Clocks

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The Beaker Water-Clock

Introduction

Al-Jazari  himself wrote the introduction to this chapter, and It makes sense to bring his opening remarks:

“The king, Salih. Abu al-Fath. Mahmud, may God assist Islam by prolonging his life, proposed that I should make for him an instrument having no chains, balances or balls, not liable to rapid change or decay, from which could be told the passage of the hours and the divisions of the hours without inconvenience. It should be of handsome design and suitable for journeys or for settled residence. I considered the matter and made, according to his suggestion, what I shall now describe. “

What follows is the water clock of the scribe (in Arabic ورّاق). The clock design required two computational parts:

  • The clock face or dial supports solar
  • The slope of the beaker radius requires some understanding of fluid mechanics.

This post is relatively heavy in mathematics, and the “blue” parts (the technical explanation) are larger than usual. I Hope you can prevail them well.

The water clock of the beaker. Probably a dispersed manuscript from Cairo, 1354

How does it work?

The technical explanation, as always, will be colored in blue, so anyone who is not interested in pulleys or balancing weight can skip those bits. The drawing below is the Beaker water clock mechanism with my captions:

This is a copper beaker divided into two parts, upper beaker and a base are connected by an onyx with a very fine hole. The beaker is filled with water at the beginning of the day. The float is raised to its maximum height, and the weight is hanging down as far as possible. During the day the water would discharge slowly through the onyx to the base. As a result, the float would sink, and the weight would rise, causing the large pulley to rotates with the scribe and his pen. The water is sufficient for 14 hours and 30 minutes for the longest day of the year. At sunset, the water is returned to the beaker from the base, and the process repeats itself.

You can watch this short YouTube video from Technology & Science In Islam” showing the beaker clock :

iframe width=”854″ height=”480″ src=”https://www.youtube.com/embed/LNpDtxpBWes&#8221; frameborder=”0″ allow=”autoplay; encrypted-media” allowfullscreen></iframe>

Two engineering issues need further discussion:

  • The clock face and the variable length of the day.
  • How did al-Jazari find a practical solution to Bernoulli’s equation which he did not know or understood?

    The clock face and the variable length of the day

    In summer the days are long and the nights are short and vice versa in the winter. We’re moving the clock one hour forward at the beginning of the summer (“DST” – Daylight Saving Time), and at the fall we set the clock back. The Idea of the “DST” is attributed to Benjamin Franklin, and the rationale is energy saving, but it was suggested that daylight saving time improves quality of sleep, as we sleep longer during the darkness that allows deeper sleep and we know that a lack of sunlight can cause Seasonal Affective Disorder. Al-Jazari also dealt with the variable length of the day. Below is a screenshot from the YouTube clip. I added some captions.

    The clock face,  “Technology & Science In Islam” with my caption.

    The clock face is divided into eighteen bands, and each band is divided into twelve equal solar hours.  The outer band covers 3600; it is designed for ten days from June 21 (the summer solstice). The solar hour will be 300, but in Diyarbakır, there are about 14.5 hours of daytime so that the solar hour will be longer by~ 12 minutes in comparison to the constant hour. The eighteenth band(innermost) is intended for the last ten days of December. Diyarbakir has only 9.5 hours light, and therefore the band was shortened:

     9.5/14.5* 360 = 2360

    Every hour will be slightly less than 200 so the hour is only 46 minutes! 

    The concept of solar hours seems very strange in the 21st century and complicates everything. Just to think that programmers will be forced to change program timings with the calendar.

    Our notion of time rests on the celestial bodies movement. The years were counted based on the Sun or the Moon and the day, hour, minutes, and seconds were all derive from it. In fact, until 1967 the second was defined as 1/86,400 of a mean solar day. Only with the development of the Atomic clock, the definition was detached from the Earth’s rotation cycle, and the second is defined to be exactly 9,192, 631,770 cycles of a Cesium atomic clock. As weird as it may sound, atomic clocks and their ridicules precision are part of our daily life, and we cannot use Waze, or any navigation software, without them. In the world of the 12th-century solar hours made perfect sense and were more connected to nature and the movement of the celestial bodies.

    Bernoulli’s equation and the “solution” of al-Jazari

    A difficult problem in any water clock is that the water flow is not constant but depends on the water level in the tank. The following diagram illustrates the problem. For simplicity the beaker is cylindrical, and the onyx was inlarge for  clarity:

It is clear that at the beginning of the day when the beaker is full of water the water flow will be much stronger in comparison to the water flow after ten hours when the water level in the tank has dropped. How can we calculate the water flow and what can be done?

