My question is only about a certain hystorical gem. The purpose of this question is to explore how vast were Archimedes's contributions to geometry that are not connected to calculus (i.e calculation of areas and volumes of curved figures), i.e to pure geometry. I know that Archimedes introduced his "twin circles" in proposition 5 of his book of lemmas, but the wiki article about the book of lemmas states that he also made use of Pappus's chain of tangential circles in this book, but i coudn't find where. So if anyone know the answer, please write it down.
No, he did not. Wikipedia's editor was most likely misled by somewhat clumsy arrangement of the text in the Heath's 1897 translation of the Book of Lemmas. Right under the statement of Proposition 8 there is a long passage in small font describing the construction of the Pappus chain. This passage however is the continuation of the footnote from the previous page, which starts as:"It may be interesting to add the enunciation of the "ancient proposition" stated by Pappus (p. 208) and proved by him after several auxiliary lemmas". In other words, the only trace of Pappus chains in the Book of Lemmas is in... Heath's retelling of the chain described in Pappus's Mathematical Collection.
What we find in the book is the construction of the arbelos, a plane figure bounded by three semi-circles resting on the same segment, with the diameters of the smaller ones adding up to the largest one. The footnote refers to Proposition 6, which describes how to find the diameter of the circle inscribed into the arbelos, i.e. the one touching all three semi-circles from the inside. In hindsight, this is the first (the largest) circle in the Pappus chain. But this is as far as it goes, the book does not inscribe any further circles into the arbelos, and the "twin circles" are unrelated to Pappus chains. Everything beyond that (iterating the inscribed circle construction, demonstrating that the centers of the circles all lie on an ellipse, etc.) is found only in Pappus's Collection.
The only reason we know that the chains are not due to Pappus is because he himself calls the result about it an "ancient lemma". Since Pappus is usually meticulous about giving credit when he knows the source (we know of many Hellenistic mathematicians' work largely due to Pappus) this probably means that in this case the lemma was already folklore. This was highly unlikely to happen to a result of Archimedes. It is worth mentioning that even the Book of Lemmas itself is not a work of Archimedes. Here is Heath:
"... we have a collection of Lemmas which has reached us through the Arabic. [...] The Lemmas cannot, however, have been written by Archimedes in their present form, because his name is quoted in them more than once. ...though it is quite likely that some of the propositions were of Archimedean origin, e.g. those concerning the geometrical figures called respectively arbelos (literally 'shoemaker's knife') and salinon (probably 'salt-cellar')..."
Pappus's Collection dates to c. 340 AD. But according to Acerbi's Two Approaches to Foundations in Greek Mathematics, Pappus's list of "moderns" in Collection IV.58 includes Demetrius of Alexandria, Phylo of Tiana, and Menelaus of Alexandria, Proclus also mentions Geminus among "moderns". These characters lived as far back as the 1st century BC or after, so the "ancient lemma" most likely originated in the late third, or second century BC, near the end of Archimedes's lifetime, or shortly after his death in 212 BC.