I am confused about the adjectives 'small' and 'short' when they are used to describe length and length scales. I think that 'The line is short.' and 'The line is of small length.' are correct, and 'The line is of short length.' is incorrect, as 'short' already implies length, and since a line is a one-dimensional object that has only a length attribute (i.e. width and height are zero), so 'short length' is redundant. Is this correct? I would appreciate a reference to some authoritative source that talks about this difference.
Interesting question. Here is my technical point of view.
Remind yourself that length is a property, just as weight, temperature, price, size, volume, terminal resistance, magnetic permeability, etc.
Generally, we use the words "low" and "high" to provide an indication of their value. Here are some examples:
This pen has a low/high weight.
This stream of air has a low/high temperature.
The laptop has a low/high price.
This motor has a low/high terminal resistance.
Sometimes, we have words that apply to specific properties (sometimes depending on the context), and we can omit the property itself:
This pen is light/heavy.
This stream of air is cold/warm.
The laptop is cheap/expensive.
Sometimes, we see that these words are combined with their property anyway:
This pen has a light/heavy weight.
This stream of air has a cold/warm temperature.
The laptop has a cheap/expensive price.
Is it incorrect? No, but it is unnecessary and some would call it sloppy. You could see them as pleonasms.
The same thing is happening to short and small. Short (or long) is used for the single-dimensional properties length (or distance) and time (i.e. duration). Small (or large) is applied to a single- or multi-dimensional dimensional property such as volume, surface area, but also length (!).
However, let me note that it is less common to apply the words low/high to length, or any other dimensional property. My guess is that this is related to the fact that "low" and "high" are words also used to indicate "altitude", which is a dimensional property itself. As such, it may lead to confusion if "high" or "low" are combined with another type of dimensional property, such as length, size or distance. It's better to stick with short and long, or if you want, short length and long length. Once again, the latter is not necessary, but not incorrect either. In fact, since short and long can also be applied to duration, the word length can be added for clarification, although, I would guess that the meaning of the word should mostly be obvious from the context.
While describing length short and small can mean the same thing.
A sawed off shotgun is just a shotgun with a small barrel
A sawed off shotgun is just a shotgun with a short barrel
However, if not constrained to length small can describe any dimension while short is always restricted to a description of length, or duration. This can prevent ambiguity.
Describing the shotguns barrel as having a small length would be correct.
Describing the shotguns barrel as having a short length would be correct, though redundant.
Describing the shotguns barrel as being short would be correct.
The dictionary definitions provide an authoritative answer to this extent: "short" is defined in terms of "length," while "small" is defined in terms of dimensions (plural). So you are correct that "short length" would be redundant (generally speaking - I can imagine a dressmaker referring to a "short length" for pants or a dress). But "of small length" is not quite right, either, because "small" really needs to refer to more than one dimension. So if you are referring only to a single dimension, say the object [the line, the person, etc.] is short; if you are referring to more than one [this will not be a line, by definition] say that the object is small.
I am not a native speaker but in technical writing "short length" sounds a bit weird to me. Something like "oily oil". For instance, the sentence "Nanotubes obtained by this method are characterized by low lengths and high diameters" can be used only with "low" and "high" because you cannot say "long diameter". In fact, saying "low length" in scientific writing you imply "low (value of) length" where "length" is a parameter and the value of a parameter cannot be short or long.