- Downloading from the internet constitutes an act of reproduction according to EU law, however
- The EU allows Member States to implement an exception permitting personal-use copies from lawful sources, and
- Germany implements such an exception, but
- Germany's personal-use exception is narrower than the base EU exception and doesn't generally apply in this case.
Note: My answer concerns whether or not the private copying exception applies to the download specified in the question. There could be other avenues to explore, such as Section 60c mentioned in the question and phoog's answer.
Downloading as an act of reproduction
As a literary work, a scientific monograph is given protection against unauthorized acts of reproduction by the EU's Copyright Directive in Article 2. This includes an end user downloading from the internet.
While it was surprisingly difficult to find an authoritative direct statement to that effect (most sources just assume this to be true), the Filmspeler ruling contains such language in paragraph 22 and paragraphs 69-72.
The EU's private copy exception
Despite the protection given by Article 2, Article 5(2)(b), henceforth the private copy exception, is one way to legally create copies without authorization. It allows Member States to enact exceptions:
(b) in respect of reproductions on any medium made by a natural person for private use and for ends that are neither directly nor indirectly commercial, [...1]
While this provides a baseline for private copy, in order to fully understand its scope, we must examine the landmark ACI Adam ruling. In it, the Court of Justice of the European Union ruled that the private copy exception cannot apply when the source of the reproduction is unlawful (paragraphs 56-58).
Additionally, according to analysis by Eleonora Rosati on the the IPKat blog, ACI Adam clarified that Member States cannot expand on the allowed copyright exceptions, they can only make them more narrow.
Following just the baseline EU private copy exception, the download specified in the question is legal provided that the source is lawful.
Germany's private copy exception
Germany transposes the baseline EU private copy exception into section 53(1) of its copyright act (German link, footnotes are mine):
(1) It shall be permissible for a natural person to make single copies of a work for private use on any medium, insofar as they neither directly nor indirectly serve commercial purposes, as long as no obviously unlawfully-produced model2 or a model which has been unlawfully made available to the public3 is used for copying. [...]
While private copying is therefore generally allowed in Germany, it implements a limiting exception to the private copy exception in section 53(4) as allowed by ACI Adam. The clause prevents unauthorized non-manual copies in the case of sheet music, periodicals, and books, unless they've been out of print for 2 years.
Therefore in Germany, the download in question is only legal under the private copying exception if the source is not obviously unlawful and the book has been out of print for at least 2 years.
Two other restrictions on private copy not particularly relevant to this question. One is not circumventing technological locks, the other is that the rightholder must receive "fair compensation." In practice, this isn't handled by the private copier, but by collection societies which get funds from levies on storage media then redistributes them to rightholders.
The edges of "obviously unlawful" are a little fuzzy. In ACI Adam, it was determined at the EU level that the private copying exception must only apply to lawful sources, and so the German law might be inconsistent with EU law. Reading through the decision, however, the logic could also apply to sources which appear to be lawful so there's still hope for the German law. This is (debatably) the approach that was taken in Filmspeler. However, that ruling didn't address Article 5(2)(b), but Article 5(1), the temporary reproduction exception. If you're interested, I have a lengthy answer discussing the Filmspeler ruling.
The translation here is poor and perhaps that's because it's not 100% clear in German either (I'm not fluent, I'd rate myself weak intermediate). Online sources conflict as to the exact meaning of this clause. Some say this clause prevents download from any public source, others only if it was made publicly available in an obviously illegal manner. I would lean towards the latter because that's closer to what the Justice Ministry thinks (German link).
Editing note: This answer has changed significantly since its original posting. This is partly because I failed to notice section 53(4) the first time around, but mostly because of the incredible amount of intelligent comments and chat the original posting generated.