Two of the five Bulgar hordes are to have longer futures.

Kubrat's son Bezmer, or Batbayan, avoids the Khazars by leading his horde far to the north, where it eventually occupies an ill-defined country around the confluence of the Volga and Kama rivers.

Subdivided here into three groups (probably through mergers with indigenous peoples or with other immigrants), the horde will maintain itself in prosperity for some six hundred years.

…Kazan' on the Volga River, thus providing the infuriated Tatars with a reason to continually attack and harass Moscow from the khanate’s new capital.

…Kazan', formed in 1438 under Ulugh Muhammad, and …

...Kazan.

Numerous Turkic tribes during the eleventh to sixteenth centuries live in what is now Russia and Kazakhstan.

The present territory of Tatarstan had been inhabited by the Volga Bulgars, who had settled on the Volga River in the eighth century and converted to Islam in 922 during the missionary work of Ahmad ibn Fadlan.

After the Mongol invasion of Europe from 1241, Volga Bulgaria had been defeated, ruined, and incorporated into the Golden Horde.

Few of the population survived, nearly all of them moved to northern territories.

According to one theory, there was some degree of mixing between it and the Cuman-Kipchaks of the Horde during the ensuing period, yet according to another theory called Bulgarism, the Bulgars did not mix with the Cuman-Kipchaks.

The group as a whole accepted the language of the Kipchaks and the ethnonym "Tatars" (although the name Bulgars persisted in some places), while the bulk of invaders eventually converted to Islam.

Mehmed I Giray, son of Meñli I Giray, had inherited power over the Crimean Tatars after his father's death in 1515.

In 1520, he had signed a temporary alliance with Sigismund, King of Poland and Grand Duke of Lithuania, against Muscovy.

The Russians have managed to briefly occupy the city of Kazan several times.

In 1521, Mehmed I Giray takes Kazan, where he puts his brother, Sahib I Giray (future Khan of Crimea, also) in power.

Leading a large Tatar army, the brothers defeat Vasili III near Moscow.

The Tatar khanate of Kazan, one of Russia’s three Tatar states, is a successor state of the Golden Horde, together with the khanates of Astrakhan and of the Crimea.

Its capital city, Kazan', had been established on the left bank of the Volga River in the early fifteenth century.

Kazan’s reinforcement of Crimea had displeased the pro-Moscow elements of the Kazan Khanate, and some of these noblemen had provoked a revolt in 1545.

The result was the deposition of the khan, Safa Giray, the son of Sahib Giray of the Crimean Khanate.

A Moscow supporter, Şahğäli, had occupied the throne.

Moscow had organized several subsequent campaigns to impose control over Kazan, but these have been unsuccessful.

Safa Giray had returned to the throne with the help of the Nogai Horde, a confederation of Turkic nomads that has occupied the Pontic-Caspian steppe from about 1500.

He had executed seventy-five noblemen, and the rest of his opposition had escaped to Russia.

Dying in 1549, his three-year old son Ütämeşgäräy had been recognized as khan; his mother Söyembikä, who acts as regent, is the khanate's de facto ruler.

The administration of the ulan Qoşçaq, head of government during the reigns of Safagäräy and Söyembikä, gains a degree of independence under her rule.

At this time Safa Giray's relatives (including Devlet I Giray) are in Crimea.

Their invitation to the throne of Kazan is vitiated by a large portion of the nobility.

Relations with Russia have continued to worsen under Qoşçaq's government.

A group of disgruntled noblemen had at the beginning of 1551 invited a supporter of Tsar Ivan the Terrible, Şahğäli, for the second time.

Qoşçaq had been forced to escape from Kazan after Şahğäli's successful coup d'etat, but Russian troops had caught him and executed him after he refused baptism.

At the same time, the Kazan lands to the east of the Volga River are ceded to Russia, and Ütämeşgäräy, along with his mother, has been sent to a Moscow prison.

Şahğäli occupies the Kazan throne until February 1552, when anti-Moscow elements in the Kazan government exile him and invite the Astrakhan prince Yadegar Moxammad, along with the Nogays, to aid them.

Kazan's water supply is blocked and the walls are breached before the final storming of the city on October 2 leads to Kazan being taken, its fortifications razed, and much of the population massacred.

The Kazan Chronicle reports about one hundred and ten thousand killed, both civilians and garrison members; sixty thousand to one hundred thousand Russians who had been kept captive in khanate are released.

The fall of Kazan has as its primary effect the outright annexation of the Middle Volga.

Kazan is the first non-Slavic state to be integrated with the Russian empire.

A guerilla war begins in the region after the city's fall.

The Tsar, in suppressing the uprising, responds with a policy of Christianization and Russification of his Tatar subjects and other indigenous peoples; this policy will not be reversed until the reign of Catherine the Great.

Muscovy is transformed as a result of the Kazan campaigns into the multinational and multi-faith state of Russia.

The city of Kazan is today the sixth largest city of Russia.

Pugachev's taking of Kazan in 1774 is the greatest victory of his insurgency.

Nikolai Lobachevsky develops non-Euclidean geometry, also referred to as Lobachevskian geometry, which consists of two geometries based on axioms closely related to those specifying Euclidean geometry.

As Euclidean geometry lies at the intersection of metric geometry and affine geometry (what remains of Euclidean geometry when not using the metric notions of distance and angle), non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one.

In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries.

Lobachevsky is known primarily for his work on hyperbolic geometry.

When the metric requirement is relaxed, then there are affine planes associated with the planar algebras that give rise to kinematic geometries that have also been called non-Euclidean geometry.

Before Lobachevsky, mathematicians had been trying to deduce Euclid's fifth postulate from other axioms.

Euclid's fifth is a rule in Euclidean geometry that states (in John Playfair's reformulation) that for any given line and point not on the line, there is only one line through the point not intersecting the given line. Lobachevsky would instead develop a geometry in which the fifth postulate was not true.

This idea is first reported on February 23 (Feb. 11, O.S.), 1826 to the session of the department of physics and mathematics of Kazan University, where Lobachevsky had become a full professor  n 1822, at the age of thirty, teaching mathematics, physics, and astronomy.

Karl Klaus, professor at Kazan State University, re-investigates Gotttfrid Osann's platinum residues and establishes the existence of the rare metal ruthenium in 1844, after a four-year investigation involving fifteen pounds of native platinum.

The Livonian chemist retains the name, which honors Ruthenia, the Latinized word for Russia, where the metal-bearing ore is first mined.