Which way did Newton find F = ma?

[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.