri‘s method, computing integrals of x to a general power, including negative powers and fractional powers.

But, the major advance in integration came with the discoveries and development by Newton and Leibniz. It was Barrow‘s student Isaac Newton, who completed the development of the fundamental theorem of calculus by providing also the surrounding mathematical theory. There, Newton makes use of the connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework developed by Gottfried Leibniz, who systematized the knowledge into a calculus for infinitesimal quantities and introduced the notation used today. Thus, the new infinitesimal calculus allowed for precise analysis of functions within continuous domains. While Newton used a small vertical bar above a variable to indicate integration, or placed the variable inside a box, Leibniz adapted the integral symbol, ?, from the letter ?, standing for summa. The modern notation for the definite integral, with limits above and below the integral sign, was first used by French mathematician Joseph Fourier.

While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour. In 1734, Irish philosopher Bishop Berkeley memorably attacked the vanishing increments used by Newton in his critical essay ‘The Analyst – A DISCOURSE Addressed to an Infidel Mathematician‘, calling them “ghosts of departed quantities“. This was a direct attack on the foundations and principles of Infinitesimal calculus and, in particular, the notion of fluxion or infinitesimal change, which Newton and Leibniz used to develop the calculus. Calculus acquired a firmer footing with the development of limits. The first, who applied limits to rigorously formalize calculus, was German mathematician Bernhard Riemann. Joseph Fourier further extended Riemann‘s concept in the context of his Fourier analysis and French mathematician Henri Lebesgue formulated a different definition of integral, founded in measure theory.

11 November 1918

Germany signs an armistice agreement with the Allies in a railroad car in the forest of Compiègne to end WWI.

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The Armistice of 11 November 1918 was an armistice during the First World War between the Allies and Germany – also known as the Armistice of Compiègne after the location in which it was signed – and the agreement that ended the fighting on the Western Front. It went into effect at 11 a.m. Paris time on 11 November 1918 (“the eleventh hour of the eleventh day of the eleventh month”), and marked a victory for the Allies and a complete defeat for Germany, although not formally a surrender.

The Germans were responding to the policies proposed by U.S. President Woodrow Wilson in his Fourteen Points of January 1918. The actual terms, largely written by French Marshal and Supreme Commander of the Allied Armies Ferdinand Foch, included the cessation of hostilities, the withdrawal of German troops to behind their own borders, the preservation of infrastructure, the exchange of prisoners, a promise of reparations, the disposition of German warships and submarines, and conditions for prolonging or terminating the armistice. Although the armistice ended the actual fighting, it took six months of negotiations at the Paris Peace Conference to conclude the peace treaty, the Treaty of Versailles.

11 November 1918

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Germany signs an armistice agreement with the Allies in Compiègne to end fighting on the Western front during World War I.

The last day of World War One was November 11th 1918, known as Armistice Day. Despite November 11th being the last day of the war, on many parts of the Western Front fighting continued as normal. This meant, of course, that casualties occurred even as the people of Paris, London and New York were celebrating the end of the fighting.

This was done at 05.10am on November 11th. However, the actual ceasefire would not start until 11.00 to allow the information to travel to the many parts of the Western Front. Technology allowed the news to go to capital cities by 05.40 and celebrations began before very many soldiers knew about the Armistice. In London, Big Ben was rung for the first time since the start of the war in August 1914. In Paris, gas lamps were lit for the first time in four years. But on the Western Front, many tens of thousands of soldiers assumed that it was just another day in the war and officers ordered their men into combat. Officially over 10,000 men were killed, wounded or went missing on November 11th 1918. The Americans alone suffered over 3,000 casualties.

11 November 1880

The Australian bushranger Ned Kelly is hanged at Melbourne Gaol for his murder of several policeman.

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The bushranger Ned Kelly is one of Australia’s greatest folk heroes. He has been memorialised by painters, writers, musicians and filmmakers alike. More books, songs and websites have been written about Ned Kelly and the Kelly Gang than any other group of Australian historical figures.Bushranging was said to have ended with the shooting of the Kelly Gang in 1880 which allowed outlawed bushrangers to be shot, rather than arrested and sent to trial.

Even before his execution, Kelly had become a legendary figure in Australia. Historian Geoffrey Serle called Kelly and his gang “the last expression of the lawless frontier in what was becoming a highly organised and educated society, the last protest of the mighty bush now tethered with iron rails to Melbourne and the world.

Ned’s real troubles with the police began when his mother, Ellen Kelly, was arrested for aiding and abetting in the attempted murder of Constable Arthur Fitzpatrick on the 15th of April. As a result of the subsequent brawl, Ned and Dan fled to the bush where they were joined by Joe Byrne and Steve Hart. They became the Kelly gang.While Ned Kelly did not try to break into Beechworth gaol to rescue his mother as planned, he offered an ultimatum to the government of the day.

11 November 1918

Germany signs an armistice agreement with the Allies in a railroad car in the forest of Compiègne, France. The fighting officially ends at 11:00 (The eleventh hour of the eleventh day of the eleventh month) and this is annually honoured with a 2 minute silence.