Hero of Alexandria'sMechanical GeometryKarin Tybjerg

Greek mathematics is often identified with the achievements of a ratherselect group of Hellenistic authors, most notably Euclid, Archimedesand Apollonius. In these authors we find a highly sophisticated form ofmathematics and fully developed practices of deductive demonstrationand it is the works of these authors and their predecessors that havereceived the majority of attention in the scholarship.

Other forms of mathematics, such as the mathematics employed byprofessionals or practitioners working with land-measurement, trade,architecture or administration, have also received some measure ofinterest, and it has become clear that Greek mathematics covers a plural-ity of practices. The methods employed by these practitioners, oftenbased on numerical calculations and approximations, have, however,often been sidelined as 'applied' or even 'lower' forms of mathematics.

These two categories of mathematics provide a strangely polarizedpicture of mathematical practices, a picture that does not accuratelyportray the mathematical landscape of antiquity. Although the dichot-omy is still alive and well, some doubt has already been cast over thepossibility of a strict division between high-level and low-level prac-tices, or between theoretical and applied mathematics. Reviel Netz hasgestured at a 'material' element in the manipulations of Euclidean-Ar-chimedean geometrical diagrams, which are, for instance, 'constructed'and 'cut' in real space.1 In the case of professional mathematicians, Jens

l Netz, 1999, Chapters 1-2; see also Russo, 2003,185-6.

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30 Karin Tybjerg

H0yrup has drawn attention to a culture of 'recreational' problemsolving and thereby shown that calculations of areas of land were notalways aimed at practical pursuits.2 Serafina Cuomo has criticized thesharp division more generally by arguing for the inclusion of 'low-brow'practices in the history of mathematics alongside advanced practices.3With this paper I hope to add nuance to this picture of Greek mathe-matics by analysing how geometry is used and presented in themechanical and mathematical writer Hero of Alexandria.

Hero of Alexandria flourished in the first century AD and his treatisesspan a wide range of topics, from geometry and land measurement topneumatics, catapults and automatic theatres.4 In his treatises we findboth elements associated with the Euclidean-Archimedean traditionand material coming from cultures of professional mathematicians.Starting with the former, Hero wrote a commentary on Euclid'sElements, which, however, is only preserved in a fragmentary form inan Arabic commentary.5 Most notably, he makes frequent referencesto Archimedean treatises throughout his work.6 Other authors of themathematical elite are also mentioned such as Eudoxus, Dionysodorus,Eratosthenes and Plato/ and Hero mentions titles of treatises that are

2 H0yrup, 1990 and 1996a

3 Cuomo, 2001

4 Of Hero's works the Pneumatics, Automaton Construction, Metrica, Dioptra andArtillery Construction are extant in the original Greek. Mechanics is preserved in anArabic translation and Catoptrics in a Latin abbreviation. I shall refer to Hero'streatises in the Teubner edition (Schmidt, Nix, Schone and Heiberg, 1899-1914),except for the Artillery Construction, which is edited in Marsden, 1971. The Teubneredition also includes the Geometry, Stereometn/ and Definitions, which are textuallydifficult and of dubious attribution.

5 Al-Nairizi's commentary is found in Codex Leidensts. There are numerous referencesto Euclid in Stereometry, but this work has been substantially edited so it is difficultto draw any conclusions from these references

6 Metrica 66.7,66.13,66.27,80.17,84 11,86.30,88.10,88.25,120.15,122.1,122.16,130.26,172.1, 184.27; Pneumatics 24.11; Mechanics 1.4, 1.24 (four references), 1.25, 1.32,1.33,II.7,11.35. Drachmann, 1963 points to Archimedean sections in Mechanics.

7 Metrica 2.14 (Eudoxus), 128.3 (Dionysodorus), 132.7 (Plato's solids); Dioptra(Eratosthenes' On the Measurement of the Earth) Brought to you by | University of Haifa

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Hero of Alexandria's Mechanical Geometry 31

associated with Apollonius and Hipparchus.8 As we shall see, a numberof Hero's treatises also contain geometrical proofs and his descriptionsof machines and measurement techniques are presented as geometricaltheorems and problems. Hero does not, however, stick to the topicsand approaches typical of the Euclidean-Archimedean tradition. Healso solves practical problems of measurement and instrument con-struction, and employs calculation and approximation methods resem-bling those found in traditions of professional mathematics. H0yrup,for instance, located material that is related to a Near-Eastern traditionof practical mathematics in Hero's Metrica and Geometry, and Neuge-bauer saw Hero's work as a Hellenistic form of a Babylonian arithmetictradition.9

