The Six-Cylinder Water Pump of Taqi al-Din: Its Mathematics, Operation and Virtual Design

The main objective of this study is to investigate into the six-cylinder water raising pump described around 1550 by the Ottoman Muslim scientist Muhammad Ibn Ma'ruf, known as Taqi al-Din, in his treatise Al-Turuq al-Saniya fi al-' alat al-ruhaniya. After an outline of the historical context and an English translation of the relevant sections of the manuscript, the focus is laid on the engineering analysis of the water pump. The result of the analysis yielded the reconstruction of the machine through a graphical model which was then used to produce a virtual 3D animation of the mechanical workings of the various parts, including the water turbine, the cam shaft, the connecting rods, the reciprocating pistons and the cylinders.

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Salim T S Al-Hassani* and Mohammed A. Al-Lawati**

Table of Contents

1. Preliminaries

1.1. Preface
1.2. Structure of the Article
1.3. Nomenclature

2. Water Machines in the Lands of Islam

2.1. The Science and Art of Water Management
2.2. Three Water-Raising Machines in Al-Turuq al-Saniya
2.2.1. The Pump with Two Opposing Cylinders
2.2.2. The Spiral Pump
2.2.3. The Pump of the Rope with Cloth Balls

3. Taqī al-Dīn and his Treatise: Historical and Textual Context

3.1. Overview on the Treatise Al-Turuq al-Saniya
3.2. Description of the Pump with Six Cylinders

4. The Functioning of the Pump: Technology and Operation

4.1. How Does it Work
4.2. Historical Assessement of the Six-Cylinder Pump

5. Mathematical Diagnostics of the Six-Cylinder Pump

5.1. Pipes, Cylinders and Pistons
5.2. The Connecting Rod, the Pivot and the Cam
5.3. Complementary Parts

6. Virtual Reconstruction of the Pump

6.1. Modelling and Animation
6.1.1. Modelling
6.1.2. Creating Objects
6.1.3. Modifying Objects
6.1.4. Assigning Materials
6.1.5. Creating Lights and Cameras
6.2. Animation
6.3. Computer Softwares
6.3.1. MathCAD
6.3.2. 3D Studio MAX
6.4. The 3D Model
6.5. By Way of Conclusion
6.6. Acknowledgments

7. Appendices

7.1. The Evidence from the Manuscript
7.2. References

* * *

1. Preliminaries

1.1. Preface

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Figure 1: Drawing of the six-cylinder pump as it was depicted in Taqī al-Dīn in Al-Turuq al-Saniya fī al-’ālat al-rūhaniya (Chester Beatty Library in Dublin, Arabic MS 5232, p. 38).


The main objective of this study is to investigate into the six-cylinder water raising pump described around 1550 by the Ottoman Muslim scientist Muhammad Ibn Ma‘rūf, known as Taqī al-Dīn, in his treatise Al-Turuq al-Saniya fī al-’ālat al-rūhaniya. Our study of this important machine consists of three major parts. The first covers some historical facts that give knowledge about the role of engineering and engineers. The second is centered on an engineering analysis of the water pump. The last obtains all the dimensions of the machine and provides a graphical model which was then used to produce a virtual 3D animation of the mechanical workings of its various parts, including the water turbine, the cam shaft, the connecting rods, the reciprocating pistons and the cylinders. The study is focused on the pump in order to verify its ability to work using modern engineering analysis.

1.2. Structure of the Article

The study is composed of seven sections. The first one is introductory whilst the layout of the remaining sections is as follows. The second chapter contains a historical background about the Islamic tradition of engineering and water raising machines. The glimpses we present about Islamic engineering give an idea of how engineering was practiced and what were the working fields for engineers. Our aim in this short survey is to know how metals and some other materials were obtained or manufactured. This will help to decide the type of the material and select the suitable mechanical properties for it.

The third section presents a concise biographical sketch about Taqī al-Dīn and the contents of his treatise, with the English translation of the relevant portions on the pump from the original manuscript. The fourth chapter explains the working of the device. The fourth and fifth sections represent the core of this study as they describe the machine and its components, explain its mechanical function and the various materials used.

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Figure 2: Parallel views of the virtual reconstruction of the pump.

The fifth section is mainly devoted to mathematical analysis to verify the mechanical adequacy of the different components. This analysis and related calculations show how robust the pump was, and whether it was capable of delivering the water to the required height. All the equations and formulas in this section were obtained from the sources (3), (4), (5), (6) and (9) [see the list of references in Appendix 7.4]. The values for some physical constants such as the coefficient of friction and the drag coefficient were found in the Data Book of the Department of Mechanical Engineering, UMIST, Manchester , UK.

