What is a “measure equation” as mentioned by this TeX Users Group guide?

Question

In this TeX Users Group (TUG) document, Typesetting mathematics for science and technology according to ISO 31/XI by Claudio Beccari, the author makes various typesetting recommendations including:

9. Measure equations should be absolutely avoided in professional scientific texts; measure equations were somewhat popular before the SI was universally adopted; now they should not be used any more. They survived in those countries where the "English system of units" is being used, but, since scientifically speaking this traditional system of units is "illegal", measure equations have no reason to be used anymore.

What are these "measure equations" he talks of? I take it that should he write say $7.25\,\text{cm}$, he would call $7.25$ the "measure" and $\text{cm}$ the "unit of measure". From this and the context of the quoted paragraph I would guess that a measure equation is a relationship between different units of the same quantity, such as " $1\,\text{in}\equiv2.54\,\text{cm}$ ". However I find such things handy, or at least not unnecessary or obsolete, especially when I'm working on atomic scales where SI isn't the most convenient.

Is this what the author means and if so, are they really that bad?

Answer 1

I suppose (or hope, for I'd agree) he means equations where the variables are plain numbers rather than physical values, and the units are written out in the equations. Like $$ F\:\mathrm{N} = \frac{E\:\mathrm{J}}{s\:\mathrm{m}} $$ i.e. "$F$ Newtons equal $E$ Joules over $s$ meters". Which is a really horrible way of writing equations, and particularly harmful when one wants to work in different unit systems (which, IMO, is not a bad thing in itself); the correct equation simply stating $$ F = \frac{E}{s} $$ holds in all unit systems, but one needs to keep track of which units are used (which you should always do anyway!) and in a unit system like the "English system" so many odd factors are involved that it can be handy to write them directly in the equation. But nowadays we have computers to keep track of the different units even in such systems, so there is really no reason any more to write equations in this way.

Simple conversion factor statements like $1\:\mathrm{in}\equiv2.54\:\mathrm{cm}$ are not considered measure equations (at least I wouldn't), as they don't include any variables but simply give the most condensed possible way of stating the relation of different units.

Answer 2

I think it has something to do with old electromagnetism units, which was vastly more complex than centimetres to inches conversion. I'm too young to have been exposed to the horrors of -cgs--units in electromagnetism so all the following is only an educated guess. My guess is due to some googling, with "measure equation" coming up in old papers associated with unit conversion in electromagnetism (this one from 1966,or this book from 1962). I didn't get the meaning of "measure equation" from those texts which I only quickly scanned : they seem to consider the notion to be too obvious to define. My guess is slightly reinforced by the $Z_0=377\Omega/\sqrt\epsilon$ example given in 8. of your TUG paper.

Basically, before the SI system existed, the centimetre-gram-second system of electromagnetic units was a nightmare (at least for modern eyes). To cite the wikipedia page linked above :

The conversion factors relating electromagnetic units in the CGS and SI systems are much more complex – so much so that formulae expressing electrical physical laws of electromagnetism are different depending on what system of units one uses.This illustrates the fundamental difference in the ways the two systems are built:

• In SI, [...] [t]he ampere is a base unit of the SI system, with the same status as the metre, kilogram, and second. Thus the relationship in the definition of the ampere with the metre and newton is disregarded, and the ampere is not treated as dimensionally equivalent to any combination of other base units. As a result, electromagnetic laws in SI require an additional constant of proportionality ([the] Vacuum permittivity) to relate electromagnetic units to kinematic units.

• The CGS system avoids introducing new base units and instead derives all electric and magnetic units directly from the centimetre, gram, and second based on the physical laws that relate electromagnetic phenomena to mechanics.

The problem comes from the fact that there are several (at least 4 in addition to SI were in common use) ways to make this link, and this changes the presence or not of some physical constants in the equation, as well as their units. I give below an example of the equation relating the attractions of two electric charges, how it changes according to the system, and how this changes the units. $$\begin{align} \text{Units}& &&\text{equation} &&\text{Charge unit}\\ \text{ESU and Gauss}&& F&=\frac{qq'}{r^2} &&1\;\mathrm{cm}^{\frac32}\mathrm{g}^{\frac12}\mathrm{s}^{-1}\\ \text{EMU} & & F&=c^2\frac{qq'}{r^2}&&1\;\mathrm{cm}^{\frac12}\mathrm{g}^{\frac12}\mathrm{s}^{-2} \\ \text{Lorentz-Heavyside}& & F&=\frac{4\pi qq'}{r^2}&&\frac{1}{4\pi}\;\mathrm{cm}^{\frac32}\mathrm{g}^{\frac12}\mathrm{s}^{-1}\\ \text{SI}& & F&=\frac{qq'}{4\pi \epsilon_0 r^2} &&1\;\mathrm{A}\mathrm{s} \end{align}$$ I guess that the last column is what was meant by "measure equation", but, as said above, it's only a guess.

Answer 3

I am not sure about this, but I think a “measure equation” is something astronomers seem to like a lot: $$ \frac\Gamma H \approx \left( \frac T{1.6\cdot 10^{10} \, \mathrm K} \right)^3 $$

Or for absolute, relative magnitude and distance (although I am sure I mixed something up): $$ m - M = 5 - 5 \log\left(\frac{R}{10 \, \mathrm{pc}}\right) $$

So equations that cancel all the units in the middle and append the final unit at the end.

Another example, I just made this one up, though: $$ T = 400 \left(\frac{M}{10 \, M_\odot} \right)^{-2} \left(\frac{d}{2 \,\mathrm{au}} \right)^{3} \mathrm{s} $$

The one nice thing is that you can say something like this: “For a mass in the ballpark of ten solar masses and a distance in the ballpark of two astronomical units, the period will the 400 seconds.”

In experimental Physics, all this is rolled up into a constant with unit of something like $\mathrm{kg^2\,s/m^2}$. And in theoretical physics, one would just set everything to 1, to get rid of it :-)