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Can someone help me here? the language is archaic. This is (translation of) Galileo

If two particles carried at a uniform rate, the ratio of their speeds will be the product of the ratio of the distances traversed The inveerse ratio of the time-intervals required

I think this is just $d = vt$ phrased in an elaborate way. Here is another one from the same section:

In the case of unequal speeds, the time-intervals required to traverse a given space are to each other inversely as the speeds.

and in the proof he draw a diagram

A |----------|
C |-----------------|E|-------------| D
B |--------|

How did these diagrams help Gaileo reason about uniform motion? Some of these propositions follow readily from his definition of uniform motion:

By steady or uniform motion I mean one in which the distances traveersed by the moving particle during any equal intervals of time, are themselves equal.

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. One could argue hardly anything moves in the way he is decribing.

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    $\begingroup$ What exactly are you asking? What Galileo's thought processes were is a matter of history (or psychology?) of science, both of which are off-topic here. $\endgroup$ – sammy gerbil Nov 21 '16 at 20:10
  • $\begingroup$ @sammygerbil I am asking about Galileo's physical argument. These are theorems in his book. Unfortunately his proofs are in arcane language. So I am asking, how did he prove it? $\endgroup$ – john mangual Nov 21 '16 at 20:25
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Is not "phrased in an elaborate way"; it is expressed in the current (at that point in time) theory of ratios, when the symbolic algebra was still in his infancy.

Uniform speed is defined in Two new sciences, Third Day :

By steady or uniform motion, I mean one in which the distances traversed by the moving particle during any equal intervals of time, are themselves equal.

This is exactly the current definition of uniform motion (i.e. constant speed) : if $t_1=t_2$ then $s_1=s_2$, that implies : $v= \dfrac s t = \text {const}$ and yes, it is an "idealization" as anything in science.

Some "obvious" axioms are set forth and then some theorems are proved :

Th.I If a moving particle, carried uniformly at a constant speed, traverses two distances the time-intervals required are to each other in the ratio of these distances.

I.e. with $v= \text {const}$ :

$\dfrac {t_1} {t_2} = \dfrac {s_1} {s_2}$, that amounts to : $\dfrac {s_1} {t_1} = \dfrac {s_2} {t_2}$.

And :

Th.III In the case of unequal speeds, the time-intervals required to traverse a given space are to each other inversely as the speeds.

I.e., given space $s$ :

$\dfrac {t_1} {t_2} = \dfrac {v_1} {v_1}$, which amount to : $v_1 \times t_1 = v_2 \times t_2$.

And also :

Th.IV If two particles are carried with uniform motion, but each with a different speed, the distances covered by them during unequal intervals of time bear to each other the compound ratio of the speeds and time intervals.

I.e.:

$\dfrac {v_1} {v_2} = \dfrac {s_1} {s_2} \times \dfrac {t_2} {t_1}$.

As you can easily verify, it is :

$\dfrac {v_1} {v_2} = \dfrac {s_1} {t_1} \times \dfrac {t_2} {s_2}$.

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Yes, all these examples are the old fashioned ways to say that $d=vt$ and $\Delta d=v\Delta t$. The reason why they did not speak of numbers and but always spoke of proportions is two-fold:

a) The notion of real number is a modern concept. For ancients a "number" was an integer. Instead of the modern theory of real numbers they had an equivalent (but very cumbersome) "theory of proportions", explained in Euclid. That's why, even a century after Galileo Newton wrote: that "attraction is inverse proportional to square of the distance" instead of just saying $F=cm/d^2$". Even if I was in high school, they taught Newton's laws this way.

b) Until the end of 18th century, there was no universal units. Every physical quantity was measured in different units in different places. Writing $d=vt$ assumes that we use some units for $d,v,t$. Instead they said is that "distances" are proportional to times, or that ratios or the distances are like rations of the times etc.

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  • $\begingroup$ I don't understand several conclusion drawn her. For instance the "That's why, Newton wrote.." How is what Newton wrote about a theory of proportions? We still say exactly the same thing "attraction is inverse proportional to..." and it means exactly the same as the equation you wrote, to me at least. Also in b) I don't think writing d=vt assumes units. If you use "dimensionfull" variables you don't need units. I know that modern mathematicians are not very aware of that, but that's not an advance compared to Newtons times. $\endgroup$ – Michael Bächtold May 8 at 6:52
  • $\begingroup$ You cannot write $d=vt$ until you specify units. With other choice of units this will be $d=cvt$, where the constant $c$ depends on your choice of units. $\endgroup$ – Alexandre Eremenko May 8 at 13:24
  • $\begingroup$ Suppose $d=2\text{cm}=0.787\text{inch}$ and $t=3\text{s}$ then the equation $d=vt$ can be written as $2\text{cm}=\frac{2}{3}\frac{\text{cm}}{\text{s}}3\text{s}$ or as $2\text{cm}=\frac{0.787}{3}\frac{\text{inch}}{\text{s}}3\text{s}$. No additional constant $c$ needed. Units are just bases for a one dimensional vector space, but the equation $d=vt$ is independent of the choice of base. $\endgroup$ – Michael Bächtold May 8 at 13:54
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If two particles carried at a uniform rate, the ratio of their speeds will be the product of the ratio of the distances traversed The inverse ratio of the time-intervals required I think this is just d=vt phrased in an elaborate way.

I would agree.

In the case of unequal speeds, the time-intervals required to traverse a given space are to each other inversely as the speeds.

I think ( or hope:) this is self explanatory, the faster "object" gets to the end point in a time inversely proportionate to its velocity.

The diagrams don't mean much to me without a scale, he seems to have taken one object to moving roughly twice as fast as the other.

By steady or uniform motion I mean one in which the distances traveersed by the moving particle during any equal intervals of time, are themselves equal.

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. One could argue hardly anything moves in the way he is decribing.

To me this is simply a definition of constant velocity, nothing more than that. Obviously constant anything depends on your level of measurement accuracy, but that's about as much as I can contribute.

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  • $\begingroup$ I can conjure up things which - in modern language - morally should be "zero" but are not, such as $\sin n x \asymp 0$ as $n \to \infty$ but it's interesting to see Galileo's logic here. $\endgroup$ – john mangual Nov 21 '16 at 17:45
  • $\begingroup$ I know Newton wrote his formal papers in Latin, and his notes in English, but I don't know about Galileo's publications. if Galileo had known about the concepts underlying calculus, perhaps he might have been more precise, but I don't know for what audience he was writing the above. I always got the impression, possibly unfairly because of his experimental work, that he was more of a Faraday than a Maxwell as regard math, but I do know he was far more clever than I am :) $\endgroup$ – CountTo10 Nov 21 '16 at 17:57
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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:

  1. The definition of Uniform Motion.
  2. An emphasis (i.e. "Caution") on how the definition provided is an improvement upon older definitions of uniform motion.
  3. The required Axioms I-IV that allow the definition to hold true.
  4. 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:

  1. time and distance for equal speeds (using definition and axioms);
  2. distance and speed for equal times (using definition and axioms);
  3. time and speed for equal distances (using Theorems 1 and 2);
  4. distances for differing speeds and times (using Theorems 1 and 2);
  5. times for differing speeds and distances (using Theorems 1 and 3); and
  6. 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

  1. if v is constant and Δt1 > Δt2, then Δs1 > Δs2;
  2. if v is constant and Δs1 > Δs2, then Δt1 > Δt2;
  3. If v1 > v2 and Δt1 = Δt2, then Δs1 > Δs2;
  4. 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):

General Statement:

"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.

Geometric statement:

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.

Proof:

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.

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