The sixth section covers the modelling and animation part. Modelling the pump and animating its motion gives a better insight of its operation and helps to properly understand its operation and the linkages and dimensions of each component. This section ends on a conclusion drawn from the research and also contains suggestions for further work.

Finally, the last section is devoted to appendixes: a reproduction of the three pages of the original manuscript and a list of the main references used in the study.

Click here to view animation.

Click here to view animation.

1.3. Nomenclature

In this section, we present the list of symbols assigned to the different physical and mathematical magnitudes involved by our description of the six-cylinder pump.


Distance between the centers of two holes


Horizontal distance between the center of pipe i to the center of collective pipe

ai, bi and g

Variable angles


Angular velocity


Diameter of delivery pipes


Diameter of piston cylinder


Diameter of Collective pipe


Diameter of suction pipes


Diameter of piston rod


Diameter of connecting rod


Diameter of wheel lever


Diameter of cam


Height of delivery pipes


Height of piston cylinder


Height of collective pipes


Height of suction pipes


Height of cylinder block


Density of water


Density of wood






Volume of collective pipe


Volume of delivery pipe


Volume of conic pieace


Volume of piston cylinder


Volume of Camshaft


Volume of wheel lever


Volume of scoop


Friction force


Heighest water force in the delivery pipes


Water force in collective pipe


Water force in conic pipe


Total water force in all three pipes


Force on cam


An old measurement; measures about 70 cm


An old measurement; measures about 1/3 of an arm (≈ 23 cm)


Length of wheel lever


Length of piston rod


Length of cam


Length of cylinder block


Weight of lead


Weight of camshaft


Weight of wheel


Radius of piston rod


Radius of camshaft


Reynolds number


Drag force




Shear stress of wood

2. Water Machines in the Lands of Islam

2.1. The Science and Art of Water Management

The use of water wheel technology was widely spread in the Middle East before Islam. It is from this long lasting heritage that Muslim engineers adopted and improved this technology and applied it everywhere. In the city of Murcia, for example, during the Islamic rule of Spain, a waterwheel was established, still known today under the name La Ñora. Although the original wheel has been replaced by one in steel, the original system applied during the Andalus period is otherwise virtually unchanged. The flywheel mechanism, which is used to smooth out the delivery of power from a driving device to a driven machine, was invented by Ibn Bassal (fl. 1038-1075) who pioneered the use of the flywheel in the chain pump (saqiya) and noria [1].

The industrial uses of watermills in the Islamic world date back to the 7th century, while horizontal-wheeled and vertical-wheeled water mills were both in widespread use by the 9th century. A variety of industrial watermills were used in the Islamic world, including gristmills, hullers, paper mills, sawmills, shipmills, stamp mills, steel mills, sugar mills, and tide mills. By the 11th century, every province throughout the Islamic world had these industrial watermills in operation, from al-Andalus and North Africa to the Middle East and Central Asia. Muslim engineers also used crankshafts and water turbines, gears in watermills and water-raising machines, and dams as a source of water. They used to provide additional power to watermills and water-raising machines. Industrial water mills were also employed in large factory complexes built in al-Andalus between the 11th and 13th centuries.

Muslim engineers used two solutions to achieve the maximum output from a water mill. The first solution was to mount them to piers of bridges to take advantage of the increased flow. The second solution was the shipmill, a type of water mill powered by water wheels mounted on the sides of ships moored in midstream. This technique was employed along the Tigris and Euphrates rivers in 10th-century Iraq, where large shipmills made of teak and iron could produce 10 tons of flour from corn every day for the granary in Baghdad [2].

Early examples of water raising machines include the Shādūf, the Sāqiya and the Noria [Nā‘ūra], all of which were known in the Muslim world [3]. At an early stage Muslim engineers were exploring new methods for increasing the effectiveness of water raising machines. Al-Jazarī and Taqī al-Dīn both described water-raising machines that show an awareness of the need to develop machines with a greater output than these traditional ones. By the 13th century, what we might call water raising machine technology lifted off with the work of al-Jazarī. In his monumental book Al-Jāmi’ bayna ‘l-‘ilm wa-‘l-‘amal nāfi‘ fī sinā‘at al-hiyal, this genius scholar was responsible for the design of five machines for raising water. The most significant is the fifth one, which was a water-driven pump. A water wheel turned a vertical cog wheel which in turn turned a horizontal wheel; the latter caused two opposing copper pistons to oscillate. The cylinders of the pistons were connected to suction and delivery pipes which were guarded by one-way clack valves. The suction pipes drew water from a water sump down below and the delivery pipes discharged the water into the supply system about 12m above the installation. This pump is an early example of the double-acting principle (while one piston sucks the other delivers) and the conversion of rotatory to reciprocating motion [4].

Taqī al-Dīn describes a slightly modified version of al-Jazarī’s fifth machine in his book on machines Al-Turuq al-Saniya.

