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Often, at the top of written work, Jews write either ב"ה, which stands for Baruch Hashem =blessed is the Name i.e., God, or בס"ד, which stands for b'siyata d'shmaya =with Heavenly help.

What is the origin of this custom and why do some people choose ב"ה over בס"ד and vice versa?

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do we have a source that ב"ה actually stands for Boruch Hashem? I was taught that in school, but it always seemed strange to me that we write B'siyata D'shamya (with the help of heaven) in aramaic, and Boruch Hashem in hebrew. It made more sense to me that it stands for B'Ezrat Hashem (with the help of G-d), so that the hebrew and aramaic match up. Also, I've seen english letters that start "By the Grace of G-d" (such as chabad.org/therebbe/article_cdo/aid/142535/jewish/…), which also matches the theme of B'Siyata D'Shmaya and B'ezrat Hashem – Menachem Jun 2 '11 at 19:28
Whenever it started, it predates the practice among some Catholics of writing the acronym for "ad majorem Dei gloriam" at the top of their papers (which could not have started earlier than 1541). R' Yizchak Karo described the practice of writing ב"ה at the top of the paper in Toldot Yitzchak (published 1517). – Fred Nov 8 '13 at 2:47
@Menachem B'Ezrat Hashem would be abbreviated like בע"ה – R. Mo Nov 4 '14 at 18:42
up vote 10 down vote accepted

I don't know the origin, but in one form or another it goes back at least to the Rambam: he began each section of his major works with the phrase בשם ה' א-ל עולם (though this has been omitted in most later printings).

There are halachic opinions that the letter ה has kedushah when it is used to represent Hashem's name (since it is one of its letters), and that therefore a paper containing such a reference must not be treated disrespectfully. (For this reason, some books, magazines, etc., use 'ד instead of 'ה when referring to Hashem.) Using בס"ד, then, avoids any such issues.

On the other hand, ב"ה has an advantage in that it is not only invoking Hashem's help (when expanded to 'בעזרת ה) but also blessing Him (when expanded to 'ברוך ה), so some people prefer that instead.

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This footnote discusses the different opinions about whether writing a ה and intending it to refer to G-d makes it a problem when it comes to throwing it out: yeshiva.org.il/midrash/shiur.asp?id=1658#2b – Menachem Jun 2 '11 at 19:42
The ה dosn't represent God's name. It represents the word השם. – Double AA Nov 4 '14 at 3:40

ב"ה stands for Be'ezras Hashem - which is technically the Hebrew translation for B'siyata Dishmaya.

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Not precisely. "Hashem" literally means "The Name" and is a direct stand-in for naming God. "Dishmaya" means "of Heaven," which also refers to God, but not in the same way. – Isaac Moses May 11 '11 at 2:05
@IsaacMoses But it is very consistent with the style of translation used by Unkelos and Yonasan ben Uzi'el, in which all references to God and His actions are stated indirectly - either as something else doing the action (such as shamayim) or the action being done in the proximity of (but not directly to) God. And we call those targumim. – WAF Jun 5 '11 at 13:39

Rav Moshe in Igros Moshe YD 2:138 writes that it is better to avoid writing bais hey and one should use beis samach dalet instead.

Rav Betzalel Stern in Batzel Hachahma 4:105 writes that one does not have to worry that a paper with bais hey will get thrown out and one can use it. He brings that the Sfas Emes and Chiddushe Harim used it on their paper(however it was divrei Torah). He brings examples from the doros harishonim that they used bais hey even on regular papers. However , he ends off saying if one uses bais samach dalet and is careful not to throw out(directly into garbage) papers with bais hey then a bracha should be upon him.

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There is a discussion amongst the ראשונים whether "והתקינו שיהא שואל שלום חבירו בשם" (Berachos 54a) is a היתר (allowance) or a חיוב (obligation).

Rav Soloveitchik discusses this in הררי קדם (collected by Rav Michel Zalman Shurkin from students' notes on the Rav's lectures), part two, number 124 (page 262). Because of the possibility that this was a היתר, combined with the fact that the Rambam does not quote this rule להלכה, convinced R' Chaim that this was a הוראת שעה (temporary rule), and that it is now forbidden to write ב"ה on letters. [The Rav respectfully disagrees, based on Rashi that there is no real איסור in (prohibition against) using Hashem's name to greet friends]

The Rav concludes that the "new מנהג" of writing בס"ד on the top of letters is a baseless and meaningless custom ("ואין טעם ומקור למנהג זה"), based on a comparison to שבועה.

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