I don't understand why in the following cases articles are omitted:
1. "This is a curve of (?) second order."
2. "Language of (?) first order"
3. "This vertex [of a graph] has (?) second degree."
4. "All the vertexes of this graph have (?) even degree."
Why is it said: "at a speed of 100km/h", "at a range of 17 miles" e.t.c.? Why is the indefinite article used whilst a certain speed and range are mentioned?
Thanks in advance.
The definite article is used to identify a specific instance of a class of things of countable things (books, oranges), or the totality of something that is uncountable (ocean, air). The indefinite article is used to refer to any instance of a class of countable things. After an instance is first referred to, the definite article is used to make further references to that specific instance.
The articles omitted in the examples you cite imply that the nouns referred to are uncountable. One does not have multiple second orders, presumably. I'd probably use the definite article in all those cases, but that's me, and without knowing exactly what is being referred to, I can't say that my usage is any more correct.
As for speed and range, the indefinite article is being used to specify one instance of a class being selected on the spot to match a specific criterion. If the speed or range had been previously mentioned, the definite article would be used. Example: "The controller can be set to a speed of low, medium, or high. If it is set to the high speed, it will run three times as fast as it does at the low speed."
OK. If I understood you correctly you are not sure about the first question.
As for the second one, the articles are omitted becouse, for example, a speed of four is not a specific element of the class of speeds but only an element of that class which satisfies the criteria that it must be equal to four. Am I right?
And one more question.
"This book offers an introduction to the theory of graphs as (?) part of (pure) mathematics." Why is the indefinite article omitted here?
I don't know the writer's original intent in the examples you gave; his choices would probably not have been mine, but I wasn't there.
Yes, you are right in the second case.
The indefinite article is often omitted; you could just as correctly say "as a part of" if you so desire. I daresay most writers would omit it, as in your example.