So Hero neither fits into the standard picture of the Euclidean-Ar-chimedean tradition, nor does he represent a tradition of professionalmathematicians and calculators. The relationship between material de-rived from these traditions is not a simple one either. Hero's work canneither be viewed as an application of Euclidean-Archimedean mathe-matics to practical problems, nor as a formalisation of practical methods.But this limbo in which Hero has been left by the scholarship is exactlywhat makes his work central to rethinking ancient mathematics. Hero'swork allows a rare view of the interaction between geometry, mechanicsand professional mathematics; it shows that these enterprises wereclosely related in the ancient world and that some demonstrative proce-dures combined elements from several traditions.

Beginning with Metrica — a treatise on measuring and dividing areasand volumes — we get a sense of how Hero situates his work.

The first geometry, as the old story teaches us, was engaged in meas-urements and distributions of land; this is also why it was calledgeometry. Since man needed this study, this type [of geometry] wasadvanced still more, so that the control of measurements and distribu-

8 Hero refers to a 'table of chords' that may have been by Hipparchus, Metrica 58.19,62.17, see Heath, 1921, Vol. II, 259, and to 'the second book of the Cuttmg-off of anArea', which is a non-extant treatise of Apollonius, Metrica 162.2, see Tannery, 1903,148.

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32 Karin Tybjerg

tions came also to deal with solid bodies. And since the first theoremsinvented were not sufficient, they were still in need of further investi-gation, so that even to this day some of them remain problematic, andyet both Archimedes and Eudoxus have applied themselves eminentlyto the study. For before Eudoxus' invention it was unfeasible to pro-duce a demonstration (άπόδειξις) that the cylinder, which has the samebase as a cone and the same height, is three times as great as this andthat the circles are to each other as the squares of the diameters are toeach other. And before Archimedes' quick-witted inventions, it wasdoubtful why the surface of the sphere is four times as large as thesurface of the greatest of its circles and that its solid is two thirds of thecylinder that contains it and likewise, many similar questions. Since theinquiries we have mentioned are necessary we think it has value tocollect as much useful material as was written before us and in additionas much as has been considered by us.10

Hero begins with a well-known topos, when he situates the origin ofgeometry in land measurement.11 This link is especially important toHero because he himself deals with both straight geometry and divisionof land. The emphasis on human needs shows the indispensability of thesubject and indicates the practical ambition of the work. Next we hearhow geometry turned to solid bodies and that Eudoxus and Archimedesproduced some important theorems. In this passage Hero continues the

10 Metrica 2.3-4.4. Ή πρώτη γεωμετρία, ως ό παλαιός ημάς διδάσκει λόγος, περί τας εν τργη μετρήσεις και διανομάς κατησχολεΐτο, όθεν καϊ γεωμετρία εκλήθη · χρειώδους δετου πράγματος τοις άνθρώποις υπάρχοντος επί πλέον προήχθη το γένος, ώστε και επί ταστερεά σώματα χωρήσαι την διοίκησιν των τε μετρήσεων και διανομών, και επειδή ουκέξήρκει τα πρώτα έπινοηθέντα θεωρήματα, προσεδέηθησαν έτι περισσότεροςεπισκέψεως, ώστε και μέχρι νυν τινά αυτών άποπεϊσθαι, καίτοι Άρχιμήδους τε καιΕύδόχου γενναίως έπιβεβληκότων tfi πραγματεία, άμήχανον γαρ ην προ της Εύδόχουέπινοίας άπόδειξιν ποιήσασθαι, δι'ής ό κύλινδρος του κώνου του την αυτήν βάσινέχοντος αύτω καϊ ϋψος 'ίσον τριπλάσιος εστί, και οτι ο'ι κύκλοι προς αλλήλους είσΐν ωςάπο των διαμέτρων τετράγωνα προς άλληλα, καϊ πρό[ς] της Άρχιμήδους συνέσεωςαπιστον ην έπινοήσαι, διότι ή της σφαίρας επιφάνεια τετραπλάσια εστί του μεγίστουκύκλου των εν αυτή καϊ οτι το στερεόν αυτής δύο τριτημόριά εστί του περιλαμβάνοντοςαυτήν κυλίνδρου καϊ οσα τούτων άδελφά τυγχάνει

11 Cf. Herodotus (fifth century BC) The Histories II109; Plato Phaedrus 274C; DiodorusSiculus (fl. 50 BC) Bibliotheca 1.69.5 and 1.81.1-2; and Strabo (64 BC — after AD 21),Geography XVII, 3. Brought to you by | University of Haifa

