Which way did Newton find F = ma?

[logics]

... you're short changing Newton ...

In Newton's words, zoobyshoe," hypotheses fingis": you are imagining things. What Newton says is clear: you are fantasising about "what he dropped", "when he dropped" and even "why he dropped". You reverse the historical facts : somebody applied, later, "momentum" to what Newton referred to as quantity of motion [measure of the same arising from velocity and quantity of matter]. And that "almost" gives you the power of deciding what suits you when it suits you. [and you cite "wiki" (!) as your authority]
[bold added]

He defines momentum "quantity of motion"... Thereafter almost every time he uses the word "motion" it is short for "quantity of motion"... for convenience. From wiki: "According to modern ideas this is understood, in modern terms, as an equivalent of:..."

But all your effort is not necessary if you read, carefully, what I said: for Newton everything was simple because he considered [as it is right] "motus" as one concept. "motion, speed, velocity, momentum" are redundant distinctions as there cannot be speed without direction nor velocity without mass, do you realize that? and cannot be matter without quantity : "pears" must be "quantity of pears", moreover there was no reason, then, to take mass into account.
It was even better for him, fabulous, because he naively thought that both Energy and velocity/momentum are proportional to force: Fg → E_k→ v,p

\textit{v, [p, E]}\propto{F}
Mass was not considered, as gravity was the only known force at the time, and its effect is not influenced by mass.

The big problems arose later, when Leibniz' theory was verified and it was accepted that E = v².
Why is it a problem for you to admit that his original formulation was re-formulated, adapted to subsequent ideas? is there anything wrong, is that something to be ashamed of? The question is only if that was a good or a bad decision.

 

[zoobyshoe]

in Newton's words "zoobyshoe, hypotheses fingis": you are imagining things. What Newton says is clear: you are fantasising about "what he dropped", "when he dropped" and even "why he dropped". You reverse the historical facts : somebody called, later, "momentum" what Newton referred to as motion and matter. And that "almost" gives you the power of deciding what suits you when it suits you. [and you cite "wiki" (!) as your authority]

[bold added]

I don't have to hypothesize about his shortening of the term, because I can read:

"Definition II

The quantity of motion is the measure of the same arising from the velocity and quantity of matter conjunctly.

The motion of the whole is the sum of all of the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple."

He defines "quantity of motion" and then, right away, drops the "quantity of" in explicating it. It's obvious from the relationships he describes he's still talking about "quantity of" motion: momentum.

"Quantity of matter" is mass, which he defines in definition I.

Paraphrased:

"Momentum arises from velocity and mass.

The momentum of the whole is the sum of the vectors. If you double a mass but keep the velocity the same, its momentum will be doubled. If you then double the velocity the momentum will be quadrupled."

If you read the definition carefully you will notice he suddenly drops "quantity of" and uses the word "motion" by itself to refer to what we call "momentum". If you happen to miss that, the rest of the Principia will sound like gibberish, because he drops "quantity of" almost everywhere else when he's referring to momentum. There, again, though, it's always clear from the relationships he's describing he's referring to momentum.

Here's another example where he begins by speaking about "quantity of motion", but then drops "quantity of" in explicating his subject:

"The quantity of motion, which is collected by taking the sum of the motions directed towards the same parts, and the differences of those that are directed to contrary parts, suffers no change from the action of bodies among themselves.

"For action and its opposite reaction are equal, by Law III, and therefore, by Law II, they produce in the motions equal changes toward contrary parts. Therefore if the motions are directed towards the same parts, whatever is added to the motion of the preceding body will be subducted from the motion of that which follows; so that the sum will be the same as before. If the bodies meet with contrary motions, there will be an equal deduction from the motions of both; and therefore the difference of the motions directed toward opposite parts will remain the same..."

I do not believe I am fantasizing that this is an explanation of conservation of momentum. If Newton were generally waving his hand simultaneously in the direction of "motion, speed, velocity, momentum", as you claim, how could he accidentally end up understanding and specifically describing conservation of momentum here?

As I said, when he means motion as 'displacement in space' he almost always qualifies it such that you understand that's what he means, by saying, for example: "uniform motion in a right line".

It was even better for him, fabulous, because he naively thought that both Energy and velocity/momentum are proportional to force: Fg → Ek→ v,p

I am not aware he ever spoke about energy, and he certainly wasn't confused about the difference between velocity and momentum:

From Law III:

"If a body impinge upon another, and by its force change the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, toward the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of the bodies."

Notice he says "...not in the velocities..."

(All my quotes of Newton come from here: http://www.archive.org/stream/Newtonspmathema00newtrich#page/n77/mode/2up)

 

[prosteve037]

He specifically defines momentum (calling it "quantity of motion) in defintion II. It's completely clear he understands it is the product of the mass and velocity.

zoobyshoe, where would/do you say this understanding arises from? The previously posted proportioning approach? Or the simpler/concise way that Whewell approached it?

I believe - yes. Actually, in Whewell's times the other value measuring amount of motion was also used: kinetic energy, having even stranger property: it is preserved in ellastic collisions and in gravitational interactions, but it gots lost in inellastic collisions.

He found that such defined 'quantity of motion' is always preserved in isolated systems, so it may be very useful to formulate laws of motion.

Yes and no. He knew that maximum velocity of the pendulum is proportional to maximum deflection angle (for a given pendulum length). But he didn't care about expressing the velocity in our modern terms (m/s or inches per second). It is yet another case, where Newton, in his euclidean approach, was focused on proportions, but not on the actual numeric values. So (you did it wrong!) he could go one step further: <br /> \theta_M/\theta_m = m/M\quad\Longrightarrow\quad<br /> v_M/v_m=m/M but he never did the next step: v_M/v_m=m/M\quad\Longrightarrow\quad Mv_M=mv_m - because for him it made no sense to multiply ounces by something else than pure number. It was also a reason why Newton never mentioned any numerical values of velocity: he could define it as a 'change of position (in time)', but he had no unit to measure it. He could measure the angle, but not the velocity: it was not only technical problem, but he lacked units of measure. He could measure angles (in degrees), but he had no unit for velocity. The idea of 'metre per second' (or rather feet per second) came 50 years later.

