Building the Elephant Clock in Lego


The elephant clock is by far the most popular of all the works by al-Jazari. There have been several modern reconstructions, multiple animations, and it has its own Wikipedia entry, and more. In the previous post, I tried to explain why millions of viewers in the mall in Dubai or the “1001 inventions” were captivated by its magic. The current post is different from anything I’ve written so far and is a record of my journey to reconstruct the elephant clock in Lego, including the difficulties and the learning on the way. The elephant is in the initial stages, and I hope to post an update every week. I’d love to hear your suggestion, ideas, or advice you may have for me.

Elephant clock, Topkapi manuscript, 1206.

Why build?

When I started this journey, I was working at the Davidson Institute of science education. I proposed to Prof. Haim Harari, founding Chairman of Davidson Institute and former President of the Weizmann Institute of science, to build al-Jazari fifty machines in the “Science Garden”, an open-air museum in Weizmann Institute. Haim refused and told me that the Science Garden was rooted in the 17th-century Newtonian physics and he would like to bring it into the 21st century. Instead, my proposal would take us back to the 12th century… This blog is my alternative exhibition.

Beyond the magic of al-Jazari machines and their value to the history of technology, It is important in undermining stereotypes about Islam that exist both in the Jewish and Arabic population in Israel. Arab Labor ( “AVODA ARAVIT”) is a racial slur that is widely used long before the amusing sitcom written by Sayed Kashua. The slur indicates low quality work. On the other hand, Arab society perceives itself as debilitated and with little contribution to the world of science. Arab students come to Weizmann Institute feeling this is the “temple of science” where they do not belong. The wonderful machines of al-Jazari emphasize the scientific achievements of the Islam golden age, which is not taught at schools at all, and offer a different perspective to both Jews and Arabs.

Previous posts included animations. To me, there is no replacement for actually building the machine. The difficulties on the road and resulting learning are priceless.

Why Lego?

Some reconstructions of al-Jazari were made, including the Elephant Clock. These are pictures of three of reconstructions:

Three reconstructions of the Elephant Clock, left Ibn Battuta Mall in Dubai, in the middle Sharjah Museum of Islamic Civilization, right the Istanbul Museum of The History of Science & Technology in Islam.

I have a lot of respect and appreciation for the restorers. Their Elephant clock is possibly more similar to the original than I am assuming. Somehow their beautiful work can’t recreate the beauty of the illustrations in the book and in the process of transferring the 2D illustration to a 3D reconstruction some of the magic is lost. I think they are too realistic. The elephant is a “real” elephant and the canopy is beautiful and very dignified. There is no room left for our imagination. Also, the Elephant Clock is static in Istanbul as well as in the 1001 inventions exhibition, more like a statue demonstrating the beauty of the design rather than a working water clock. In Dubai, I think, the Elephant Clock is powered by electric motors. This reduces the experience, at least in my eyes. The reconstruction should rely on 12-century technology and the ability of al-Jazari to perform complex control scheme before we had electronics and controllers. The magic of al-Jazari is contemporary (strange but true).

I have no reason to assume that I know something the restores did not know. The move to LEGO simplifies the aesthetic choices and offers playfulness and sense of contemporary at the same time. No LEGO work is a part of the Turkish tradition of miniature art. However, Lego is free to correspond with this art. Lego has its own design language, and even adults works in Lego and Hyper-technology Lego works are fun and playful, also this is a great excuse for me to build in Lego…

Previous al-Jazari machines in Lego

I know only one of al-Jazari machine made in LEGO. Interestingly enough, it is the Elephant Clock. You can see it here:

I Think the design is charming and the use of LEGO is both clever and quite sophisticated. There’s breathing space in the Lego elephant, and I like the human figures, which are an entertaining use of LEGO components. But although the design follows al-Jazari, its operation is based on the “smart brick” (a programmable, lego controller which serves as the brain of LEGO robots) and electric motors. This, in my mind, defies the point. Or at least the challenge I am hoping to meet: reconstruct, and on the way test and learn,al-Jazari engineering from the 12th century.

Where am I

I made a plan to build a  LEGO elephant, 60  bricks high(about 60 cm). There is no precise measurement in the Book of Knowledge, but this is roughly 2:1 scale in relation to the book. The giant elephant you see in the Dubai restoration, for example, is elephant real size (~3.5 m) rather than the dimensions extracted from the original work. I used Tinker Cad, I found it very user-friendly. I inserted a 3D a model of an Asian elephant and filled it with LEGO bricks:

At this point, I did not insert to the model the other components of the clock, such as the canopy, Mahout, the dragons, etc. l will see as I go if I prefer hands-on experimentation or the use the CAD software.