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Hero of Alexandria's Mechanical Geometry 33

history of mathematics that Archimedes began in the prefaces of hisMethod and On the Sphere and the Cylinder where he refers to the exactsame Eudoxan demonstration as part of his own mathematical prehis-tory.12 By adding Archimedes' own famous findings — also found in Onthe Sphere and the Cylinder13 — Hero continues the story in the track setout by Archimedes and he is able to place himself at the end of thisformidable line-up. Hero leaves room for his own contribution by notingthat there are still some outstanding problems and that he will add hisown material to the collection of earlier material. He thus claims a jointlegacy of practical land measuring skill and the highest level of Greekgeometry.

Another important feature of this passage is the reference to Eudoxus''demonstration'. This is the first, but not the last, time that Hero uses theterm demonstration (άπόδειξις) in Metrica. Investigating the contextswhere Hero uses the term, we find that he uses it most frequently whenhe refers to demonstrations by Archimedes.14 By employing the term inconnection with the names of Eudoxus and especially Archimedes Heroassociates the term άπόδειξις with Archimedean-style proofs. Thuswhen Hero uses it elsewhere in his work it implies the authority andreliability of Archimedean demonstrations. He can therefore rely on thestahis of geometrical proof when he moves on to the division of land inthe last book of Metrica; here he states emphatically that geometry issuperior to all other arts and sciences for distributing land because 'the

12 Archimedes Method 430.1-9; Archimedes On the Sphere and the Cylinder 4.2-9

13 Sphere and Cylinder Book I (especially 1.33 and 1.34). The theorem that links thevolumes of the sphere and the cylinder was emblematic of Archimedes' achieve-ment as indicated by Cicero's claim that his tombstone showed a sphere inscribedin a cylinder, Tusculan Disputations V.23.

14 Hero employs the terminology of demonstration (άποδείκνυμι or δείκνυμι) in six-teen out of nineteen references to Archimedes in Metrica. He refers mainly todemonstrations in specific treatises, chiefly On the Sphere and the Cylinder, On theMeasurement of the Circle and Method, e.g., 84.11: 'But Archimedes demonstrated inthe Method ...' (άπέδειξεν δε Αρχιμήδης εν τω έφοδικ"). Hero's Mechanics containssix references to Archimedean demonstrations, e.g., I. 24: 'Archimedes has shownin the Equilibrium of Planes ...'; the Arabic term used in the examples from theMechanics is bayana, which means 'prove' or 'show'. There is also a single referenceto an Archimedean demonstration in Pneumatics 24.11.Brought to you by | University of Haifa

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34 Karin Tybjerg

proof of these things is incontrovertible'.1? So although Hero neitherincludes many of the demonstrations by Archimedes to which he refers,nor develops many purely geometrical demonstrations of his own, theterm maintains its association to rigorous deductive demonstration.

As will become clear, Hero adapts the Archimedean legacy andapplies it to a new range of problems of a more mechanical and numeri-cal kind. In contrast to Archimedes, who in the Method showed cautionconcerning the application of numerical and mechanical methods, andexplicitly privileged deductive geometrical proof, Hero appears unham-pered by such concerns.16 He makes the power of Archimedean deduc-tive proof continuous with his own broader project.

In the following we shall consider ways in which Hero's geometricalpractices deviate from the Archimedean ones with which they are asso-ciated. The strategies broadly fall into three categories:

1) Hero makes numerical examples an integral part of his demon-strative practices.

2) Mechanical methods and instruments are made legitimate toolsfor demonstration.

3) Geometry is applied to physical space and mechanical devicesso that the boundary between geometrical and mechanical ob-jects is blurred.

Treating these three strategies in turn we shall see how Hero employsspecific strategies to eliminate distinctions between high-level geomet-rical proof and the methods of practical mathematics. By blurring theboundaries between Archimedean and practical mathematics he enableshimself to combine the power of geometrical proof with the capabilitiesof numerical and mechanical techniques. At the end I shall also showhow the amalgamation of mechanics and geometry supports Hero'sclaims that mechanics is useful. In general, the techniques employed byHero show that it is not possible to maintain the notion that Euclidean-Archimedean geometry was sealed off from traditions of professional

15 Meinen 140.22-142 1

16 See for instance Archimedes Method 428 24-430.1.Brought to you by | University of HaifaAuthenticated | 142.58.101.27

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Hero of Alexandria's Mechanical Geometry 35

problems and calculation techniques. It moreover shows that practicalconsiderations and a focus on measurement inform Hero's geometry,but it is not a simple relation of practical application of geometry.

