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Galileo is known to have studied kinematics through his work with projectiles. How did he first consider researching motion and velocity? Was he inspired by previous work done by earlier scientific minds such as Archimedes, or was this work of his own volition?

If there are any links that highlight the source of Galileo's motivations, please leave an answer below.

Thank you in advance!

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Kinematics was distinguished from dynamics by the Merton school (a.k.a. Oxford calculators) of scholastics in 14th century, who worked out kinematics of uniformly accelerated motion. In particular, they formulated the mean speed theorem (a.k.a. Merton rule) (distance traveled is half the sum of the initial and final velocities, times the elapsed time), which was proved by Oresme using geometry and simple function graphs.

The influence on Galileo is controversial. He seemed to insist on his originality in a letter to Sarpi, and he initially thought that speed is proportional to distance traveled despite being likely familiar with the Merton rule from the university years. What he wanted, unlike Mertonians or Oresme, who explicitly described his work as secundum imaginationem, was an experimental demonstration one way or the other. In Dialogue Concerning the Two Chief World Systems (1632) he finally accepted Oresme's postulate that speed is proportional to time traveled, and in Two New Sciences (1638) stated a version of the mean speed theorem, along with the "law of odd numbers" (consecutive distances traveled in equal times are in the same ratios as consecutive odd numbers) that follows from it. Both the statement and the demonstration are reminiscent of Oresme's.

Here is from How original is Galileo’s work on kinematics? by Nagai:

"There is evidence of Galileo knowing the Merton rule and Oresme’s theory. We can find Heytesbury, Oxford Calculators, the mean speed theorem, “uniformly difform (uniformiter difformis)" quality and “Parisian doctors (Doctores Parisienses)" in manuscripts that Favaro, chief editor of the National Edition of the works of Galileo Galilei, called Youthful Writings[17]. Judging from the contents of the manuscript, the “Parisian doctors" must include not only Jean Buridan (1295 – 1358), who developed the concept of impetus close to the modern concept of inertia, but also his disciple, Oresme. Their theories were made so popular by Albert of Saxony (Albertus de Saxonia; ca. 1320 – 1390)and Domingo de Soto (1494 – 1560) that it is natural that Galileo could hear of them.

[...] He finally left the University in 1585. His later works show his contempt for and hostility toward the Scholastics and the Aristotelians. It is no wonder that he did not accept all of the theories by the “Parisian doctors" that he considered Scholastic and Aristotelian. His contempt for and hostility toward the Scholastics and the Aristotelians is the key to the riddle of Galileo’s theory of falling bodies... we can find Oresme’s influence on Galileo, but he never mentioned his name, although he often heaped the highest praise on the works of Archimedes. Maybe he did not want to credit the achievement to the Aristotelian Scholastics."

For more on Oresme and Merton school see Medieval Quantifications of Qualities by Sylla. For a broader perspective see The Case of Kinematics, the Genesis of a Discipline by Koetsier.

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