So because the concept of momentum was considered long before energy, can we just disregard any connection between the formulation of \textit{mv} and the concept of kinetic energy? I just don't want to dive into a whole different concept and focus on momentum :P

Also, it's very interesting that you said that Newton never thought it would be reasonable to multiply two different kinds of values together (like ounces to meters per second as you said). I read somewhere else that he gave a name to this idea of his and named it something like the Principle of Similitude or something...

Anywho, it seems that we have a conflicting argument here in terms of zoobyshoe's post and xts'. From what I can see, (and please correct me if I'm wrong) while xts says that Newton had no clear-cut mathematical definition for momentum (\textit{mv}) zoobyshoe is saying that Newton knew full well that momentum had to be \textit{mv} (Hence why I ask zoobyshoe about his view on how Newton came to see it this way).

 

[zoobyshoe]

Anywho, it seems that we have a conflicting argument here in terms of zoobyshoe's post and xts'. From what I can see, (and please correct me if I'm wrong) while xts says that Newton had no clear-cut mathematical definition for momentum (\textit{mv}) zoobyshoe is saying that Newton knew full well that momentum had to be \textit{mv} (Hence why I ask zoobyshoe about his view on how Newton came to see it this way).

I've greatly enjoyed all of xts' posts in this thread and wouldn't want to think there was any conflict.

My reasoning goes like this: given the proportional relationships he described, he couldn't be talking about anything other than momentum when he uses the term "quantity of motion".

Definition II

The quantity of motion is the measure of the same arising from the velocity and quantity of matter conjunctly.

The motion of the whole is the sum of all of the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple.

The only relationship where doubling the mass but keeping the velocity the same results in a doubling of an unknown third thing (quantity of motion?) is multiplication. If that statement ( "in a body double in quantity, with equal velocity, the motion is double") is true, he must have arrived at the original (undoubled) "quantity of motion" by multiplying the mass times the velocity. Addition, subtraction, division would not result in the same conclusion.

You may well ask what units of mass and velocity he would have used. He wouldn't need fixed units, just "parts". 2parts mass times 3parts velocity = 6parts motion. Let's double the mass: 4parts mass times 3parts velocity = 12parts motion. The relationship: "in a body double in quantity, with equal velocity, the motion is double" holds true. To find how many "parts" a mass has he has to compare it to another mass. If mass A is twice mass B (a fact he could discover by weighing them) then mass A is two parts of mass and mass B is one part of mass. The same holds for velocity. Given any two pendula he can describe the mass or velocity of both in terms of the number of "parts" they sum up to. One will have so many "parts" of the total mass or velocity, and the other will have what's left over of the total.

All that being the case, he could well have said, "quantity of motion is collected by multiplying the number of parts of quantity of matter by the number of parts of velocity".

Following his logic requires squarely facing what xts asserted earlier:

At Newton's times proportionality was perfectly known and understood - it was used even more frequently than nowadays. Just contrary - those times the numerical values, measurement units, etc. were rather neglected. It was the effect of geometrical representation. As in Euclid "Elements" - the line segment represent the value, and twice longer section represent twice bigger value, but no one cares to say that 1 inch on the drawing represents 1 pound of mass. You may make the same Euclidean construction in different scale, and all conclusions will be the same.
Such approach is really difficult to understand for modern people, who learn on numbers, rather than on Euclodean constructions.
Newton was one of the very last scientists using such geometrical representations (but it was common till his times, Copernicus did the same). It was 18th century when numerical approach (started by Rene Descartes even a bit earlier than Newton worked) finally won popularity.

I would add that Galileo worked everything out geometrically, as well.

Descartes was new. I am currently reading a bio of Newton in which the author asserts that the concept of Cartesian coordinates was vital to Newton in the development of his "fluxions", which would put him in the vanguard of those who caused the shift of emphasis from geometry to algebra.

So because the concept of momentum was considered long before energy, can we just disregard any connection between the formulation of mv and the concept of kinetic energy?

As far as I can see Newton worked everything out in terms of conservation of momentum, and there's no indication so far in my reading that kinetic energy will be brought into it.

 

[PatrickPowers]

I have read that Newton worked out his mathematics using algebra, analytic geometry, and calculus, then translated it all to the then-more-familiar geometry so it could be understood and accepted.

 

[D H]

I have read that Newton worked out his mathematics using algebra, analytic geometry, and calculus, then translated it all to the then-more-familiar geometry so it could be understood and accepted.

"Citation needed."

 

[PatrickPowers]

"Citation needed."

By the help of the new Analysis Mr. Newton found out most of the Prepositions in his Principia Philosophiae: but because the Ancients for making things certain admitted nothing into Geometry before it was demonstrated synthetically, he demonstrated the Propositions synthetically, that the System of the Heavens might be founded upon good Geometry. And this makes it now difficult for unskilful men to see the Analysis by which these propositions were found out.

--- Sir Isaac Newton, 1715

Our specious algebra is fit to find out, but entirely unfit to consign to writing and commit to posterity. -- Sir Isaac Newton, 1694.

 

[zoobyshoe]

By the help of the new Analysis Mr. Newton found out most of the Prepositions in his Principia Philosophiae: but because the Ancients for making things certain admitted nothing into Geometry before it was demonstrated synthetically, he demonstrated the Propositions synthetically, that the System of the Heavens might be founded upon good Geometry. And this makes it now difficult for unskilful men to see the Analysis by which these propositions were found out.

--- Sir Isaac Newton, 1715

Newton is talking about himself in the third person here? What's this quote from? A letter? A publication of the Royal Society?

 

[PatrickPowers]

Newton is talking about himself in the third person here? What's this quote from? A letter? A publication of the Royal Society?

Third person, yes. A letter. You can easily find this and all of his letters online. He also wrote that algebra was for "bunglers."

 

[zoobyshoe]

Third person, yes. A letter. You can easily find this and all of his letters online. He also wrote that algebra was for "bunglers."

Any way you could just link me to these two letters?

It surprises me because he refers to himself in the first person in the Principia.

