I often encounter a problem when writing sentences in mathematical contexts, where one "does something" in order to "obtain something." These sentences typically have the form "using X, one can do Y," where generally X is a mathematical rule or function, and Y is its result. I never know if I should put "by" before the present participle. I have this problem mainly with the verb "using," but this is not the only verb that gives me trouble.
Here are some examples:
- (by) using Theorem 2, one can prove that…
- We obtain the result (by) expanding the function …
- (by) differentiating w.r.t. the 2nd variable, we obtain …
- (by) taking the integer part of F, one can compute…
Best Answer
All the (fuller versions of the) OP's examples use by in the sense of by means of, so, if we find a difference in acceptability of omission of the preposition, semantic factors cannot be the only ones operating here.
Consider
1a. Through the expedient of using Theorem 2, one can prove that the triangle is equilateral.
1b. By means of using Theorem 2, one can prove that the triangle is equilateral.
1c. By using Theorem 2, one can prove that the triangle is equilateral.
1d. Using Theorem 2, one can prove that the triangle is equilateral.
These are all equivalent in meaning, though considerations of style would probably suggest that a shortened version be chosen. 1d is an obviously ellipted version of 1c (and perhaps 1b).
1e. One can prove that the triangle is equilateral through the expedient of using Theorem 2.
1f. One can prove that the triangle is equilateral by means of using Theorem 2.
1g. One can prove that the triangle is equilateral by using Theorem 2.
1h. One can prove that the triangle is equilateral using Theorem 2.
So far, not too many problems (except with style), though some might prefer a comma after equilateral in 1h. (This actually gives a slightly different meaning - making 'using Theorem 2' a free rather than a bound modifier)
1x. Analysing DNA samples, we see that chihuahuas are dogs.
1y. We see that chihuahuas are dogs analysing DNA samples. OOPS!
So, we have to be careful here not to make an ellipsis that results in an unintended reading (not always an absurdity!)
I wouldn't just add a comma after dogs in 1y; if the main point being made is that chihuahuas are dogs, I'd use 1x, but if the main point is that DNA analysis is used, I'd insert by or by means of or replace the by-clause with with by using DNA analysis.