In my analysis I shall draw on Hero's more mathematically-orientedtreatises: Metrica, which concerns the measurement of planes, threedimensional objects, and ways of dividing them; Dioptra, which solvesvarious problems of land measurement and construction with a devicethat measures distances and angles; Catoptrics, which concerns reflec-tions in mirrors; and Mechanics, which deals with lifting, sizing andmoving heavy objects.

Numbers and Measurement

Metrica is the most mathematical of the treatises thought to be Hero'swith some certainty, and it provides a clear example of how Herocombines traditions. A prominent feature of the style of the propositionsin Metrica is that Hero often assigns specific numbers to the geometricalobjects under investigation. He then calculates the specific areas andvolumes of these figures rather than producing a general method fordoing so. He writes, for instance, 'Let ABG be an obtuse angled triangle,which has the AB 13 units, the BG 11 units, and the AG 20 units.'17 Whatis the role of these numerical examples?

The numbers are given in units, and to understand their significancewe need to look at Hero's definitions of units. In the introduction to thefirst book of Metrica Hero explains why surfaces are measured in termsof areas with right angles and straight sides—i.e., squares. His argumentis geometrical. A straight line, he says, is always equal to another straightline and a right angle equal to another right angle; curved figures, bycontrast, are not always equal. Measurement therefore consists in 'com-paring' the area under investigation to square ells or feet, or simply to'units'. The units are introduced, Hero explains, partly for ease so as notto name the specific units for each measurement, and partly for general-ity; Hero assures his reader that a unit can stand for any measure Onewants'.18 He thus gives a geometrical argument for the possibility of

17 Meinen 14.18-20

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36 Karin Tybjerg

measurement based on a generalized unit of measurement, but at thesame time he links measurement to specific, physical measures. Thedouble nature of Hero's project is also reflected in the wording wherethe term for 'area' (χωρίον) can also mean 'landed property'.

The physical nature of Hero's measurements comes through evenmore clearly when he deals with solid bodies in Book II. The 'demon-stration' (απόδειξις) that the volume of a cube is found by multiplyingits sides is, Hero writes, manifest, and if we think about the cube being'sawed into unit volumes' we get the requisite number.19 By using thestandard term for 'sawing' he provides a highly physical analogy for theact of measuring; and by saying it is 'manifest' or 'visible' (φανερός) Heroemphasizes that the proof is based on the senses. Thus, when Hero usesnumerical examples referring to units of area or volume he links thegeometrical situation to the physical and thereby maintains the connec-tion to practical measurement.

The numerical examples, however, do not just establish a connectionto practical situations of measurement. Nor is their function solely totrain the reader to do the calculations through examples. In the sectionon the measurement of triangles that comes early in Book I, it becomesclear that the numerical examples play a real demonstrative role. Herobegins the section with simple cases of measuring different kinds ofMangles such as the right-angled, the isosceles, and the acute- or obtuse-angled triangles. In each proposition he gives the dimensions of a specifictriangle and derives a calculation procedure for finding its area based onminimal geometrical argument. Having derived a procedure Heroshows how to find the area, this time without reference to the geometricalsituation. Hero is, however, quite explicit that the calculations are notjust included as exercises:

Until now we produced the geometrical demonstrations through calcu-lation; next we shall produce the measurements according to analysisthrough the synthesis of the numbers.20

Hero states surprisingly that calculation is a means for producing 'geo-metrical demonstrations'. Geometry and calculation are thus inter-

19 Metrica 94.5-6: the term used is καταπρίω, whose primary meaning is 'to saw', LSJ.

20 Metrica 16.11-14: Μέχρι μεν τούτου έπιλογιζόμενοι τας γεωμετρικός αποδείξειςέποιησάμεθα, έξης δε κατά. άνάλυσιν δια της των αριθμών συνθέσεως τας μετρήσειςποιησόμεθα. Brought to you by | University of Haifa

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Hero of Alexandria's Mechanical Geometry 37

twined with calculation responsible for producing the crowning gloryof geometry — the demonstration.

