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, Two New Sciences contains, 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]
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.
Hans, can you recommend any texts that give an idea of the developments of physics, and the relevant mathematics, in the eighteenth century? I can read any of English, French, Italian, Spanish and German. I can also work through Latin and Russian, but that is much harder.
Carl Boyer’s A History of Mathematics is an excellent overview of the subject (https://amzn.to/3S8levE).
The most influential was the brilliant and unorthodox Clifford Truesdell, whose writings are as excellent as they are difficult to find at a reasonable price. He wrote Essays in the History of Mechanics, explaining the development of Newtonian physics. See: https://amzn.to/2O6Autv.
Truesdell said "It is easy to forget that matters which are now clear and simple first had to be made so by someone." And then he goes and explains how that process worked. In particular, he demonstrated how Newton's work was perverted by his successors, disconnected from its empirical roots, and turned into a purely deductive system. All that's in Essays on the History of Mechanics.
This brings back the old days for me. Equation-based physics problems are relatively easy to solve since one only needs to apply the proper pre-existing equations. The most difficult problems for students are "word problems" or language based physics problems, where the natural language description of the physical phenomenon must be converted into a set of mathematical equations in order to solve it. This translation is the most difficult part. I knew math majors on scholarship who got stumped by these. Now, imagine perceiving a physical phenomenon for which no mathematical interpretation has yet been developed, then going on to develop it. Even Einstein couldn't do this. I have always been in awe of Newton and Leibnitz.
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).
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.
Hans, can you recommend any texts that give an idea of the developments of physics, and the relevant mathematics, in the eighteenth century? I can read any of English, French, Italian, Spanish and German. I can also work through Latin and Russian, but that is much harder.
The four historians of the period who most influenced me are:
Thomas L. Hankins Science and the Enlightenment (https://amzn.to/3UdSzb4),
J.L. Heilbron, Elements of Early Modern Physics (https://amzn.to/3SpLX7l), and
A. Rupert Hall who wrote both
From Galileo to Newton (https://amzn.to/4b0w13H) and
The Scientific Revolution 1500-1800 (https://amzn.to/3tXYDdk).
Carl Boyer’s A History of Mathematics is an excellent overview of the subject (https://amzn.to/3S8levE).
The most influential was the brilliant and unorthodox Clifford Truesdell, whose writings are as excellent as they are difficult to find at a reasonable price. He wrote Essays in the History of Mechanics, explaining the development of Newtonian physics. See: https://amzn.to/2O6Autv.
Clifford Truesdell was definitely a colourful writer! His writings, I think, will be very useful for me.
Truesdell said "It is easy to forget that matters which are now clear and simple first had to be made so by someone." And then he goes and explains how that process worked. In particular, he demonstrated how Newton's work was perverted by his successors, disconnected from its empirical roots, and turned into a purely deductive system. All that's in Essays on the History of Mechanics.
Hans, many thanks for the suggestions. I have a few nights' worth of reading in front of me.
Mind if I re-post this and some of your other pieces to my Substack?
Yes, by all means. Thank you.
Posting on Sunday. Thanks! I'll probably help myself to more posts as they become relevant.
Sure! I'll run through my library and make a list of recommendations in the next couple of days. Ping me if I don't get back to you..
This brings back the old days for me. Equation-based physics problems are relatively easy to solve since one only needs to apply the proper pre-existing equations. The most difficult problems for students are "word problems" or language based physics problems, where the natural language description of the physical phenomenon must be converted into a set of mathematical equations in order to solve it. This translation is the most difficult part. I knew math majors on scholarship who got stumped by these. Now, imagine perceiving a physical phenomenon for which no mathematical interpretation has yet been developed, then going on to develop it. Even Einstein couldn't do this. I have always been in awe of Newton and Leibnitz.
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).
Did you mean "= x1(tm)" at the end?
Yeah!