Hero of Alexandria
Hero of Alexandria
Hero of Alexandria
(fl. Alexandria, a.d. 62)
mathematics, physics, pneumatics, mechanics.
Hero (or Heron) of Alexandria is a name under which a number of works have come down to us. They were written in Greek; but one of them, the Mechanics, is found only in an Arabic translation and another, the Optics, only in Latin. Apart from his works we know nothing at all about him.
His name is not mentioned in any literary source earlier than Pappus (a.d. 300), who quotes from his Mechanics.1 Hero himself quotes Archimedes (d. 212 b.c). which gives us the other time limit. Scholars have given different dates, ranging from 150 b.c. to a.d. 250, but the question has been settled by O. Neugebauer, who observed that an eclipse of the moon described by Hero in his Dioptra (chapter 35) as taking place on the tenth day before the vernal equinox and beginning at Alexandria in the fifth watch of the night, corresponds lo an eclipse in a.d. 62 and to none other during the 500 years in question.2 An astronomical date is the most reliable of all, being independent of tradition and opinion. The rather minute theoretical possibility that Hero might have lived long after this date I have discussed and dismissed, while I have elsewhere reviewed the whole controversy about his dates, which is now of historical interest only.4
The question of what sort of man he was has also been debated. H. Diels found that he was a mere artisan.5 I. Hammer-Jensen took him to be an ignorant man who copied the chapters of his Pneumatics from works which he did not understand.6 Although E. Hoppe attempted to defend Hero.7 Hammer-Jensen maintained her opinion.8 In 1925 J. L. Heiberg wrote; “Hero is no scientist, but a practical technician and surveyor. This view, which has been challenged in vain, was first put forth by H. Diels: [who called him] ‘Ein reiner Banause.’”9
Such adverse judgment was based on a study of the Pneumatics at a time when neither the Mechanics nor the Metrica was known; and the Pneumatics, although by far the largest work (apart from the elementary textbooks) was neither by its contents nor form apt to inspire confidence in a serious scholar. The contents are almost exclusively apparatuses for parlor magic, and there is no discernible plan in the arrangement of the chapters. Apart from the introduction, there is no theoretical matter in the book, which consists entirely of practical descriptions.
But since then, the Mechanics has been published in Arabic, and a manuscript has come to light giving the Metrica in its original form; thus the image of Hero has changed. The Mechanics shows nothing of the disorder of the Pneumatics, consisting of an introduction, a theoretical part, and a practical part; the Metrica shows that Hero possessed all the mathematical knowledge of his time, while a chapter of the Dioptra indicates that he was familiar with astronomy. We also find that he quotes Archimedes by preference and has copied many chapters of a lost work of his on the statics of plane figures.
In the introduction to the Pneumatics, Diels found a quotation from Strato of Lampsacus (fl. 288 B.C.) and suggested that it was taken from Philo of Byzantium (fl. 250 b.c.), who probably took it from Ctesibius (fl. 270 b.c.);10 but Philo’s Pneumatics, which was discovered later, does not contain this passage, and a strictly accurate quotation is most likely to have been taken from the original work. The form of this theoretical introduction led I. Hammer-Jensen to assume that Hero was an ignoramus who did not understand what he copied from diverse sources; yet to me the freely flowing, rather discursive style suggests a man well-versed in his subject who is giving a quick summary to an audience that knows, or who might be expected to know, a good deal about it.
This discursive style, so very different from the concise style of the technical descriptions, is found again in the Mechanics, in which Hero, before giving the propositions from Archimedes’ book On Uprights, presents the theory of the center of gravity as explained by Archimedes, not by Posidonius the Stoic, whose definition was not good enough.11 Here again there is a strong suggestion of a teacher repeating swiftly a piece of knowledge which his students ought to know. Since we know the author as Hero of Alexandria, it seems reasonable to assume that he was appointed to the museum, that is, the University of Alexandria, where he taught mathematics, physics, pneumatics, and mechanics, and wrote textbooks on these subjects.
The Pneumatics can best be regarded as a collection of notes for such a textbook, of which only the introduction and the first six chapters have been given their final shape. All the chapters are uniform in style, even those taken from Philo, and eminently clear, so the idea of an ignorant compiler cannot be upheld. But there is more to be learned from the Pneumatics. While there is no order at all in the general arrangement of the chapters, we find here and there a short series of related chapters in which it is clear that Hero is searching for a better solution to a mechanical problem. This shows unmistakably that he was an inventor; it is therefore probable that he himself invented the dioptra, the screw-cutter, and the odometer, as well as several pneumatic apparatuses. This is all that can be learned about Hero himself.
