I know I'm a few years late to this party, but I stumbled across your post in search of a related question. After reading your question and the responses below, I figured I could fill in a few gaps left unanswered.
How did these diagrams help Gaileo reason about uniform motion?
It may help you to see the context of where your passages are pulled from. The first passage you quote is Uniform Motion Theorem VI, Proposition VI. The second passage you quote is Uniform Motion Theorem III, Proposition III.
What Galileo is doing in Theorems I-VI is illustrating the consequences of his proposed definition of uniform motion. You need to consider that Galileo's "Two Sciences" is written as a sequential series of proofs that builds up in logical order starting from a set of assumed true and simple axioms. Simply put, in the section of Uniform Motion, the order of his argument is as follows:
- The definition of Uniform Motion.
- An emphasis (i.e. "Caution") on how the definition provided is an improvement upon older definitions of uniform motion.
- The required Axioms I-IV that allow the definition to hold true.
- Logical deductions (i.e., Theorems I-VI) resulting from the definition and its required axioms.
So to answer your question about the theorem's diagrams, Galileo is using them to provide Eucleadian (geometrical) arguments in comparing magnitude lengths corresponding to the time, distance, and speed qualities of the motion of two particles. He starts off investigating how the difference (or more accurately, ratio) in one quality of motion results in a difference of another while holding all others equal, and then builds up to show how the ratio of a quality results when all the qualities are different. In other words, Theorems I-VI (1-6 for formating reasons) demonstrate how a quality of motion is proportions related:
- time and distance for equal speeds (using definition and axioms);
- distance and speed for equal times (using definition and axioms);
- time and speed for equal distances (using Theorems 1 and 2);
- distances for differing speeds and times (using Theorems 1 and 2);
- times for differing speeds and distances (using Theorems 1 and 3); and
- speeds for different distances and times (using Theorems 2 and 3).
So you can see he covered all the scenarios for differing variables.
Some of these propositions follow readily from his definition of uniform motion:
That's literally what Galileo is arguing in his section of Uniform Motion: that these relationships are the result of the definition of uniform motion. Notice how the logic of Theorems 1-6 above can each be traced back to the fundamental definition of uniform motion and its associated axioms.
This is a bit of an idealization, no? Even a car starts and stops and
exhibits small amounts of acceleration driving up a hill or down a
road. Perhaps on a very smooth highway and for short periods of time.
Yes, it's idealization, but the purpose is to apply a mathematical model that accurately describes reality with an acceptable degree of error. Are you able to draw a perfect circle or a parallel set of lines? Of course not, but we can still deduce further truths from these idealization from our mind's eye and then apply them to reality. Air resistance, friction, and road bumps are only inhibitions to the true undisturbed motion, but these can be added later. We want to know how motion works within a metaphorical and literal vacuum.
One could argue hardly anything moves in the way he is decribing.
The philosophy and the mathematics of motion have been lengthly discussed long before Galileo. The "Mean Speed Theorem" (aka The Merton Rule) was mathematically demonstrated by 14th-century philosopher Nicole Oresme, who predates Galileo by about 300 years, and Galileo originally set out to prove him wrong, only to realize that Oresme was correct (much to his frustration, as discussed in his "Two Sciences"). If you read the rest of his book, you will see that Galileo was able to build upon the definition of uniform motion along with the definition of uniform acceleration to further deduce a hypothesis that objects fall with a distance proportional to a square of their times.
What makes Galileo unique over his predecessors is that he took it a step further and experimentally measured it to show that data matched his hypothesis, which is something Oresme never bothered to do, and that's what makes Galileo the "father of modern science." This is why the scientific method has us develop a logically reasoned hypothesis first, then experimentally test a measurable deduction from our hypothesis to see if the data matches our theory. If the data doesn't match our reasoning, then there's a flaw somewhere in our initial assumptions that must be routed out. This is also why Flat Earthers fail at performing legitimate science: they develop "theory" AFTER any flat/round earth experiment is performed. No deduction is made nor is it further tested, and any data that doesn't conform to their model is tossed out. For example, a scientist would say, if we assume the earth is spherical with poles and equator, and if we assume Newton's laws are true, then we would expect that an oscilating pendulum will precess at the poles (with a period of 24-hrs, no less or more) but none at all at the equator -- "Foucault's Pendulum" then confirms this. Conversely, a Flat Earther will see the precession of Foucault's Pendulum first and then argue that it's because of "invisible vortices" with no further deduction and experimentation. To turn this back around to your comment, Galileo effectively demonstrated that the definitions of uniform motion and acceleration are accurate based on his supported by data, so really, it does "move in the way he is describing."
