Talk:Dimensional analysis/Archive 2

"substantial mass"

 * , I am left wondering what "substance" and "substantial mass" mean precisely in Huntley's terminology. I see that you added a note about "amount of substance" in the SI, but I feel that this is likely only a coincidence of naming, as this could not even remotely be confused with "mass" (of any kind) and rather unlikely that Huntley would have been at pains to distinguish it.  I surmise that what Huntley chose to make a distinction between inertial mass and gravitational mass, which is appropriate in the context and a distinction not made even today in the SI.  If you have access to the source, it would be good if you could clarify this.  —Quondum 00:27, 18 July 2020 (UTC)

, I have Huntley's book, yes. I used the term "substantial mass" because that was in line with the terminology already in the discussion. The quantity is introduced on pg 81 of the book, where he calls it "quantity of matter", and he introduces the symbol $$M_\mu$$ as the dimension for it. Gravity is not mentioned. He then works through a fluid flow problem on pp 82-83, where he treats the dimension of mass flow rate $$m$$ as $$ M_\mu T^{-1}$$. From his discussion it is clear that in today's terminology we would call $$m$$ the "molar mass flow rate", with SI dimension $$N T^{-1}$$ and unit mole s^{-1}. It would be a good exercise to rework his example here in today's terminology, but then it would lose the direct link to his authorship, so I am not proposing to do that. But it is clear that it could be done. Recall that in SI terminology quantities measured in mole are considered "amount of substance", hence by calling Huntley's dimension $$M_\mu$$ "substantial mass" it provides a bridge to today's terminology. I wouldn't be dogmatic about terminology if someone can think of something better, but the link to the SI dimension $$N$$ is very clear so I wouldn't want to give up on that. BTW I have the original hardback edition published by Rinehart in 1951. The pages might be different in the Dover edition referred to in the main article.—antiskid56 18 Jul 2020 11:46 AM EDT


 * SI's amount of substance as a stand-alone concept is ill-defined, and is primarily of interest in the context of chemical reactions. I don't find the terminological bridge you mention strong enough to be sure of this interpretation, and so the underlying concepts and context must be looked into.  Do you think there is a short passage that we could quote to make his meaning clear, or even discuss on the talk page?  Referencing the original edition will be no problem.  —Quondum 16:22, 18 July 2020 (UTC)

If "amount of substance" were not well defined it would probably not be in an internationally accepted standard in daily use around the world. Molar masses are used in many contexts that do not require chemical reactions, such as in fluid statics and dynamics. Look at the ideal gas law - what do you see? Be that as it may, I recognize that since Huntley does not introduce atomic weight or Avogadro's number, the link between "quantity of matter" and "amount of substance" may seem tenuous. Here is what he has to say:

"In physics, mass is commonly regarded from two quite different points of view: (1) as quantity of matter and (2) as inertia. While it is true that there is a strict proportionality between these two, that does not make them identical. They are in fact quite different in nature and should therefore be differentiated by distinctive dimensional symbols. Examples so far considered have been concerned with mass as inertial, but in problems involving heat and temperature it is mass as quantity of matter that must be taken into account. In writing the dimensional formula of thermal capacity, for example, it is with mass as quantity of matter, not as inertia, that we are concerned.

"Let us make this distinction in our dimensional formulae, writing $$[M_\mu]=\text{quantity of matter}$$, $$[M_\iota]=\text{inertia}$$".

He then discusses switching his definition of specific heat capacity so that its dimension involves $$[M_\mu^{-1}]$$ rather than $$[M_\iota^{-1}]$$. Now if you look at tabulated heat capacities, you will find massic, volumic, and molar. Which is he discussing here? He already ruled out massic in the classic sense of inertial mass. He is certainly not discussing volumic - he never mentioned volume. What is left? Molar. He is defining specific molar heat capacity. He didn't mention moles because he didn't really need define the unit of "quantity of matter" to carry out his analysis, and perhaps also because the unit was probably still in the process of being defined in 1951. But it is quite certain that his "quantity of matter" is a molar quantity, corresponding identically to the SI quantity "amount of substance".

I recommend, if you feel this is not definitive, that you get the book, read it for yourself, and then write here why you have a different opinion of what he meant. You will notice, if you read it, that he is not necessarily up to date, even for 1951, in what he is writing about. For instance, when discussing electromagnetic units, he never discusses the rationalized Giorgi units, which were already well known by then, being part of the MKSA standard adopted in 1946.

I do not claim that he anticipated SI in every particular related to "quantity of matter". I do claim that SI's "amount of substance" is equivalent in every relevant particular to his "quantity of matter" and that because this is now internationally accepted as a base quantity with recognized unit and dimension in science, commerce, and technology, Huntley's innovation is in a very good position to be adopted as a standard approach in dimensional analysis. It is "mainstreamed" so to speak.