The following works have survived under the name of Hero: Automata, Barulkos, Belopoiica, Catoptrica, Cheirobalistra, Definitiones, Dioptra, Geometrica, Mechanica, De mensuris, Metrica, Pneumatica, and Stereometrica. These can be divided into two categories, technical and mathematical. All the technical books, except the Cheirobalistra, seem to have been written by Hero; of the mathematical books only the Definitiones and the Metrica are direct from his hand. The others are, according to J. L. Heiberg, Byzantine schoolbooks with so many additions that it is impossible to know what is genuinely Heronian and what is not.12
The Pneumatics is by far the longest book, containing an introduction and two books of forty-three and thirty-seven chapters, respectively; but it is merely a collection of notes for a textbook on pneumatics. Only the introduction and the first six or seven chapters are finished. The introduction treats the occurrence of a vacuum in nature and the pressure of air and water; although it is written in a very prolix style with occasional digressions, the train of thought is never lost. It seems to have been written by a man very well versed in his subject, who is summarizing for students of pneumatics matters already known to them from their textbooks. Some of the theory is right, some is wrong (for instance, the horror vacui of nature), but it was the best theoretical explanation to be had at the time; a real understanding of the phenomenon had to wait for the experiments of Torricelli.
The first chapters, most of them taken from Philo’s Pneumatics, describe experiments to show that air is a body, and that it will keep water out of a vessel unless it can find an outlet and will keep water in if it cannot enter. Hero goes on to siphons; but soon all order is lost, and the chapters appear haphazardly. Yet there is nothing haphazard about the chapters themselves, each of which—whether taken from Philo or a description of an apparatus seen by Hero—is written in the same concise style and according to a fixed plan, beginning with a description of the apparatus, with letters referring to a figure, then a description of how it works, then last (if necessary) an explanation. With very few exceptions it is evident that the chapters were written by Hero himself, and without exception they are very clear: each instrument can be reconstructed from the description and the figure.
The contents, on the other hand, have always been a source of puzzlement and despair for serious-minded scholars. Certainly Hero describes some useful implements—a fire pump and a water organ—but all the rest are playthings, puppet shows, or apparatuses for parlor magic. Trick jars that give out wine or water separately or in constant proportions, singing birds and sounding trumpets, puppets that move when a fire is lit on an altar, animals that drink when they are offered water—how can one respect an author who takes all these frivolities in earnest?
But Hero’s treatment of these childish entertainments is quite matter-of-fact; he is interested in the way they work. In 1948 I explained this by the assumption that he was writing a handbook for the makers of pneumatic instruments, but this is not necessarily correct.13 Hero was a teacher of physics, of which pneumatics is part. The book is a text for students, and Hero describes instruments the student needs to know, just as a modern physics textbook explains the laws governing the spinning top or the climbing monkey. Playthings take up so much of the book because such toys were very much in vogue at the time and the science of pneumatics was used for very little else. (Among the many toys of the Pneumatics there are even a few that use hot air or steam as a moving power, which has given rise to ill-founded speculations that the steam engine could have been invented at this time.) To this we must add that Hero was an inventor; and to a real inventor any clever apparatus is of interest, regardless of its purpose.
There is a slightly different text, found only in four manuscripts, that is generally designated Pseudo-Hero. of seventy-eight chapters, seven have been radically changed; elsewhere the changes are only verbal corrections to clarify an already quite clear text. This text cannot have been written later than a.d. 500; therefore when the two texts agree, neither of them has been changed since then. For every chapter there is a figure, and the text in most cases begins with a reference to it, such as “Let ABCD be a base....” Since Pseudo-Hero has the same figures as Hero, the figures cannot have been changed after a.d. 500; and there is every reason to believe that they were drawn by Hero himself. A complete set of these illustrations has been published in a reprint of Woodcroft’s translation of the Pneumatics.14 The Pneumatics was by far the most read of Hero’s works during the Middle Ages and the Renaissance; more than 100 manuscripts of it have been found.
The Mechanics, preserved only in an Arabic version, was published in 1893 with a French translation and in 1900 with a German translation. A textbook for architects (that is, engineers, builders, and contractors), it is divided into three books. Book 1 deals with the theoretical knowledge and the practical skill necessary for the architect: the theory of the wheel, how to construct both plane and solid figures in a given proportion to a given figure, how to construct a toothed wheel to fit an endless screw, and the theory of motion. Drawing largely upon Archimedes, Hero then presents the theory of the center of gravity and equilibrium, the statics of a horizontal beam resting on vertical posts, and the theory of the balance.
Book 2 contains the theory of the five simple “powers”: the winch, the lever, the pulley, the wedge, and the screw. The five “powers” are first described briefly, then the mechanical theory of each is presented and the results of a combination of the powers are calculated. Next is a chapter with answers to seventeen questions about physical problems, evidently inspired by Aristotle’s Mechanical Problems, followed by seven chapters on the center of gravity in different plane figures and on the distribution of weight on their supports, once more from Archimedes. Book 3 describes sledges for transporting burdens on land, cranes and their accessories, other devices for transport, and wine presses: the last chapter describes a screw-cutter for cutting a female screw in a plank, which is necessary for direct screw presses.
