Synthesized answer
Archimedes’ “Eureka!” moment happened when he stepped into a bath and saw water overflow. He realized he could test King Hiero’s crown by putting the crown and an equal weight of pure gold separately into a full water vessel and measuring the difference in overflow. If the crown had silver mixed in, it would be bulkier and push out more water. He was so excited he ran home naked shouting “I have found it” [1].
Beyond this simple test, Archimedes uncovered the fundamental principle of hydrostatics: that a floating or submerged object displaces its own weight or volume of fluid. This discovery is the “foundation of that whole science” of hydrostatics [2]. It was significant because it allowed him to measure density and purity of materials by comparing weight and displaced water, and it led to his later work on floating bodies, such as the stability of a paraboloid segment in fluid [4]. The passages do not give a formal statement of the principle (like Archimedes’ principle), but they clearly state that this discovery founded the science of hydrostatics [2].
Synthesized from the book passages below. Chat with the book on Feynman for follow-up.
From the book
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,…
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;…
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…
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…
death of so illustrious a person, directed an honourable burial to be given him, and befriended his surviving relatives. In accordance with the expressed desire of the philosopher, his tomb was marked by the figure of a sphere inscribed in a cylinder, the discovery of the relation between the volumes of a sphere and its circumscribing cylinder being regarded by him as his most valuable achievement. When Cicero was quaestor in Sicily (75 B.C. ), he found the tomb of Archimedes, near the Agrigentine gate, overgrown with thorns and briers. “Thus,” says Cicero ( Tusc. Disp. , v. c .
More questions about this book
- Archimedes explicitly valued mathematical research over his famous mechanical contrivances. If you were explaining his perspective, why might someone consider practical, world-changing inventions "beneath the dignity of pure science," and what does this reveal about his ultimate intellectual priorities?
- How do the "Eureka!" story (hydrostatics) and the "Give me a place to stand and I will move the earth" statement (mechanics) both illustrate Archimedes' unique approach to understanding and leveraging physical principles, even though they concern different domains of science?
- The text notes that the story of the burning mirror is "discredited" by historians. What is the value of critically evaluating popular stories against historical accounts when studying a figure like Archimedes, and what does this process tell us about the nature of scientific and historical truth?
- 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?