Though in your defense, you COULD argue it doesn't reflect reality to some degree. As we've learned from Einstein, we don't live in Euclidean space, so our axioms in Euclidean geometry are called into question. For instance, under special relativity, two different observers moving at different speeds won't agree on the speed of a third object (e.g., speed of light is always constant for all observers). This is because time and space measurements will differ for each observer. Does that make everything Galileo and Newton found to be garbage? No, because it's more practical to use these Newtonian/Galilean assumptions when the errors caused by relativity are small. The take away here is that it's important to understand the underlying assumptions for a particular theory in order to understand its limitations.
As the last point to wrap things up, I saw some of your other comments that particularly stuck out to me as being the real question behind your post:
Unfortunately his proofs are in arcane language. So I am asking, how did he prove it?
I'm going to make two guesses in you mean by "it."
First, if "it" you mean the definition of uniform motion, then look to his axioms for support. As an axiom, they're generally so fundamental, they're assumed to be undeniable -- they're the initial assumptions. They don't necessarily need to be true, but they are assumed to be true for argument's sake. Here are my interpreation of his axioms for uniform motion:
For a particle moving with velocity v through space s and time t such that time intervals Δt1 and Δt2 correspond to displacements Δs1 and Δs2, wherein such intervals may take on any size, and wherein such intervals can be chosen at any point during the motion, then
- if v is constant and Δt1 > Δt2, then Δs1 > Δs2;
- if v is constant and Δs1 > Δs2, then Δt1 > Δt2;
- If v1 > v2 and Δt1 = Δt2, then Δs1 > Δs2;
- If Δs1 > Δs2 and Δt1 = Δt2, then v1 > v2.
From there, considering all the inequalities above, it's safe to deduce that a special condition exists where simultaneously Δt1= Δt2 and Δs1= Δs2, which is defined to be "Uniform Motion." With axioms as simple as that, it's hard to argue against it without tossing out all of Euclidean geometry.
If "it" you mean the theorems, then walk through the theorems in sequential order and follow the proofs he provides within. Theorem 1 was hard for me because I didn't understand the Theory of Ratios. For that, I had to look up the mathematical definition of a ratio. If I recall correctly, Galileo borrows a proof from Euclid's Elements Book 5, definition 5.
For each Theorem, he first states the theorem, followed by a geometric statement of the theorem, and lastly followed by the proof. For example, to match the diagram you originally provided (i.e., Theorem III):
"In the case of unequal speeds, the time-intervals required to
traverse a given space are to each other inversely as the speeds."
This is a just a broad conclusory statment of which he will elaborate.
Let the larger of the two unequal speeds be indicated by A; the smaller, by B; and let the motion corresponding to both traverse the given space CD. Then I say the time required to traverse the distance CD at speed A is to the time required to traverse the same distance at speed B, as the speed B is to the speed A.
Here, the first sentence sets the stage, then the second sentence details the theorem in context of the figure.
For let CD be to CE as A is to B; then, from the preceding (he's refering to theorems 1 and 2), it follows that the time required to complete the distance CD at speed A is the same as the time necessary to complete CE at speed B; but the time needed to traverse the distance CE at speed B is to the time required to traverse the distance CD at the same speed as CE is to CD, that is, as speed B is to speed A.
Decyphering this can be a challenge, but if you understand how ratios are written in prose (i.e., "let CD be to CE as A is to B" means CD/CE = A/B), and if you follow it step by step, you'll see his argument unfold. Also, you can see in his proof above that he references prior proofs. Sometimes he won't even say why a relationship is invoked simply because it just "is" because he showed it earlier in the section. This is why it's important to understand the context of each theorem in relation to prior thoerems, definitions, and axioms. Jumping straight into a random theorem in a book like this is much like jumping to the middle of the novel: you won't understand what's happening in the plot or why. I learned the hard way when I tried doing that with Newton's Principia and I was absolutely baffled as to what the hell he was doing until I understood the context of the theorem.
Anyways, I know it's been 4 years since you posted this, but I hope you ended up finding your answer since then, and I hope that maybe this'll be useful for anyone else who stumbles here.