I ordered 5420 dark gray Lego bricks of various sizes from eight different suppliers, from Denmark to Croatia, from the U.S. to France. Thank you bricklink (Internet market for LEGO) I wouldn’t manage without you. The reason for so many suppliers is simple. No one supplier had so many gray parts. Last week I began to build. It looks like this:

About one-third of my LEGO elephant. The trunk rests in our cereal Bowl. Otherwise, it would fall. I began experimenting with the buoy and the mechanism of the scribe, but this will be in my next post.


The Elephant Clock – Multiculturalism or a Circus?


The elephant clock is by far the most popular of all al-Jazari’s works. There are few modern reconstructions of it, some in different exhibitions and museums, but also in the Dubai Mall. It has a variety of animations in 2D and 3D, and it has a Wikipedia entry of its own. Due to the complexity of the mechanism, I divided this post to two; in the first part, I will explain what the viewer sees and try to explore the sources of the magic. The second part will be more engineering oriented, and I will explain how the mechanisms work in the backstage, and what is so unique in this clock.

The Elephant clock, manuscript from 1315, Syria

What does The viewer see?

An elephant,  approximately one meter and twenty centimeters long, carries on its back a canopy with four pillars and a castle. On top of the castle’s dome, there is a bird. Inside the elephant, there is a hidden water reservoir and a sinking float(a float with a hole which sinks slowly) during half an hour. More details in the next post. In the canopy sits a scribe holding a pen pointing at semi-circle with tick marks. During this half an hour the scribe rotates and his pen indicating the minutes passed. At the end of the every half an hour, the scribe will return to its original position. At the same time, there will be quite an impressive show. Between the elephant shoulders, ride a mahout (the elephant keeper and driver). In his right hand, there is an ax and a mallet in his left. Every half an hour the mahout will strike the copper elephant, first, with the ax and then with the mallet. In the castle balcony sits a man, his hands are on the heads of two falcons like he keeps them from opening their beaks. Once every half an hour, he would raise his hand, right or left and the Falcon will emit a ball. The ball falls to the Dragon’s maw, make the dragon swing on its axis and lay the ball on the cymbal in a jar. Also, the bird on the dome will go spinning. Above the head of the Falconer (falcon trainer), there is a semi-circle with fifteen black holes. Every full hour one hole will be colored white, thus indicating the passing hours since sunrise.

You can see a short video demonstrating the elephant clock and explaining the mechanism. A fuller explanation in the next post.

Why an elephant?

In Wikipedia, the Elephant water clock entry, it says:

“The elephant represents the Indian and African cultures, the two dragons represents ancient Chinese culture, the phoenix represents Persian culture, the water work represents ancient Greek culture, and the turban represents Islamic culture” signifying the multicultural mentality of the intellectual al-Jazari. “

The quote is attributed to al-Jazari himself. I am afraid that the whole concept of multiculturalism is completely foreign to the 12th century and al-Jazari. The quote certainly is not by al-Jazari, but whoever wrote it (who?) explained the wealth of the clock. It made me think, and first of all about the elephant.

The Middle Ages and strange elephants

The trade routes in the middle ages were spread over Europe and the Middle East but also in India, China, and  Africa. On the East coast of Africa, they traded ivory, gold,  ebony, and slaves. China exported silk and porcelain and India spices and drugs. It means that rumors about elephants, giraffe, and other exotic animals reached Europe but the artists that drew the manuscripts had never seen an actual elephant drew them based on his imagination. There is an entire site dedicated to the weird drawings of elephants. I give just two examples:

Thomas of Cantimpré, Liber de natura rerum, France 1290

A hoofed wooly elephant “Livre des simples médecines” a manuscript from the 15 century.

The original manuscript was unfortunately lost, but the manuscript from Topkapi is from 1206, I wrote about it here. This is the year al-Jazari died, so it is probably “firsthand”  copy. You can see that the elephant looks like an Asian elephant and the mahout and the canopy are located right. The acquaintance of al-Jazari with elephants is not necessarily surprising, although I couldn’t find any evidence for elephants in Diyarbakir. Arab rulers held menageries or collection of exotic animals. In addition to the curiosity and pleasure they provided, they demonstrated the wealth and the power of the ruler and demonstrated the impact of the sovereign from India to Africa. Offerings of rare animals were part of the diplomatic process and sometimes part of the tax system. Until the 13th century, the agreement between the Nubian Kingdom and the Muslim rulers of Egypt demanded from Nubian people to provide Egypt with three hundred and sixty slaves annually and wildlife. It This was the primary source for giraffes in the Sultan of Cairo menagerie.

