I read that several principles of Al-Jazari's monumental water clocks were based upon earlier designs of water clocks by Archimedes, for example the use of valves, feedback system and flow control regulator. Apparently, Archimedes even wrote a treatise On the Construction of Water Clocks. So what innovations did Archimedes introduce in his hydraulic clock?
Jazari is referring to "an Arabic treatise of unknown date and authorship" that describes a monumental water-clock. It is not listed among Archimedes's works in any ancient sources, and according to Hill, most of it was written by several Arabic authors. The "author" is now referred to as Pseudo-Archimedes. Ridwan al-Saati built Jayrun water clock based on the Pseudo-Archimedes's design. He also mentions some Hormuz, who invented a water clock used by his father in the construction of the Damascus clock:“the design continued in the land of Fars for a long time, and was transmitted from there to the land of the Greeks, and its construction spread out in the land until it was transmitted to Damascus, where it was constructed up to the days of the Byzantines and after that in the days of Banu Umayya, according to what is mentioned in the histories". For details see Hill's Arabic Water Clocks,
It is known that a public water clock was erected in Gaza in fifth century AD, so Hormuz might have lived before the rise of Islam, and possibly re-invented the water clock independently. But simple water clocks were already used by ancient Babylonians and Egyptians. A perfected version, clepsydra ("water thief") is due to Archimedes's contemporary Ctesibius (c.285-222 BC), a renowned engineer, the founder of pneumatics, and possibly the first head of the Alexandrian Museum. Unfortunately, none of his works survived to our times, but his inventions are described by Vitruvius, Athenaeus, Philo of Byzantium, Hero and Proclus. A detailed analysis of Ctesibius's work is given in Russo's Forgotten Revolution.
Some modern engineers credit clepsydra as the first automatic feedback device. The problem Ctesibius faced was that the speed of water escaping a container through a hole depends on the water level in it. To make it work for keeping time he needed a container where the water level remains fixed. This is accomplished by a float with a valve that blocks the inflow when the level rises, and lets it through when it falls, a feedback controller in modern terms. The clock involves three containers, the first one serves as a water reservoir, and empties into the second, which has the float with a valve ensuring that its water level remains constant. So the speed at which it empties into the third container is also constant, and a float in the third container will rise uniformly. A pointer attached to it indicates time. See a nice illustration at Lahanas's site.
al-Jazari (and Ridwan too) explicitly says that Archimedes is the source for several of the components of his clock(s); he also says that one component of Archimedes' clock (the flow regulator) does not work, and proposes a solution where the regulator is a full circle rather than a semi-circle.
The Arab treatise on the construction of water clocks that is attributed to Archimedes describes a clock that is very similar, from a mechanical, pneumatic and hydraulic point of view, to al-Jazari's and Ridwan's clocks, so it is very probable that both those engineers based their own designs on this treatise (or on the tradition it is based onto).
Scholars such as Hill believe that the work might have been originally written in Greek, but that the treatise was heavily reworked by Islamic authors; others, such as Drachmann, believe that it might be a compilation of several Greek authors.
More recently, Anette Schomberg recognised the treaty as (mostly) a translation of a treatise on water clocks by Philo of Byzantium; the clock described by Philo, therefore, should be imaged as similar to the Gaza clock described by Procopius, or the water clock described by Cassiodorus, which later were adopted by the Islamic engineers (see al-Jazari Castle clock for a very similar example).
Furthermore, Schomberg recognizes both the clock and the flow regulator in the description of Ctesibius' clock written by Vitruvius: this flow regulator is the same as that described by al-Jazari, so the basic idea should be:
- Philo of Byzantium invents a water clock based on a floater sinking in a vessel that provides the motion for several mechanical, hydraulic and pneumatic automata (including the hour indication). Philo's flow regulator is semi-circular
- Ctesibius improves Philo's clock on a couple of points, including a circular flow regulator
- This clock is know to Vitruvius and some implementations are known to other authors (Procopius of Gaza, Cassiodorus and Boethisu, maybe Teophanes)
- Arab authors translate a treatise that describes Philo's clock (including the semi-circular flow regulator)
- al-Jazari and Ridwan receive this tradition and build several clocks on the same principles; they either receive the tradition that links directly back to Philo (maybe the same Arab treatise we have), and re-invent the circular flow regulator, or they receive Ctesibius' tradition and present this innovation as their own (the second possibility has a higher probability, due to the presence of another of Ctesibius' improvements into al-Jazari's and Ridwan's clocks)
So to sum it up:
What were Archimedes' innovations to the water clock? Probably none, he did not build any as far as we know.
What were Philo's innovations to the water clock? He probably built the original one, with all the automata and the sounds, and also wrote the treatise that is currently attributed to pseudo-Archimedes.
What were Ctesibius' innovations to the water clock? He definitely produced a better outflow device (Vitruvius explicitly says so) and he probably invented the circular flow regulator. "Ctesibius clock" was, in any major sense, similar to al-Jazari's Castle clock
How did the flow regulator work? It was composed by two parts, one to make the inflow to the tank regular (through a cone-shaped floater that matches a cone-shaped spout) and one to make it vary with the day of the year (the semi-circular/circular stuff I mentioned above)
What about the syphon in Ctesibius' clock? Very probably never existed, it was probably due to the imagination of Perrault, who produced a lavish and accurate edition of Vitruvius' in the 17th century, but proposed the wrong reconstruction for Ctesibius' clock
Source: Schomberg, A. (2017): "To amaze the world" - A contribution to the shape and mean-ing of the water clock in antiquity. In: Wellbrock, K. (ed.): Schriften der Deutschen Wasserhistorischen Gesellschaft, Band 27-1, Siegburg, 2017, 301-340. (you can find it on academia.edu)