Again, I would not insist on the phrase "substantial mass", but it was already being used in the discussion, and seems reasonable, so I adopted it. I don't think it is misleading with respect to the original source material, and it does provide a bridge to the present-day SI terminology.

antiskid56 18 Jul 2020 07:24 PM EDT —Preceding undated comment added 23:57, 18 July 2020 (UTC)


 * I did qualify my statement: "as a stand-alone concept". By that I meant that without further qualification, "amount of substance" is ambiguous in the SI.  For example, given a solution of sugar in water, what is the amount of substance?  Only once the substance is specified (e.g. water molecules, sugar molecules, total molecules, total atoms, electrons, nucleons, etc.) does it become well-defined, and this is highlighted in the SI brochure.  Anyhow, this is neither here nor there.
 * Your mentioned tabulated molar heat capacities is more persuasive, though he would have to have defined a unit of amount of substance (equivalent to the mole, just not of the same size) to create such a table, unless the table listed relative values (with respect to a reference substance). He would then also have had some idea of what determined a unit in different contexts (e.g. whether it was with respect to a molecule, or the simplest sub-unit (e.g. for diamond that would have been a carbon atom rather than the entire diamond as a single molecule, and for similarly for a salt, which does not form molecules, but this leaves other ambiguities such as for ).  Relative values would have been a bit antithetical to an exposition of dimensional analysis.
 * Obtaining a copy of the book is a non-trivial exercise: they are not readily available (at least where I am), and I do not have ready access to research facilities. Trying to dissect the meaning by remote is somewhat challenging, but it will probably make sense to hang on until I happen upon a copy.  —Quondum 01:54, 19 July 2020 (UTC)

I believe you will find little more in any reading of Huntley to make the concept more specific than what I have outlined here. It seems likely to me that Huntley did not assume anything beyond having another independent dimension which is proportional to inertial mass, but does not implicate inertial properties. This may not precisely be molar mass - but it is precisely like it in the only relevant sense, which is that molar mass is proportional to inertial mass, yet does not implicate inertial properties. This is all that is required for molar mass to be used in the same way Huntley used his "quantity of matter". I do not think you can dispute this. Once molar mass (i.e. amount of substance) becomes a recognized independent quantity with unit and dimension, that accomplishes exactly what Huntley set out to do here (assuming one uses the dimension appropriately in a problem of interest). I believe this is all that really matters.Antiskid56 (talk) 03:18, 19 July 2020 (UTC)


 * The thing is that introducing the SI concept of amount of substance here makes an implication of relevance in the context, albeit obscure. We cannot make statements in Wikipedia's voice that are no more than vague inferences of an author's intent.  Molar mass is clearly not what you mean (it has unit kg/mol and is an intensive quantity); amount of substance may be what you mean (it has unit mol and is an extensive quantity).  I do dispute the connection with amount of substance, since, contrary to what you say, it is unequivocally not proportional to mass in general, but I do accept that Huntley intended an independent quantity proportional to mass.  He did not apparently mean gravitational mass (though this could be relevant as a further new dimension for analysis), so it seems to me that he was simply proposing a way of defining an independent quantity that makes sense and adds power to dimensional analysis.  In all, what is currently in the article is a sensible reporting of what he said, with the exception of the statement "It may be noted that the quantity 'amount of substance' was added to the SI in 1971 as an independent base quantity, making this less of an innovation and more of a standard approach", which I would suggest should be removed.  —Quondum 13:01, 19 July 2020 (UTC)

I stipulate that molar mass is an intensive quantity and that I erred in my usage above. Let's just stick with the extensive quantity "amount of substance". Please explain how this is unequivocally not proportional to the extensive quantity mass in general.Antiskid56 (talk) 14:19, 19 July 2020 (UTC)

I add that your critique of the reference as vague is probably accurate. I think I should strengthen the reference to remove the vagueness rather than remove it. This discussion has helped me see how to do that. But first I prefer to hear your answer to my previous question, so I can meet your objections head-on in any revision.Antiskid56 (talk) 16:03, 19 July 2020 (UTC)


 * Saying that amount of substance is proportional to mass in general is equivalent to saying that for all samples there exists a single proportionality constant that relates those two quantities in all samples. That is, that $m = αn$ holds for a universal constant $α$, where $m$ is the mass of the sample and $n$ is the amount of substance in the sample.  If we look at some simple substances, we find that $α ≈ 0.018 kg/mol$ for water, $α ≈ 0.002 kg/mol$ for molecular hydrogen, $α ≈ 0.154 kg/mol$ for carbon tetrachloride, etc.  Even if one allows the substance-specific constants of proportionality, this would imply that the substance density (with unit mol/m$3$) cannot be determined even approximately without an understanding of the chemical composition of the substance, a somewhat unattractive and unnecessary complication for a fluid dynamics engineer.  This becomes more complicated when mixtures of different substances are considered, something that occurs routinely in fluid dynamics.