Book of the animals, Syria, 15th Century.

The use of animals as a diplomatic gesture is well documented. For example, Baybars, the Mameluke Sultan of Egypt and Syria in the 13th century gave elephants, giraffes, and zebras to the King of Spain, Emperor of Byzantium and the Mongol Khan. In the 10th century, Cordoba sent a giraffe to Tunisia and a story I particularly like about the elephant, Abul-Abbas. Harun al-Rashid,  the Abbasid caliph in Baghdad, sent an Asian elephant to Aachen, Germany to the Carolingian emperor Charlemagne. It happened in the 9th century, and surprisingly enough, there’s a Jewish angle to this story. The elephant was brought by Yitzhak the Jew. There is a historical novel “The travels of Isaac the Jew and Abu Alabas the elephant”(in Hebrew). It is interesting to note that other presents including an elaborate water clock made of brass, described in the Royal Frankish Annals were sent with the elephant. The water clock marked the 12 hours with balls of brass falling on a plate every hour, and also had twelve horsemen who appeared in turn at each hour.  Perhaps al-Jazari knew the story as part of his extensive knowledge of water clocks?

The Elephant Clock and the Circus

The diplomatic delegations and the royal gifts indicate that the Elephant was a symbol of power and wisdom, but in the context of the elephant clock, I think more about the circus and exotic acts. The elephant is made of copper and is just a stage for the show, but the swinging dragons, the Mahout with his fearsome tools, the Falcons and the spinning bird form a great circus number. An exciting circus act has, in my opinion, four components, not in binding order, not always all of them, and certainly not of the same weight:

  • Freshness (something new)
  • High skills
  • Sense of danger
  • Magical picture

I looked at several iconic circus shows such as Jules Léotard, a French acrobatic performer that made history as the first man ever with the aerial act on a trapeze. It must certainly meet the requirement for ” Freshness or something new.” The elephant clock is also the first of its kind, no clock ever, before or after is similar, and everyone who watches it, even today, is amazed. Secondly, high skills – Léotard practiced its aerobatic stunts over his parents’ pool before he revealed them in 1859  in Cirque Napoleon in Paris. Then he appeared in London before an audience that went crazy because of his aerial act and flips between five trapezes with only a pile of mattresses to protect him. The elephant clock also demonstrates high proficiency and skills both to the innocent and skilled observer: The control of the timing using the sinking float, the complex movement of the three characters(the mahout, the scribe and the man in the balcony), the virtuoso swing of the dragons. All are innovative engineering tricks demonstrating al-Jazari skills in water clocks and automatons. The crowd in the circus was afraid for Léotard life, and the sense of danger, which automatons be definition lack, intensified the experience.

Jules Léotard, a French acrobat, 19 century

Last but not least, the image. Léotard, like all circus performers, could be dressed in a sports suit. But as you can see in the picture he’s wearing theatrical shorts, bracelets emphasize his wrists, and the collar of his shirt reminds us of a royal necklace. All this help to imprint his image in our mind.  The picture that al-Jazari created is a lot more than the sum of its components and is intriguing audiences till this very day.

The Castle Clock


Al-Jazari opened  “The Book Of Knowledge Of  Ingenious Mechanical Devices”  with a monumental clock, perhaps the most complex of all ten water clocks and candle clocks explained in the book: The Castle Clock.

Sometimes you know you read a wonderful book the second  you read the first paragraph:

“Call me Ishmael. Some years ago – never mind how long precisely – having little or no money in my purse, and nothing particular to interest me on shore, I thought I would sail about a little and see the watery part of the world. It is a way I have of driving off the spleen and regulating the circulation.”

Moby Dick by Herman Melville

In the right hands, the beginning of a novel can make you feel like you were abducted from reality and you are drifting down a river which will take you to other worlds. Not only engineers who open al-Jazari’s book are captured immediately by its magic of the machines he designed eight hundred years ago. We will never know if al-Jazari wanted a powerful opening to demonstrate his ability at its best, or he positioned machines at random order and was surprised by the very question? This post hopes to explain the Castel Clock as well as discuss what we can about al-Jazari from the text.