 

[zoobyshoe]

For anyone who still doubts Newton understood that momentum was the product of mass times velocity, I found a passage where he directly says it is:

...we are to take the product (if I may so say) of the body A, by the chord of the arc TA (which represents its velocity), that we may have its motion in the place A immediately before reflexion; and then by the chord tA, that we may have its motion in the place A immediately after reflexion. And so we are to take the product of the body B by the chord of the arc Bl, that we may have the motion of the same immediately after reflexion. And in like manner, when two bodies are let go from different places, we are to find the motion of each, as well before as after reflexion; and then we may compare the motions between them, and collect the effects of reflexion.

http://www.archive.org/stream/Newtonspmathema00newtrich#page/n95/mode/2up Page 91

Terminology: I've already demonstrated the word "motion" is to be understood as "momentum".

"Body" means "mass", which we learn from the very first definition:

It is this quantity that I mean hereafter everywhere under the name of body or mass.

http://www.archive.org/stream/Newtonspmathema00newtrich#page/n77/mode/2up

The chord that represents velocity is a chord drawn from the point to which you pull the pendulum back before you let it go, to its bottom dead center position. The length of that chord represents the magnitude of the pendulums velocity when it is going fastest i.e. when it is swinging through that bottom dead center position. Newton says this is well known,

For it is a proposition well known to geometers, that the velocity of a pendulous body in the lowest point is as to the chord of the arc which it has described in its descent.

http://www.archive.org/stream/Newtonspmathema00newtrich#page/n95/mode/2up Page 90

So, by saying that the momentum is the product of the mass times the velocity (or vectorial representation of the magnitude of the velocity) he is saying ρ=mv.

 

[prosteve037]

Thanks for the direct links zoobyshoe! Great post! :]

The only relationship where doubling the mass but keeping the velocity the same results in a doubling of an unknown third thing (quantity of motion?) is multiplication. If that statement ( "in a body double in quantity, with equal velocity, the motion is double") is true, he must have arrived at the original (undoubled) "quantity of motion" by multiplying the mass times the velocity. Addition, subtraction, division would not result in the same conclusion.

So then did Newton identify the "unknown third thing" as the quantity of motion because there was no other similar quantity between the two?

I just wanted to explicitly clarify because you used "unknown third thing" as if to indicate that Newton merely declared this unknown third quantity as a body's quantity of motion, without noting the comparability between two bodies with said attributes (1 body w/ double the mass of the other, equal velocities).

I think it's safe to assume though, that what you're saying is what I was inferring; that Newton knew that the only thing that the two bodies had in common were their mass-velocity products, which had to denote the bodies' quantities of motion.

You may well ask what units of mass and velocity he would have used. He wouldn't need fixed units, just "parts". 2parts mass times 3parts velocity = 6parts motion. Let's double the mass: 4parts mass times 3parts velocity = 12parts motion. The relationship: "in a body double in quantity, with equal velocity, the motion is double" holds true. To find how many "parts" a mass has he has to compare it to another mass. If mass A is twice mass B (a fact he could discover by weighing them) then mass A is two parts of mass and mass B is one part of mass. The same holds for velocity. Given any two pendula he can describe the mass or velocity of both in terms of the number of "parts" they sum up to. One will have so many "parts" of the total mass or velocity, and the other will have what's left over of the total.

All that being the case, he could well have said, "quantity of motion is collected by multiplying the number of parts of quantity of matter by the number of parts of velocity".

Are these "parts" of a whole, intrinsic to each? Or are they treated as being synonymous to units?

 

[zoobyshoe]

So then did Newton identify the "unknown third thing" as the quantity of motion because there was no other similar quantity between the two?

I just wanted to explicitly clarify because you used "unknown third thing" as if to indicate that Newton merely declared this unknown third quantity as a body's quantity of motion, without noting the comparability between two bodies with said attributes (1 body w/ double the mass of the other, equal velocities).

No, the third thing is not unknown to Newton, it is unknown to us. We are trying to decipher Newton's use of language, because there's a confusing mystery. He shifts from speaking of "quantity of motion" to just speaking about "motion". We start out with a list of three known things: mass, velocity, quantity of motion. Suddenly he says "motion", and not "quantity of motion". The word "motion" now becomes an unknown third thing. Unknown to us, not Newton. It could well be he's now referring to a different concept. Maybe, for example, he's shifted to make a point about displacement in space. Its meaning is, for the moment, unknown to us. Its an "unknown third thing", unknown to us, that is. How do we figure out what it might mean? We look at the relationship he describes, and which he now calls by the word "motion" and we realize he's still talking about "quantity of motion". The word "motion", therefore, in the context of the Principia, can often be understood as short for "quantity of motion".

That's important to realize if someone wants to read the Principia. When you read the word "motion" you should always suspect he means "momentum", and try that meaning out to see if it fits the context in which it's used. Rule of thumb: when he does not mean momentum he almost always qualifies the word "motion" with something like "uniform motion in a right line".

At any rate, my post #61 supercedes all previous posts because it reports my discovery of a passage where he directly says "motion" (momentum) is the product of mass and velocity.

I think it's safe to assume though, that what you're saying is what I was inferring; that Newton knew that the only thing that the two bodies had in common were their mass-velocity products, which had to denote the bodies' quantities of motion.

What he knew, and what exited him, was that the mass-velocity product was a conserved quantity. He didn't discover this, it was an insight that arose from the experiments with pendulums of the members of the Royal Society:

By the same, together with the third Law, Sir Christ. Wren, Dr. Wallis, and Mr. Huygens, the greatest geometers of our times, did severally determine the rules of the congress and reflexion of hard bodies, and much about the same time communicated their discoveries to the Royal Society, exactly agreeing among themselves as to those rules. Dr. Wallis, indeed, was something more early in the publication; then followed Sir Christopher Wren, and lastly, Mr. Huygens. But Sir Christopher Wren confirmed the truth of the thing before the Royal Society by the experiment of pendulums, which Mr. Mariotte soon after thought fit to explain in a treatise entirely upon the subject.

By "the rules of the congress and reflexion of hard bodies" he means "conservation of momentum in elastic collisions". Newton repeated many of those experiments himself with many different pendulums of different lengths and different materials.

Are these "parts" of a whole, intrinsic to each? Or are they treated as being synonymous to units?

