CHAPTER VII.

HYDROSTATICS.

The science of hydrostatics is, even more than that of statics, the original creation of Archimedes. In hydrostatics he seems to have 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 Aristotle. This is the proposition that the surface of a fluid at rest is that of a sphere with its centre at the centre of the earth.

Archimedes founds his whole theory on two postulates, one of which comes at the beginning and the other after Prop. 7 of Book I. Postulate 1 is as follows:--

"Let us assume that a fluid has the property that, if its parts lie evenly and are continuous, the part which is less compressed is expelled by that which is more compressed, and each of its parts is compressed by the fluid above it perpendicularly, unless the fluid is shut up in something and compressed by something else."

Postulate 2 is: "Let us assume that any body which is borne upwards in water is carried along the perpendicular [to the surface] which passes through the centre of gravity of the body".

In Prop. 2 Archimedes proves that the surface of any fluid at rest is the surface of a sphere the centre of which is the centre of the earth. Props. 3-7 deal with the behaviour, when placed in fluids, of solids (1) just as heavy as the fluid, (2) lighter than the fluid, (3) heavier than the fluid. 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 difference between the weight of the solid and that of the fluid displaced. If the solid is heavier than the fluid, it will, if placed in the fluid, descend to the bottom and, if weighed in the fluid, the solid will be lighter than its true weight by the weight of the fluid displaced (Prop. 7).

The last-mentioned theorem naturally connects itself with the story of the crown made for Hieron. It was suspected that this was not wholly of gold but contained an admixture of silver, and Hieron put to Archimedes the problem of determining the proportions in which the metals were mixed. It was the discovery of the solution of this problem when in the bath that made Archimedes run home naked, shouting [Greek: eureka, eureka]. One account of the solution makes Archimedes 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, (2) by the same weight of pure gold, and (3) by the same weight of pure silver. This gives an easy means of solution. Suppose that the weight of the crown is W, and that it contains weights w1 and w2, of gold and silver respectively. Now experiment shows (1) that the crown itself displaces a certain volume of water, V say, (2) that a weight W of gold displaces a certain other volume of water, V1 say, and (3) that a weight W of silver displaces a volume V2.

From (2) it follows, by proportion, that a weight w1 of gold will displace w1/W . V1 of the fluid, and from (3) it follows that a weight w2 of silver displaces w2/W . V2 of the fluid.

Hence V = w1/W . V1 + w2/W . V2;

therefore WV = w1V1 + w2V2,

that is, (w1 + w2)V = w1V1 + w2V2,

so that w1/w2 = (V2 - V)/(V - V1),

which gives the required ratio of the weights of gold and silver contained in the crown.

The last two propositions of Book I. investigate the case of a segment of a sphere floating in a fluid when the base of the segment is (1) entirely above and (2) entirely below the surface of the fluid; and it is shown that the segment will in either case be in equilibrium in the position in which the axis is vertical, the equilibrium being in the first case stable.

Book II. is a geometrical _tour de force_. Here, by the methods of pure geometry, Archimedes investigates the positions of rest and stability of a right segment of a paraboloid of revolution floating with its base upwards or downwards (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) according to the specific gravity of the solid in relation to the fluid; where the position of rest and stability is such that the axis of the solid is not vertical, the angle at which it is inclined to the vertical is fully determined.

The idea of specific gravity appears all through, though this actual term is not used. Archimedes speaks of the solid being lighter or heavier than the fluid or equally heavy with it, or when a ratio has to be expressed, he speaks of a solid the weight of which (for an equal volume) has a certain ratio to that of the fluid.

BIBLIOGRAPHY.

The _editio princeps_ of the works of Archimedes with the commentaries of Eutocius was brought out by Hervagius (Herwagen) at Basel in 1544. D. Rivault (Paris, 1615) gave the enunciations in Greek and the proofs in Latin somewhat retouched. The _Arenarius_ (_Sandreckoner_) and the _Dimensio circuli_ with 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 years; it is now superseded by the definitive text with Latin translation of the complete works, Eutocius's commentaries, the fragments, scholia, etc., edited by Heiberg in three volumes (Teubner, Leipzig, first edition, 1880-1; second edition, including the newly discovered _Method_, etc., 1910-15).

Of translations the following may be mentioned. The Aldine edition of 1558, 4to, contains the Latin translation by Commandinus of the _Measurement of a Circle_, _On Spirals_, _Quadrature of the Parabola_, _On Conoids and Spheroids_, _The Sandreckoner_. Isaac Barrow's version was contained in _Opera Archimedis_, _Apollonii Pergoei conicorum libri_, _Theodosii Sphoerica_, _methodo novo illustrata et demonstrata_ (London, 1675). The first French version of the works was by Peyrard in two volumes (second edition, 1808). A valuable German translation, with notes, by E. Nizze, was published at Stralsund in 1824. There is a complete edition in modern notation by T. L. Heath (_The Works of Archimedes_, Cambridge, 1897, supplemented by _The Method of Archimedes_, Cambridge, 1912).

CHRONOLOGY.

(APPROXIMATE IN SOME CASES.)

B.C. 624-547 Thales 572-497 Pythagoras 500-428 Anaxagoras 470-400 / Hippocrates of Chios \ Hippias of Elis 470-380 Democritus 460-385 Theodorus of Cyrene 430-360 Archytas of Taras (Tarentum) 427-347 Plato 415-369 Theaetetus 408-355 Eudoxus of Cnidos / Leon fl. about 350 < Menaechmus | Dinostratus \ Theudius fl. 300 Euclid 310-230 Aristarchus of Samos 287-212 Archimedes 284-203 Eratosthenes 265-190 Apollonius of Perga