How should I cite something learned second-hand (eg, from Wikipedia) when I haven't seen the primary source?

What you're referring to is an indirect source. In general, you should always work as hard as you can to find the original source. If that is not possible, all of the major style guides include a way to cite indirect sources. Note that you should not cite Wikipedia (see the "do not cite Wikipeida" note at the end of this answer). If an indirect citation is absolutely necessary, it should come from a reputable, peer-reviewed journal or other academically respected source.

1. According to Purdue University, the MLA rule is to name the author of the indirect source in the text and cite the work you have in-hand:

   For such indirect quotations, use "qtd. in" to indicate the source you actually consulted. For example:

   Ravitch argues that high schools are pressured to act as "social service centers, and they don't do that well" (qtd. in Weisman 259).

   Note that, in most cases, a responsible researcher will attempt to find the original source...

   Williams College further clarifies that the indirect work should be included in your Works Cited list:

   ...include the indirect source in the Works Cited.

2. The APA rule (also from Purdue University) is to exclude the indirect source (called the "original source", below) from your reference list and only include the work you have in-hand (called the "secondary source"):

   ...name the original source in your signal phrase. List the secondary source in your reference list and include the secondary source in the parentheses.

   Johnson argued that...(as cited in Smith, 2003, p. 102).

   [...] Also, try to locate the original material and cite the original source.

3. The Chicago rule (once again, from Purdue) is to cite the indirect source, followed by the in-hand resource:

   ...Chicago discourages the use of [indirect sources]. In the case that an original source is utterly unavailable, however, Chicago recommends the use of "quoted in" for the note:

   7. Ian Hacking, The Social Construction of What? (Cambridge, MA: Harvard University Press, 1999), 103, quoted in Manuel DeLanda, A New Philosophy of Society (New York: Continuum, 2006), 2.

That said, do not cite Wikipedia in a formal document (unless, perhaps, you are actually writing about Wikipedia or collaborative editing techniques). I love Wikipedia, and I believe it is reasonably well-maintained and has a lot of good information. However, you have no way to verify if the information in an article is true -- or, if you do have a source to verify it, you would just cite that source. Aside from the tired, "Anyone can edit it!" complaint, two severe issues with Wikipedia as a citation source are:

• You get whatever version of an article stands at the exact moment your web browser fetches the page. No matter how hard Wikipedia's editors work, they can't stop a bad edit from reaching your web browser if it was made seconds before you fetched the page. Wikipedia doesn't undergo any kind of pre-publication review; all review is post-publication, which means you may see totally unreviewed information. (You can mitigate this by citing a specific past revision, but it still stands that a post-publication review process means that any given revision of an article could have claims that have been reviewed by absolutely no one except the author.)
• In order for a reader or reviewer to ascertain the usefulness of a source, it must have an identifiable set of authors (or, for anonymous works, at least a consistent, reasonably small set of authors). Wikipedia makes that requirement incredibly difficult to satisfy. (Again, it's possible to satisfy this requirement if you cite a specific revision of a page and find out what contributors wrote each part of a page, but it is still difficult since a potentially huge number of contributors have helped build that revision.) It's hard for a Wikipedia article to be reputable where there are no clearly identifiable authors to which a reader could attach a reputation.

First, I would try harder to get the primary source. Really. But, if that isn't possible (price, availability, etc.), you may have to do without. In that case, a few solutions:

• Find another secondary source, possibly one that is more “academically acceptable” than Wikipedia. For example, try to find a textbook on the topic that make mention of the fact you want to source, or a review article, a book, etc.

• If not possible, what I have usually seen people do is cite the primary source anyway. That's bad, but people do it. If you write for a journal, where the reviewers might not allow a reference to Wikipedia, you might not have any other choice.

• What I would recommend, if the format and/or editor allow it, is to cite both the primary source and the secondary source, possibly indicating the relationship:

  J. Doe, Journal of Failed Experiments 10, 1024-1028 (1971); as cited by secondary source

Note: This is written from the perspective of a postgraduate student in applied mathematics.

1) Do not cite Wikipedia.

This is not about perception or laziness but rather, your thought process as a researcher. Suppose I read about a mathematical fact that might be useful to my research. I need to verify that the fact is true and have some ideas about why this is true.

By stating that the mathematical fact has been published in a reputable peer-reviewed journal or in a reputable textbook, I demonstrate that I have at least verify its authenticity, and perhaps even read technical details about it.

However, if I cite Wikipedia, it demonstrates that I accept facts off the internet without verifying or having technical understanding about it (Wikipedia usually don't go into deep technical details). This does not bode well for my reputation as a researcher.

2) Try to find an academically acceptable source to cite the same information from.

Suppose I want to use an equation. Random example: Kullback-Leibler Divergence. But lets pretend there is no source or citations on Wikipedia.

What I will do is to search directly for "Kullback-Leibler Divergence" using search engines like Google, Google Scholar or Google Books. I will also try to search for the term in my university or local library's search tool.

Assuming this fails. Then, I would look at topics that the Kullback-Leibler Divergence is in or is related to. For this specific example, I would look for textbooks or materials on "Information Theory" and look up their index or table of contents for "Kullback-Leibler Divergence". If this fails, I will dig deeper: think about what this equation does and search for similar topics. For this example, it compares two probability distributions. I will then look for ways to compare two distributions in Information Theory.

Once I find a paper or textbook talking about it, it shouldn't be too difficult to locate the source or pick a suitable paper/textbook to cite the equation from. If after all these searching and perhaps asking my supervisor/professor, I cannot find anything acceptable to cite from, I would ask myself these questions: Is this equation valid? Why should I believe in the authenticity of this equation?