Archimedes

by

The only measurement tool in the book and why al-Jazari is the first engineer

Introduction

Al-Jazari tells that when he mentioned to some people that any(not colinear) three points could be position on the circle they didn’t believe him, so he built the only measurement instrument in the book to find the center-point of three points of unknown position. The device is quite straight forward, but we can learn quite a bit from his choice to solve, what is clearly a mathematical problem, with an “engineering” solution.

An Instrument for finding the center of a circle, Topkapi, 1206

How does it work?

The technical explanation is so short that I decided to make an exception and not color it in blue. I hope you can forgive me. Al-Jazari took a ruler and built a vertical on the center point. He placed his instrument between the two points; found the center point and drew a perpendicular segment. He repeated the process for two more points. The intersection of the perpendicular segments is the center of the circle and the distance to each point is the radius. Besides, similarly to angle measuring instruments, there is an ark, which allows to measure and mark different angles.

Some Math

One can prove that any three points that are not colinear (al-Jazari was aware of this point and specify it explicitly) are on a circle in two approaches:

  • Euclidean geometry
  • Analytic geometry.

In Euclidean geometry, three points which are not colinear form the vertices of a triangle. All triangles can be within a circle. The center of the circle is the intersection of the three perpendicular bisectors. It is relatively easy to prove. If you want to practice your Euclidean geometry, look at the diagram below, build the three radiuses BO AO CO and prove they are identical using triangle congruence theorems. Euclid’s “Elements” was translated into Arabic relatively early in the House of Wisdom in Baghdad (بيت الحكمة )). There is no direct reference in al-Jazari’s book to Euclid, but his device is based on this theorems:

Right side drawing from my Euclidean geometry book, on the left a drawing by al-Jazari.

In a different approach, you can find the center circle with analytic geometry:

Circle equation, Analytic geometry

When:

r is the circle radius

a,b are the coordinates of the center point

Since the triangle has three vertices, we have three equations in three unknowns (a, b, r) and an immediate solution. Analytic geometry has roots in ancient Greece and Persia of the 11th century, but the breakthrough was made by René Descartes, philosopher, scientist and mathematician. We remember Descartes mostly because of the proposition “I think, therefore I am.” Descartes was a remarkable mathematician and the first to offer a system of axes (x, y), as in the diagram above, which is named after him: Cartesian coordinate system. It allows the graphical representation of functions. Generations of mathematics students were, are and will be very grateful. Also, he took advantage of the Cartesian system to connect geometry and algebra, creating analytical geometry. Descartes was an impressive polymath, his contributions to philosophy and mathematics are the pillars of the two disciplines, but he also was a key figure in the Scientific Revolution and made a contribution to optics.

Polymath and al-Jazari, the first engineer

A polymath (Greek: πολυμαθής) literally “having learned much” is an individual whose knowledge spans a significant number of subjects. Both in English and Hebrew we often use the term “Renaissance man” although all the “engineers” before al-Jazari were actually polymath long before the Renaissance:

Archimedes was a gifted mathematician, scientist, and engineer, who invented the “Archimedes Screw” (a pump, still used to this day), he has improved the power and the accuracy of the Catapult, made a giant crane known as “Archimedes Claw” not to mention the myth (?) of burning the Roman fleet using mirrors. All this pales in comparison to his contributions to mathematics and physics. Archimedes anticipated modern calculus and analysis by applying concepts of infinitesimals, developed the concept of buoyant force in “On Floating Bodies” and gave the mathematical explanation to the lever.

Hero of Alexandria was an engineer, mathematician, and physicist.  Hero may have been either a Greek or a Hellenized Egyptian. It is almost certain that Hero taught at the famous Library of Alexandria because most of his writings appear as lecture notes. He is known for his research in hydrostatics, but I have already written about Hero concerning his book on automata, he also built the Aeolipile, the first steam engine. In mathematics, Hero described a method for iteratively computing the square root of a number, but his name is most closely associated with Hero’s formula for finding the area of a triangle from its side lengths.

The Banū Mūsā (“Sons of Moses”) were three 9th-century Persian scholars who lived and worked in Baghdad. The Banu Musa wrote almost 20 books, the majority of which are now lost. They are known for their Book of Ingenious Devices on automata and mechanical devices. I wrote about them in the context of the fountains, but in the context of a polymath, we can mention their contribution to mathematics, The most important work of theirs is the Book on the Measurement of Plane and Spherical Figures, a foundational work on geometry that was frequently quoted by both Islamic and European mathematicians.

Al-Jazari is not like that. His contribution to engineering is diverse. I mention already the automata and the use of the camshaft, the significant advances in candle clocks [Hebrew] including the invention of the bayonet connection, the thermal insulation, the double-action pump but he was not involved in science or math or other fields outside engineering.

The concept of the Renaissance man was coined by Leon Battista Alberti ” A man can do all things if he but wills them”, a manifestation of the deep humanism in the roots of the Renaissance. The basic premise is that the infinite human ability to evolve, and we must embrace all knowledge in our way to develop our abilities. The world has expanded so that it is just impossible. Thomas Young, an English polymath in early 19th century, regarded by many as the last man who “knew everything” was skilled in medicine, physics, Linguistics, harmony (music) and even accounting. The web site of the Israeli medical association includes thirty-two different major specialties and more, numerous subspecialties. It is not possible, even theoretically, to complete all medical specialties during one life, let alone in other areas.

We live in a more skeptical and concerned world. We ask our children, already at a young age, “What do you want to be when you grow up? We narrow the field in high school and ask the students to find majors area of study where they excel. We have institutions, counseling centers, and tests to help young people choose their profession. A physics student will get the necessary mathematical background but will not receive academic credit for courses in Assyrian or typography. In second degree studies, we reduce the field of study further, and in Ph.D., we focus on one question only. As a society, we look at people that change profession with concern, maybe as less stable who lack the ability to focus.

Following Donald Hill, The book translator, and annotator and somewhat because of my own training, I thought that using an instrument (instead of a formal proof) indicates a limited background in mathematics. This may be true. My Love M. commented that mathematical proofs are less approachable to most people and lack the magic of the instrument al-Jazari built. Al-Jazari was the first “pure engineer” not because of lack of mathematical background, or ability in math and science, but because of his passion for engineering and his ability to translate abstract and formal issues to instruments.

by

The Castle Clock

Introduction

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.