The Polymers Center | Plastic Resin Manufacturing

NEWTON AND THE WATER CLOCK

We think of mistletoe in terms of Christmas celebrations, but for centuries it was associated with an adhesive thread derived from its berries, called viscin, proving that even parasites may provide some measure of utility.¹ To the ancients, viscin was related to viscosus, which referred to anything sticky and was also related to that which melts or flows, especially a malodorous or foul fluid.² That, of course raises questions of fluid flow. Given that such flows carry with them the possibility of a measure of time, viscous enjoyed a long association with the clepsydra (Greek for “water thief”), a device whose origins are lost in pre-Babylonian history, but whose original function was to permit one to draw clean water from a well. A vessel with a hole at the bottom, the idea was to insert it into the standing water in a fountain and permit it to fill with sub-surface water, avoiding the uppermost, viscin layer. It became apparent that, instead of water flowing into such a vessel, that which issued from it emptied at a known rate, making the clepsydra ideal as a timing device. It found hundreds of uses, for example in court cases where it constrained bombastic orators, “whose time, literally, ran out.”³

But the clepsydra suffered from an obvious failing since the water dripping out of it ran at a non-uniform rate. It ran faster at the beginning when the pressure or head was high, then more slowly as the vessel drained. If it were asked to divide its time into halves or quarters, it failed. Worse, it could not cycle itself, like a clock, so it could not tell the time of day. One solution was to keep it filled, such that the head of the column remained nearly constant. Babylonian diviners originally developed such a vessel. Termed a dibdib, it measured time by tracking the weight of the water discharged, such that when the weights balanced, the time was up.⁴ That solution relied on a constant head to supply the clepsydra—but what if one could design a vessel such that it could show equal times by equal heights of the water column? The Egyptians developed a rough approximation of such a device, known as the Karnak Clepsydrae, in 1400 BC. Its conical shape improved the accuracy, but it still suffered from a known inaccuracy that eluded correction.

The Karnak Clepsydrae, from the Egyptian Museum

Water clocks found their highest expression in the “Tower of the Winds,” in the Roman Agora of Athens. There, a complex series of storage tanks and reservoirs provided a nearly constant head, water from which drove a floating rack that, as it rose, turned an axle. That axle in turn drove a revolving anaphoric disk that showed the hours, sun, and stars, and which could even be adjusted for the seasons. It would take until the twelfth century and the development of the verge and foliot escapement for mechanical clocks, and another 350 years after that, with Galileo’s pendulum clock, to improve their accuracy enough to enjoy universal adoption as time-keeping devices.⁶

Building on Galileo’s work on falling bodies, Evangelista Torricelli in 1644 developed his formulation for the velocity of water “issuing with violence” from the bottom of a vessel. He showed that it was proportional to the square root of the height or head times a constant (we now know to be gravitation). Torricelli was part of the “Italian School of Hydraulics,” who, over the next fifty years experimented with the resistance of various shaped bodies to fluid flow and established that a relationship

existed between that resistance and the square of the velocity of the body that moved in through it.⁸

At about the same time as the Italian School developed, in 1654, twelve-year-old Isaac Newton started building clocks. According to a biographical account of him by William Stukeley, Newton was unimpressed with a mechanical clock that he had built, and after reading a description of a clepsydra in John Bates’s Mysteryes of Nature and Art, decided to build his own. He constructed it himself, including a specifically designed face, which rendered the time remarkably accurately.⁹ He even referenced it in a paper read before the Royal Society years later—and one suspects that it was his early experience with the clepsydra that informed, in part, his investigations of fluid flow. He sought to understand it though his model of impact, in which he conceptualized flow in terms of solid bodies under acceleration.¹⁰ To determine the velocity of water escaping through a hole in the bottom of a vessel, he imagined the rotated parabola, the top of which being the diameter of the vessel and the bottom that of the discharge hole, and at one point posited that one could consider the funnel shape to be made of ice, and to offer no resistance to the flow of water through it.¹¹ In such a case, he stated, the water would reach the original height of the top of the column, a conclusion that conformed with observation but that ignored any resistance to flow of that water in the column as it traversed those layers adjacent to it—the Newtonian fluid.

Newton’s solution to measure

the velocity of a fluid¹²

Accordingly, one might think that he had no idea of viscosity, but that would be wrong. In his time that word still meant “sticky,” and accordingly he wrote (in translation) in his Principia, “The resistance arising from the want of lubricity in the parts of a fluid, is, other things being equal, proportional to the velocity with which the parts of the fluid are separated from one another.”¹³ Since he wrote in Latin, it is doubly hard to translate, but his “want of lubricity” seems to refer to a kind of stickiness, and since Newton’s mechanical philosophy was grounded in a corpuscular concept of matter, he was thinking of those corpuscles sticking together as opposed to those that slid around each other easily. From 1687 to 1726 Newton continued to refine his fluid dynamics, dividing the resistance into inertial and “inviscid” fluids in his Propositions 36-39.¹⁴ Euler and others would develop those relationships over the next century.