Not sure what you're asking. Do this: make a mixture of 2 parts water to 7 parts flour by volume. You may literally do this or do it as a gedanken. If you're at a loss as to where to start, then let me explain that "parts" is short for "equal parts". Since I specified that the parts must be equal in volume (as opposed to say, weight), the recipe is asking for two volumes of water to be mixed with 7 volumes of flour. What "volume" do you use? The recipe doesn't say. The only stipulation, which is an implicit one, is that you use the same volume for each "part". If you use a "cup" you use that same cup to measure out all the parts. You fill it up once with water, pour that into your mixing bowl, then you fill it again with water and pour that into your mixing bowl. There's your two "parts" of water. Do the same with the same cup for the flour, but do it 7 times. There's your 7 "parts" of flour. You could do the same thing using a teaspoon as your volume, or you could use a 55 gallon drum as your volume. The proportion of equal parts is what matters.

So, that being explained, I don't know how to link it to the particular questions you asked.

 

[prosteve037]

No, the third thing is not unknown to Newton, it is unknown to us. We are trying to decipher Newton's use of language, because there's a confusing mystery. He shifts from speaking of "quantity of motion" to just speaking about "motion". We start out with a list of three known things: mass, velocity, quantity of motion. Suddenly he says "motion", and not "quantity of motion". The word "motion" now becomes an unknown third thing. Unknown to us, not Newton. It could well be he's now referring to a different concept. Maybe, for example, he's shifted to make a point about displacement in space. Its meaning is, for the moment, unknown to us. Its an "unknown third thing", unknown to us, that is. How do we figure out what it might mean? We look at the relationship he describes, and which he now calls by the word "motion" and we realize he's still talking about "quantity of motion". The word "motion", therefore, in the context of the Principia, can often be understood as short for "quantity of motion".

That's important to realize if someone wants to read the Principia. When you read the word "motion" you should always suspect he means "momentum", and try that meaning out to see if it fits the context in which it's used. Rule of thumb: when he does not mean momentum he almost always qualifies the word "motion" with something like "uniform motion in a right line".

At any rate, my post #61 supercedes all previous posts because it reports my discovery of a passage where he directly says "motion" (momentum) is the product of mass and velocity.

What he knew, and what exited him, was that the mass-velocity product was a conserved quantity. He didn't discover this, it was an insight that arose from the experiments with pendulums of the members of the Royal Society:


By "the rules of the congress and reflexion of hard bodies" he means "conservation of momentum in elastic collisions". Newton repeated many of those experiments himself with many different pendulums of different lengths and different materials.

Not sure what you're asking. Do this: make a mixture of 2 parts water to 7 parts flour by volume. You may literally do this or do it as a gedanken. If you're at a loss as to where to start, then let me explain that "parts" is short for "equal parts". Since I specified that the parts must be equal in volume (as opposed to say, weight), the recipe is asking for two volumes of water to be mixed with 7 volumes of flour. What "volume" do you use? The recipe doesn't say. The only stipulation, which is an implicit one, is that you use the same volume for each "part". If you use a "cup" you use that same cup to measure out all the parts. You fill it up once with water, pour that into your mixing bowl, then you fill it again with water and pour that into your mixing bowl. There's your two "parts" of water. Do the same with the same cup for the flour, but do it 7 times. There's your 7 "parts" of flour. You could do the same thing using a teaspoon as your volume, or you could use a 55 gallon drum as your volume. The proportion of equal parts is what matters.

So, that being explained, I don't know how to link it to the particular questions you asked.

I think I may have inquired the wrong question to you through my last post :P Apologies!

I understand and agree with your argument from your post completely. However, I feel I may have unintentionally steered attention away from my main, underlying question; regarding how \textit{mv} was declaratively chosen to be the definition of momentum.

With that said I did ask you and xts about your thoughts on this, namely on your opinions of the method by which \textit{mv} was conceptualized (either through proportioning techniques or the "simpler logic" I referred to in an earlier post). From what I understand, the consensus so far (between xts, you, and I) is that, yes, \textit{mv} was conceived through a simpler logic. Now I guess the next "section" to my question is "What were the steps taken in this 'simple logic'?".

I've read that Descartes was the one to propose \textit{mv} as a quantity for motion, having reasoned that 2 bodies with masses \textit{m} and \textit{2m} and respective velocities \textit{v} and \textit{2v}, will move another body the same amount. And following the reasoning you described, multiplication would be the only appropriate operation in which this holds.

With all that said, I hope we're on the same page

Again, in regards to your post zoobyshoe, I never meant to ask on the context of "quantity of motion"/"motion" and I'm sorry if that confused you or anything :P

 

[zoobyshoe]

I think I may have inquired the wrong question to you through my last post :P Apologies!

I understand and agree with your argument from your post completely. However, I feel I may have unintentionally steered attention away from my main, underlying question; regarding how \textit{mv} was declaratively chosen to be the definition of momentum.

With that said I did ask you and xts about your thoughts on this, namely on your opinions of the method by which \textit{mv} was conceptualized (either through proportioning techniques or the "simpler logic" I referred to in an earlier post). From what I understand, the consensus so far (between xts, you, and I) is that, yes, \textit{mv} was conceived through a simpler logic. Now I guess the next "section" to my question is "What were the steps taken in this 'simple logic'?".

I've read that Descartes was the one to propose \textit{mv} as a quantity for motion, having reasoned that 2 bodies with masses \textit{m} and \textit{2m} and respective velocities \textit{v} and \textit{2v}, will move another body the same amount. And following the reasoning you described, multiplication would be the only appropriate operation in which this holds.

With all that said, I hope we're on the same page

Again, in regards to your post zoobyshoe, I never meant to ask on the context of "quantity of motion"/"motion" and I'm sorry if that confused you or anything :P

OK. Yes, I did not understand what you were asking.

Looking back through the thread I find you asking for an explanation of where p=mv comes from to begin with. In other words, who first looked at motion and decided there was an important aspect of it different from mere mass at a given velocity, not just mass + velocity, but mass x velocity.

I don't know. No one is credited with this in the Principia. The wiki article on momentum says it was a notion that cropped up here and there throughout history without taking root or leading anywhere till Newton wrote it down. My guess is that specifically focusing on it as an important concept arose from the combined experiments and papers of the members of the Royal Society who experimented with pendulums before Newton. Newton defines it confidently, and fairly briefly, in his opening definitions, almost as if he takes it to be self evident.

