I've recently read T.W. Körner's "The Pleasures of Counting" and I consider it a true gem, unlike anything I've read before.
Largely, it is a book about the historical uses of applied mathematics
which manages to be accessible to a clever high-school student yet
rewarding to a math/science major with some courses under her belt.
Among topics covered are fighting cholera epidemics in Europe using
statistics, and the birth of operations research in WW2 (the organization of convoys, how not to train pilots, why you should repair your
planes less often). He explores predator-prey models through
epidemiology as well as Darwinism. Dimensional analysis is used
to derive power laws from mathematical biology and physics (humans
and fleas can jump roughly the same height, how on earth can that be
can that be??). He discusses Richardson and the genesis of weather modeling.
Euclid and Galileo are in there too, as is as the obligatory
section on the Enigma machine. It's an amazing book.
The math content of the stories is relished rather then elided. There
are occasional many exercises fueled by the history (Requiring anything
common sense to trig, probability, calculus, ODEs, group theory).
Occasionally, there's a choose-your-own-adventure moment right before
a crucial point in the historical narrative, when you're given a
chance to reflect on a problem before learning how it was actually
solved and lives saved.
It's filled with example after example of real life crisis where
the use of simple mathematical tools coupled with ingenuity
lead to insights that made a huge, undeniable difference. Examples
showing that actually measuring things can prove common-wisdom to
be wrong. The book manages to do this without resorting to synthetic,
trivialized problems such as would be found in your average
calculus book these days.
I found it to be a truly inspirational book; it got me curious and excited
about math, and made me want to learn more. It was also a thoroughly
good read. I'm convinced it has the potential to be a life-changing read to
a certain kind of person at the right age.
One thing to point out is that the exercises are positively
riddled with errors, so you better arm yourself with a copy of the
(lengthy) errata and maintain vigilance. Other then that, I cannot
recommend it enough.