In a similar vein calculation and geometry are presented as two sidesof the same solution when Hero states that measurements are producedthrough 'analysis' and 'synthesis'. The pair, analysis and synthesis,refers to a two-part method of presenting geometrical problems that wasused by geometers such as Apollonius and Archimedes. In the analysisthe solution is assumed and its consequences investigated; the synthesissolves the problem in the light of the understanding obtained in theanalysis.21 The concepts, however, did not have stable, well-definedmeanings, and this flexibility allowed Hero to adjust the concepts to hisown priorities. He uses this freedom to make geometrical proof andcalculation equal and necessary parts of solving a problem, a problemthat in Hero's context concerns producing a measurement.

This approach to measurement is applied in the propositions thatfollow, beginning with what is now known as 'Hero's formula' forcalculating the area of a triangle from the lengths of its sides. The methodis introduced by a complex numerical example. Hero then proclaims that'[t]he geometrical demonstration for this is the following'22 and deliversa Euclidean-Archimedean style proof based on a lettered diagram withno reference to particular values for the dimensions of the triangle.23

Following the proof, he writes, 'it is synthesized like this' and givesanother numerical example of the calculation of the area. In subsequentpropositions Hero frequently writes after finishing a geometrical proofthat 'it is synthesized in accordance with the analysis' and then offers anumerical example. He thus establishes a pattern that is followedthroughout the treatise: a geometrical proof, characterized as the 'analy-sis' followed by a calculation characterized as the 'synthesis'.24 Thesynthesis — a standard component of the solution to a geometricalproblem — is here turned into a numerical calculation.

21 On analysis and synthesis in Greek mathematics, see, e.g., Hintikka and Remes,1974; Behboud, 1994; and Netz, 2000.

22 Metnca 20.6: ή δε γεωμετρική τούτου άπόδειξίς εστίν ήδε

23 The same demonstration is found in Dioptra 280.16-284.10.

24 Metrica 30.5: συντεθήσεται δε ακολούθως τρ αναλύσει ούτως. See 24 22, 32.15,34.15-18, 38.26-7,42.4-5,48.24,... (there are twenty-seven further occurrences).Brought to you by | University of Haifa

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38 Karin Tybjerg

Hero's presentation mirrors Archimedes' solutions to complex geo-metrical problems in On the Sphere and Hie Cylinder II. Like Hero, Ar-chimedes mainly uses synthesis and analysis when dealing withproblems, and he introduces the synthesis with the formulation thatHero reuses: 'It is synthesized like this'.2' But when Hero replaces thegeometrical problem with one of measurement, he reduces the geomet-rical part of the solution to the analysis, and bases the synthesis purelyon calculation. Sometimes the geometrical component is left out alltogether and Hero simply refers to a demonstration by Archimedes,which is then used for the calculation. This is the case when Hero refersto Archimedes' measurement of the area of a parabola segment or thevolumes of figures created by inserting cylinders into a cube — all fromthe mechanical treatise Method.26 In keeping with his interest in calcula-tion Hero views Archimedes' results as ways to calculate areas andvolumes, rather than as relations between geometrical figures.

Hero thus sketches two approaches: one where geometrical demon-strations are based on calculation, and one where measurement prob-lems are solved by combining geometrical demonstration andcalculation. In this way physical measurement and calculation are givendemonstrative power and geometry is made part of measurement. Herois thus able to draw on a tradition of numerical problem solving used byprofessional mathematicians and at the same time inscribe calculationand measurement of physical bodies into the Archimedean legacy pre-sented in the introduction to Mctrica.

Recent scholarship recognizes that Metrica contains both Near-East-ern and Euclidean-Archimedean material, but it has focused mainly onevaluating the relative contribution of each. Vitrac noted in his analysisof the first section of Metrica, that although some of Hero's calculationprocedures were informed by Near Eastern practices, the treatise wasfirmly inscribed in the Euclidean-Archimedean tradition.27 The organi-sation of the treatise is deductive in character, problems are described ingeometrical terms and the calculation procedures are justified geomet-

25 Archimedes On the Sphere and the Cylinder 172.7,184.21,192.7,198 13,204 11,208.15

26 The parabola segment is treated by Hero in Metrica 82.25-84 2 and by Archimedesin Method 434.14-438.15. The cylinders in Ihe cube are treated by Hero in Mctrica130.12-132 1 and by Archimedes in Method 426 8-428 7 and 484.26-506.31.

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Hero of Alexandria's Mechanical Geometry 39

rically. His analysis captures the combination of methods well, butperhaps overemphasises the division between the geometrical and cal-culation-centred parts of the treatise. It interprets Metrica as a practicalwork aiming for Euclidean-Archimedean standards of demonstration.

