From τὰ φυσικά (ta physika) to physics – XXXII

As should be clear by now the renaissance in awareness and study of the works of Archimedes, which had been largely ignored in the medieval period, played a major role in the new developments in physics in the sixteenth century. Benedetti (1530–1590), Stevin (1548–1620), and Galileo (1564–1642) all turned to Archimedes’ On Floating Bodies, which had been published in print for the first time, both in Italian and Latin by Tartaglia (1499–1557), in order to develop an alternative concept for the laws of fall to that of Aristotle. However, as might be expected, this important work by Archimedes not only played a central role in the development of a new dynamics but naturally in new developments in hydrostatics. 

As with many works by mathematicians from antiquity Federico Commandino (1509–1575) commented on and  corrected the printed version of Tartaglia’s Latin translation (1565), which he published in Bologna in 1565. 

On Floating Bodies is in two book. Book I establishes general principles. His concept of hydrostatic pressure is presented as a basic postulate and is somewhat obscure:

Let it be granted that the fluid is of such a nature that of the parts of it which are at the same level and adjacent to one another that which is pressed the less is pushed away by that which is pressed the more, and that each of its parts is pressed by the fluid which is vertically above it, if the fluid is not shut up in anything and is not compressed by anything else.

 He spells out the law of equilibrium of fluid and proves that water with adopt a spherical form around a centre of gravity, such as the water on the surface of the Earth. Most famous is, of course, Proposition 7, in Book I, the so-call Principle of Archimedes:

“Solids heavier than the fluid, when thrown into the fluid, will be driven downward as far as they can sink, and they will be lighter [when weighed] in the fluid [than their weight in air] by the weight of the portion of fluid having the same volume as the solid.”

The more succinct modern version is:

Any body wholly or partially immersed in a fluid experiences an upward force (buoyancy) equal to the weight of the fluid displaced.

In addition to the principle that bears his name, Archimedes discovered that a submerged object displaces a volume of water equal to the object’s own volume (upon which the story of him shouting “Eureka” is based). This concept has come to be referred to by some as the principle of flotation

Book II of On Floating Bodies is a detailed study of the stable equilibrium positions of floating right paraboloids of various shapes and relative densities when floating in a fluid of greater specific gravity, according to geometric and hydrostatic variations. It is restricted to the case when the base of the paraboloid lies either entirely above or entirely below the fluid surface.

It is assumed that this study is an idealisation of the shape of ship’s hulls. The study is generally regarded as one of Archimedes greatest mathematical achievements.

A copy of Commandino’s Latin text of On Floating Bodies found its way into the hands of Simon Stevin (1548–1620), who then in his De Beghinselen des Waterwichts (Principles on the weight of water) published in 1586 wrote the first systematic treatise on hydrostatics since Archimedes.

Like Archimedes’ work, Stevin’s book is in two parts, a theoretical part, The Elements of Hydrostatics, and a practical part, Preamble to the Practice of Hydrostatics. An intended more complete practical part was apparently never written. 

In part one, Stevin argued on practical grounds that water surfaces could be taken to be flat, for Archimedes they were part of a sphere, and vertical lines were parallel to each other, for Archimedes they converge to the centre of the sphere. As with his discussion of the inclined plane, Stevin used a perpetual motion argument to show that water keeps its position in water. He argued that if a portion of water moved then the portion that took its place must also move leading to perpetual motion, the third portion replacing the second and so on ad infinitum, and perpetual motion is an absurdity. (Meli, see footnote 1, p. 43)

Stevin rephrased Archimedes’ principle thus:

The gravity of any solid body is as much lighter in water than in air as is the gravity of the water having the same volume. 

This is Theorem 7 in The Elements of Hydrostatics, with his Theorem 8, Stevin breaks new ground in hydrostatics contributing the first major advance in the discipline since Aristotle with the so-called hydrostatic paradox. In its modern form this reads: 

The barometric formula depends only on the height of the fluid chamber, and not on its width or length. Given a large enough height, any pressure may be attained. This feature of hydrostatics has been called the hydrostatic paradox. As expressed by W. H. Besant (Elementary Hydrostatics, George Bell & Sons, 1900, p. 11)

Any quantity of liquid, however small, may be made to support any weight, however large.

In 1916 Richard  Glazebrook (1854–1935) mentioned the hydrostatic paradox as he described an arrangement he attributed to Pascal (1623–1662): a heavy weight W rests on a board with area A resting on a fluid bladder connected to a vertical tube with cross-sectional area α.

Pouring water of weight w down the tube will eventually raise the heavy weight. Balance of forces leads to the equation

W = wA/𝛼

Glazebrook says, “By making the area of the board considerable and that of the tube small, a large weight W can be supported by a small weight w of water. This fact is sometimes described as the hydrostatic paradox.” (Wikipedia)

Stevin studied the problem of calculating the weight at the bottom of vessels with different shapes. It is here that he extended Archimedes doctrine to new areas, studying both containers with their bottoms parallel to the horizon and those with their bottoms inclined to the horizon. He examined the first case in theorem 8, stating that the weight the weight on the bottom is given by the gravity of the water whose volume is equal to that of the prism whose base is that bottom and whose height is the vertical from the plane through the water’s upper surface to the base” (fig. 2.3).

Stevin illustrate the problem in several corollaries. Wherein the water is replaced by a solid body with a peculiar shape and equal specific gravities to the water. Subsequent theorems deal with the weight of water in containers with an inclined base. In the Preamble of the Practice of Hydrostatics Stevin provided a number of ingenious examples involving scales and vessels so that “anyone may test and see with his own eyes.”^[1]

In a related issue several researchers provided values for the ratios between the weights of equal volumes of different substances, among them Tartaglia, Giambattista della Porta (1535–1615), and the Jesuit Juan Bautista Villalpando (1552–1608).

The most extensive analysis was presented in tables by Marino Ghetaldi (1568–1626) in book on hydrostatics according to Archimedes, Patricii Ragvsini promotvs Archimedis sev de varijs corporum generibus grauitate et magnitudine comparatist published in Rome in 1603.

Using a hydrostatic balance to measure the decrease in weight of bodies of different materials in water. Ghetaldi worked with great care and skill to ensure maximum accuracy. For example, he used cylinders rather than spheres because they could be made with greater precision then converting then calculating the weight of a sphere inscribed in a cylinder using Archimedes theorem. 

These new approaches to Archimedes’ work saw a relaunch of hydrostatics at the beginning of the seventeenth century, which would be taken up and extended in the following decades by Torricelli (1608–1647) and Pascal (1623–1662) amongst others.

^[1] Domenico Bertoloni Meli, Thinking with Objects: The Transformation of Mechanics in the Seventeenth Century, The Johns Hopkins University Press, 2006, p. 44