Archimedes

e had no predecessors. Only one of the facts proved in his work On Floating Bodies, in two books, is given with a sort of proof in A

, one of which comes at the beginning and the other

h is less compressed is expelled by that which is more compressed, and each of its parts is compressed by th

pwards in water is carried along the perpendicular [to the su

uid. It is proved (Props. 5, 6) that, if the solid is lighter than the fluid, it will not be completely immersed but only so far that the weight of the solid will be equal to that of the fluid displaced, and, if it be forcibly immersed, the solid will be driven upwards by a force equal to the

imedes use the proposition last quoted; but on the whole it seems more likely that the actual discovery was made by a more elementary method described by Vitruvius. Observing, as he is said to have done, that, if he stepped into the bath when it was full, a volume of water was spilt equal to the volume of his body, he thought of applying the same idea to the case of the crown and measuring the volumes of water displaced respectively (1) by the crown itself

will displace w1/W · V1 of the fluid, and from (3) it follows

= w1/W · V1

??? WV = w

(w1 + w2) V =

1/w2 = (V2 ?

o of the weights of gold and

e segment is (1) entirely above and (2) entirely below the surface of the fluid; and it is shown that the segment will in

(but completely above or completely below the surface) for a number of cases differing (1) according to the relation between the length of the axis of the paraboloid and the principal parameter of the generating parabola, and (2) acc

the solid being lighter or heavier than the fluid or equally heavy with it, or when a ratio has to be expres

IOGR

ith Eutocius's commentary were edited with Latin translation and notes by Wallis in 1678 (Oxford). Torelli's monumental edition (Oxford, 1792) of the Greek text of the complete works and of the commentaries of Eutocius, with a new Latin translation, remained the standard text until recent year

row's version was contained in Opera Archimedis, Apollonii Perg?i conicorum libri, Theodosii Sph?rica, methodo novo illustrata et demonstrata (London, 1675). The first French version of the works was by Peyrard in two volumes (second edition, 1808). A

ONO

ATE IN SO

.

547

7 Pyth

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as of

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ytas of Tara

347

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Eudoxus

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eu

300

ristarchu

2 Arch

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