How does it work?

The Castle Clock had a complicated movement throughout the day, and it is on the boundary between a clock and an automaton(a machine that performs a function according to a predetermined set of instructions). There is something theatrical in many automata. Sometimes it is by design, like the automata in Greek theater used for “Deus ex machina”, literally “god from the machine”. Sometimes there are other objectives like the lion automaton built by Leonardo da Vinci for François Ier, king of France. When the King tapped the lion with his sword, its body opened and presented lilies, a symbol associated with the French royalty. The clock by al-Jazari is also very theatrical.

The Castle Clock from a dispersed copy, 1315.

At the beginning of the day all twenty-four doors, in two rows, are closed and the Golden Crescent,  which is a little hard to see in the picture, is positioned to the left. During the day, the half-moon is moving right, and  every hour three things are happening:

  • The upper doors open and a figure comes out and stands as if he had suddenly emerged.
  • The lower door is rotating on its axis, and the text “Allah al-Malik” meaning ” God is The King and Owner of “
  • The Two falcons with outspread wings lean forward and cast a bronze ball into a vase, inside the vase a cymbal is hung, making a sound which can be heard from afar.

The picture of the falcon is taken from a dream or myth. Horus is one of the most significant ancient Egyptian deities. He was most often depicted as a falcon. Horus had many battles with Seth, the god of the desert, in which he lost his left eye, then a new eye was created for him called “the eye of the Moon” or “the diamond” and symbolizes an endless vision. I have no reason to assume that al-Jazari was familiar with Egyptian mythology, but who knows?

Above the upper row of doors, we can see the Zodiac sphere. At the beginning of the day, the sun will be on the eastern horizon, about to rise. The sun climbs until noon, then descends until nightfall and the six signs that have been visible will disappear, and the six that have been hidden will appear. At noon the drummers drum, the trumpeters blow, and the cymbalist plays his cymbals for a while.

Al-Jazari does not write anything about the reason for multiple mechanisms to display the time. The crescent actually functions as a modern analog clock hand, and the rest are just “decoration” and maybe a resonance box. In the world of modern engineering, it could be considered excessive and even wasteful, but there is magic that passes through centuries of the Falcons even if there is no additional information.

Erich Kästner, the wonderful author of Pünktchen und Anton(Dot and Anton in English), was concerned:   ” By the children who would prefer to eat porridge for three days than deal with such complex issues as his reflections [my translation from Hebrew]. He came up with a different font “so if you see something like that you can skip it altogether…” It seems to me this even more needed for technical explanations of engineers that will be in blue.

The Castle Clock is a sophisticated version of the classical water clock or clepsydra where time is measured by the regulated flow of out a vessel where the amount is then measured.  The difficulty is that the water flow rate is not uniform and depends on pressure (altitude) of the water in the vessel. To overcome this problem, al-Jazari used a conical plug and the float chamber.

Conical plug, the Castle clock, Topkapi, 1206

The main reservoir is feeding the float chamber through a conical plug thus whenever the water level drops the valve (a float that is a plug in a cone shape) goes down with the water level allowing the chamber to be refilled. Every time the chamber is full of water, the conical plug will seal the chamber isolating it from the main reservoir. In this way, the float chamber is always full of water and therefore the water flow at a constant rate and does not depend on the height of the water in the main reservoir.

A drawing of the clock mechanism, Topkapı manuscript, 1206, my captions


At Sunrise a servant makes sure that all doors are closed and the time cart is on the right side (looking from the back). During the day water will flow at a rate determined by the flow regulator and the main float would drop with the water level at the main reservoir. The main float is made of copper, and it is quite heavy.  When it drops, it pulls the rope, which through the pulley would turn the main disk and pull the time cart attached to the golden crescent which would move to the left at a constant velocity indicating the time passed from sunrise. Every hour the cart will progress one door, and a smart mechanism would open the doors while dropping down two bronze balls. The balls would roll down and reach an opening above the heads of the Falcons. The curving claws of the Falcons are welded to a copper tube that can rotate on its axis. The falcon stands upright because of a balancing weight. When the bronze ball drops down, it changes the balance, and the falcon would lean forward, and the falcon wings, attached to a body on a hinge will spread open, and the ball will fall on the cymbal hidden in the vase. Now that the falcon head is light again, the balancing weight will bring him to its original position. The clock is packed with similar invention and  “patents”.

