ONE of the continuing surprises of the 15th century is that Florence - a city no larger than medieval York - produced such a rich inheritance of art and of artists, which changed the way in which Western art developed. How could so small a city produce such a wealth of material?

The veteran social economist Richard Goldthwaite has long made the argument that this was because the city was a republic at the time of economic expansion, and it is against this background that much of this splendid exhibition is currently staged in the palace that Benedetto da Maiano built for the Strozzi in the 1490s.

But angels and florins are only part of the story, as Hans Belting has demonstrated in his 2008 study of Renaissance art and Arab science, Florence and Baghdad. The revolution that we call the Renaissance crucially drew on a theory of perspective, first formulated in Baghdad by the 11th-century mathematician Ibn al-Haytham (Alhazen), when it became known on the banks of the Arno.

Giotto di Bondone (d. 1337) is sometimes still credited with introducing perspective into painting, as Florentines themselves claimed for him in the 15th century on his monument (1489) in the Duomo. But although Giotto could see the difference in "forms", if by that we mean the visual impressions created by objects, mathematical linear perspective was not invented until the 15th century. In turn, that defined afresh how sculpture was to be seen, as well as crafted.

A more comprehensive exploration of the sculptural transition from the medieval Gothic world of the 1260s and 1270s, with Arnolfo di Cambio and Nicola and Giovanni Pisano, is unlikely, and a more beautiful show than this is inconceivable.

Surveying 60 years of this phenomenon from 1400, this is the 15th century of Brunelleschi, Donatello, della Robbia, and Desiderio da Settignano. It opens with two of the trial plaques made for a contest for the great north doors of the Florentine Baptistery, which attracted half a dozen sculptors.

Seeing them side by side, we now get to judge between the Lorenzo Ghiberti and Filippo Brunelleschi competition entries. Fortunately, we have already been offered a textbook of how to look at sculpture with the painted poplar wood St Stephen by Francesco di Valdambrino (c.1375-1435) from Empoli, and Jacopo della Quercia's elegant St Ansanus (c.1410) for a church in Lucca. Both works date from ten years after the competition, but give a strong idea of the quality that the commissioners sought.

The 20-year-old Ghiberti was the outsider (he was working in Mantua at the time that the competition was announced), but his partly gilt bronze of The Sacrifice of Isaac won the day. His prize-winning panel is set next to that by the slightly older Brunelleschi, who went on to design the great dome for the cathedral, the wooden model for which is one of the great treasures on show here.

Brunelleschi's Abraham looks up doubtfully at the angel, his sword at Isaac's throat. The boy's mouth is wide open as he shouts out in an unheard silence, but his eyes are shut as he swoons in the expectation of death at his father's hand. A more vigorous Isaac appears in the Ghiberti piece with his classically moulded six-pack, derived straight from a Hellenistic statue, but known to us now, thanks to Calvin Klein and others. His father stands at a distance, the generations separated, as YHWH commands an end to Israelite child-sacrifice.

Classical references abound, and in both the sacrifice is taking place at a pagan altar. Brunelleschi interpreted both of Abraham's servants as classical figures (a copy of the famous Il Spinario, in which a seated lad pulls a thorn from his sole, is placed nearby), but their use is wooden as they act as mere supporters in the framing device of the whole composition. Ghiberti, daringly, has both caught in indifferent conversation, talking across the back of their packhorse like two Florentine traders, wholly unaware of the profound revelation of God behind them on the rocks.

Florence became the centre of Christianity in the 1430s, when Eugenius IV, the Venetian-born pope Gabriele Condulmer (1383-1447), fled there. In the political struggle for Rome, which briefly became a republic under the opposing Colonna family in 1434, he abandoned the city, being rowed down the Tiber to the port at Ostia, disguised as a Benedictine monk, as his enemies pelted him with eggs from both banks of the river.

It was in Florence that he condemned slavery in the Bull Sicut Dudum, in response to aggressive Portuguese colonisation of the Canary Islands; and, on Lady Day in 1436, he consecrated the Duomo, when Brunelleschi's great dome was finished, "rising high into the skies, vast enough to cover the entire Tuscan population with its shadow", as the architect in the papal entourage, Leon Battista Alberti, marvelled.

An illuminated page from one of the choirbooks provided for the cathedral by Francesco d'Antonio del Chierico in 1471, vividly recaptures the moment when the Pope was about to enter, thronged with cardinals and prelates.

But, even in 1436, no one could have predicted the extraordinary events that would unfold within three years when the Bull Laetentur Coeli was read out under the great dome, in Latin by Cardinal Giuliano Cesarini, and in Greek by Archbishop Bessarion. Eugenius called an ecumenical council at Ferrara in January 1438, in opposition to one in Basel with its own pope. When plague broke out, he removed it to Florence.

It was here that the Eastern Orthodox and Latin Churches came to be reconciled after the Great Schism (1054) with an agreement about the nature of the Trinity, concerning an argument about the procession of the Holy Spirit from the Father and the Son, in the so-called filioque clause, and about the dignity of the pope.

