Galileo’s last work, *Discorsi e Dimostrazioni Matematiche, intorno a due nuove scienze* (*Two New Sciences*), published in 1638 while he was in house arrest, is his most important work. According to Alessandro De Angelis,1

Two New Sciencescontains, to mention only some of its main discoveries, the principle of inertia, the description of the motion of falling bodies, the observation that bodies of different weight fall with the same acceleration in vacuo, a demonstration (correct only at the first order) of the isochronism of pendulum oscillations, a demonstration of the parabolic motion of projectiles, and innovative considerations related to acoustics and music. [p.xx]

Unfortunately, the most commonly available English-language translation, undertaken by Henry Crew and Alfonso da Salvio, published in 1914,2 is problematic. In one of my first posts, How Two English-language Translators of Galileo Permanently Changed His Words, I presented Alexandre Koyré’s find that this text includes an egregious mistranslation, confusing “I have discovered” with “I have discovered by experiment.”

So when I decided to study the *Two New Sciences* in more detail, I went looking for something better. It turns out that Stillman Drake, the English-language biographer of Galileo, published a translation of *Two New Science*s in 1974.3 This is much easier to read.

But then I came across something completely different. Alessandro De Angelis, professor of physics at the University of Padua, where Galileo himself was professor for 18 years, recently published an English-language adaptation of the complete *Two New Sciences*, using modern algebraic notation.

The result is quite striking. Consider, for example, this theorem from Drake’s translation:

PROPOSITION I. THEOREM I

The time in which a certain space is traversed by a moveable in uniformly accelerated movement from rest is equal to the time in which the same space would be traversed by the same moveable carried in uniform motion whose degree of speed is one-half the maximum and final degree of speed of the previous, uniformly accelerated, motion. [Third Day, p.165]

In De Angelis’s version, this becomes: [Day Three, p.105]

For the modern reader, it is a lot easier to read De Angelis’s version. However, the structure of the two is very different, and Galileo’s thoughts are more easily followed reading Drake’s translation. So, in my opinion, a careful reading in English of the *Two New Sciences* should be undertaken by placing side-by-side Drake’s translation and De Angelis’s adaptation, and reading them simultaneously.

Alessandro De Angelis. *Galileo Galilei’s “Two New Sciences” for Modern Readers*. Springer, 2021.

Galileo Galilei. *Dialogues Concerning Two New Sciences*. Translated from the Italian and Latin into English by Henry Crew and Alfonso de Salvio. New York: MacMillan, 1914.

Galileo Galilei. *Two New Sciences, Including Centers of Gravity & Force of Persuasion*. Translated by Stillman Drake. University of Wisconsin Press, 1974.

It's a challenge reviewing the old masters and avoiding reading into them too much with the benefit of subsequent interpretation and development. A good example is Newton's Second Law: "The alteration of motion is ever proportional to the motive force impress’d; and is made in the direction of the right line in which that force is impress’d." Yes, that became "F = m a," but only decades after Newton's death as d'Alembert, Euler, and others developed the mathematical formalism to apply Newton's thinking.

Reading the original side-by-side with the modern adaptation is an excellent idea.

Interesting puzzle to think about! The second body will only reach half of its final speed at over half of the time.

In high school physics we denoted acceleration a = dv/dt, velocity v = dx/dt, and x as location. For the first body the acceleration a is constant (c), so v1 is linear (c*t) and x1 is quadratic (1/2*c*t^2) in time. For the second body a2 is zero, so v2 is constant and x2 is linear (v2*t) in time. Let tm be the final time. Given that v2 is half of v1(tm), x2(tm) = v2*tm = 1/2*v1(tm)*tm = 1/2*c*tm^2 = x2(tm).