One final battle is fought west of modern Hanover, repeating the pattern of high German fatalities and forcing their flight.

With his main objectives reached and with winter approaching, …

Hanover, which includes the western portion of the North German Plain, extending from the border with the Netherlands and the North Sea to the Harz Mountains, comes under the rule of the Welf family.

Hannover, located on the south bank of the Leine River some eighty miles (one hundred and thirty kilometers) south of Hamburg, was a small village of ferrymen and fishermen that became a comparatively large town in the thirteenth century due to its position at a natural crossroads; it is chartered in 1241.

As overland travel is relatively difficult, its position on the upper navigable reaches of the river has helped it to grow by increasing trade.

It is connected to the city of Bremen by the Leine, and is situated near the southern edge of the wide North German Plain and northwest of the Harz mountains, so that east-west traffic such as mule trains pass through it.

Hanover is thus a gateway to the Rhine, Ruhr and Saar river valleys, their industrial areas which grew up to the southwest, and the plains regions to the east and north, for overland traffic skirting the Harz between the Low Countries and Saxony or Thuringia.

Gottfried Leibniz had begun working on the calculus in 1674; the earliest evidence of its use in his surviving notebooks is 1675.

He had had a coherent system in hand by 1677, but has waited seven years to publish it.

A German philosopher of Sorbian origin, educated in law and philosophy and writing primarily in Latin and French, Leibniz serves as factotum to the House of Hanover.

The Brunswicks have tolerated the enormous effort Leibniz has devoted to intellectual pursuits unrelated to his duties as a courtier, pursuits such as perfecting the calculus, writing about other mathematics, logic, physics, and philosophy, and keeping up a vast correspondence.

Being able to calculate rates of change also allows one to determine where maximum and minimum values occur—the title of Leibniz's first calculus publication, in the recently founded journal Acta Eruditorum, is “ova Methodus pro Maximis et Minimis, Itemque Tangentibus, qua nec Fractas nec Irrationales Quantitates Moratur, et Singulare pro illi Calculi Genus (“A New Method for Maxima and Minima, as Well as Tangents, Which Is Impeded Neither by Fractional nor by Irrational Quantities, and a Remarkable Type of Calculus for This”).

A branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, calculus constitutes a major part of modern university education.

Leibniz, in 1686, first defines the curvature of a curve at each point in terms of the circle that best approximates the curve at that point.

Mathematicians from antiquity had described some curves as curving more than others and straight lines as not curving at all.

The gambit to bring the Germans to terms, although militarily successful, as well as strengthening the Turkish resolve in the Balkans, fails.

Just after the initial attack in October 1688, Hanover and ...

With the imminent formation of a single Hanoverian state, and the Hanoverians’ continuing contributions to the Empire's wars, Ernest Augustus is made elector of Brunswick-Lüneburg (i.e., Hanover), in 1692 in return for lavish promises of assistance.

Indignant protests follow this proceeding.

A league is formed to prevent any addition to the electoral college; France and Sweden are called upon for assistance.

This agitation, however, soon dies away, and his son is confirmed as elector by the imperial diet.

George's prospects are now better than ever, as the sole heir to his father's Electorate—the ninth of the Holy Roman Empire—and his uncle's duchy.The Welf family has divided into several branches over the past five centuries.

Chief among these are those of Brunswick-Luneburg and Brunswick-Wolfenbuttel.

Ernest Augustus had in 1679 become reigning Duke of Lüneburg-Calenburg (which becomes known as the duchy of Brunswick-Lüneburg or, more popularly, because of its capital city, the duchy of Hanover).

His wife, Sophia of the Rhineland Palatinate, is the last surviving offspring of the marriage between Frederick V, elector Palatine, and Elizabeth (Stuart) of Bohemia, daughter of James I of England.

Two of Ernest Augustus’s three brothers had died childless and the third, George William of Celle, had married his mistress in order to legitimize his only daughter, Sophia Dorothea of Celle, but looked unlikely to have any further children.

Under Salic law, where inheritance of territory was restricted to the male line, the succession of Ernest Augustus’s sons to his father's and uncle's territories now seemed secure.

