The Grand Duke offered Viviani a monthly salary to continue his education and introduced him to Galileo, whose deteriorating health required a capable and competent assistant.

[2] Their frequent exchange over the subject inspired Galileo to elaborate on the mathematical theory and complete his demonstration on free bodies that would establish its connection to nature.

Viviani was also one of the first members of the Grand Duke's experimental academy, the Accademia del Cimento, when it was created a decade later.

[4] In 1654, Cardinal Leopoldo de’ Medici tasked Viviani with gathering material in order to write a biography about Galileo's life and work.

Timing the difference between the seeing the flash and hearing the sound of a cannon shot at a distance, they calculated a value of 350 meters per second (m/s), considerably better than the previous value of 478 m/s obtained by Pierre Gassendi.

[1] From 1666 to 1667, Viviani guided Nicolaus Steno on the geometrical propositions of anatomical figures and dissections of shark muscle fibers.

[3] In 1687, he published a book on engineering, Discorso intorno al difendersi da' riempimenti e dalle corrosione de' fiumi.

The monument that was created in the church of Santa Croce was constructed with the help of funds left by Viviani for that specific purpose.

In Florence, Viviani had Galileo's life and achievements written in Latin on the façade of his palace, on huge stone scrolls.