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This varies substantially, depending on the sort of book, how well the author negotiated (each book involves an individually negotiated contract), the price of the book, etc. Based on one Springer contract I'm familiar with, here's a first approximation. I can't say for sure how representative it is, but it's at least one data point, and I'd guess it's typical for Springer and competing publishers.

For regular sales, the author may get 9-12% of "net cash receipts", defined as what the publisher made on the sale (excluding taxes), with the percentage depending on things like how well the book has sold. Note that "regular sales" excludes some special cases, for which there may be lower royalty rates.

Then the question is how much Springer charged the bookseller for the book. You can download a 6 megabyte zip file of Springer wholesale prices. I glanced at a few math books, and the prices look like they are about 60% of what bookstores are charging. That's certainly the right ballpark, but I haven't computed any real statistics.

So a first approximation is that the author's royalty is probably 5-7% of the price you pay in a bookstore. Considering that academic books are expensive, that's not bad, but nobody's going to grow rich off it. For advanced books, selling 1000 copies in total is very good and selling 10,000 is amazing.

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I would be interested in the hourly pay for writing such a book :) I guess it would be rather sobering. –  xLeitix Apr 15 at 7:35
    
Also, realtime.springer.com shows download statistics for e-books. Do authors get royalties for e-books, too? –  Geremia Apr 15 at 15:18
    
@Geremia in a word, yes. Individual eBook sales count a normal sales from the royalty point of view. Bulk access from bundles works differently but the author still gets something. –  BSteinhurst Apr 15 at 19:31
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Based on my 1 contract with Springer, I would like to add that the way to get rich (presumably) is translations. The author royalty on those is 50%. Other than that there is a 33% discount on Springer titles, though I haven't used it. Oh and you get a few free copies of your book, in my case I got 5. I haven't seen a cent other than that –  Ivar Persson Apr 16 at 15:00
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@IvarPersson: It's 50% of Springer's net proceeds if they license the translation rights to another publisher, right? (At least that's what I've seen.) Basically, Springer collects royalties from licensed translations and then splits them 50-50 with the original author, but this isn't as lucrative as it sounds since these royalties are only a modest fraction of the price of the translated book (most goes to the translation's publisher and the bookseller). Also, net proceeds are after production costs, so any expenses on Springer's part are deducted before the 50-50 split. –  Anonymous Mathematician Apr 16 at 15:48

Here's what Springer says:

§ 6 Royalty

For each sold and paid copy of the Work or part thereof, print or electronic, sold as an individual book or individual part thereof, Springer shall pay to Author a royalty of 6% based on the net income received by Springer (list-price minus discount and VAT, withholding tax and any other applicable taxes). For the avoidance of doubt, this includes (but is not limited to) such individual sales to digital and non-digital libraries.

If there is a legal requirement for Springer to withhold any taxes on a royalty payment, the taxes will be deducted from the royalty payment to Author. If the Work is sold electronically as part of a Springer e-book package, Author will receive an equitable share of royalties from the income generated by Springer from the e-book package. The share formula for each individual title within the e-book package will be determined by Springer no later than April for the preceding calendar year. This amount will be paid in addition to the royalty described above and shown separately on the annual royalty statement.

In the case of Bulk Sales, Author's exclusive royalty for the respective Bulk Sale will be 0 % of the related Net Proceeds, with “Bulk Sale” being defined as the production and/or distribution of a special number of copies of the Work or parts thereof at a special price regardless of whether it be performed by Springer or by third parties, regardless of whether in print or electronic format and further regardless of whether in a layout identical to the normal normal edition or not. For the purpose of this Clause 6 Royalty, the term "Net Proceeds" shall be defined as the net amount Springer actually receives after deduction of all discounts, minus production costs incurred by Sprnger or any Springer Group Company and minus VAT, withholding tax and any other applicable taxes.

If Springer grants licenses to use the Work or derivative works thereof or parts of either in non-Springer products and the related use is not covered by the above subsections of this clause 6 “Royalty” (e.g., a license to translate the work and to distribute the translation, or a license to distribute parts of the Work in a third party book), Author’s exclusive royalty for the respective license and any related use will be a share of the Springer’s Net Proceeds according to industry standards (currently 50%).

The aforesaid royalties shall be the sole compensation to be paid to Author with respect to the Work and the rights granted. In case the Work contains or links (e.g. through frames or in-line links) to media, social or functional enhancements, the royalties, complimentary copies and/or access rights granted under this contract, are deemed to be adequate consideration. For the avoidance of doubt, the aforesaid royalties will be paid as a total to the joint group of authors if “Author” comprises several individual authors. Each co-author will receive an equal share of any payment.

Accounts will be settled annually in April for the preceding calendar year, with payment to follow soon thereafter.

Authors are responsible for the taxation of their royalties. Springer is entitled to report related information (including personal and financial data) to the respective authorities.

Free copies which are provided by Springer for the purposes of review, promotion, sample or otherwise free of charge are not subject to royalty payments. Likewise, copies that are damaged and cannot be sold are not subject to royalty payments.

Any publisher's proceeds from rights managed by national copyright organizations (collecting societies including but not limited to societies such as Copyright Clearance Center) are the sole property of Springer. Any such author’s proceeds are the sole property of Author, and if applicable, registration and taxation of such proceeds is the Author’s sole responsibility. This subsection shall have precedence over any other subsection of this Clause 6 Royalty.

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