Apart from the first chapter of book 1, which contains the Barulkos, the work proceeds in an orderly fashion; it shows nothing of the disorder of the Pneumatics, but the style is equally clear and concise, with a single exception. In book 1, chapter 24, Hero gives the theory of the center of gravity, and there he uses the same prolix and discursive style as in the introduction to the Pneumatics. This chapter would also seem to be a summary for students who should already know the subject. There are figures for most of the chapters; that they go back to the original Greek text can be seen from a mistake in the translation of a Greek work in one of the figures.15 Editions of the work give only an interpretation of the figures; facsimilies have been published, with an English translation of many chapters, by A. G. Drachmann.16 The fragments from Archimedes have been published in English with the manuscript figures.17
The Dioptra contains a description of an instrument for surveyors; it consists of a pointed rod to be planted in the ground, with two interchangeable instruments: a theodolite for staking out right angles and a leveling instrument. The description, which unfortunately is imperfect owing to a lacuna in the manuscript, covers six chapters; chapters 7–32 contain directions for the use of the two instruments in a great number of tasks. In chapter 33 Hero criticizes the groma, the instrument then used for staking out lines at right angles; chapter 34 describes an odometer actuated by the wheel of a car, used for measuring distances by driving slowly along a level road. Chapter 35 indicates the method for finding the distance between Alexandria and Rome by simultaneously observing a lunar eclipse in the two cities; this chapter has been thoroughly studied by O. Neugebauer.18 There is no chapter 36, and chapter 37 is the Barulkos, which is also chapter 1 of book 1 of the Mechanics; it is out of place in both. Chapter 38 describes a ship’s odometer and is certainly not by Hero.19
The Belopoiika contains the description of the gastraphetes, or stomach bow, a sort of crossbow in which the bowstring is drawn by the archer’s leaning his weight against the end of the stock, and two catapults worked by winches; two bundles of sinews provide the elastic power to propel the arrow, bolt, or stone. The catapults are shaped like those described by Vitruvius and Philo.20
The Automata, or Automatic Theater, describes two sorts of puppet shows, one moving and the other stationary; both of them perform without being touched by human hands. The former moves before the audience by itself and shows a temple in which a fire is lit on an altar and the god Dionysus pours out a libation while bacchantes dance about him to the sound of trumpets and drums. After the performance the theater withdraws. The stationary theater opens and shuts its doors on the performance of the myth of Nauplius. The shipwrights work; the ships are launched and cross a sea in which dolphins leap; Nauplius lights the false beacon to lead them astray; the ship is wrecked; and Athena destroys the defiant Ajax with thunder and lightning. The driving power in both cases was a heavy lead weight resting on a heap of millet grains which escaped through a hole. The weight was attached by a rope to an axle, and the turning of this axle brought about all the movements by means of strings and drums. Strings and drums constituted practically all the machinery; no springs or cogwheels were used. It represents a marvel of ingenuity with very scant mechanical means.
The Catoptrica, found only in a Latin version, was formerly ascribed to Ptolemy, but is now generally accepted as by Hero. It deals with mirrors, both plane and curved, and gives the theory of reflection; it also contains instructions on how to make mirrors for different purposes and how to arrange them for illusions.
Barulkos, “the lifter of weights,” is the name given by Pappus to his rendering of the Dioptra, chapter 37, and the Mechanics, book 1, chapter 1.21 It is an essay describing how one can lift a burden of 1,000 talents by means of a power of five talents, that is, he power of a single man. The engine consists of parallel toothed wheels and is derived from the Mechanics, book 1, chapter 21; however, it is only a theoretical solution: parallel toothed wheels were not used for cranes during antiquity.22 L Nix takes Barulkos to be the name of the Mechanics, even though Pappus mentions the Barulkos and the Mechanics in the same sentence, because the Arabic lame of the Mechanics is “Hero’s Book About the Lifting of Heavy Things.”23 But since the essay is found as the first chapter of the Mechanics (where it does not belong), the translator would seem to have taken this title to be the title of the whole work, The Cheirobalistra was published in 1906 by Rudolf Schneider, who regarded it as a fragment of a dictionary dealing with catapults; it consists of six items, each describing an element that begins with the letter K.24 E. W. Marsden has interpreted these chapters s a description of a sort of catapult, which he has reconstructed.25 It is unlikely, however, that the Cheirobalistra is actually a work by Hero.
1. Pappus of Alexandria, Collectionis quae supersunt..., Friedrich Hultsch, ed., III. pt. 1 (Berlin, 1878), 1060–1068.
2. O. Neugebauer, “Über eine Methode zur Distanzbestimmung Alexandria-Rom bei Heron,” in Kongelige Danske Videnskabernes Selskabs Skrifter, 26 , no, 2 (1938). 21–24.
3. A. G. Drachmann, “Heron and Ptolemaios,” in Centaurus, 1 (1950), 117–131.
4. A. G. Drachmann, Ktesibios, Philon and Heron, vol. IV of Acta historica Scientiarum naturalium et medicinalium (Copenhagen, 1948), pp. 74–77.
5. H. Diels, “Über das physikalische System des Straton.” in Stizungsherichte der k. Preussischen Akademie der Wissenschaften zu Berlin, no. 9 (1893), 110, n. 3.
6. I. Hammer-Jensen, “Die Druekwerke Herons von Alexandra,” in Neue Jahrbücher für das klassischen Altertum, 25 , pt. 1 (1910), 413–427, 480–503.
7. Edmund Hoppe, “Heron von Alexandrien,” in Hermes (Berlin), 62 (1927), 79–105.
8. I. Hammer-Jensen, “Die heronische Frage,” ibid., 63 (1928), 34–47.
9. J. L. Heiberg, Geschichte der Mathematik und Naturwissenschaften im Altertum, which is in Iwan von Müller, ed., Handbuch der Altertumswissenschaft, V, pt. 1, sec. 2 (Munich, 1925), 37.