A drawing of the falcon mechanism, Topkapi manuscript, 1206

The book contains almost 50 pages explaining the various mechanisms with detailed construction instructions. Readers who are interested in the details can learn them here and see the simulation here


What did I learn about Al-Jazari?

We have no information about al-Jazari except what is in the text itself. We can “pick” the book to learn about al-Jazari and his world. Consider the adjustable flow regulator intended to ensure that the clock movement fits the changing length of the day. This controller is a small engineering marvel itself, but I am interested in it because of the triple encounter it offers with al-Jazari and his world:

  • First, al-Jazari is a man who is familiar with the literature of his time. The opening lines of the Castle clock chapter are: “I followed the method of the excellent Archimedes in distributing the twelve signs of the Zodiac. Al-Jazari is probably referring “On the construction of water clock” – كتاب أرشميدس في عمل البنكامات. This book was attributed to Archimedes, but its source is unclear. This reinforces al-Jazari statement in the introduction:

“I have studied the books of the earlier [scholars] and the works of the later [craftsmen] –masters of ingenious devices with movements like pneumatic [movements], and water machines … I considered the treatment of this craft for a period of time and I progressed, by practicing it, from the stage of book learning to that of witnessing, and I have taken the view on this matter of some of the ancients and those more recent [scholars]. “

The question of openness or seclusion to the world for people of faith is a relevant question even today for Jews or Muslims.    Maimonides, Rabbi Moshe Ben Maimon, the most important rabbinical arbiters in Jewish history, and polymath, scientist, and  physician lived almost in the same time frame in Cordoba, far away from Diyarbakir in Anatolia but he was a part of the same  Muslim world. During his medical studies, he was introduced to the writings of Aristotle in natural science and did not feel any threat to his faith. He even wrote:

” Consequently he who wishes to attain to human perfection, must therefore first study Logic, next the various branches of Mathematics in their proper order, then Physics, and lastly Metaphysics.” Guide for the Perplexed

It’s amazing to read when today Orthodox Jewish children are forbidden to learn mathematics or natural sciences. Al-Jazari, more engineer than a philosopher, does not deal with matters of faith directly, but his faith is embedded in the text. This doesn’t bother him at all to read and learn from pagan scholars.

  • Secondly, in Diyarbakir in eastern Turkey, there are little more than fourteen hours of daytime in the summer and approximately nine hours of daytime in winter. Al-Jazari made a considerable engineering effort ensuring that there would be twelve hours between sunrise and sunset in summer and winter. This is the purpose of the flow regulator which adjust a short hour in the winter compared to the longer hour in the summertime. Time is not an illusion or a pure man-made concept. The Earth orbited the sun before there were humans around and the sunrise and the sunset, as well as summer and winter, were here before we gave them their name. But the perception of time and its measurement are human inventions. If I would have met al-Jazari and told him that a second that was impossible to measure in his time is the basic unit of time and its scientific definition is approximately 9 billion (for those who want precision 9,192,631,770) cyclic switching between two energy levels of the atom cesium. Not only would that he would not understand a word but also would think me really He did not need such precision that did not fit his daily experience. But I use Waze, a navigation application, and we need accurate atomic clocks at this level of precision to bring me to my destination on time. In today world, the concept of time which varies according to the seasons seems far-fetched, but in the world of al-Jazari who knew sundials and water clocks, it made perfect sense.
  • Thirdly al-Jazari made detailed measurements of the water regulator attributed to Archimedes and found it insufficient. Then he explains in detail how he tries to solve the problem without success through trial and error. It’s ridiculous to compare a modern engineer to al-Jazari, but it is delightful to read the report of a very talented engineer more than eight hundred years ago. It turns out that his concerns are not very different from the concerns of a current engineer. From the text, it turns out he did a “literature review” and theoretical calculations (in this case unsuccessful), and plan and perform the experiments. He was also a skilled man who knows copper, bronze and wood and their processing. When al-Jazari explains, for example, how to prepare the main water reservoir, he’s not satisfied with a drawing and selecting material (copper) but explains how to get a perfect cylinder with a precise wooden disk and how to ensure that the cylinder would have the same diameter all along. For the technical reader, it is easy to sympathize with the difficulties and solutions. There is something appealing in this combination of a man of the books, an engineer, a craft master and an artist who we can meet through the pages and the hundreds of years that passed.

The Beaker Water-Clock


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 :

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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.