As in OTL, in 1250 the people of Florence elected for the first time the 36 caporali di popolo, a political counterweight to the nobles. The use of cannons in Europe was first mentioned 1326 in a Florentine document.
Since 1327, after the fall of Constantinople many Byzantines fled to Florence. The knowledge they brought to Western Europe helped spawning the "Rinascita" (it wasn't called that at this time) that already started in the last century. 1348, the displaced Constantinopolitans helped founding a Platonic Academy in Florence. The knowledge of Greek started to spread throughout Italy.
In 1330, Florence acquired Lucca for 75 000 florin (gold coins). 1376, Giovanni de Medici got dictatorial power in the city; since the republic felt threatened by the beggars roaming in Italy, the Medici could keep and strengthen their power. They also projected their power outwards: 1380, after having taken over other city states like Pistoia and Arezzo, now they conquered Ghibelline Pisa, acquiring a harbor (which was later replaced by Livorno). In 1387, Florence defended successfully against various mercenary groups who roamed through the Italian peninsula after the first Aquitainian War had ended. The family degli Albizzi tried to cooperate with the mercenaries to oust the Medici from power, but were banned in the end.
The luck of Florence
It was a very lucky coincidence that Florence (as Savoy) was spared by the Black Death, which hit the other Italian states 1395. In 1399, Florence bought Corsica from Genoa because the current ruler, Cosimo de Medici, planned to build up a fleet and becoming a naval power in the Mediterranean. 1404, Florence invaded the beggar's republic of Parma, annexing it. The arts flourished too: 1415, the inventor and artist Lorenzo del Vacca started his work as a studio boy. Until his death in 1463, he'd invent many things way ahead of his time, like the "movable fortress" (a primitive tank), a hang glider, improved guns and bombs, better cryptography, and many others. In addition, he further improved the Rinascita art, leading it to its apex.
Florence took a further step to dominance in 1407 when Maffeo Servitore, a cunning Florentine diplomat, saw the weakness of the divided Northern Italy and devised a plan. Meeting with the rulers of Savoy and Venice, all of Northern Italy except Genoa was divided into spheres of influences, which said three states were allowed to conquer. Otherwise, the big three were supposed to live in peace. Until the 1430s, this was what happened: The little city states (which were near collapse after the difficult 14th century) of Northern Italy were "mopped up". As a result, many Italians were leaving their country (especially from Pisa and Milan), going to France and Aragon, and some other states too, spreading Italian art.
In 1444 Florence suffered a setback however, when Venice and Aragon defeated it, because they feared its economical competition. The victors destroyed its fleet, forced it to sign an "everlasting treaty" forbidding Florence's ships to go to ANY harbor in Europe or the Mediterranean (except those on its own territory). In addition, Florence had to pay a big sum of money. The republic offered the winners to give them Corsica instead, but they were not interested. This treaty turned out to help Florence, however: In 1456, Florentine traders arrive in Portuguese Tangiers (which is neither in Europe nor in the Mediterranean ...), where they saw the new sailing ships. 1468, the first Florentine caravel under captain Buonarotti crossed the Atlantic, found the way to Nystad. They discovered the Potomac, and since the area was yet unclaimed by Sweden, they claimed it for Florence.
From Republic to Duchy
Shortly after, in 1477 Piero de Medici became the first duke of Florence after defeating millenialist sects which had spread after the infamous Sacco di Roma. Times were difficult for Florence: 1516-18, it was invaded by the Rum-Seljuks. Although the latter stayed supreme in open battle, they didn't manage to take the new "Italian fortresses". When the Janissaries became discontent because they couldn't plunder the cities, the Sultan had to cancel the attack, went into the Marches instead. Then the other Christian powers mobilized their forces, and 1519-22 the alliance of the Quadruple Monarchy, Florence, Venice and some German princes fought the Seljuks to a stalemate along the Po river, which became the new northern border of the Seljuk empire.
In 1533, religious uprisings shook up Florence. The radicals were suppressed, but the duke gave in insofar as he had the bible translated into the vernacular. Otherwise, Florence still stood firmly at the side of the pope.
1536, during the Great Occidental War, Florentines took OTL Saint Vincent, Grenada, Barbados (which they called Elba Nuova). 1543, with Scandinavia in Civil War / unrest, Florence like other powers used the opportunity. The South and West of the Chesapeake peninsula went to Italia Nuova, and Florence made a treaty with the Netherlands, ruling the border between their colonies in Atlantis, and promising to respect their borders.