10. Diels, op. cit., pp. 106–110.
11. Hero, Mechanics, ch. 24.
12. Heiberg, loc. cit.
13. Drachmann, Ktesibios..., p. 161.
15. A. G. Drachmann, The Mechanical Technology of Greek and Roman Antiquity, vol. XVII of Acta historica Scientiarum naturalium et medicinaliuin (Copenhagen, 1963), p. 110, text for fig. 44.
16.Ibid., pp. 165 ff.
17. A. G. Drachmann, “Fragments from Archimedes in Heron’s Mechanics,” in Centaurus, 8 (1963), 91–146.
18. Neugebauer, op. cit.
19. Drachmann, The Mechanical Technology....
20. Vitruvius, De architectura, X, ch. 11; and Philo, Belopoiika, Greek and German versions by H. Diels and E. Schramm, in Abhandlungen der Preussischen Akademie der Wissenschaften for 1918, Phil.-hist. K.I., no. 16 (1919).
21. Pappus, op cit., pp. 1060 ff.
22. Drachmann. The Mechanical Technology..., p. 200.
23. Hero, Mechanics, introduction, pp. xxii ff.; Pappus, op. cit., p. 1060.
24. Rudolf Schneider, ed. and trans., “Herons Cheirohalistra,” in Mitteilungen des kaiserlich deutschen archaeaologischen Instituts. Römische Abt., 21 (1906), 142–168.
25. E. W. Marsden, Greek and Roman Artillery. Technical Treatises (Oxford, 1971), pp. 206–233.
I. Original Works. Heronis Alexandrini Opera quae supersunt omnia, 5 vols. (Leipzig, 1899–1914), contains all Hero’s works except the Belopoiica. Automata is published with Pneumatica, Opera, 1. Belopoika appeared as “Heron’s Belopoiica Griechisch und Deutseh von H. Diels und E, Schramm,” in Abhandlungen der K. Preussischen Akademie der Wissenschaften, Phil.-hist. K1. no. 2 (1918); and in E. W. Marsden. Greek and Roman Artillery. Technical Treatises (Oxford, 1971), with English trans, and notes. Catoptrica is published with Mechanica, Opera, II, pt. 1. Cheirobalistra, edited and translated by Rudolf Schneider, is in Mitteilungen des kaiserlich deutschen archaeologischen Institut s, Römische Abt., 21 (1906). 142ff.; and in Marsden, above. Dioptra is published with Metrica, Opera, III. Definitiones and Geometrica appear as Heronis definitiones cum variis collectionibus Heronis quae feruntur Geometrica, J. L. Heiberg, ed., Opera, IV. Mechanica is available as Carra de Vaux, “Les mécaniques ou l’élévateur de Héron d’Alexandrie,” in Journal asiatique, 9th ser., 1 (1893), 386–472, and 2 (1893), 152–269, 420–514, consisting of Arabic text and French translation; and as Herons von Alexandria Mechanik und Katoptrik, edited and translated by L. Nix and W. Schmidt, Opera, II, pt. 1. De mensuris is published with Stereometrica, Opera, V. Metrica is available in three versions: Herons von Alexandria Vermessungslehre und Dioptra, Greek and German versions by Hermann Schöne, Opera, III: Codex Constantinopolitanus Palatii Veteris, no. 1. E. M. Bruins, ed., 3 pts. (Leiden, 1964)—pt. 1, reproduction of the MS; pt. 2, Greek text; pt. 3, translation and commentary; and Heronis Alexandrini Metrica..., E, M. Bruins, ed. (Leiden. 1964). Pneumatica can be found as Herons von Alexandria Druckwerke und Automalentheater, Greek and German versions edited by Wilhelm Schmidt, Opera, I; and The Pneumatics of Hero of Alexandria, translated for and edited by Bennet Woodcroft (London, 1851) and fac, ed. with intro. by Marie Boas Hall (London-New York, 1971). Stereometrica appears as Heronis quae feruntur Stereometrica et De mensuris, J. L. Heiberg, ed., Opera, V. Fragmenta, the commentary on Euclid’s Elements, is found as Codex Leidensis 399, l. Euclidis Elementa ex interpretatione Al-Hadschdschadschii cum commentariis Al-Narizii, Arabic and Latin edited by R. O. Besthorn and J. L. Heiberg, 3 vols. (Copenhagen, 1893–1911).
II. Secondary Literature. See A. G. Drachmann, Ktesibios, Philon and Heron, vol. IV in Acta historica Scientiarum naturalium et medicinalium (Copenhagen, 1948), on Pneumatics; and The Mechanical Technology of Greek and Roman Antiquity, vol. XVII in Acta historica Scientiarum naturalium et medicinalium (Copenhagen, 1963), on Mechanics; J. L. Heiberg, Geschichte der Nathematik und Kantrwissenschaften un Altertum, in Iwan von Müller, ed., Handbuch der Altertumswissenchaft, V, pt. 2, sec. 1 (Munich, 1925), on the mathematical works; and O. Neugebauer, “Über eine Methode zur Distanzbestimmung Alexandria-Rom bei Heron,” in Kongelige Danske Videnskabernes Selskab Meddelelser, 26 , no. 2 (1938), on Dioptra.
