Talk:Parallelogram of force

Controversy Section

I have a problem with the whole controversy section in that it implies there is nothing wrong with the proof provided. I was reading Spivak - Mechanics, when coming across this article, and he clearly does not agree that the provided proof is valid, and implies that most others do not. Furthermore, while reading the provided citation, it doesn't appear that even the author of that article believes the proof is entirely valid (I haven't finished the last sections that provide his work). I've provided an updated link, as it appears UNC has restructured their website since that link was posted.

First, the controversy in that article doesn't even appear to be about the validity of Newton's proof, but about whether statics is simply a corollary of dynamics, or if certain statics laws are stronger than the dynamics version given by Newton. If you make it past the first couple sentences of the abstract, you'll find:

If the parallelogram law is explained statically, then the laws of statics are separate from and (in an important sense) "transcend" the laws of dynamics. Alternatively, if the parallelogram law is explained dynamically, then statical laws become mere corollaries to the dynamical laws.

Also, in his introduction (where I presume the text about the nineteenth century comes from), he claims his interest is more in the metaphysical controversy

Despite the routine treatment it receives today, the parallelogram of forces was the subject of considerable controversy throughout the nineteenth century. Its truth was unquestioned. The controversy concerned its explanation (that is to say, why it is true) and also, I shall argue, its metaphysical status.

The next couple sentences are particularly damning. Does this sentence mean that the parallelogram of forces cannot be deduced via philosophy or mathematical proof, but only by experiment, as Spivak claims?

That is, philosophy alone should not suffice to settle an essentially empirical question: What is the scientific explanation of the parallelogram law?

His next section, provides a form of the proof given on this page, as well as its proponent's view points...but these are view points tailored to whether or not it's the foundation of statics:

The standard objection to the dynamical derivation disputes not its cogency but its explanatory power – that is, whether (as its proponents contend) it contains "the most philosophical foundation for statics" (Cockle 1879, 12, characterizing Thomson and Tait 1888) or whether (as its critics charge) it is "unnatural and a defect in method" (Anon. 1829, 314).

His next couple sections, provide two statical (his word, not mine) proofs from Duchayla and Poisson, neither particularly elegant like Newton's, but you can find them in there. Then it appears he switches to metaphysics and how to show which way his initial point should go (This is about where I stopped reading his work). I believe he ends up with a statical point of view.

I will attempt to rewrite the controversy section with a better citation. Spivak provides long sections about the parallelogram law, the strength of Newton's version and a bit of why it fails mathematically, but very unconventional for a physics or even a mathematical physics book. Spivak also provides two beautiful algebraic proofs, which I will attempt to recreate as they are fairly easy to follow.

I ask that a new, better citation be added to the controversy section, if any can be found, before reverting it back to saying their is no current controversy over Newton's Corollary I.

I also believe this article should be of higher importance, but I (and Spivak) might be in the minority. This law, that resultant of two forces is the diagonal of a parallelogram, is one of the powerful, empirical axioms that are needed to support Newton's laws. It's also, in another form, one of the main laws of statics (see Hartog - Mechanics as well as the current erroneous citation in this article). And finally it led forces to having the vector structure they have today, which leads to (along with Galileo's work on the invariance of a velocity "boost"), Hamilton's quaternion and Gibb's vector mechanics, which leads to vector spaces and all of linear algebra!! (Hyperbole in talk pages doesn't need citations right?)

I'm fairly new to Wikipedia editing, but the default citation system seems odd for a Talk Page.

Spivak's book which I referenced

Spivak, Michael (2010). Mechanics I. Physics for Mathematicians. Publish or Perish, Inc. ISBN 0-914098-32-2.

There's also copies of his lecture notes that got formulated into this book available online

Spivak, Michael. Elementary Mechanics from a Mathematician's Viewpoint.

MixaNikos (talk) 21:48, 1 October 2017 (UTC)MixaNikos

Algebraic proof is utterly unclear

The "Algebraic proof" section is essentially a copy-paste from Spivak's Mechanics, and it is completely unclear what exactly the proof is (it's not much clearer to me in the original text). As written, it looks as if what is being proven is that any associative, commutative binary operation on 2d vectors must be standard vector addition, which is false. This section needs at minimum a careful statement of what is being proven. Gaiacarra (talk) 18:30, 26 May 2021 (UTC)