Against the Seljuks
In 1542, a great uprising of the Carbonari in South Italy started, which soon spread to Rome itself. Florence used the opportunity and invaded Latium. Many volunteers went to Italy to fight against the Seljuks; some pious nobles, mainly from Castille-Portugal, supported Florence with money. 1543, Florentine troops stood in Naples and the Marches. Now however, the main army of the Seljuks arrived, and the Florentine army was defeated several times. The chaos allowed many carbonari to leave South Italy and flee to safer places. Many would even settle in Italia Nuova, which soon included all of OTL Virginia and Maryland (although thinly settled). In 1544 the Seljuks made peace, again having failed to take the Florentine fortresses, although Florence had to pay some tribute. The people of Rome had to leave their city, which the sultan wanted to settle anew with Muslims. Some of them went to Atlantis too, but others swore to retake Rome ASAP.
The opportunity came in 1556, when the Triple Monarchy of England-Castille-Portugal declared war on the Seljuks, enraged over the fate of Rome. Savoy and Florence joined the war. 1559, Rome was reconquered by Christians; all the Muslims found there were massacred. 1565, the peace of Ostia was made. The Seljuks had to give up Latium (to Florence) and the Marches, Algiers, Tunis and Sicily (to the Triple Monarchy of England-Castille-Portugal). The Romans displaced ~20 years ago returned in triumph. The victorious Italians demanded that the Pope returned to Rome, which the French king denied however.
Then in 1567, France, Venice and the Holy Roman Empire declared war on the weakened Seljuks. 1572, after another uprising of the Carbonari, Florence joined the war too, attacked Naples. And in the peace of Krems (Austria) 1574, the Seljuks had to cede Dalmatia to Venice, the county of Naples to Florence, Salzburg to Bavaria, Austria to Württemberg-Carinthia and western Hungary to France. After the Great Seljuk War, they also acquired Pescara and Benevent in the peace of Naples. And after the Second French-Seljuk War (Grand Duke Piero III died in this war fighting the Seljuks), the peace of Györ gave them the northern half of South Italy.
1599, Florence bought Aragonian Sardinia, when the kings Henry V of the Triple Monarchy of England-Castille-Portugal and François III of France agreed to solve their conflict in Aragon by dividing the country.
1616-19, the "Island uprising" in Sardinia and Corsica happened after trying to reform the administration too fast (after French example). Florence proved unable to overwhelm them again, but got the promise from France that they wouldn't try to conquer them. (Instead, Sicily would take them later.)
The relations to the Triple Monarchy of England-Castille-Portugal were strained however when in 1624 an incident in the Caribbean lead to the latter occupying the Florentine islands in the Caribbean (except Elba Nuova). As a consequence, Florence retaliated by taking Ceuta in 1628, which was very important for them to protect their connection to Atlantis.
1631, the Florentines finally parted with the Triple Monarchy of England-Castille-Portugal, since the Florentines wanted to annex the Duchy of the Marches. They made a compromise with France: While the former Papal States would still be nominally under the pope, Florence would "administrate" it for him. This was also seen as a compensation for the pope staying in Avignon, while many Italians still wish for his return to Rome. In addition, Florence had to allow France's ally Venice to annex the more valuable North: Bologna, Ferrara and Ravenna.
1652-56, Florence allied with Denmark-Braunschweig against king Humphrey I of England in the "Atlantean Coalition" and surprisingly defeated the English fleet. New England stayed a Danish-German protectorate.
During the years 1660-67, Florence and Venice waged war against the Seljuks, despite that France didn't fight with them, but they expected that the Seljuks were beaten after their defeat by Russia. Instead, They were defeated, losing their conquests from the peace of Györ. The humbled Florence even needed Sicilian help to evacuate their troops from Greece. As a consequence, in 1667 Florence made a dynastic marriage with Sicily, between king Francisco II and the only surviving child of grand duke Lorenzo III, Condolcessa. (Even besides the political consequences, the marriage is considered a scandal - there are rumors that Condolcessa was the mistress of both her father and her uncle, late grand duke Giovanni III.)
In 1688, Sicily and Florence united, formed the Italian kingdom.
Part of Italy