A. G. Drachmann
Hero of Alexandria
HERO OF ALEXANDRIA
(fl. Alexandria, maybe first century CE),
mathematics, optics, pneumatics, mechanics. For the original article on Hero see DSB, vol. 6.
The main points in the present postscript are the problem of Hero’s dating, returned to scholarly attention; an updated account of the treatise ascribed to him and known as De speculis; the extent, contents and aims of his commentary on Euclid’s Elements; and his peculiar approach to the procedure of analysis and synthesis. Scholarship is beginning to assess Hero’s mathematical works in a way unbiased by his traditional renown as the mechanician. It appears that he was a skilled and original author who played a key role in the development of Greek mathematics, not only as a “vital link in a continuous tradition of practical mathematics” (DSB, p. 314), but also in mastering and adapting purely theoretical tools.
Hero’s Dating . A lunar eclipse was described in Dioptra 35 as observable in Alexandria at the fifth hour (and in Rome at the third hour) of the night ten days before the vernal equinox. The eclipse was identified as the one that occurred on 13 March 62 CE, although with some latitude, because the vernal equinox fell on 20 March 62 CE. This provides a terminus post quem. The argument was completed by observing that the eclipse must have really been observed by Hero because its proximity to the equinox makes it particularly ill-suited to the graphical solution he is proposing for the analemma-problem solved in Dioptra 35. The identification has been challenged by two kinds of arguments. The first argument is a statistical one. References to Hero in Pappus of Alexandria and to Archimedes in Hero leave a span of approximately five hundred years as the one in which Hero must have lived. Within such a time span there is a probability of about three over four that at least one lunar eclipse actually fits the date, and even more if some margin for error is allowed. The probability reduces to about 10 percent if one inserts the time datum, but remains nonnegligible. Therefore, the fact that one eclipse fitting Hero’s data has been found does not authenticate his description as a reference to a historical eclipse rather than a textbook example. The second argument concerns the identification of the eclipse itself. Two other lunar eclipses fit the temporal data as well as the one of 62, and the date actually better than that. In any of the three cases, however, considerable latitude must be allowed for the determination both of the date of the equinox and of the hours in which the eclipse took place in Alexandria and in Rome: one might well wonder how much can be tolerated for Hero’s description to keep the status of an accurate observational report. The argument that the eclipse is ill-suited to a graphical solution does not stand, because it can be shown that the same problem can be solved exactly, and of course also the other two eclipses are ill-suited as well.
To these arguments it can be added that Hero’s reference to the eclipse was worded in a very peculiar way. He asserted that if simultaneous observations of a lunar eclipse cannot be found in the almanacs, then the reader can provide one, because they occur at intervals of five or six months. After that, the mere existence of the eclipse and its temporal data are set out, by using imperatives, in the typical format of geometrical propositions. This puts the assertions under the hypothetical mode of expression and strengthens the suspicion that the data of the eclipse in Dioptra 35 have been invented by Hero. Finally, care should be taken in handling astronomical phenomena for dating purposes. They are easily transformed from termini post quem into absolute chronological determinations by a very questionable application of a principle of economy of hypotheses. The same holds for chronological determinations that have been proposed on the basis of other passages in Hero’s works. What is more, all of these concur to a year near 62 and are in fact intended to support the dating by means of the eclipse. The search for textual clues is thus biased by the assumption that this is correct and hence they are of no independent value.
The De speculis . A new edition led to an improved assessment of this short monograph, preserved only in the Latin translation of William of Moerbeke. The ascription to Ptolemy in the title was long ago recognized as a copyist’s mistake: for the late optical author Damianus ascribed to Hero a proof that straight lines reflected at equal angles are minimal among those that inflect on the same (straight) line. This is precisely the proof found in the De speculis, even if Damianus’s testimony does not guarantee that the short treatise is originally Heronian. It is now clear that the Greek text translated by Moerbeke was a late compilation. A first clue comes from the phrase in rymis sive in plateis (in the streets or lanes) in the enunciation of section 22: it transliterates a Greek expression in Luke 14:12 and otherwise found only in Christian writers. This presupposes in the author an acquaintance with Christian literature that suggests a late dating. The very initial portion of the introduction refers to Plato and develops considerations on the harmony of the spheres that are at odds with the typically direct argumentations of Hero’s introductions.
