Borgia, Da Vinci and Machiavelli. How much do we know about their relationships?

The relationship these men had were all very different.

Cesare Borgia was at the height of his military career during the Italian Wars at the end of the 15th century. Leonardo da Vinci was not just one of the top artists in the peninsula, but also one of the most distinguished engineers as well (evidence of this is seen in his large project to change the course of the Arno river, which we will address later since it ties in to your question.) Leonardo was in charge of designing/improving military machines as well as foreseeing the construction and repair of various keeps and castles in the Italian North. The relationship they both held was very business oriented. Despite Borgia's military desires, he was a well paying patron for da Vinci. He gave him a lot of creative control to explore the campaign trail and analyze potential future scenarios. Ultimately, da Vinci was a man of creativity, his attention turned elsewhere, mostly towards anatomical studies, and their relationship ended. Borgia's murderous rages did not help paint him as a hero in the eyes of Machiavelli and da Vinci either. Machiavelli and da Vinci radically changed their opinion after Borgia killed some of his men and raided Sinigaglia.

Rafael Sabatini described the occurrences in one of his books:

"Immediately upon the arrest of the condottieri Cesare had issued orders to attack the soldiers of Vitelli and Orsini, and to dislodge them from the castles of the territory where they were quartered, and similarly to dislodge Oliverotto's men and drive them out of Sinigaglia. This had been swiftly accomplished. But the duke's men were not disposed to leave matters at that. Excited by the taste of battle that had been theirs, they returned to wreak their fury upon the town, and were proceeding to put it to sack, directing particular attention to the wealthy quarter occupied by the Venetian merchants, which is said to have been plundered by them to the extent of some 20,000 ducats."

Machiavelli had been traveling with Borgia as an envoy from Florence. That included sending direct reports to the people of Florence as to the plans and ideas that Borgia was carrying. Borgia knew that Machiavelli was sending out reports and would intercept them. Borgia would obviously pick which ones would reach Florence. Borgia's ruthless military campaigns would go on to influence Machiavelli's writings, including "The Prince."

Machiavelli and da Vinci shared a much more successful relationship which was fairly work oriented. When Machiavelli, a public servant of the city of Florence, was tasked with commissioning an individual to change the flow of the Arno, he picked da Vinci. The project was ultimately a failure due to various reasons, mostly structural. The event however, would catapult both da Vinci and Machiavelli to complete two of their most important works. Machiavelli, frustrated with the political world, went on to face treason accusations which led to his time in exile..which he used to write "The Prince." Leonardo returned to painting, possibly exhausted by the difficulties of war and politics, and completed the Mona Lisa. Notice the river in the background? it is speculated by many that it is the Arno. The same river that he tried to change the flow of with the support of Machiavelli and the Florentine government.

Ultimately, their relationships were mostly secured through work. Their interactions tho, helped to shape their lives. Not much is known about honest opinions when it comes to the three figures. As stated above, correspondence was often intercepted and could be dangerous. Leonardo was known to constantly "shut down" emotionally and avoid writing of every day occurrences in his journal.