And although we grant Newton priority in the conceptualization of viscosity, the concept had not penetrated western scientific consciousness a century after Newton’s death. When Alexander von Humboldt visited the Mission of St. Balthasar along the Atabapo River in Venezuela he encountered a curious substance, known to the natives as dapicho, which when heated, softened and became elastic and somewhat viscous. He went on to compare it to the juice of pinnate leaves, which he stated was milky, “and very thin, and almost destitute of viscosity.”¹⁵ One might think that Humboldt was referring to the resistance of the milky juice to flow, but no, he was pointing out that it was sticky. And while my grandfather’s edition of The American Civil Engineer’s Pocket Book, c. 1916, describes viscosity as the resistance to flow, it makes no mention of non-Newtonian fluid flow, probably because it was not much under consideration at that time.

As for us, we formalize the concept by comparing a stationary fluid between two fixed plates. When the bottom plate begins to move, how does that movement get transmitted to the fluid between it and the upper plate? The “parts of the fluid” are layers, with the bottom most moving nearly at the speed of the bottom plate and the topmost layer being almost stationary.

Idealized shear stress diagram for flow between one moving and one stationary plate.

With the development of plastics and paints it soon became plain that for those materials an increase in force did not proportionally increase the flow of the fluid—instead they thinned—and as the force (shear) increased the flow (rate) increased even more. Such observations become more common in the first half of the twentieth century, and accordingly more complex. (Yet another example of technology leading investigation.) Rapidly growing interest in rheology led to an unexpectedly high turnout for a “Plasticity Symposium” at Lafayette College in 1924, and five years later the formation of The Society of Rheology.¹⁷ Today, viscosity measurements at different pressures and temperatures are essential for the design of dies and molds, something that we consider both necessary, unremarkable, and dissociated from mistletoe.

Footnotes:

¹ Jack Tamisiea, ”Mistletoe Sticks Around,” Scientific American, Vol. 327, No. 4 (October 2022), 15. See also, Horbelt, N., Fratzl, P. and Harrington, M.J. “Mistletoe viscin: A hygro- and mechano-responsive cellulose-based adhesive for diverse materials applications.,”(2022) PNAS Nexus. 1, pgac026. DOI: 10.1093/pnasnexus/pgac026.

² https://www.etymonline.com/word/viscous

³ A. A. Mills, “Newton’s Water Clocks and the Fluid Mechanics of the Clepsydre,” Notes and Records of the Royal Society of London Vol. 37, No. 1 (August 1982), 35-61.

⁴ David Brown, John Fermor and Christopher Walker, “The Water Clock in Mesopotamia,” Archiv für Orientforschung Bd. 46/47 (1999-2000), 130-148.

⁵ https://egypt-museum.com/clepsydra-of-karnak/

⁶ Donald Cardwell, Wheels, Clock, and Rockets: A History of Technology (New York: W. W. Norton, 1995), 39.

⁷ Joseph V. Nobel and Derek J. de Solla Price, “The Water Clock in the Tower of the Winds,” American Journal of Archaeology, Vol. 72, No. 4 (October 1968), 345-355.

⁸ Julián Simón Calero, trans. Veronica H. A. Watson, The Genesis of Fluid Mechanics, 1640-1780, (Dordrecht: Springer, 1996),13.

⁹ Mills, “Newton’s Water Clocks, 46.

¹⁰ Richard Westfall, Force in Newton’s Physics: The Science of Dynamics in the Seventeenth Century, (New York: Science History Publications, 1971), 499.

¹¹ Newton concluded that a column of water would acquire only enough velocity to rise to half the height of the top of the water in the vessel, which it doesn’t. Realizing his error, he amended his solution in the second edition of his Principia.

¹² Westfall, Force in Newton’s Physics, 503.

¹³ Newton, Principia Book II Section 9.

¹⁴ Peter Rowlands, Newton: Innovation and Controversy, (London: World Scientific Publishing, 2018) 157.

¹⁵ “Indian Rubber,” The Guardian, July 7, 1821, p. 4 col. 5; taken from, Alexander von Humboldt, Personal Narrative of Travels to the Equinoctial Regions of the New Continent, during the years 1799–1804, (London, 1821), 405.

¹⁶ https://www.iitg.ac.in/kartha/CE203FM/Lectures/Week1/Lecture_3%20Fluid%20Properties.pdf

¹⁷ Deepak Doraiswamy, “The Origins of Rheology: A Short Historical Excursion,’ White Paper, DuPont iTechnologies, Experimental Station, Wilmington DE. http://www.rheology.org/sor/publications/Rheology_B/Jan02/Origin_of_Rheology.pdf