Red dwarfs, depending on your definition, can range from 2.5 to 150 times more dense than the Sun.
What is the cause of this discrepancy?
They give no calculations, so I can only guess.
- The article is from 1946 and we've gotten a lot better at science.
- It's 1946 and information exchange is limited. No internet, no TV, and long distance calls are expensive.
- Red dwarfs are tricky to observe and we've gotten a lot better at that.
- Their hypothesis is that red dwarfs do not fuse hydrogen which is incorrect.
Edgeworth's other contemporary works might provide insight.
There is no strict definition of a red dwarf. Some use class M stars below a certain temperature and mass. Others include some class K stars. Fortunately they give a definition.
low luminosity (say, not more than one tenth that of the sun), small mass (say, not more than three quarters that of the sun) and high density (perhaps 30–100 times the density of the sun).
Luminosity and mass are basically correct for the modern definition of a red dwarf, M-class, but red dwarf density runs a considerably larger range from 2.5 to 150 times more dense. Still, they're inside the range.
They give no rationale for their density calculations. Being 1946 its probable their mass and radius calculations were considerably off. Red dwarfs, being so faint, are very difficult to observe.
Red dwarfs fuse hydrogen
In the case of stars belonging to the main sequence, it seems to be necessary to assume that the general conditions in the interior of the star are such that convection currents occur which are of sufficient importance to convey hydrogen in adequate amounts from the outer layers to the central core...
Main sequence stars are convective, but it's complicated.
...although this does not imply that the convection currents are of
sufficient magnitude to be the dominating factor in determining the
distribution of temperature.
Incorrect. The lower the mass the more convection is the "dominating factor" over radiation.
...red dwarfs probably contain a normal proportion of hydrogen...
...but that convection currents in these stars have never been of
sufficient magnitude to convey the hydrogen in adequate amounts from
the outer layers to the central core...
Incorrect. The lower the mass the more convection. Low-mass red dwarfs are fully convective with no helium accumulating in the core.
When the mass of the star is less than [0.75 M☉], the hydrogen which
the star contains never reaches the core in sufficient quantity to
initiate and sustain the process of transforming hydrogen into helium,
and dynamical equilibrium is secured only by contraction involving
comparatively high densities.
Incorrect. Red dwarfs fuse hydrogen. What they're describing is more like a brown dwarf.
On this theory the red dwarfs are not stars of great age which have consumed their substance in riotous living; they are simply feeble things which have never been able to make use of the supplies of fuel which they actually possess.
Red dwarfs are of great age, but they aren't bigger stars which have run through their fuel. They also are not "feeble things" which can't fuse hydrogen. They are low-mass stars which use their fuel very efficiently and so live to a great age.
With an incorrect model they will get incorrect answers.