Late Medieval is a narrative of notable…

Regiomontanus’s important work on trigonometry, De triangulis planis et sphaericis libri quinque, completed in 1464, is first published in 1533.

Much of the material on spherical trigonometry in Regiomontanus' On Triangles had been taken directly and without credit from the twelfth-century work of Jabir ibn Aflah otherwise known as Geber, as noted in the sixteenth century by Gerolamo Cardano.

Gerolamo Cardano reveals Tartaglia's solution of the cubic equation—a betrayal—in his Ars magna (“The Great Art”), the first great Latin treatise on algebra.

Niccolò Tartaglia is the first to apply mathematics to the investigation of the paths of cannonballs; Galileo's studies on falling bodies will validate his work.

Lodovico Ferrari, a twenty-eight-year-old pupil of Italian physician and mathematician Gerolamo Cardano, a lecturer and writer on mathematics, medicine, astronomy, astrology, alchemy, and physics, succeeds him as public lecturer on mathematics in Milan in 1540.

Gerolamo Cardano, two years after accepting a professorship in medicine at Pavia in 1543, reveals Niccolo Tartaglia's solution of the cubic equation—which he had promised not to do—in his Ars magna (“The Great Art”), the first great Latin treatise on algebra.

He also reveals the solution of the quartic equation found by Cardano's former servant, Ferrari.

Nicolas Gombert, one of the most famous and influential composers between Josquin Desprez and Palestrina, best represents the fully developed, complex polyphonic style of this period in music history.

While most composers of the next generation will not continue to write vocal music using Gombert's method of pervasive imitation, they will continue to use this contrapuntal texture in instrumental works.

Forms such as the canzona and ricercar are directly descended from the vocal style of Gombert; Baroque forms and processes such as the fugue are later descendants.

Gombert's music represents one of the extremes of contrapuntal complexity ever attained in purely vocal music.

Gombert vanishes from chapel records in 1540, during the height of his career.

According to contemporary physician and mathematician Gerolamo Cardano, writing in Theonoston (1560), Gombert had been convicted in 1540 of sexual contact with a boy in his care and was sentenced to hard labor in the galleys.

The exact duration of his service in the galleys is not known, but he was able to continue composing for at least part of the time.

Most likely he was pardoned sometime in or before 1547, the date he sent a letter along with a motet from Tournai to Charles' gran capitano Ferrante I Gonzaga.

The Magnificat settings preserved uniquely in manuscript in Madrid are often held to have been the "swansongs" that according to Cardano won his pardon; according to this story, Charles was so moved by these Magnificat settings that he let Gombert go early.

An alternative hypothesis (Lewis 1994) is that Cardano was referring to the highly penitential First Book of four-part motets; however, in neither case is it clear how Gombert was able to compose while rowing in the galleys as a prisoner.

Ferrari, the discoverer of the solution, or root, of the quartic equation (fourth-degree polynomial equation), defends his teacher Cardano in a debate in Milan with Tartaglia in 1548.

The debate was provoked by Cardano’s publication three years earlier of Tartaglia's solution of the cubic equation (third-degree equation), which Tartaglia, having solved it in 1535, had confided to Cardano on the condition that it not be published.

This debate brings Ferrari public attention and gains for him a position as tax assessor in Mantua.

Florentine native Philip Neri, who studied in Rome as a young man and also worked among the poor and sick of the city, was ordained in 1551 and becomes noted as a confessor and for bringing about a religious revival among the Romans.

Pope Gregory XIII in 1575 approves Neri’s Congregation of the Oratory, where priestly members lead a devotional life without vows.

The charismatic Neri, whose good judgment and friendly, playful disposition earn him love and respect, becomes known as the Apostle of Rome.

Italian mathematician Gerolamo Cardano, a pioneer in the study of probability, publishes one of the first modern psychological autobiographies, De vita propria liber (“Book of My Life”) in 1575.

Spanish composer Tomas Luis de Victoria, who may have studied with Bartolomé Escobedo and Palestrina, gains experience in Rome, where he first studies, as an organist and choirmaster, having competent Catholic church choirs for the performance of his works.

He enters the priesthood in 1575 but continues to compose and direct music in Rome.

Stukley allies with Fitzmaurice and moves to Rome in 1575, where he walks about the streets and churches barefoot and bare legged (which causes the lord deputy of Ireland, Sir William Fitzwilliam, to write sarcastically of his holiness, remarking that it had caused the people of Waterford, where he had put on a similar performance while awaiting favorable winds for five weeks prior to his departure, to believe in his piety).

He has an interview at Naples with Don John in June, when he gives details of the plans hatched with the pope for an October expedition.

The intention is to deliver Mary Stewart from prison and take possession of England.

He has corresponded with Nicholas Sanders at this stage.

Don John, who is now in charge of the Spanish forces in Flanders, says the king would have to approve and that three thousand men are too few, but is cautiously optimistic that the expedition would help to contain the rebellion in the Netherlands.

The prospect of a major invasion has been growing, and detailed proposals are put forward for Ireland.

Friar Patrick Healy arrives at Rome in 1575, bearing a letter from the king and announcing that he seeks sanction for an unnamed Irish gentleman to revolt and to request assistance; he insists Philip II has given his blessing.

Gregory stresses that the crown ought not to go to a French or Spanish claimant, but to a native Catholic, i.e. Mary Stewart, lest the king gain too much power and territory, and is opposed to Don John being crowned in Ireland.

The king denies O'Healy's authority to enter discussion on the Irish matter and queries the pope's opposition to the increase of Spanish authority; he is willing to guarantee six months pay for two hundred men and their shipping expenses to go to England in the pope's name, and wonders if a personal attempt might be made against Elizabeth.

Later, it is suggested that five thousand go to Liverpool and free Mary before possessing the country, or go to Ireland.

Gregory bargains for Philip II to defray the entire expense of the expedition, and suggests that if the Vatican is to pitch in then it should receive some benefit in Italy by way of material return.

The Spanish think the leader of the expedition should be married, so as to prevent papal approval of a match with Mary.