Most physics texts introduce you to it the same way. It's pretty much evident that, given two masses at the same velocity, the larger will have more of something important. Likewise it's evident that the difference between throwing a bullet at someone and shooting it at them is drastic. Somehow that difference has to be quantifiable, measurable. I'm betting we'd find a detailed discussion of that in the works of Wallis, Wren, or one of the other pendulum experimenters Newton names.

So, 10 kgs at 10m/s = 20kgs at 5m/s = 50 kgs at 2 m/s. It is a proportional consideration, and the realization it is conserved is really what makes it important.

 

[prosteve037]

OK. Yes, I did not understand what you were asking.

Looking back through the thread I find you asking for an explanation of where p=mv comes from to begin with. In other words, who first looked at motion and decided there was an important aspect of it different from mere mass at a given velocity, not just mass + velocity, but mass x velocity.

I don't know. No one is credited with this in the Principia. The wiki article on momentum says it was a notion that cropped up here and there throughout history without taking root or leading anywhere till Newton wrote it down. My guess is that specifically focusing on it as an important concept arose from the combined experiments and papers of the members of the Royal Society who experimented with pendulums before Newton. Newton defines it confidently, and fairly briefly, in his opening definitions, almost as if he takes it to be self evident.

Most physics texts introduce you to it the same way. It's pretty much evident that, given two masses at the same velocity, the larger will have more of something important. Likewise it's evident that the difference between throwing a bullet at someone and shooting it at them is drastic. Somehow that difference has to be quantifiable, measurable. I'm betting we'd find a detailed discussion of that in the works of Wallis, Wren, or one of the other pendulum experimenters Newton names.

So, 10 kgs at 10m/s = 20kgs at 5m/s = 50 kgs at 2 m/s. It is a proportional consideration, and the realization it is conserved is really what makes it important.

Yes, thank you. This was the kind of reply that I was hoping to see

Though it's not what I wanted to hear, I agree that the topic was vaguely put forward as if it was indeed self-evident.

I Google'd to see if I could find any documents or readings on the matter involving Wren, Huygens, and Wallis and found a book that hinted at the location of \textit{mv}'s origin.

If you click the hyperlink you can see where it says this, at the bottom-left side of the page. It hints that the 1669 issue of Philosophical Transactions of the Royal Society may have the works/writings of the pendulum experimenters' findings.

Naturally I looked for an English translation of this issue, but sadly to no avail. I was only able to find the original Latin version. https://docs.google.com/open?id=0B8alG534jxFGYzBlNjAxMWEtZDg1NC00YTc3LTgyYzMtM2ViNjljZDUwYWI2's the segment of the issue that Huygens wrote if anyone wants to see it. There you can see a demonstration of the geometric methods that were used for calculations.

It would be great though if an English translation of this publication was available. That or if there was an article that explained the formula's origin more thoroughly :P

 

[zoobyshoe]

Yes, thank you. This was the kind of reply that I was hoping to see

Excellent. I'm glad we're now on the same page.

Though it's not what I wanted to hear, I agree that the topic was vaguely put forward as if it was indeed self-evident.

I Google'd to see if I could find any documents or readings on the matter involving Wren, Huygens, and Wallis and found a book that hinted at the location of \textit{mv}'s origin.

If you click the hyperlink you can see where it says this, at the bottom-left side of the page. It hints that the 1669 issue of Philosophical Transactions of the Royal Society may have the works/writings of the pendulum experimenters' findings.

I read the whole chapter of that book and it looks like a great history of the subject of momentum. It's very interesting to me that William of Occam, of Occam's Razor fame, was one of the early contemplaters of the subject. It's clear that Descartes got closest to our present understanding before the Royal Society took it up and, collectively, made all the final corrections. Newton received and reports it as a vector quantity, and not a Cartesian scalar. Your book also restores some credit to Hooke for his experiments, credit that Newton expunged from the Principia in his recap of the Societies' research (which I quoted earlier). Your book, by the way, relates that Huygens, at the same time, but independently of the rest, began to work out the concept of conservation of energy, which should be of interest to you because you seem to favor analyzing things that way if possible. I wonder if you haven't already read forward into the chapter on conservation of energy to see how that played out.

Naturally I looked for an English translation of this issue, but sadly to no avail. I was only able to find the original Latin version. https://docs.google.com/open?id=0B8alG534jxFGYzBlNjAxMWEtZDg1NC00YTc3LTgyYzMtM2ViNjljZDUwYWI2's the segment of the issue that Huygens wrote if anyone wants to see it. There you can see a demonstration of the geometric methods that were used for calculations.

It would be great though if an English translation of this publication was available. That or if there was an article that explained the formula's origin more thoroughly :P

I took latin in high school over 35 years ago and took a stab at this with a Latin-English dictionary. I have just forgotten too much, though.



I can tell you that the first part of that article is a tedious history of who in the society made what contribution when to the experiments and the understanding. Then we get down specifically to Huygen's Regulae de Motu Corporum ex mutuo impulsu - "Rules of the Motion of Bodies after striking against each other" (I'm pretty sure).

Rule #1 seems to be: "If a hard body runs into any other equal hard body at rest, that hard body will come to rest after contact and, at the same time, however, the one that was at rest will acquire the speed of the one that struck against it."

That is the only one I'm reasonably sure of. All the rest have elements I'm not sure how to tackle, so I'm only grasping fragments.

You're quite right that the chart is a geometric analysis. Fusing the fragments of the explanation I could make out with what I learned of their methods from Newton, I see that each line represents a specific, illustrative case of an interaction between two bodies designated A and B. The speed of body A is always represented by the length of the segment AD and segment BD the speed of body B (There's some discussion of different cases this applies to, but I can't sort it out. Vel...vel seems to mean "either...or"). Point C in each case, represents the "center of weight" (centrum gravitatis) of the two bodies. (In other words, the lengths of segments AC and BC represents the relative proportions of their masses.) With points A, B, C, and D established, he introduces a point E. Point E is always relative to point C, such that segment EC will be equal in length to segment DC (if you look down the chart you'll see EC is always equal to DC in length, though it's hard to sort that out in some cases because the label E or D is crowded right together with A or B, the points coinciding, I guess.) Segments AE and BE, are the results it was intended to find, the final speeds of A and B: "I say: EA has (represents) the speed of body A after collision, EB that of body B..." The final direction is also indicated, but I can't make out exactly how.

