Most of the books I've read that introduce Archimedes' achievements so far are mostly exaggerated, so it's difficult to objectively understand his achievements.
Therefore, I would appreciate it if you could recommend a book that describes Archimedes' achievements as objectively as possible.
A nice book that discusses Archimedes and his works is "Archimede – alle radici della scienza moderna" by Pier Daniele Napolitani. It is in Italian; however, the author has made it available for free and it is searchable: you can select the text and use an automatic translator. It might be a bit inconvenient, but don't look a gift horse in the mouth.
Another well-known book about Archimedes is Archimede, Carocci, 2019, by Lucio Russo, the author of The Forgotten Revolution.
It is in Italian, however, and unfortunately, it has not been translated in English, as far as I know.
But it is surely rigouros and not hagiographic or anecdotal. The presentation of the book says:
Several works of Archimedes have come to us, but only few specialists read them. Some of his results [...] are taught at school, but without mention of the way Archimedes reached them. So, the thought of the most important exponent of the Hellenistic school continues to be ignored also by classicists and scientists, while its fame with the public is left to the recollection of some legendary anecdotes.
The critical analysis of the episodes transmitted by the tradition is here accompanied by the exposition of a substantial part of Archimedes' results, sufficient to see at work his scientific method.