H0yrup, by contrast, argued that Metrica is basically a collection ofreceived approximations and that Hero's contribution is restricted toadding the proof of 'Hero's formula' (also found in Dioptra2*) to analready complete text.29 While Heyrup is right that the geometrical proofinterrupts the flow, we have seen how it introduces the style found inthe rest of Metrica where a geometrical analysis is followed by a numeri-cal synthesis. By contrast this form is found only twice in Dioptra and ina set of proofs that are not integrated into the main flow of text.

To regard Metrica as literally the product of two different treatises oras a set of calculation procedures with added geometrical justificationbelies the way Hero integrates measurement and calculation procedureswith geometrical practice and presentation. Hero both generalizes thecalculation procedures by introducing units and reinterprets the methodof analysis and synthesis to suit his own problems of measurement. Thework is not simply an application of geometrical methods for practicalpurposes such as the measuring of vaults or basin mentioned by Hero.30

Hero produces a form of geometry and demonstration suitable formeasurement.

Instruments of Geometry

Hero does not just broaden the scope of geometrical demonstrativepractice by including calculation and measurement, he also incorporatesmechanical language and methods into geometrical investigations. Anexample of a change towards a more mechanical language is found inHero's description of the Archimedean problem of finding the volumeoccupied by two cylinders inscribed in a sphere. If we compare theformulations of Archimedes and Hero, Hero 'pushes' or 'forces' (διωθέω)

28 Dioptra 280.16-284.10

29 H0yrup, 1996b, 15 η 32

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40 Karin Tybjerg

the cylinders into the cube where Archimedes' cylinders are 'inscribed'in the cube. The term 'pushes' is one that Hero uses elsewhere in histreatises about mechanical parts of catapults or automatic theatres.31 Inthis way, Hero makes an indirect statement that geometrical objects arephysical in nature and can be treated accordingly.

The introduction of mechanical methods in geometry is clearly seenwhen Hero extends his range of standard areas and volumes to includeirregular (άτακτος) shapes, i.e., shapes that cannot be measured withstandard geometrical methods. In direct continuation of his accounts ofhow to measure geometrically well-defined areas and volumes, Herointroduces methods for dealing with irregular figures that appear highlysurprising within the context of a geometrical treatise. Hero ensures,however, that the subject matter has an Archimedean pedigree by ascrib-ing the discovery of irregular figures to Archimedes.32

Hero starts softly with a geometrical approximation for dealing withan irregular plane area. The curve is to be approximated with straightlines and the resulting polygon measured by dividing it into triangles.When Hero gets to the non-planar surfaces, however, geometry is rele-gated to the back seat. The surface of a statue may be measured, heexplains, by covering its surface with small pieces of textile. The piecesare then taken off the statue and fitted into a square whose area is easilymeasured.33 With equal disregard for the conventional methods ofEuclidean-Archimedean geometry Hero suggests methods for measur-ing irregular volumes.34 He first explains how to determine the volumeof smaller objects by submerging them into water and measuring theamount of water they displace. This method reinforces the connectionbetween Archimedes and irregular bodies because it relates to the well-known story of how Archimedes exposed a craftsman, who tried to cheathis king by replacing some gold for a commissioned crown with cheapermetal. The resulting crown weighed the same as the original amount of

31 Artillery Construction 77.8; Pneumatics 78.14; Automaton Construction 24.3

32 Metrica 92.7-9 and 138.6-9. Heiberg took Hero's references as evidence that Ar-chimedes had written a treatise entitled Surfaces and Irregular Bodies, Heiberg,1910-15,543-5.

33 Metric« 90.4-23

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Hero of Alexandria's Mechanical Geometry 41

gold, but Archimedes revealed the fraud by showing that it had a largervolume.35 For objects with larger volumes Hero recommends the equallyhands-on method of packing the object into a cube of wax that can easilybe measured. The volume of the object is then the volume of thiswax-square minus the volume of a square formed of the wax alone.

What is surprising here is that Hero at no point remarks on thedifference between the methods he employs in the regular and irregularcases. His main concern seems to be to provide a complete account since,as he emphasizes, it is necessary to include irregular volumes 'so thatthe material is in no way incomplete for those who wish to pursuethem'.36

We find a similar extension of the methods and objects of geometry inMechanics, where Hero poses the problem of how to reduce or enlarge agiven plane or solid with a given ratio. He concentrates on the unit planeand the unit solid (which, of course, we recognize as the unit of meas-urement introduced in the opening sections of the Metrica) and heintroduces his survey with plane surfaces and volumes. Doubling a unitof area is a famous problem solved by Plato, Euclid and Vitruvius, butHero treats it very cursorily and does not even supply a proof.37 Theproblem does not seem to have his interest, and Hero uses it simply topresent his own mechanical geometry as part of a systematic geometricalprogression from the unit area to the irregular volume. Solving theproblem of doubling of an area does not require any instruments eitherand these are — as will soon become clear — of central importance toHero's geometry.