I know a girl who is a medieval scholar who reads Latin (expert level: she teaches medieval studies at UCLA). Unfortunately she is on and extended tour of Russia and Eastern Europe, and I'd rather wait till she gets back before I show this to her. In the meantime, I might dig someone else up who can certainly do a better job than me.

I accidentally and incidentally ran across another interesting tidbit which says something about the history of Newton II as he actually wrote it vs. F=ma, which I'll post later.

 

[sahil_time]

Force was not derived. It is a new quantity, and F=kma or F=ma will not
make any difference to solving mechanics or work energy or anything
for that matter because Force being the most fundamental. Work, Power,
Impulse etc etc are derived from force. You can re-create whole mechanics
with F=kma as much as you can create with F=ma.

 

[prosteve037]

Excellent. I'm glad we're now on the same page.


I read the whole chapter of that book and it looks like a great history of the subject of momentum. It's very interesting to me that William of Occam, of Occam's Razor fame, was one of the early contemplaters of the subject. It's clear that Descartes got closest to our present understanding before the Royal Society took it up and, collectively, made all the final corrections. Newton received and reports it as a vector quantity, and not a Cartesian scalar. Your book also restores some credit to Hooke for his experiments, credit that Newton expunged from the Principia in his recap of the Societies' research (which I quoted earlier). Your book, by the way, relates that Huygens, at the same time, but independently of the rest, began to work out the concept of conservation of energy, which should be of interest to you because you seem to favor analyzing things that way if possible. I wonder if you haven't already read forward into the chapter on conservation of energy to see how that played out.


I took latin in high school over 35 years ago and took a stab at this with a Latin-English dictionary. I have just forgotten too much, though.



I can tell you that the first part of that article is a tedious history of who in the society made what contribution when to the experiments and the understanding. Then we get down specifically to Huygen's Regulae de Motu Corporum ex mutuo impulsu - "Rules of the Motion of Bodies after striking against each other" (I'm pretty sure).

Rule #1 seems to be: "If a hard body runs into any other equal hard body at rest, that hard body will come to rest after contact and, at the same time, however, the one that was at rest will acquire the speed of the one that struck against it."

That is the only one I'm reasonably sure of. All the rest have elements I'm not sure how to tackle, so I'm only grasping fragments.

You're quite right that the chart is a geometric analysis. Fusing the fragments of the explanation I could make out with what I learned of their methods from Newton, I see that each line represents a specific, illustrative case of an interaction between two bodies designated A and B. The speed of body A is always represented by the length of the segment AD and segment BD the speed of body B (There's some discussion of different cases this applies to, but I can't sort it out. Vel...vel seems to mean "either...or"). Point C in each case, represents the "center of weight" (centrum gravitatis) of the two bodies. (In other words, the lengths of segments AC and BC represents the relative proportions of their masses.) With points A, B, C, and D established, he introduces a point E. Point E is always relative to point C, such that segment EC will be equal in length to segment DC (if you look down the chart you'll see EC is always equal to DC in length, though it's hard to sort that out in some cases because the label E or D is crowded right together with A or B, the points coinciding, I guess.) Segments AE and BE, are the results it was intended to find, the final speeds of A and B: "I say: EA has (represents) the speed of body A after collision, EB that of body B..." The final direction is also indicated, but I can't make out exactly how.

I know a girl who is a medieval scholar who reads Latin (expert level: she teaches medieval studies at UCLA). Unfortunately she is on and extended tour of Russia and Eastern Europe, and I'd rather wait till she gets back before I show this to her. In the meantime, I might dig someone else up who can certainly do a better job than me.

I accidentally and incidentally ran across another interesting tidbit which says something about the history of Newton II as he actually wrote it vs. F=ma, which I'll post later.

Great post, this was immensely helpful and informative! Thanks a bunch zoobyshoe!

Well I've been digging deeper since my last post and I think I'm finally getting warmer and warmer to a (possibly) definite answer.

In accordance with your post, this says that Descartes had introduced the modern notion of momentum, basing his theory on the context of collisions.

However it also shows (pg. 105) what Galileo had written before Descartes, saying how the weight × velocity of one body is equal to the weight × velocity of another body in a certain case. Algebraically:

\textit{p = } Weight

\textit{v = } Velocity

\frac{p_1}{p_2}\textit{ = }\frac{v_2}{v_1} → \textit{p}_1{v_1}{ = p_2}{v_2}

I don't understand exactly what case this pertains to as it is vaguely stated in the book; the book says that it pertained to a specific case regarding an "oscillating balance" (pg. 106). It is also noted too that Galileo's definition of momentum is quite confusing since he uses momentum, power, and force in the same context :P

I think it's important to realize here too that Galileo was the first to declare the quantity "\textit{pv}" as the representation of a body's quantity of motion. This, to me, is fundamental because although the relation was only present in the specific case of equilibrium (with the "oscillating balance"), he was able to logically deduce that this quantity "\textit{pv}" must represent the only similar property between the two bodies involved; their "quantities of motion", as defined by Galileo. Though it's obviously not \textit{mv}, it's the first fundamental step that was taken in the right direction.


With that said, I think it's safe to say that Galileo's conception of "quantity of motion" was either ill-defined, proven right with the wrong experiment, or a combination of the two. Though he had a formula created, what exactly was he trying to measure? This is unclear, but I'm sure if I understood the context of his formulation with respect to the experiment and the data he collected, the formula would make sense.

At the same time, however, I think Galileo was the one who completed all the required algebra (assuming he used algebra) needed to arrive at the correct quantification of motion; he was just measuring in a context outside of collisions (it says on pg. 106 that momentum was viewed in the context of collisions only after Descartes).

What are your thoughts on this, zoobyshoe?

 

[zoobyshoe]

Great post, this was immensely helpful and informative! Thanks a bunch zoobyshoe!

You're very welcome!

What are your thoughts on this, zoobyshoe?

Allow me to read that link over a few times and cogitate.

