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"This can be done using the technique of Peters, and using the technique of Matthews, both which involve mathematics"

Having searched both which and both of which in Google, it appears both of which is more often used.

1) Why?

2) What does the of do?

3) If the of is necessary, why is it not necessary in: "This can be done using the technique of Peters, which involves mathematics"?

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Welcome to ESL&U! –  tchrist May 23 '12 at 1:27
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a third alternative is "which both". This actually sounds more natural to me in the context that you've supplied. –  user16269 May 23 '12 at 6:44
    
Perhaps you are less interested in the specific case of "both" than you are in the general rules governing the usage of indefinite pronouns (both, either, neither, several, each, etc.). If so, I refer you here. –  jeffclef May 23 '12 at 14:48
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4 Answers

'Of' is linked to both, not which. In the same way you would say "Both of my brothers like sport".

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But you do not have to say “both of my brothers”; you can just say “both my brothers”. In contrast, you must say “both of which”; leaving out the “of” there makes it no longer seem a grammatically well-formed construct, at least to me. –  tchrist May 23 '12 at 1:38
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I fully agree with you that 'both my brothers' is acceptable (and more compact). So why is 'both which' not "grammatically well-formed" ? –  user1271772 May 23 '12 at 1:46
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added: If you understand English the simple answer is, OF is used when "which" is not used as a question. Otherwise you can tell by the context or content of the sentence if it implies a question such as, where were you? Otherwise it suggests a comment which is "relative" to the previous comment. Here, OF adds style, formality, poetic or professorial quality to English in both written or spoken form.

Which is better, to start a question with which, the pronoun to start an interrogative question with multiple choices or to make a statement of which might offer a statement. {of reason or proof or opinion, for example}

Both of *which* are two examples of different meanings.

Here, there is no confusion, as to which is which, or am I wrong?

Can you see the difference in structure and where "of" is necessary to distinguish easily a question from a statement or relevance or relative to a noun, now being an adjective?

So, in short use "of which", when it is not a question, to reduce ambiguity.

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Thanks for answering Tony, but with all due respect, that was not easy for me to follow! –  user1271772 May 23 '12 at 11:13
    
English is not easy nor was this question. Or let me make that perfectly clear. "OF WHICH" adds clarity, but "OF" is redundant, due to the context within the sentence or the inflection in your voice using a question. IN fact MANY english words are REDUDANT to express an idea. Emphasis is only for Style. YOu might not be able to tell , but I prefer BREVITY. My conclusion says it all, my example was over the top with double meanings. Sorry but I dare you will forget the meaning now that you understand. –  Tony Stewart May 23 '12 at 14:16
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The "of" is optional; both constructions are grammatically correct. Note, however, that an article or pronoun is mandatory with "both of." "Both of the techniques" (article), "both of their techniques" (possessive pronoun), "both of these techniques" (demonstrative pronoun), and "both of which" (relative pronoun referring to "techniques") are all acceptable. "Both of techniques", on the other hand, is not.

The reason "both of which" may show up on Google more times than "both which" has to do with rhythm--not grammar. English is said to flow in iambic rhythms, forming a general pattern of alternating weak and strong stresses, much like your heartbeat. The "of," in this case, creates a weak beat separating what would be two strong stresses in "both which."

The question of whether a particular syllable constitutes a strong or weak stress depends on the immediately surrounding context. A pattern that our vocal apparatus has difficulty pronouncing ("both which") will slow things down, resulting in a series of consecutive strong beats (spondaic rhythm), while reversing the pattern of stresses or introducing filler words can create something more iambic ("which both" or "both of which"). Most native speakers of English don't even think about these matters of prosody, but I know from experience that they can be annoying to students learning English. Good writers and speakers develop a sense for how they can shape meaning and emphasis by manipulating speech rhythm.

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I am sure the following excerpt from an 'Oxford University Press' book will clear all your doubts:

We can put both (of) before nouns and pronouns. Before a noun with a determiner (for example: the, my, these), both and both of are both possible.

Both (of) my parents like riding.

She's eaten both (of) the chops.

We can also use both without a determiner.

She's eaten both chops. (=...both of the chops.)

Only both of is possible before a personal pronoun (us, you, them).

Both of them can come tomorrow.

Mary sends her love to both of us.

We can put both after object pronouns.

I've invited them both.

Mary sends us both her love.

I've made you both something to eat.

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..and therefore, (since which stands for them), 'of' is required. –  TimLymington Jun 30 '12 at 22:27
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