López de Legazpi governs the Manila colony…

The last thirty years of the sixteenth century see the rapid onset of a decline in Ottoman power, symbolized by the defeat of the Turkish fleet by the Spanish and Portuguese at the Battle of Lepanto in 1571 and by the unbridled bloody succession struggles within the imperial palace, the Seraglio of Constantinople.

Setthathirat's military strategy in repelling Burmese invasions in 1565 and 1570 has preserved the autonomy of his kingdom.

A prolific builder, he has erected many Buddhist monuments including Wat Xieng Thong in Louang Phrabang and the That Luang in Vientiane.

A conspiracy in 1572 between a Lord Phya Nakhon and the former abbot of Wat Maximavat, who holds personal grudges against Setthathirath, leads to the murder of the thirty-eight-year-old king in the southern frontier of the country.

Because Setthathirath leaves only a toddler as his heir, the child's maternal grandfather, a military commander of common birth named Saensurin (or Sene Soulintha), declares himself king.

This begins a period of turbulence, with various men ruling unsteadily for short periods.

Takeda Shingen decides, at the height of the anti-Nobunaga alliance, to rise against the Oda clan.

Nobunaga, his forces tied down in siege warfare at Ishiyama Hongan-ji and Nagashima, sends insufficient aid to Tokugawa.

As Takeda Shingen drives south, seeking to attack Tokugawa at his fortress at Hamamatsu, his forces meet Tokugawa's on a high plain called Mikata, just north of the fortress.

According to the Kōyō Gunkan, the contemporary Takeda military history, Takeda outnumbers Ieyasu three-to-one, and organizes his men in the gyōrin (fish-scale) formation, enticing his opponent to attack.

Tokugawa's men are arranged in a line, so they can make the best use of their arquebusiers.

Tokugawa may have expected his superior weaponry to defeat the tactical attempts of Takeda, who now looses a cavalry charge, his mounted spearmen racing towards the gunners, and overrunning them; some of Tokugawa's men stand firm, but many withdraw or are killed.

A fresh set of horsemen initiates a new charge, led by Takeda Katsuyori and Obata Masamorim and are soon joined by the main body of the Takeda army, which drives their adversaries into full retreat.

Tokugawa seeks to re-engage the Takeda army, to free his trapped generals, but is persuaded by Natsume Yoshinobu to retreat; Tokugawa's life was too important.

Natsume is killed leading a charge into the Takeda line.

Tokugawa returns to the Hamamatsu fortress, reportedly accompanied by only five men, and commands that the fortress gates remain open, and that braziers be lit to guide his retreating army back to safety.

When the Takeda vanguard, led by Baba Nobuharu and Yamagata Masakage hears the drums sounding retreat, and see the braziers and open gates, they assume that Tokugawa is planning a trap; they stop and make camp for the night.

A small band of Tokugawa warriors attacks the Takeda camp in the night and forces the Takeda into a ravine, where they lie helpless.

Shingen and his army withdraw the following morning.

The Crimean Tatars, undertaking another assault against Moscow in 1572, encounter Russians returning from the Russo-Turkish War of 1568-69, who help their countrymen defeat the Tatars at the battle of Molodi, about twenty-five miles from Moscow, after which the Tatars withdraw to the Crimea.

Devlet Giray's attempts have been unsuccessful, but still he manages to impose annual monetary and fur tributes on some of the Russians and Ukrainians living in the south.

The Oprichnina has been a total failure.

Ivan is in autumn of 1572 forced to disband it, ending his domestic policy of political police, mass repressions, public executions, and confiscation of land from Russian aristocrats.

Saying the word Oprichnina is now punishable by death.

Dávid, influenced by Johannes Sommer, rector of the Kolozsvár gymnasium, abandons the worship of Christ in about 1572.

Akbar completes the conquest of Rajasthan from 1561 through 1570, then pushes westward against the heavily defended kingdom of Gujarat, an opponent of Muslim rule since the time of the Delhi Sultanate.

He systematically attacks the kingdom’s network of fortresses and fortified cities, easily reducing Ahmadabad in November 1572.

Akbar seizes the Gujarati fortress at Cambay in December 1572, a month after his successful siege of Ahmadabad. Annexing the sultanate, he then vanquishes the forces of a hostile relative at Sarnal before returning to his new court city of Fatehpur Sikri.

Tirumala retires in 1572, and his son Sriranga I tries to continue the process of rebuilding while struggling to maintain his place among the Muslim sultanates without any support from the major Telugu houses.

An invasion by Bijapur is repulsed with the aid of Golconda.

Rafael Bombelli feels that none of the works on algebra by the leading mathematicians of his day provide a careful and thorough exposition of the subject.

Instead of another convoluted treatise that only mathematicians could comprehend, the Bolognese-born mathematician decides to write a book on algebra that can be understood by anyone.

His text is to be be self-contained and easily read by those without higher education.

Bombelli treats cases involving imaginary roots in his L'Algebra, published in 1569.

He introduces the representative symbols +i and -i and describes how they both work.

Complex numbers, used by Bombelli in connection with the cubic, become gradually accepted about this time.

Bombelli dies in 1572 in Rome after being crushed by a large calf.