both, both the, both of....

你這問題是關乎Both (of)（注：括號代表可省略）的用法
Both和Both of都可以置於與限定詞（好像：the, my, their, these等)連用的名詞前面。在美語中，通常用both of
如：
She's eaten both (of) the apples.
Both (of) my sisters like swimming.
Both (of) these apples are good.
從例子中，我們看到both或both of沒有分別
但若我們用both時，後面的the，或his, her, their通常會省略（但也可以不省略），如：
She's eaten both apples.
She lost both parents when she was a child.
但當我們用both of時，就不會省略限定詞
所以，你四句中，除了I bought both of books外，意思一樣，文法也對
希望幫到你！

2009-05-30 01:34:08 補充：
回應樓下

若是代名詞，英文只會說：

Both of them......，而沒有Both them或Both they的說法

但若both置於代詞後面則可以，如：

She's invited us both.

They both wanted to marry him.

I think the answer given by 001 is excellent and should be chosen as the best answer.

I would like to add that the word "both" in "I bought both books." is an adjective. The word "both" in "I bought both of the books." is a promoun.

I have noticed that there might be some other differences between the two of them. Hope 001 would enlighten us again.

Both my sisters like swimming. (correct)
Both of my sisters like swimming. (correct)
Both of them like swimming. (correct)
Both them/they like swimming. (?)
Both he and his sister like wimming. (correct)
They both like wimming. (correct)

2009-05-30 16:46:36 補充：
Thank you 001.