It cannot be "ban" because it went on to say the "led to modern science".

Yes, it is suspicious because in this case those people don't like curious people. Lukewarm means without interest, in this case is not disinterest, there is an interest, but with bad prejudices.

In earlier times people were suspicious of thinkers so the last blank can only be sanction as they started to tolerate, and if they were suspicious of idle then they considered it childish

"For much of human history, priests, politicians, and philosophers cast a (i) ______ eye on curious folks. It wasn‘t just that staring at rainbows all day or pulling apart insects‘ wings seemed weird, even (ii) ______. It also represented a colossal waste of time, which could be better spent building the economy or reading the Bible. Only in the 1600s did society start to (iii) ______ or at least tolerate the pursuit of idle interests, which led to the rise of modern science."

Meaning- Priests, politicians, and philosophers cast a doubtful or skeptical eye on curious folks (must be a -ve word). It wasn't just that staring at rainbowns seemed weird or (another -ve word needed here), it was also considered a waste of time. Only in 1600s did society start to approve or tolerate these pursuits.

A- Trustful is a +ve word and therefore incorrect.
B- Lukewarm means indifferent. Illogical in context.
C- Suspicious means doubtful or skeptical. Works well.
D- Childish also works well as its a good -ve word.
E- Impressive means spectacular. +ve word so incorrect.
F- Promising is another +ve word. Incorrect.
G- Ban means to prohibit. Opposite to what we are looking for.
H- Sanction means to allow. Works well.
I- Admonish means to scold. Illogical.

For much of human history, priests, politicians, and philosophers cast a (i) ______ eye on curious folks.
it means -
human history, priests, politicians, and philosophers consider ____ about curious folks.
what type of consideration we will get it in 3rd line.

It also represented a colossal waste of time, which could be better spent building the economy or reading the Bible.
here its clearly described that this work isn't something good or useful we can say its negative.
so in blank 1 clearly we have to find a negative word - only negative word here is suspicious

It wasn‘t just that staring at rainbows all day or pulling apart insects‘ wings seemed weird, even (ii) ______.

here even gives us a notation of same meaning of previous word weird
so we will find a word of similar weird or negative something.

only negative word in blank 2 is childish

Only in the 1600s did society start to (iii) ______ or at least tolerate the pursuit of idle interests, which led to the rise of modern science.

here or at least tolerate gives us same connotation of tolerate or positive word
only positive word in blank 3 is Sanction

Answer suspicious,childish,sanction

hope this will help

Why couldn't we use "Admonish" for blank 3. They did something or at the least they did tolerate. If thy have tolerated other's actions, then at the least they could have done is tolerate them. Then they could have actually done is advise or scold them. So, according to my conscience admonish is correct. Correct me if I'm mistaken.

Admonish is to scold.

In the last line, it is mentioned that Only in the 1600s did society start to _______ or at least tolerate the pursuit.

Hence, it means that society started to approve the practice. So the word should mean being supportive and that's why 'admonish' can't be there.