In the meantime, this is the tidbit I found that I thought was interesting:

I picked up a book of Poe stories last week. In one of them, The Purloined Letter, one of the characters is trying to analyze the limits of the intelligence of the Police detective he's just spoken with. He draws an analogy between a massive body and the policeman's mind:

"The principle of the vis inertiae, for example, seems to be identical in physics and metaphysics. It is not more true in the former, that a large body is with more difficulty set in motion than a smaller one, and that its subsequent momentum is commensurate with this difficulty, than it is, in the latter, that intellects of the vaster capacity, while more forcible, more constant, and more eventful in their movements than those of inferior grade, are yet the less readily moved, and more embarrassed, and full of hesitation in the first few steps of their progress."

What I noticed right away is that this character speaks of the overcoming of inertia as resulting in momentum, rather than acceleration. The story was written in 1844, over a hundred years after Newton's death. It's likely Poe's formal education in mechanics happened during his brief time at West Point, around 1830-31, which means, at that time (a hundred years after Newton), Newton II was not yet generally identified as F=ma. It was still being taught as Newton wrote it. I'd like to find out when the transition started and when we could consider it complete.

 

[zoobyshoe]

However it also shows (pg. 105) what Galileo had written before Descartes, saying how the weight × velocity of one body is equal to the weight × velocity of another body in a certain case. Algebraically:

\textit{p = } Weight

\textit{v = } Velocity

\frac{p_1}{p_2}\textit{ = }\frac{v_2}{v_1} → \textit{p}_1{v_1}{ = p_2}{v_2}

I don't understand exactly what case this pertains to as it is vaguely stated in the book; the book says that it pertained to a specific case regarding an "oscillating balance" (pg. 106). It is also noted too that Galileo's definition of momentum is quite confusing since he uses momentum, power, and force in the same context :P

…two weights of unequal size are in equilibrium with each other, and have equal moment, whenever their gravities are in the inverse ratio to the velocities of their motion".

What Galileo is talking about is very interesting. I'll go into it in detail. It's a mathematical device that goes back to the Greeks. Archimedes lays it out it in his Treatise, On the Equilibrium of Planes.

Some preliminary explanation is needed:

A "plane" as discussed by Archimedes is defined by Euclid in The Elements, Book 7. But, before he defines a plane, he must first define multiplication, in definition 15:

A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.

Multiplication having been rigorously defined, he then defines plane in definition 16:

And, when two numbers having multiplied one another make some number, the number so produced is called plane and its sides are the numbers which have multiplied one another

This is important: to the students of Euclid, including all from Archimedes up to and including the members of the Royal Society, a plane was a rectangle (or square) that represented a multiplication. The length of its sides represented the magnitudes of the two numbers multiplied, and its area the product.

This is why we have Mariotte representing momentum ("quantity of Motion") this way at the bottom of p. 106 and top of page 107, of your current link, as a rectangle. He's representing it as a Greek "Plane". (That seems to confuse the author of that book, who seems to think its some idiosyncratic thing Mariotte invented himself.)

Back to Archimedes:

Archimedes begins his book discussing normal, physical balances (by which I mean devices for weighing things) and then demonstrates the same principles can be used to "weigh" abstract things, like the products of multiplications, or planes. In particular, he's searching for cases where planes can be said to be in equilibrium.

Speaking of physical balances he says:

Unequal weights will balance at unequal distances [from the fulcrum], the greater weight being at the lesser distance

Of planes he says:

Two magnitudes...balance at distances reciprocally proportional to the magnitudes

Which should sound familiar. It's the same as Galileo's statement above, in different guise. Galileo has substituted velocity times weight for distance times magnitude.

This implied that a form of equilibrium existed between two bodies on an oscillating balance, when the products of the numbers representing their weights and velocities were equal. This became widely recognized.

When a balance is in equilibrium it will oscillate when pushed (sometimes it seems impossible to get them to stop oscillating, which you may know if you've ever weighed things on a balance). In the case of a balance in equilibrium with unequal weights and, consequently, with arms of unequal length, the velocities of the ends of the two arms will, of course, not be the same: they are covering unequal distances as they oscillate up and down, in exactly the same time. The ratios of those velocities to the weights is in the inverse proportion, as Galileo says, at the same time the weights will be in the inverse ratio of the distances. The greater weight will move the slower simply because it's on the shorter arm, and will consequently cover less distance in any given time interval. If the greater weight is twice the other (2/1), it's velocity will be the inverse, 1/2 of the smaller weight's velocity, etc.

Newton and all his pals were equally aware of this principle (both weight times distance and weight times velocity), having received it from the Greeks:

So those weights are of equal force to move the arms of a balance; which during the play of the balance are reciprocally as their velocities upwards and downwards...

http://www.archive.org/stream/Newtonspmathema00newtrich#page/n97/mode/2up

and on the next page:

The power and use of machines consists only in this, that by diminishing the velocity we may augment the force, and the contrary…

That latter statement about augmenting force should sound familiar. Should ring a bell from way back when you learned about simple machines, the function of all of which, is to multiply force. They can all do this, but at the expense of something else: the augmented force cannot be applied for the same distance or at the same speed as the smaller input force. You can lift a huge stone with a small force and a lever, but you cannot lift it at the same speed the small force moves the long arm of the lever, nor can you lift it as far as the distance moved by the long arm. The "plane" of the input = the "plane" of the output. By multiplying force you decrease both distance and speed.

Galileo, Archimedes, and Newton here, are all talking about: The Law of the Lever

http://math.nyu.edu/~crorres/Archimedes/Lever/LeverLaw.html

The Geometers, incidentally, make no distinction between a balance and a lever. When calculating for a lever they looked for equilibrium. Having found that, they knew anything beyond that would move the object.

Archimedes is more interested in the abstract uses of this law than in the physical, mechanical uses, and uses it as a mathematical device to perform non-physical things. (In a different book, clever man that he was, he uses a lever to measure the area of a parabola. But that's a different story.)

So, that's what Galileo is talking about. Here again, the author of the book at your link seems not to be aware of Euclid/Archimedes, which puts him at a disadvantage in discussing people who knew them well.

What are your thoughts on this, zoobyshoe?

Galileo certainly knew the Law of the Lever, and he knew that, given two equal bodies in motion, the faster had more of something important than the slower, but he did not, apparently, think to represent that fact as a "plane" which could be connected to the conservation of weight x velocity that he recognized in the LotL. It didn't seem to occur to him that that "more of something important", as I put it, might be quantified as weight times velocity. He doesn't seem to have connected the Law of the Lever with individual bodies in motion or collisions. He also doesn't seem to have sorted mass out from weight.

