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