I solved this one by the process of elimination.

Holmes's delightful book about the history of hot air balloons is not just a tale of (i) ___ visionaries of those with dash and (ii) ___; it‘s an utterly (iii) ___ history of attempts to harness the power of air.

Let's break it apart and start with the first sentence.

Holmes's delightful book about the history of hot air balloons is not just a tale of (i) ___ visionaries of those with dash and (ii) ___;

The book is delightful, and it's about the history of hot air balloons. It's just not a tale of _______ visionaries of those with dash. It seems like the author has effusive praise for this book. So out of the choices available, only quixotic would make sense. There's no context for chivalrous and gruesome is too negative.

it‘s an utterly (iii) ___ history of attempts to harness the power of air.

Notice, in the first sentence, it was said "not just", so the third blank should also be positive. Hence, engrossing is the answer.

The word "dash" is a hint for the second blank. My understanding is here dash means a kind of style, so I picked eccentricity as the option. Conformity would not be consistent with dash and lassitude would not make the book "delightful".

I hope the explanation makes sense.