2.2. Three Water-Raising Machines in Al-Turuq al-Saniya.

2.2.1. The Pump with Two Opposing Cylinders

This pump, already dealt with by Al-Jazarī, is a unique mechanical device that may be considered as a major achievement in the history of mechanical engineering. Historians considered that al-Jazarī’s machine was the archetype from which the steam engine was developed [5].

Because all figures and drawings in the manuscripts of Al-Jazarī’s book contained mistakes done by scribes, historians of technology could not reach an agreement on the correct design of the machine. This is why Taqī al-Dīn’s description of this pump has a greater historical value, as the figure which he drew himself was accurate and helped remove obscurity that surrounded parts of this machine.

Taqī al-Dīn’s description of this machine in his treatise is accompanied by a drawing (fig. 3) that combines both the tools of conic sections and perspective drawing. This is why drawings are difficult to read and understand, and it becomes imperative to read texts carefully to arrive at a correct conception of the model.

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Figure 3: Drawing of the reciprocating pump with two opposing cylinders (Chester Beatty Library in Dublin, Arabic MS 5232, p. 32).

2.2.2. The Spiral Pump

The description of the spiral pump in Al-Turuq al-Saniyah is very significant because it was not mentioned in the Arabic books of engineering before Taqī al-Dīn. The use of spiral in mechanical devices is attributed to Archimedes and it was largely used around the Mediterranean. We do not, however, find any description of a spiral pump machine in the available references written before Taqī al-Dīn (fig. 4) [6].

Taqī al-Dīn’s spiral pump is worked by a waterwheel through two cogwheels each fitting into the other. It seems, according to historians of technology, that the earliest description of this kind of machines in the West goes back to Cardan in 1550 and Ramelli in 1588. This means that Taqī al-Dīn was amongst the first to describe this water machine as he finished his manuscript in 1551-52 (959 H). The use of spiral pumps flourished in Europe after this period, especially in drainage projects, before they came into common use in the 17th and 18th centuries and were turned by wind as well [7].

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Figure 4: Drawing of the spiral pump (Chester Beatty Library in Dublin, Arabic MS 5232, p. 34).

2.2.3. The Pump of the Rope with Cloth Balls

No description of this pump was given before Taqī al-Dīn. Moreover, as far as we know Taqī al-Dīn preceded Western engineers in describing it. Agricola mentioned a similar device in De re metallica which (completed in 1553 and published three years later), namely few years after that that our Muslim engineer completed Al-Turuq al-Saniya.

The pump of the rope could get water raised from great depths up to 72 metres (fig. 5). In contrast, pumps with pistons can raise water to small heights only, as they are limited by air pressure. Therefore, the alternative is to use a rope or chain that carries buckets or to use a pump of a robe with cloth balls. The cloth balls move tightly in a vertical tube. These balls are spaced at equal distances and fastened to a rope or chain. When the balls move upwards inside the tube, they behave like a piston in cylinder as it pulls water up [8].

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Figure 5: Drawing of the pump of the rope with cloth balls (Chester Beatty Library in Dublin, Arabic MS 5232, p. 35).

3. Taqi al-Din and his Treatise: Historical and Textual Context

The real name of our scholar is Muhammad Ibn Ma‘arūf, but he was nicknamed Taqī al-Dīn. He is more known under this kunya. Al-Rāsid is often added to his name to attest his main scientific interest in astronomy and his fame as founder and director of the Istanbul observatory (1575-1580). Taqī al-Dīn was born in Damascus as he mentioned it in his book Al-Turuq al-Saniya. He was born in 1525/1526, and he died in Istanbul in 1585 [9].

3.1. Overview on the Treatise Al-Turuq al-Saniya

The book of Taqī al-Dīn Al-Turuq al-Saniya fi’1-ālāt al-rūhaniyya was written in Damascus about 1551. Its manuscript copy held in Chester Beatty Library in Dublin is dated back to 1551-52 (959 H), and the title page indicates that it was reviewed a year later by a student. The treatise, which is not yet translated into English, contains descriptions and illustrations of clocks, weight-lifting equipment, pumps and various other machines. It survived in several manuscript copies preserved at different libraries throughout the world: Dār al-kutub al-misriya in Cairo (MS Falak 3845, Miqat 557/4), Chester Beatty Library in Dublin (MS 5232), the library of Kandilli Observatotry in Istanbul (MS 96, autograph). In 1976, Ahmad Yusuf al-Hassan published an offset reproduction of the Dublin’s manuscript copy, that he accompanied with a thorough study[10].

The treatise includes six sections dealing in order with:

  • clepsydras,
  • devices for lifting weights,
  • devices for raising water,
  • fountains and continually playing flutes and kettle-drums,