The family had agreed in 1682 to adopt the principle of primogeniture, meaning George, the eldest, would inherit all the territory, and not have to share it with his brothers.

The same year, George married his first cousin, Sophia Dorothea, thereby securing additional incomes that would have been outside Salic laws requiring male inheritance.

The marriage of state was arranged primarily as it ensured a healthy annual income, and assisted the eventual unification of Hanover and Celle.

Sophia was at first against the marriage, looking down on Sophia Dorothea's mother (who was not of royal birth) and concerned by Sophia Dorothea's legitimated status, but was eventually won over by the advantages inherent in the marriage.

George and his brother, Frederick Augustus, served in the Great Turkish War at the Battle of Vienna in 1683, and Sophia Dorothea had borne George a son, George Augustus.

Frederick Augustus had been informed the following year of the adoption of primogeniture, meaning he would no longer receive part of his father's territory as he had expected.

It led to a breach between father and son, and between the brothers, that lasted until Frederick Augustus's death in battle in 1690.

Sophia Dorothea had borne a second child, a daughter named after her, in 1687, but there have been no other pregnancies.

George Louis of Hanover and his wife Sophia Dorothea have become estranged—George prefers the society of his mistress, Melusine von der Schulenburg, by whom he had two daughters in 1692 and 1693, respectively; and Sophia Dorothea, meanwhile, had her own romance with the Swedish Count Philip Christoph von Königsmarck.

Threatened with the scandal of an elopement, the Hanoverian court, including George's brothers and Sophia, urges the lovers to desist, but to no avail.

According to diplomatic sources from Hanover's enemies, in July 1694, the count is killed, possibly with the connivance of George, and his body thrown into the river Leine weighted with stones. (The murder was claimed to have been committed by four of Ernest Augustus's courtiers, one of whom—Don Nicolò Montalbano—was paid the enormous sum of one hundred and fifty thousand thalers, which was about one hundred times the annual salary of the highest-paid minister.)

Sources in Hanover itself, including Sophia, deny any knowledge of Königsmarck's whereabouts.

George's marriage to Sophia Dorothea is dissolved, not on the grounds that either of them had committed adultery, but on the grounds that Sophia Dorothea had abandoned her husband.

Ernest Augustus of Brunswick-Lüneburg dies on January 23, 1698, leaving all of his territories to his son George Louis with the exception of the Prince-Bishopric of Osnabrück, an office he had held since 1661.

George, who has received a thorough education in the diplomatic and military arts, thus becomes Duke of Brunswick-Lüneburg (also known as Hanover, after its capital) as well as Archbannerbearer and a Prince-Elector of the Holy Roman Empire.

His court in Hanover is graced by many cultural icons such as the mathematician and philosopher Gottfried Leibniz and the composers George Frideric Händel and Agostino Steffani.

Ernest Augustus was born at Herzberg am Harz, the son of George, Duke of Brunswick-Lüneburg and Anne Eleonore of Hesse-Darmstadt (Ernest Augustus is thus descended from the Albertinian line of the Habsburg family and also the House of Hohenzollern).

He married Sophia of the Palatinate in Heidelberg in 1658.

As the fourth son, he had little chance of succeeding his father as ruler, and so, in 1662, his relatives had appointed him administrator of Osnabrück; according to the Peace of Westphalia, every second incumbent of the Osnabrück see was to be appointed by the dukes of Brunswick-Lüneburg.

However, after two of his elder brothers had died without sons, Ernest Augustus had inherited part of his father's territories in 1679, namely Calenberg (including Göttingen).

Against the protestations of his five younger sons, Ernest Augustus in 1683 had instituted primogeniture, so that his territory would not be further subdivided after his death, and also as a precondition for obtaining the coveted electorship.

He participated in the Great Turkish War on the side of Holy Roman Emperor Leopold I.

He was in 1692 appointed prince-elector by the Emperor; however, the electorship will not come into effect until 1708.

Whig politicians believe Parliament has the right to determine the succession, and bestow it on the nearest Protestant relation of the Queen, while many Tories are more inclined to believe in the hereditary right of the Stuarts.

George announces in 1710 that he will succeed in Britain by hereditary right, as the right had only been removed from the Catholic Stuarts, and he retains it.