Ancient Babylonians Used Geometry That Anticipated Calculus

...

Galileo stated a theorem similar to the mean speed theorem in his book Two New Sciences published in 1638.

The time in which any space is traversed by a uniformly accelerated body starting from rest is equal to the time in which that same space would be traversed by the same body moving at a uniform speed whose value is the mean of the highest and last speed of the prior uniformly accelerated motion.

There is a small difference between this formula and the Merton rule. The Merton school and Oresme used the speed at the middle moment, while Galileo used half of the highest and last speed of the constant acceleration motion to apply the mean speed theorem. To explain Galileo’s rule by reference to the Fig.MR, the area of the triangle OAB is equal to that of the square OACD whose height is half of OA. Galileo’s rule is true of a special case when the initial velocity is zero and the acceleration is positive, while the Merton rule holds true even when the initial velocity is not zero and the acceleration is negative. Galileo’s rule, however, has its own merit of being easy to verify experimentally. Measuring the meantime speed of constant acceleration motion was difficult at that time, but the last speed was easily measured by converting the constant acceleration motion into constant velocity motion.

...

Did Galileo discover the mean speed theorem or the double distance rule independent of the Oxford Calculators or Oresme? This is a point in dispute among historians of science. Takahashi Kenichi, a Japanese historian of science, denies the influence, taking up a letter from Galileo to Paolo Sarpi dated 16 October 1604 as a counterevidence. ...