The mathematical solution to the problem was given by Daniel Bernoulli, a Swiss mathematician of the 18th century and a winner of the French Academy Award ten times. The first, to my surprise, was for a clepsydra (water clock) to measure time at sea. (I’m looking for specs of the clock and any assistance would be welcomed.) The many awards were not always a source of happiness. In 1734 he won the Academy Award with his father, Johann Bernoulli, a mathematician in his own right. The father couldn’t bear the shame of being equivalent to his son and banned Daniel from his house and did not reconcile with him until his death. I doubt that Joseph Cedar (Israeli movie director) was aware of the Bernoulli’s story, but the similarity to the movie “Footnote” is striking. The most important work of Daniel Bernoulli is hydrodynamics released in 1738:

Despite extensive research (I found six different studies!) that indicates that students of Physics and Engineering have conceptual difficulties to understand Bernoulli’s equation, I will challenge my readers with the solution of the water clock problem.

Bernoulli equation states:

Where :

P is the pressure.

rho is the water density.

g  is the gravitational acceleration~ 9.8 m/s2

h is the water height  above a reference plane.

v is the water velocity.  

He/she who wants to go deeper can go here and there are four lessons which I recommend at khan academy. Our problem looks like this:

We can write the Bernoulli equation:

 

Where  P1 is the pressure in the beaker, h1 is the height of the water in the beaker and v1 is the water flow velocity in the beaker. Respectively the pressure in the onyx is P2, h2 is the water height in the onyx, and v2 is water flow velocity in the onyx.  However, the beaker and the onyx are both open to the atmosphere. Thus P1 = P2 = 1 atm and can be removed. The water level in the beaker is h(t) and depends on time because when the water flows through the onyx to the base, h will be reduced. However, the onyx water height was determined as the reference plane and hence h2 = 0. Rearranging:

Since the onyx is very narrow in comparison with the beaker, we can assume that the flow in the onyx is much faster relative to the water velocity in the beaker  and can be neglected for the calculation of the water velocity in the onyx:

 

If this looks somewhat familiar, it is because this is Torricelli law and I used to run some very nice experiments with my middle school students at Beit Hashmonai:

Torricelli law, three identical holes at different heights

The amount of water through the onyx must be equal to the amount of water lost by the beaker:

Where A2 is the cross-section of the onyx  and A1 is the cross-section of the beaker:

Where r2 is the radius of the onyx. However, A1 is a function of time since the radius of the beaker is not constant but gets narrower at the bottom:

The velocity v1 is the change in the beaker water height:

We combine the last five equations:

Rearrange and make sure that the rate is constant (This is the reason for the whole exercise!) or:

For dh/dt to be constant, the radius of the beaker must be equal to the fourth root of the water height.

These mathematical tools were not available to al-Jazari. There is no evidence in the “Book of Knowledge of Ingenious Mechanical Devices” to the extensive mathematical knowledge that was available in the Muslim world of the 12th century.  I suspect that the mathematical education of al-Jazari was rather limited. This is a different topic and I hope to write a separate post in the future.

However al-Jazari was very resourceful, he developed a practical technique that allowed him to overcome the lack of mathematical tools. While preparing the beaker, he filled it with water and observed the outflow of the water with a reliable clock. If the float sank to the second mark, then the beaker radius is correct else al-Jazari hammered the beaker to widen it or make it narrower. Then the water is emptied from the beaker. The process was repeated for each mark. It is a pity that we do not have the beaker al-Jazari hammered to compare it to the theoretical calculation. One must admire the practicality of al-Jazari solution.

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The Scribe Candle clock, on clock face and hands

Introduction

The Scribe candle clock is the second scribe holding a pen out of three scribes that appear in the book. The scribe rotates continuously and passes fifteen degrees every hour, so one degree (one marking) is approximately four minutes. We already met a scribe holding a pen in the elephant water clock (in Hebrew), and soon I hope to write on the beaker water clock that has a different mechanism, but a very similar scribe. The scribe and his pen are used as a hand in a clock. It reminded me “modern” analog clocks and made me go back and examine the development of concepts such as minutes and seconds and the development of the clock dial.

The candle clock of the scribe ” Book of Knowledge of Ingenious Mechanical Devices” Topkapi manuscript, 1206.

How does it work?

Al-Jazari opens this chapter:

“I came upon a clock made by Yunus al-Asturlabi which had the appearance of the clock I described in the first chapter[ meaning the candle clock of the sword men]. A cross-beam which had a hole in its center for the wick replaced the cap which I used to hold the candle down, and I discovered that the wax flowed into the interior of the sheath and over the instruments inside the sheath. .. This gave much trouble; for this reason the design was useless. “

We do not know who Yunus al-Asturlabi was. Eilhard Wiedemann, a German physicist, one of the first researchers of science in Islam, who did much to bring the work of the al-Jazari to the west, suggested the astronomer and mathematician Ibn Yunus. Probably we will never know for sure. Correct identification or not, it is quite interesting because we have no evidence of any sophisticated candle clocks before al-Jazari’s.

