On the Sphere and Cylinder

Question

How did Archimedes' personal relationships, particularly with King Hiero and the mathematician Conon of Samos, likely influence not only the types of problems he investigated but also the dissemination and eventual legacy of his discoveries?

Synthesized answer

The passages indicate that Archimedes was on intimate terms with King Hiero and studied at Alexandria, where he met Conon of Samos, whom he admired as a mathematician and cherished as a friend [3]. He was in the habit of communicating his discoveries to Conon before publication [3]. This relationship likely influenced the types of problems he investigated, as Hiero directly posed practical challenges—such as testing the gold content of a crown (leading to the principle of hydrostatics) and demonstrating how a small force could move a great weight (illustrating mechanics) [1][4]. These royal commissions spurred foundational work in physics and engineering.

Regarding dissemination and legacy, Archimedes sent problems to mathematicians at Alexandria, including a letter to Eratosthenes with the Cattle-Problem [5]. However, the passages do not detail how Conon or Hiero specifically affected the wider spread of his discoveries. The text notes that Archimedes himself valued pure mathematics over mechanical inventions and left few written records of them [3]. His death during the Roman capture of Syracuse, despite Marcellus’s orders to spare him, and the subsequent honorable burial and…

Synthesized from the book passages below. Chat with the book on Feynman for follow-up.

From the book

ning to leave any written record of them except in the case of the σφαιροποιἶα ( Sphere-making ), as to which see below. As, however, these machines impressed the popular imagination, they naturally figure largely in the traditions about him. Thus he devised for Hiero engines of war which almost terrified the Romans, and which protracted the siege of Syracuse for three years. There is a story that he constructed a burning mirror which set the Roman ships on fire when they were within a bowshot of the wall. This has been discredited because it is not mentioned by Polybius, Livy or Plutarch;…
Passage [3]
Hiero asked him to give an illustration of his contention that a very great weight could be moved by a very small force. He is said to have fixed on a large and fully laden ship and to have used a mechanical device by which Hiero was enabled to move it by himself: but accounts differ as to the particular mechanical powers employed. The water-screw which he invented (see below) was probably devised in Egypt for the purpose of irrigating fields. Archimedes died at the capture of Syracuse by Marcellus, 212 B.C. In the general massacre which followed the fall of the city, Archimedes, while…
Passage [5]
← Archimandrite 1911 Encyclopædia Britannica , Volume 2 Archimedes by Thomas Little Heath Archimedes, Screw of → See also Archimedes on Wikipedia ; and our 1911 Encyclopædia Britannica disclaimer . 359765 1911 Encyclopædia Britannica , Volume 2 — Archimedes Thomas Little Heath ​ ARCHIMEDES ( c. 287–212 B.C. ), Greek mathematician and inventor, was born at Syracuse, in Sicily. He was the son of Pheidias, an astronomer, and was on intimate terms with, if not related to, Hiero, king of Syracuse, and Gelo his son. He studied at Alexandria and doubtless met there Conon of Samos, whom he admired as…
Passage [2]
the question whether a crown made for him and purporting to be of gold, did not actually contain a proportion of silver. According to one story, Archimedes was puzzled till one day, as he was stepping into a bath and observed the water running over, it occurred to him that the excess of bulk occasioned by the introduction of alloy could be measured by putting the crown and an equal weight of gold separately into a vessel filled with water, and observing the difference of overflow. He was so overjoyed when this happy thought struck him that he ran home without his clothes, shouting εὒρηκα,…
Passage [4]
and stability of a right segment of a paraboloid of revolution floating in a fluid. (8) The Psammites ( Ψαμμίτης , Lat. Arenarius , or sand reckoner), a small treatise, addressed to Gelo, the eldest son of Hiero, expounding, as applied to reckoning the number of grains of sand that could be contained in a sphere of the size of our “universe,” a system of naming large numbers according to “orders” and “periods” which would enable any number to be expressed up to that which we should write with 1 followed by 80,000 ciphers! (9) A Collection of Lemmas , consisting of fifteen propositions in…
Passage [12]

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