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In mathematics he is mostly remembered for Heron's formula, a way to calculate the area of a triangle using only the lengths of its sides.
Much of Hero's original writings and designs have been lost, but some of his works were preserved--mostly in manuscripts from the Eastern Roman Empire, and a smaller part in Latin or Arabic translations.
Hero described the construction of the aeolipile (a version of which is known as Hero's engine) which was a rocket-like reaction engine and the first-recorded steam engine (although Vitruvius mentioned the aeolipile in De Architectura some 100 years earlier than Hero). It was created almost two millennia before the industrial revolution. Another engine used air from a closed chamber heated by an altar fire to displace water from a sealed vessel; the water was collected and its weight, pulling on a rope, opened temple doors. Some historians have conflated the two inventions to assert that the aeolipile was capable of useful work.
The first vending machine was also one of his constructions; when a coin was introduced via a slot on the top of the machine, a set amount of holy water was dispensed. This was included in his list of inventions in his book Mechanics and Optics. When the coin was deposited, it fell upon a pan attached to a lever. The lever opened up a valve which let some water flow out. The pan continued to tilt with the weight of the coin until it fell off, at which point a counter-weight would snap the lever back up and turn off the valve.
A wind-wheel operating an organ, marking the first instance in history of wind powering a machine.
Hero also invented many mechanisms for the Greek theater, including an entirely mechanical play almost ten minutes in length, powered by a binary-like system of ropes, knots, and simple machines operated by a rotating cylindrical cogwheel. The sound of thunder was produced by the mechanically-timed dropping of metal balls onto a hidden drum.
The force pump was widely used in the Roman world, and one application was in a fire-engine.
A syringe-like device was described by Hero to control the delivery of air or liquids.
In optics, Hero formulated the principle of the shortest path of light: If a ray of light propagates from point A to point B within the same medium, the path-length followed is the shortest possible. It was nearly 1000 years later that Alhacen expanded the principle to both reflection and refraction, and the principle was later stated in this form by Pierre de Fermat in 1662; the most modern form is that the optical path is stationary.
A standalone fountain that operates under self-contained hydro-static energy; now called Heron's fountain.
A programmable cart that was powered by a falling weight. The "program" consisted of strings wrapped around the drive axle.
Automata, a description of machines which enable wonders in temples by mechanical or pneumatical means (e.g. automatic opening or closing of temple doors, statues that pour wine, etc.); see Automaton and Bernardino Baldi's translation
Mechanica, preserved only in Arabic, written for architects, containing means to lift heavy objects
Metrica, a description of how to calculate surfaces and volumes of diverse objects
On the Dioptra, a collection of methods to measure lengths, a work in which the odometer and the dioptra, an apparatus which resembles the theodolite, are described
^ abResearch Machines plc. (2004). The Hutchinson dictionary of scientific biography. Abingdon, Oxon: Helicon Publishing. p. 546. Hero of Alexandria (lived c. AD 60) Greek mathematician, engineer and the greatest experimentalist of antiquity
^Marie Boas, "Hero's Pneumatica: A Study of Its Transmission and Influence", Isis, Vol. 40, No. 1 (Feb., 1949), p. 38 and supra
^ abA.G. Drachmann, "Heron's Windmill", Centaurus, 7 (1961), pp. 145-151
^ abDietrich Lohrmann, "Von der östlichen zur westlichen Windmühle", Archiv für Kulturgeschichte, Vol. 77, Issue 1 (1995), pp. 1-30 (10f.)
^T. D. De Marco (1974). "Gas-Turbine Standby-Power Generation for Water-Treatment Plants", Journal American Water Works Association66 (2), p. 133-138.
^Victor J. Katz (1998). A History of Mathematics: An Introduction, p. 184. Addison Wesley, ISBN0-321-01618-1: "But what we really want to know is to what extent the Alexandrian mathematicians of the period from the first to the fifth centuries C.E. were Greek. Certainly, all of them wrote in Greek and were part of the Greek intellectual community of Alexandria. And most modern studies conclude that the Greek community coexisted [...] So should we assume that Ptolemy and Diophantus, Pappus and Hypatia were ethnically Greek, that their ancestors had come from Greece at some point in the past but had remained effectively isolated from the Egyptians? It is, of course, impossible to answer this question definitively. But research in papyri dating from the early centuries of the common era demonstrates that a significant amount of intermarriage took place between the Greek and Egyptian communities [...] And it is known that Greek marriage contracts increasingly came to resemble Egyptian ones. In addition, even from the founding of Alexandria, small numbers of Egyptians were admitted to the privileged classes in the city to fulfill numerous civic roles. Of course, it was essential in such cases for the Egyptians to become "Hellenized," to adopt Greek habits and the Greek language. Given that the Alexandrian mathematicians mentioned here were active several hundred years after the founding of the city, it would seem at least equally possible that they were ethnically Egyptian as that they remained ethnically Greek. In any case, it is unreasonable to portray them with purely European features when no physical descriptions exist."
^Hero (1899). "Pneumatika, Book , Chapter XI". Herons von Alexandria Druckwerke und Automatentheater (in Greek and German). Wilhelm Schmidt (translator). Leipzig: B.G. Teubner. pp. 228-232.
^Hero of Alexandria (1851). "Temple Doors opened by Fire on an Altar". Pneumatics of Hero of Alexandria. Bennet Woodcroft (trans.). London: Taylor Walton and Maberly (online edition from University of Rochester, Rochester, NY). Archived from the original on 2008-05-09. Retrieved .
^For example: Mokyr, Joel (2001). Twenty-five centuries of technological change. London: Routledge. p. 11. ISBN0-415-26931-8. Among the devices credited to Hero are the aeolipile, a working steam engine used to open temple doors and Wood, Chris M.; McDonald, D. Gordon (1997). "History of propulsion devices and turbo machines". Global Warming. Cambridge, England: Cambridge University Press. p. 3. ISBN0-521-49532-6. Two exhaust nozzles...were used to direct the steam with high velocity and rotate the sphere...By attaching ropes to the axial shaft Hero used the developed power to perform tasks such as opening temple doors
^Humphrey, John W.; John P. Oleson; Andrew N. Sherwood (1998). Greek and Roman technology: A Sourcebook. Annotated translations of Greek and Latin texts and documents. Routledge Sourcebooks for the Ancient World. London and New York: Routledge. ISBN978-0-415-06137-7., pp. 66-67
Schellenberg, H.M.: Anmerkungen zu Hero von Alexandria und seinem Werk über den Geschützbau, in: Schellenberg, H.M./ Hirschmann, V.E./ Krieckhaus, A.(edd.): A Roman Miscellany. Essays in Honour of Anthony R. Birley on his Seventieth Birthday, Gdansk 2008, 92-130 (with a huge bibliography of over 300 titles and discussion of the communis opinio on Hero).