The technical explanation, as always, will be colored in blue, so anyone who is not interested in pulleys or balancing weight can skip those bits. The drawing below is by the book translator and annotator Donald R. Hill with my captions:

A drawing of the mechanism by Donald Hill with my captions

The candle is placed on a holder inside a brass sheath, and only the wick protrudes through a hole in the cap. A long rod is soldered to the bottom of the holder. The rod runs through the main weight so that the weight is free to move up and down. Two strings are connected to the bottom of the rod and through two pulleys to the main weight. The latter is relatively heavy, slightly more than one kilogram. At nightfall the wick is lit, at that time the candle is in full size, the rod reaches its lowest point and the main weight its highest. As the candle is consumed, the main weight will descend exerting force, through the pulleys, on the holder upward and the holder and rod will go up at a constant rate depending on the rate of the combustion. A string which turns the scribe is attached to the bottom of the weight. Every hour the scribe and his pen will cover 150, so one can tell the time within 4 minutes. The holder pulls the ball’s channel up and every hour the highest ball in the channel has risen until it is level with the hole in the back of the falcon’s head, at which point it rolls out and falls from the falcon beak.

Minutes and their measurement

The globe and the clock face owe their divisions to a numerical system which is four thousand years old. The Babylonians made astronomical calculations using Sexagesimal (base 60) numeral system.  We can only conjecture why people of the ancient Middle East (Assyrians were also Sexagesimal ) adopted the use of base 60. One assumption is that the number 60 was chosen because it is the first number divisible by all the numbers 1 to 6. Alternatively, base 60 was preferred because the lunar year contains three hundred and sixty days. There are more suggestions. Hipparchus of Nicaea already mentioned here(Hebrew), as well as other Greek astronomers, used the tools previously developed by the Babylonians astronomers.  Hipparchus used the geometry of a sphere to find locations on Earth. There were attempts to use grid lines before, but he was the first to apply rigorous mathematical principles to the determination of places on the Earth’s surface, by specifying their longitude and latitude in terms of 3600 running South to North(longitude) and parallel to the equator(latitude).

Claudius Ptolemy considered the most famous astronomer of antiquity. His book the Almagest, from Arabic  (المجسطي) is considered to be one of the most influential scientific texts of all time. Its geocentric model whereby planets revolve around Earth was accepted for more than twelve hundred years until the work of Nicolaus Copernicus in the 16th century. Ptolemy used and expanded the work of Hipparchus by subdivisions of 3600 of longitude and latitude into smaller sections. Each degree was divided into sixty parts called “partes minutae primae” literally “the first small part.” This was later reduced to minutes. The minutes were further divided into sixty “partes minutae secundae” or “second small parts.” Later reduced to seconds.  Interestingly enough the time units in Hebrew “DAKA” and “SHNIYA” reflect the historical names.

Clock still didn’t show minutes and seconds for hundreds of years after the Almagest, partly because of technology limitations and partly because there was no need. In the middle ages, the meaning of an hour as sixty minutes was not understood by most people. Not many mechanical clocks from the fourteenth century are left, but those I could find do not have hands, in most cases, and ring a bell to indicate the hours.

The Salisbury cathedral clock is said to be the oldest working clock in the world. It is dated to 1386 (not certain). It is a large iron-framed clock without a dial and obviously with no hands. There are other clocks competing for this title. None of them has minutes’ hand:

The Salisbury cathedral clock

The Forchtenberg clock tower in a small town in south Germany is one of the oldest surviving mechanical clock towers. In contrast to the controversial dating of the Salisbury cathedral clock, the year 1463 is carved in iron. The only uncertainty; was the clock made at this date? Or could it be older and this is the first repair date? This clock has only an hour hand:

The Forchtenberg clock tower

Who was the first to install the minute hand? It is not clear, but the second hand has a story we know. Jost Bürgi was a Swiss clockmaker, a maker of astronomical instruments and a mathematician. He was employed at the Court William IV, Landgrave of Hesse-Kassel, a mathematician and astronomer by himself. Although now forgotten he was an outstanding astronomer, his observations, particularly those of the fixed stars, were at least as accurate as those by Tycho Brahe. Bürgi was brought to the court to develop scientific instruments, and assist in the observation that could confirm the heliocentric model by Copernicus. He built various instruments. In 14th April 1586, the count wrote to Tycho Brahe about a highly accurate clock which Bürgi had built which, for the first time, had a minute hand, a seconds hand and had an error of less than a minute in 24 hours! Christoph Rothman, another astronomer wrote about the new amazing clock:

“The duration of a second is not very short but resembles the length of the shortest note in a moderately slow song.“

This quote commemorates a time when science and technology produce a new reality.

Bürgi precision clock

Epilogue

I read today about a new exhibition of Christian Boltanski in the Israel Museum called “life”. He wrote: [my translation from Hebrew]

“a major part of my job is the fact that each person is special, one-of-a-kind and important, each will finally vanish. Most of us will be forgotten in two generations, with the passing of those close to us. “

It’s certainly not true for al-Jazari but probably true for most of us. The exhibition combines early works of Boltanski alongside new works and includes a digital timer continually counting the seconds from the moment of birth of the artist. I found a photo of a timer installation of Boltanski at the Biennale. I don’t know if the installation in the Israel Museum is identical.

Christian Boltanski, the Venice Art Biennale, 2011.