Hero first introduces an instrument in his solution to the problem ofthe duplication of the cube, the problem of finding the length of the sidesof a cube that is double the size of a known cube. This famouslytroublesome problem cannot be solved by ruler and compass, so it isnecessary to employ methods that go beyond standard geometrical

35 Vitruvius On Architecture, 9.praef.9-12

36 Metrica 92.7-13

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42 Karin Tybjerg

practice.38 Hero's solution consists of a Euclidean-Archimedean styleproof, based on a diagram, but the construction of the diagram involvesan instrument: a sliding ruler. Hero can, however, do this with someimpunity, as he was not the first to use instruments in this context.Several earlier solutions to the problem by famous mathematicians suchas Eratosthenes and Apollonius involved similar instruments,39 andHero is correspondingly upfront about his use of a mechanical device.He announces that he will 'show this with the aid of an instrument'.40 Hethus links the use of an instrument to a problem that was already part ofthe mathematical canon, but one that allowed for an instrumental solu-tion. Now Hero has an instrumental foot in the door, and he does notleave it at that.

In the last sections — how to enlarge irregular planes and solids —Hero can freewheel into demonstrations where instruments play morecentral roles. After a geometric proof that similar plane figures exist,Hero launches into a proof of how to construct reduced or enlargedfigures with an instrument:

Let us now prove, with the aid of an instrument, how to find for a givenplane figure a similar one that is in a given ratio to it. Let us make tworound discs (ac, ab), that are cogged regularly, around the same center(a),-. .4 1

The proof of mathematical existence is thus seamlessly followed by thephysical construction of the figures. Both are given the status of proofand again there is no indication of a change in subject matter. Likewisethe diagrams that show the geometrical situations in the case of thegeometrical proofs now represent the mechanical devices. The style of

38 The problem of the duplication of the cube can be reduced to finding two meanproportionals between the volumes of the cubes. If the relationship between thevolumes of the original and the enlarged cubes are a b, then the length of the sidesof the enlarged cube must be c, where a-c c:d · d.b. For a discussion of the problemsee Heath, 1921, and Knorr, 1986

39 Eratosthenes' and Apollonius' solutions are described in Eutocius' On Archimedes'Sphere and Cylinder 88.3-96 27 and 64 15-66 7; Netz, 2004b, 278-9, 294-8.

40 Mechanics 111

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Hero of Alexandria's Mechanical Geometry 43

both the diagrams and the propositions, however, remains the same,thus blurring the boundary between mechanical and geometrical objects.When Hero lastly addresses the problem of enlarging irregular solids theaccount is given completely over to a long and detailed description ofhow an irregular solid such as a statue might be copied using aninstrument.

There is no comparison between the space that Hero dedicates toregular and irregular figures in Mechanics. While Hero pays lip serviceto the doubling of the area and the cube — the problems that haveoccupied Euclidean-Archimedean mathematicians — it is the irregularcases that steal the show. The introductory sections on the plane area andthe cube are essential, however, as they inscribe Hero's mechanicalproject into a geometric tradition.

Both in Mechanics and in Metrica we find Hero integrating instrumentaland practical methods with problems associated with Euclidean-Ar-chimedean geometry. Hero's inclusion of practical methods has ledscholars to classify Hero as a so-called 'practical mathematician'.Thomas Heath — the grand old man of ancient mathematics — statedthat Hero aimed at 'practical utility rather than theoretical complete-ness.'42 But considering what we have just seen it would be more correctto say that Hero prioritizes completeness over purity of method. In fact,Hero shows that instruments are necessary to provide a complete ac-count.

Hero associates his work closely with the Euclidean-Archimedeantradition and takes his starting point in demonstrations derived fromtheir work. This background makes it credible for Hero to draw on theauthority of Archimedean demonstrations, but at the same time toextend the area of validity to include irregular figures measured withmechanical methods. We have seen how Hero makes this transitioncontinuous. First in Metrica, where Hero employs an Archimedeanmethod for measuring irregular volumes; and second in Mechanics,where Hero includes a famous mathematical problem for which instru-ments had been used before and thus legitimises further use of instru-ments.

