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?

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…

lic segment, (2) of the area included between any two parallel chords and the portions of the curve intercepted by them. (6) The Quadrature of the Parabola ( Τετραγωνισμὸς παραβολῆς ) is a book in twenty-four propositions, containing two demonstrations that the area of any segment of a parabola is 3 of the triangle which has the same base as the segment and equal height. The first (a mechanical proof) begins, after some preliminary propositions on the parabola, in Prop. 6, ending with an integration in Prop. 16. The second (a geometrical proof) is expounded in Props. 17-24. (7) On Floating…

b of Archimedes, near the Agrigentine gate, overgrown with thorns and briers. “Thus,” says Cicero ( Tusc. Disp. , v. c . 23, § 64), “would this most famous and once most learned city of Greece have remained a stranger to the tomb of one of its most ingenious citizens, had it not been discovered by a man of Arpinum.” Works .—The range and importance of the scientific labours of Archimedes will be best understood from a brief account of those writings which have come down to us; and it need only be added that his greatest work was in geometry, where he so extended the method of exhaustion as…