What may (or may not) be original to Galileo is the notion of restating the LotL as velocity x weight, though it's clear from the quote at my link the Aristotelians were aware of the velocity difference between the two arms. By Archimedes using the term "magnitude" he allows for either weight or velocity to be multiplied by distance. Galileo saw that weight times velocity works, too.

"Absolutely equal weights, moving with unequal velocities, have unequal powers, virtues, momenti, the most powerful is the one which is most rapid…

I dug up this other book which shows that Galileo is not using "powers, virtues, moments" as interchangeable. They each mean something specific. "Momenti" doesn't refer to "momentum" but to "moments", as in "moment of force". When Galileo says "momenti" here:

…two weights of unequal size are in equilibrium with each other, and have equal momenti whenever their gravities are in the inverse ratio to the velocities of their motion".

he is saying they have equal moments of force about the axis of rotation, they produce equal torque on the lever arms.

Despite that being specific, his terminology is ultimately vague, as this book describes in detail:

http://books.google.com/books?id=CZ...g#v=onepage&q=galileo moment of force&f=false

Your link says that Huygens specifically stated mv as "quantity of motion" in an unpublished manuscript in 1652. That predates the formation of the Royal Society by eight years, so it certainly predates the Society's experiments with pendulums. Huygens may well have arrived at it first, independently of all the British: Christopher Wren, Hooke, and Wallis who don't seem to have taken the subject up till the society was formed. Can't really say without a translation of that tedious history that precedes Huygens rules. But I'm thinking, he's the guy who first got it completely right.

 

[jetwaterluffy]

The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed. — If a force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.

-Newton.

 

[zoobyshoe]

-Newton.

Yes, but Newton received this from the collective experiments of the Royal Society members. He checked it all thoroughly by replicating their experiments for himself (see the Scholium to the chapter your quote comes from), but he didn't originate it. Between Descartes and Newton some third person saw the importance of the part about it acting "in a right line", i.e. that it was a vector, rather than the scalar Descartes proposed (D. conceived of it as mass times speed).

prosteve037 would like to find out exactly who figured this out and how.

 

[prosteve037]

A "plane" as discussed by Archimedes is defined by Euclid in The Elements, Book 7. But, before he defines a plane, he must first define multiplication, in definition 15:

Multiplication having been rigorously defined, he then defines plane in definition 16:

This is important: to the students of Euclid, including all from Archimedes up to and including the members of the Royal Society, a plane was a rectangle (or square) that represented a multiplication. The length of its sides represented the magnitudes of the two numbers multiplied, and its area the product.

This is why we have Mariotte representing momentum ("quantity of Motion") this way at the bottom of p. 106 and top of page 107, of your current link, as a rectangle. He's representing it as a Greek "Plane". (That seems to confuse the author of that book, who seems to think its some idiosyncratic thing Mariotte invented himself.)

I'm back! I'm so, so sorry for the delay; I finally finished classes for this semester and now have enough time to reply! :]

Anyways, that was a beautiful post, zoobyshoe!

Before reading your post, I had not seen the innards of Euclid's Elements nor had I a general sense of how influential the work was/is. But after reading your post I began to look to the Elements and its influence on the works of subsequent mathematics/physics works more carefully.

With that said, I've found that I have trouble understanding a few points regarding the use of figures/shapes to represent arithmetic operations, and why the definitions of some operations are as they are.

Now if my understanding is correct, Euclidean Geometry prohibited the adding/subtracting of 2 different kinds of magnitudes while allowing them to be set in proportion to one another (can't add/subtract \textit{m} to/from \textit{v} but you can set \frac{m_{1}}{m_{2}} equal to \frac{v_{2}}{v_{1}}).

But why is the latter permitted and the former not? Also, why is the "plane" defined as such? I've tried reading on this but couldn't find an explanation as to why these definitions were finalized as such. Perhaps I'm too pre-disposed into thinking in terms of real numbers...

The Geometers, incidentally, make no distinction between a balance and a lever. When calculating for a lever they looked for equilibrium. Having found that, they knew anything beyond that would move the object.

Archimedes is more interested in the abstract uses of this law than in the physical, mechanical uses, and uses it as a mathematical device to perform non-physical things. (In a different book, clever man that he was, he uses a lever to measure the area of a parabola. But that's a different story.)

This is very interesting. I'd love to read on how Archimedes did this :]

But was the Law of the Lever significant at all in helping members of the Royal Society to determine the formulas of momentum? Besides demonstrating the idea of comparing ratios of different "kinds" (ratio of mass to ratio of velocity), I don't think the Law of the Lever had any real impact on the Royal Society's efforts.

So, that's what Galileo is talking about. Here again, the author of the book at your link seems not to be aware of Euclid/Archimedes, which puts him at a disadvantage in discussing people who knew them well.

Galileo certainly knew the Law of the Lever, and he knew that, given two equal bodies in motion, the faster had more of something important than the slower, but he did not, apparently, think to represent that fact as a "plane" which could be connected to the conservation of weight x velocity that he recognized in the LotL. It didn't seem to occur to him that that "more of something important", as I put it, might be quantified as weight times velocity. He doesn't seem to have connected the Law of the Lever with individual bodies in motion or collisions. He also doesn't seem to have sorted mass out from weight.

What may (or may not) be original to Galileo is the notion of restating the LotL as velocity x weight, though it's clear from the quote at my link the Aristotelians were aware of the velocity difference between the two arms. By Archimedes using the term "magnitude" he allows for either weight or velocity to be multiplied by distance. Galileo saw that weight times velocity works, too.

I bolded that segment out because of the implication that it conveys, showing how the arithmetical restrictions applied to certain figures in Euclidean Geometry guided the conclusions that were made. Again, this begs the question "Why are arithmetical operations between certain figures restricted in Euclidean Geometry?".

I think the rest of what you say here though really encapsulates Galileo's treatment in implementing Euclidean reasoning to actualize his empirical data. I think I may have posted about this book earlier on in the thread but in "The Mathematics of Measurement: A Critical History" by John J. Roche, he talks about how Galileo advocated the use of Euclidean principles to demonstrate calculations and how his advocacy led to subsequent physicists to follow suit.