The original Heronian work appears, as far as can be discerned from a critical assessment of the extant De speculis, to have aimed at providing tighter foundations for the theory of reflection than the ones found in the Catoptrics ascribed to Euclid. To this end, the equal-angle rule for reflection is based on a less contrived assumption than the one in the Catoptrics. The argument leading to this basic assumption is a combination of mathematical and physical arguments that is typically Heronian. Visual rays move with infinite speed, and what moves with the greatest speed moves along a straight line, for, because of the speed, it tends to move along the shortest path and straight lines realize the shortest path between two fixed points. In case of reflection, direct vision being excluded, the assumption naturally reduces to the requirement of realizing the shortest path through broken straight lines. The compiler completed the original theoretical part, that possibly contained also a proof of the contrived assumption of the Catoptrics, with a rather arbitrary choice from an initial segment of the Catoptrics itself. The subsequent description of the several mirror devices was shortened to an extent that cannot be determined, as the introduction mentions arrangements that are not found in the extant text. The bad status of many of the excerpted descriptions must more likely be ascribed to accidents in the transmission of the text than to ineptitude of the compiler. An exception is section 23, where a defective trivialization of Catoptrics 14 is expounded. Marks of original Heronian conception surface here and there, however: for instance, the proof in section 22 is shaped in the format of analysis and synthesis and a dioptra is employed to perform a part of the construction.
The Commentary to the Elements . Extensive and literal quotations from Hero’s commentary to Euclid’s Elements are found in Proclus and in a number of Arabic commentators, most notably in an-Nayrizi. The title of the commentary as reported in the Fihrist and in other Arabic sources is Book of the Solution of the Difficulties in Euclid. It appears that Hero was one of the first authors of technical commentaries. He possibly originated the genre by fixing some canonical critical attitudes toward the commented text:
addition of missing cases, of lemmas and corollaries or of mathematical complements;
proposal of alternative proofs suited to strengthen the deductive structure or to simplify it;
replacement of proofs by reduction to impossible with direct proofs or, less frequently, the opposite;
structural adjustments, such as inversion of the order of certain propositions, suppression or addition of definitions and axioms;
even rewriting entire segments of the text, such as propositions 2–10 of Book 2.
As is clear, such a commentary was less exegetical and more aggressive than the ones compiled in late antiquity. It speaks for an author concerned with issues of logical coherence and deductive structure, adapting demonstrative techniques from other fields, as in the case of the rewritten proofs of 2.2–10. If a comparison can be made, it is more to the Alexandrian commentaries of the literary genre, where difficult passages of the edited author were explained but on the same grounds substantial modifications to the text were proposed. However, it is unlikely that Hero procured a new text of the Elements. Proposals of modifications rather than direct changes are found there and the Arabic title at least suggests that the work might have had a format and a circulation independent from the Euclidean treatise. Minor changes in the text cannot be detected with the available critical methods.
What is unquestionable is that Hero’s commentary, in the hands of later editors, interfered with the transmission of the Elements, producing modifications that affected both the received Greek and Arabic versions, often in independent ways. For instance, the alternative proof to 3.10 proposed by Hero was is now attested as such in the Greek manuscripts of the Elements, whereas the case added to 3.11 directly became 3.12. Additional cases of 3.20, 25, and 30 found their way into the text, whereas the proposed displacement of 3.25 affected a part of the Arabic tradition only. It is not clear whether Hero’s commentary aimed at didactical purposes, as is usually assumed for any such work. The fact that it belonged to a well-established literary genre appears to provide enough reasons for its existence, and its aims are more properly described as scholarly than as pedagogical.
Analysis and Synthesis . Two heterodox variants of the technique of analysis and synthesis are found in the Heronian corpus. The first occurs in the Metrica, the second in the alternative proofs to Elements 2.2–10. The proofs in the Metrica concern problems that are set out assigning numerical values to the relevant quantities, such as, for instance, the sides of a triangle whose area is sought. Typically, the sought quantity is obtained at the end of an analysis, consisting of a series of steps framed in the language of givens. The analysis is followed by a calculation. The latter is termed synthesis by Hero, who overcame in this way any concern about constructions. The synthesis–calculation repeats step by step the chain of the givens. The calculation is performed with specific numbers, but they must be taken as paradigmatic, as always in Hero, and hence the calculation must be read as a description of an algorithm. Proofs similar to those in the Metrica can be found in other treatises of applied mathematics, for instance in Ptolemy’s Almagest, where the format of the chain of givens is employed to show how to single out one magnitude from an expression in which it is involved. The unifying feature of such proofs is that the calculation is legitimated once the analysis has shown that the magnitude in question is uniquely determined by the givens of the problem.
The alternative proofs to Elements 2.2–10 are reported by an-Nayrizi. They form a strict deductive sequence. Each proposition is considered as proved insofar as it is reduced to previous results. Such a feature makes these proofs similar to the kind of analysis of complex deductive chains developed within the framework of Stoic logic. Both analysis and synthesis are framed as reductive sequences, in which two expressions, combining squares and rectangles formed from suitable lines, are transformed until it is manifest that they are equal. As a consequence, the analysis already ends with the proof of the sought equality. No chain of givens is displayed. Analysis and synthesis really become two independent and full-fledged ways of proof, as Hero himself is reported, in the short account preceding the alternative proofs, to have already stressed in his definition of the method. Such an approach can be found also in treatises dealing with number theory, such as Diophantus’s De polygonis numeris. To the Heronian proofs of 2.2–10 the alternative ones to 13.1–5 attested in the manuscript tradition of the Elements are usually but not properly compared.
