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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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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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