There are 196 sovereign nations in the world (more or less); each of them has its own copyright laws. 168 are signatories of the Berne Convention which requires them to extend to foreign copyright holders at least the same rights that they would have domestically. It relies from this on the concept of country of origin. Assuming that the book was originally published in the UK (as it has a London based publisher) then the applicable copyright law is that of the UK.
Exceptions to copyright can be found here. The relevant exemption is the first part of the Teaching section:
the copying of works in any medium as long as the use is solely to illustrate a point, it is not done for commercial purposes, it is accompanied by a sufficient acknowledgement, and the use is fair dealing.
Acknowledgement is pretty straightforward but fair dealing as a legal concept means:
There is no statutory definition of fair dealing - it will always be a matter of fact, degree and impression in each case. The question to be asked is: how would a fair-minded and honest person have dealt with the work?
Factors that have been identified by the courts as relevant in determining whether a particular dealing with a work is fair include:
- does using the work affect the market for the original work? If a use of a work acts as a substitute for it, causing the owner to lose revenue, then it is not likely to be fair
- is the amount of the work taken reasonable and appropriate? Was it necessary to use the amount that was taken? Usually only part of a work may be used
The relative importance of any one factor will vary according to the case in hand and the type of dealing in question.
If the work had been simultaneously or previously published in the USA the relevant consideration is fair use rather than fair dealing. Fair use typically gives the copier much greater latitude than fair dealing does.
As to the specifics of posting on stack exchange. Extracts on SE are unlikely to affect the original market (they may even enhance it by serving as advertising) so providing that you have used the minimum amount necessary to illuminate your answer it is probably fair dealing (and fair use). This would probably be true even if, across tens of thousands of questions, the entire book were quoted.
Notwithstanding, if the copyright owner issued SE with a take down notice they would take it down because they are not in the business of fighting protracted copyright cases even if they are in the right.