 * I was not critiquing the reference as vague. I was saying that the inference being made in our article is vague.  (But yes, the reference does not seem to define "substantial mass" adequately, so I guess that too is vague.)  —Quondum 17:04, 19 July 2020 (UTC)

I am afraid that you are introducing several considerations that are not germane to the discussion. For use in dimensional analysis, as Huntley's "quantity of matter", it is not necessary to know the precise composition and molecular weights of the substance (matter) in question - the constant of proportionality between "amount of substance" and mass. It is only necessary to know that the constant of proportionality exists. That is all that Huntley assumes - a constant of proportionality that exists. From a metamathematical perspective, SI's "amount of substance" is a model of his postulates for "quantity of matter", which are (1) that his new dimension "quantity of matter" is proportional to mass, but (2) does not implicate inertial properties. If it models his postulates then it is an example of what he had in mind, no matter what else may be true about it. We do not need to look at concrete examples of simple substances, or mixtures, or to consider the convenience of using "amount of substance" in fluid dynamics. All theses issues are not germane to the real question, which is whether SI's "amount of substance" is a model of Huntley's postulates for "quantity of matter". It is. Antiskid56 (talk) 18:24, 19 July 2020 (UTC)


 * There is no problem introducing an additional dimension with a proportionality constant with reference to an existing dimension. This happens with amount of substance (which is proportional to count), with temperature (which is proportional to energy/count), and with luminous intensity (which is proportional to an existing derived dimension, but with frequency-weighting).  Historically they weren't known to have such a direct proportionality, but in the current SI, the proportionality is explicit.  So the model you refer to is there.  Some rewording will be necessary to make this clear that this is what is meant, especially with the potential confusion between Huntley's "quantity of matter" and SI's synonym-sounding "amount of substance".  One could include temperature as an example.  It should be made clear that "quantity of matter" relates to mass in the same way that "amount of substance" relates to entity count and that "temperature" relates to energy per available degree of freedom.  The way it is currently stated, it can easily be interpreted as quantity of matter being more closely related to amount of substance than only through the same model of introducing a new dimension – which is the inference that I had thought that you intended.  —Quondum 20:03, 19 July 2020 (UTC)

All right. I appreciate you helping me sharpen my thinking on this. Let me think about it for a day or two and I will make another pass at editing the text in question.

I have made other significant changes to the article related to Siano's contribution, but they haven't been challenged. I believe they are quite accurate though.Antiskid56 (talk) 20:17, 19 July 2020 (UTC)


 * Your changes to Siano's extension were, IMO, a definite improvement in the clarity/correctness – as with Huntley's extension, but for the confusion discussed above. —Quondum 20:34, 19 July 2020 (UTC)

Please explain terms
Under "Mathematical Examples" occurs the phrase "m is the rank of the dimensional matrix" There is no explanation of this phrase and no link to any other article explaining it. Please could an explanation be provided? Lucy Skywalker (talk) 20:51, 4 February 2019 (UTC)

Follow the link at the beginning of the sentence to Buckingham π theorem for more information on the dimensional matrix.Antiskid56 (talk) 18:32, 20 July 2020 (UTC)

Notation of Units
The units should not be put squared brackets. The correct notation is: m = (m) * [m] [m] = Unit (m) = Scalar if unit of m is kg, we have: [m] = kg 192.102.17.49 (talk) 08:50, 11 October 2019 (UTC)

It is hard to get perfect consistency in notation when different authors relying on different references are involved. Probably the well-known "Green book" has the most consistent notation, along with the "Guide to the SI". But neither of these references uses Maxwell's bracket notation at all.

This type of inconsistency goes all the way back to Maxwell, whose use of bracket notation was not entirely consistent. But I agree someone might usefully edit the entire article for better consistency in this area.Antiskid56 (talk) 18:40, 20 July 2020 (UTC)Antiskid56 (talk) 19:19, 20 July 2020 (UTC)

Use of "Compared"
In this passage: "The rule implies that in a physically meaningful expression only quantities of the same dimension can be added, subtracted, or compared." A ratio is a comparison and thus contrary to this stipulation. It seems like a better wording would be equated. — Preceding unsigned comment added by 73.2.57.25 (talk) 16:03, 29 December 2021 (UTC)