The gathering of richly dressed oriental clergy, including both the Emperor John VIII Palaeologus and his brother and the Patriarch of Constantinople (who died on 11 June 1439, and is buried in Santa Maria Novella, where the Council met), must have seemed exotic to contemporaries, and at odds with any perceptions of "classical" Greek dress and decorum.

The Bull of Union of 6 July 1439, the signed copy of which is still kept in the Laurentian library in Florence, was housed in a silver casket commissioned for it by Cardinal Cesarini. Richly worked enamels and inlaid semi-precious stones adorn it; the inscription records that the concordat also included the Armenian Church (22 November). The Patriarch of Kiev and Moscow had also been present throughout.

The workmanship of the Florentine guilds of gold- and silversmiths would have seemed familiar to the Eastern visitors, but within 15 years their own world, with all the imperial riches of Constantinople, would be destroyed by the forces of Islam. Eugenius himself returned in triumph to Rome in September 1443, but died before the Bull of Union had become a dead letter.

The exotic sight of the Eastern potentates and visitors certainly made an immediate impression on contemporaries. Carlo Ginzburg has convincingly shown that Piero della Francesca's enigmatic painting of the Flagellation (Urbino) is a later commentary on the event, as is The Baptism in the National Gallery in London. The Eastern delegates brought rich vestments among diplomatic gifts, and Ghiberti dressed God the Father in just such robes when he designed a gilt door for Bernardo Rossellino's tabernacle for a hospital chapel.

The exhibition gives little prominence to the realisation of a united Christendom, but visitors will see that the Trinity became important, and can at least read the excellent catalogue essay by Paolo Viti on this critical moment in the Church's history. It came to mind in Passion Week with the graciously accepted invitation to the Ecumenical Patriarch to attend the first mass of Pope Francis.

In the 1454 fresco by Andrea del Castagno for the Florentine basilica of SS. Annunziata, St Jerome with his two female disciples, mother and daughter Paula and Eustochium, gaze up at a vision of the Trinity. This is rare enough in iconography, and the catalogue strains to link the subject to the first name of the potential donor, since the Girolo-mites, an order with which the Trinity was associated, had no known connection to the church. But the depiction of the Godhead clearly shows the Holy Spirit proceeding from the Father alone.

The profound sense of a common link between civic and Christian "Romanitas" emerges in the statuary that deliberately echoed Roman models. Prophets and saints are cast in the mould of orators and senators, and the exhibition allows us to make direct comparison.

From Herculaneum comes the so-called Pseudo-Seneca, a bronze portrait head that is set alongside Donatello's Head of a Prophet, thought to date from between 1435 and 1443. Almost identical in scale (the Roman head measures 36 × 29 × 25cm) they appear at first sight to be by the same studio. In fact, they stand some 1500 years apart. At least 18 centuries separate the great horse heads of theMedici protomeand Donatello's rival, the Carafa protome, that he cast for an equestrian statue of the King of Naples which was never fully realised.

Lorenzo Ghiberti's study for a statue of St Stephen, in a niche for the exterior of the wool guild's church of Orsanmichele, and Donatello's 1423-35 monumental figure of St Louis of Toulouse, which stands 2.85m, recall the ancient statuary that must once have dominated the cities of the Roman Empire. It was originally commissioned by the Guelph party, and it is suggested that Donatello achieved its striking realism by casting the bronze over a manikin clad in fabric. The folds of his cope and the texture of his soft leather gloves are remarkable.

TheSt Louiswas restored by Bruno Bearzi in 1946-48 after it had been brought back into the city from the safety of a railway tunnel near Incisa Valdarno by the so- called "Venus-fixers". These were teams of British and American arts specialists who worked alongside the army to recover and restore as much as they could at the time, in the wake of the Germans' slow withdrawal.

The Allies did not always agree with Bearzi's methods or approach. Ilaria Brey, in her fascinating 2009 account of how the Allied soldiers saved Italy's art during the Second World War, recounts a famous row with him over how best to move the ponderous bronze statue of Cosimo I back from safe-keeping at Poggio a Caiano to Florence. The statue of St Louis was damaged in the 1966 flood, and Bearzi restored it again. It has now been extensively restored for the exhibition.

Men such as the politician Niccolò da Uzzano, who died in 1433, and whom we meet in a painted terracotta bust by Desiderio da Settignano (working from a death mask), and the noblewoman Marietta Strozzi, in whose palace we meet, might well have regarded Florence as the new Rome. We can be grateful that, for all the depreda-tions of July and August 1944, so much remains, and has been restored to such magnificent glory.

"The Springtime of the Renaissance" is at the Palazzo Strozzi, Piazza Strozzi, Florence, until 18 August. www.palazzostrozzi.org. Phone 00 39 055 2645155. Booking: Sigma CSC: phone 00 39 055 2469600.

The exhibition transfers to the Musée du Louvre, Paris, on 23 September, and runs until 6 January next year. www.louvre.fr/en. Phone 00 33 1 40 20 53 17.