WORKS BY HERO OF ALEXANDRIA
Jones, Alexander. “Pseudo-Ptolemy De Speculis.” SCIAMVS 2(2001): 145–186. This is the new edition of the De Speculis, based on a fresh reading of the autograph of William of Moerbeke’s translation.
an-Nayrizi. Anaritius’ Commentary on Euclid. The Latin Translation I–IV. Edited by P. M. J. E. Tummers. Nijmegen, Netherlands: Ingenium Publishers, 1994. The Latin version of an-Nayrîzî’s commentary is now read in this critical edition, superseding, as far as the first four books are concerned, Ernst Curtze’s edition published as a Supplementum to Johan l. Heiberg and Heinrich Menge’s Euclidis Opera Omnia.
Argoud, Gilbert, ed. Science et vie intellectuelle à Alexandrie (Ier–III e siècle après J.-C.). Saint-Étienne, France: Publications de l’Université de Saint-Étienne, 1994. Much scholarly attention is being paid to the mechanical works of Hero. Besides the entire volume containing Souffrin’s note cited below, several contributions specifically dealing with Heronian matters may be found here.
Cambiano, Giuseppe. “Automaton.” Studi Storici 35 (1994): 613–633. This and the following article investigate the relationships between Hero’s mechanics and the philosophical background.
Guillaumin, Jean-Yves. “L’Éloge de la géométrie dans la préface du livre 3 des Metrika d’Héron d’Alexandrie.” Revue des etudes anciennes 99 (1997): 91–99.
Høyrup, Jens. “Hero, Ps.-Hero, and Near Eastern Practical Geometry. An Investigation of Metrica, Geometrica, and other Treatises.” In Antike Naturwissenschaft und ihre Rezeption, edited by Klaus Döring, Bernard Herzhoff, and Georg Wöhrle, 67–93 Band 7. Trier, Germany: Wissenschaftlicher Verlag Trier, 1997. This text focuses on Hero as a link between Babylonian and post-Greek mathematics.
Keyser, Paul. “Suetonius Nero 41.2 and the Date of Heron Mechanicus of Alexandria.” Classical Philology 83 (1988): 218–220. A textual finding that supports (with the reservations expressed in the text) Neugebauer’s proposal is found here.
———. “A New Look at Heron’s ‘Steam Engine.’” Archive for History of Exact Sciences 44 (1992): 107–124. The author proposes to interpret a device described by Hero in the Pneumatics as an experimental refutation of one Aristotelian thesis on self-motion.
Knorr, Wilbur R. “Arithmêtikê Stoicheiôsis: On Diophantus and Hero of Alexandria.” Historia Mathematica 20 (1993): 180–192. Knorr makes an attempt to challenge the traditional ascription to Hero of the core of the Definitiones.
Sidoli, Nathan. “Heron’s Dioptra 35 and Analemma Methods: An Astronomical Determination of the Distance between Two Cities.” Centaurus 47 (2005): 236–258. Neugebauer’s identification of the eclipse is criticized in this article.
Souffrin, Pierre. “Remarques sur la datation de la Dioptre d’Héron par l’éclipse de lune de 62.” In Autour de la Dioptre d’Héron d’Alexandrie, edited by Gilbert Argoud and Jean-Yves Guillaumin, 13–17 Saint-Étienne, France: Publications de l’Université de Saint-Étienne, 2000. This short note contains the statistical argument concerning Hero’s dates.
Tybjerg, Karin. “Doing Philosophy with Machines: Hero of Alexandria’s Rhetoric of Mechanics in Relation to the Contemporary Philosophy.” PhD diss., University of Cambridge, 2000. A rich and updated bibliography may be found here.
Vitrac, Bernard. “A Propos des démonstrations alternatives et autres substitutions de preuves dans les Éléments d’Euclide.” Archive for History of Exact Sciences 59 (2004): 1–44. A preliminary assessment of aims and extent of Hero’s commentary on the Elements is given here.
———. “Peut-on parler d’algèbre dans les mathématiques grecques anciennes?” Ayene-ye Miras n.s. 3 (2005): 1–44. The algorithmic character of some proofs in the Heronian corpus is analyzed in this article.
Hero of Alexandria
Hero of Alexandria
First century a.d.
Greek Inventor and Mathematician
Hero (or Heron) of Alexandria was a prolific writer of mathematical and technical textbooks. His best known works are Pneumatics and Metrica. Credited with the invention of an early form of steam engine, Hero created a number of technical devices, including the odometer, dioptra (surveying tool), and screw press.
Little is known about Hero and his life. In fact, the time during which he lived is subject to debate, with speculation ranging from 150 b.c.to a.d. 250. The most accurate estimate appears to be around a.d. 62. Even less is known about his personal life. Due to the number of books he wrote, and the content of these books, it has been suggested that he was appointed to the Museum or the University of Alexandria, where he probably taught mathematics, physics, pneumonics, and mechanics. Many of Hero's books were likely intended to serve as textbooks for his classes. What kind of person Hero was has also been the subject of debate. While some considered him to be incompetent and uneducated, simply copying the works of different scientists, others believed him a skilled mathematician and creative inventor.
