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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...
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If you are making an -ing form the object of a preposition, it must necessarily be a noun not an adjective. Therefore, it is a gerund not a participle. If that matters. Compare and contrast: “I wash my sink by running water in it first” with “I sleep more soundly by running water’s white gurgling noise” with “More paths were trodden by sandals than by running shoes.” –  tchrist Mar 22 '13 at 11:04
    
By and large, by is elided. Whether it is expressly stated or not, it is supposed to exist in its place functionally and as such it does not make a difference in grammaticality or semantics. Both constructions are prevalent in mathematical as well as other domain literature. –  Kris Mar 22 '13 at 11:19
    
In some cases, by can alter the emphasis, or even the meaning. Differentiating result 1, we obtain result 2 [though you might have expected result 3] as against By differentiating... [this is the way we obtain result 2, which we obviously need]. –  TimLymington Apr 27 '13 at 9:33
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2 Answers

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.

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This is a question I find fairly tricky to answer, but here is something that might be helpful.

Consider that "by" means "accomplished through the process, agency, or effect of." (That's an awfully long way of saying it, but I think you'll see why this works if you try my upcoming suggestion.)

Now substitute that phrase for the word "by" and see if the sentence makes sense or not. Sometimes it will make sense whether you use the phrase or not, and in those cases, you can put in "by" or leave it out, but the structure of the sentence will usually be more effective and elegant if you leave it out.

Sometimes you will find that the sentence doesn't really make sense when you put the phrase in, and sometimes it doesn't make sense when you take it out, and in each case you will then see whether you need "by" or not.

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