Hero wrote many books, with Pneumatics being the longest and perhaps the most read. It was very popular during medieval times and during the Renaissance. The book outlined various pneumatic devices, and shared descriptions of how they worked. Most were no more than toys used for magic and amusement, and has led some scholars to believe he was not a serious scientist or inventor. Hero indicated that some of the inventions were his and that others were borrowed, but did not clarify which ones were actually his, giving the impression that he was merely collecting the knowledge of others. Most formed this opinion before some of Hero's works, such as Metrica and Mechanics, had been found. Metrica, his most important work on geometry, was lost until 1896 and contained formulas to compute the areas of things like triangles, cones, and pyramids. The area of the triangle is often attributed to Hero, but it is likely he borrowed it from Archimedes (287?-212 b.c.) or the Babylonians. Mechanics deals with machines, mechanical problems of daily life, and the construction of engines. Though these books have been criticized for their preoccupation with child-like toys and disorganization, they were likely used as textbooks. The attention to popular toys was probably employed to explain the principles of physics and pneumatics to students, and the lack of proper organization in his books may result from the fact that they were never completed. Other books by Hero include Dioptra, Automata, Barulkos, Belopoiica, Catoprica, Definitiones, Geometrica, De mensuris, and Stereometrica.
One of Hero's greatest achievements was the invention of the aeolipile, considered by many to be the first steam-powered engine. The plans for this machine are found in Pneumatics. Also described in Pneumatics were siphons, a fountain, a coin-operated machine, a fire engine, and other steam-powered machines. In Dioptra, Hero described the diopter or dioptra, a surveying instrument similar to the theodolite. Hero also displayed a familiarity with astronomy in a chapter of Dioptra, in which he described a method for finding the distance between Rome and Alexandria using a graphical formula based on the position of the stars. Another notable invention of Hero's was the screw press; at the time it was a new and more efficient way to extract juice from grapes and to extract oil from olives.
Hero's contribution to science was varied, though his tireless devotion to the collection of ideas and knowledge was significant in itself. Several of Hero's machines, such as the steam engine, are often cited as his most important contributions. While Hero did not invent the steam engine as we now know it, he did contribute to its eventual creation. The steam engine had a major impact on society, allowing physical, time-consuming labor to be completed by a machine, and freeing people to concentrate on other things, like exploration and discovery. The field of mathematics also benefited from Hero. His books chronicled the mathematical knowledge of his day and allowed others who came later to build on that work.
Heron of Alexandria
Heron of Alexandria
The engineer, mathematician, and inventor Heron of Alexandria (active ca. 60) ranks among the most important scientists of the ancient Roman world in the tradition of Aristotelian experimentation.
Heron, about whose personal life virtually nothing is known, resided in Alexandria, Egypt, among the scientists and men of letters of the late Ptolemaic and Roman eras who dwelled around the famed library and museum. A brilliant theoretical scientist and a prolific writer, Heron wrote with clarity and insight. The knowledge of his writings and scientific investigations was preserved in the writings of the late Roman, Byzantine, and Arabic scientists and encyclopedists.
One of Heron's outstanding treatises was the Metrica, a geometrical study, in three volumes, on the measurement of simple plane and solid figures from polygons to hendecagons. It approximates the areas of triangles, polygons, quadrilaterals, ellipses, spheres, circles, and cones, and the volumes of various solids, including the cone, cylinder, and pyramid. In developing the mathematical studies, Heron solved complex quadratic equations arithmetically, approximated the square roots of nonsquare numbers, and calculated cube roots. Heron's other mathematical works include the Definitions, Geometrica, Geodaesia (Land Measurements), Stereometrica (Solid Measurement), Mensurae (Measures), and Liber geëponicus (Book on Agriculture).
In the Mechanica, preserved only in Arabic, Heron explored the parallelograms of velocities, determined certain simple centers of gravity, analyzed the intricate mechanical powers by which small forces are used to move large weights, discussed the problems of the two mean proportions, and estimated the forces of motion on an inclined plane. The Pneumatica, possibly derived from the works of Philo of Byzantium and Ctesibus, describes mechanical devices operated by compressed air, water, or steam. Included are the steam engine, siphon, fire engine, water organ, slot machines, and water fountains. Other works by Heron dealing with the problems of mechanics and engineering are the Barulcus (On Raising Heavy Weights), Belopoeica (Making Darts), On Automaton-making, Catoptrica (On Mirrors), and On the Dioptia. In the last treatise Heron describes a machine called the "Cheirobalistra," which depended on a refined screw-cutting technique and could be used to bore a tunnel through a mountain. He also described an instrument called the hodometer for measuring distances traveled by wheeled vehicles.
Beyond gadgetry, the practical application of Heron's ideas in antiquity was minimal, although they did influence Arabic and Renaissance construction of fountains, clocks, and automated objects. In Heron's time the widespread use of slave labor throughout the Roman world negated most interests in labor-saving devices.
Some fragments of Heron's writings appear in English in Morris R. Cohen and Israel E. Drabkin, A Source Book in Greek Science (1958). Heron's scientific ideas are best presented in A. G. Drachman, The Mechanical Technology of Greek and Roman Antiquity (1963), and Robert S. Brumbaugh, Ancient Greek Gadgets and Mechanics (1966). General books which discuss Heron and ancient science are Marshall Clagett, Greek Science in Antiquity (1955), and L. Sprague de Camp, The Ancient Engineers (1963). □