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The problem of the duplication of the cube was at the centre of along-standing debate about the legitimacy of alternative methods: whichis the lesser evil when it is necessary to move beyond the traditionalmethods based on ruler and compass? Solutions offered or criticized bydifferent authors indicate various ways in which geometrical practicecould be extended;43 so when Hero offers his own solution he steps intoa tradition where the fronts are already drawn up.

The problem supposedly originated when an oracle told the peopleof Delos that they had to construct a new altar double the size of theirexisting one. Baffled by the problem they went to Plato for advice. ButPlato — not famous for his practical advice — replied that the oracle wasnot concerned with the bigger altar; rather it wanted to shame the Greeksfor their lack of mathematical knowledge and make them dedicate moretime to the study of geometry. The incident is recounted by Eratosthenes(before Hero) and Plutarch (a generation after Hero), but their attitudesto the problem are very different.44

Eratosthenes does not include the part of the story concerning Plato'sreply; he simply proceeds to comment on various solutions to theproblem. He rejects Archytas' and Eudoxus' solutions and suggestsinstead a 'mechanical way' which he recommends for its ease. Thismethod is, according to Eratosthenes, not just useful for enlarging altars,but also for sizing measurement vessels and for enlarging catapults andstone-throwers. Eratosthenes' presentation of the problem thus fitsclosely with Hero's interests: Hero investigates the measurement andenlargement of architectonic objects in Metrica and Mechanics and theenlargement of catapults is a central topic in his Artillery Construction,

43 See Cuomo, 2000,127-51, for an account of the history of the problem and an analysisof how Pappus (early fourth century AD) uses the varied meanings associated withthe problem to support his own mathematical agenda in his Collection.

44 Eratosthenes' account has not been preserved directly, but it is recounted by Theonof Smyrna (fl. c. AD 115-140) in Aspects of Mathematics Useful for Reading Plato 2.3-12,and by Eutocius (sixth century AD) in On Archimedes' Sphere and Cylinder. Plutarchtells the story in Moralia 386E, 579Α-D and 718E-F. The story is also told by Vitruviuswho refers to Archytas solving it T^y a diagram with cylinders' and Eratosthenessolving it 'by means of an instrument', On Architecture 9 praef.13-14 Vitruviusrecounts the story in the same section that deals with the doubling of the square andthe story of Archimedes and the gold crown. All but the last of these referencesderive from Knorr, 1986, 3-4. Brought to you by | University of Haifa

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Hero of Alexandria's Mechanical Geometry 45

which actually contains the full solution to the problem of duplicatingthe cube.45 By placing his account and purpose so close to Eratosthenes',Hero indirectly uses the fact that a known mathematician produced aninstrumental solution to a geometrical problem and recommended it forits practicality, to take instrumentation further.

Plutarch's Moralia puts a very different spin on the story. Plutarchdistances himself from the whole idea of an applicable solution byemphasizing that the oracle was not concerned with the altar at all, butonly with the study of geometry. Like Eratosthenes he criticizes Ar-chytas' and Eudoxus' solutions, but this time the criticism concerns theuse of instruments. Plutarch finds instrumental methods unacceptableand derides them for abandoning reason in favour of whichever methodworks.46 Plutarch's antipathy to instrumental method also surfaces in hisLife ofMarcellus which describes Archimedes' mechanical achievementsduring the siege of Syracuse. Here Archimedes is praised for his mathe-matical and mechanical ability in defending the city, but Plutarch disas-sociates him from mechanics by saying that he thought it too vulgar andtainted by the needs of life to be the subject of a treatise.47 Here, Plutarchalso comments on the use of instruments:

For this admired and famous art of instrumentation was first gotmoving by the followers of Archytas and Eudoxus: they embellished(ποικίλλοντες) geometry with subtleties (γλαφυρω) and used it as sup-port for problems where there was no ready proof by argument anddiagram by means of perceptible and instrumental examples. Forinstance the problem of finding two mean proportional lines, a neces-sary element for many diagrams, both mathematicians reduced toinstrumental constructions .. ,4*

45 Artillery Construction 117-19

46 Plutarch Moralia 718E

47 Plutarch Life ofMarcellus ΧΥΠ.4

48 Plutarch Life of Marcellus XIV.5: Την γαρ άγαπωμένην ταύτην και περιβόητονόργανικην ήρξαντο μεν κινεϊν ο'ι περί Εϋδοξον και Αρχύταν, ποικίλλοντες τω γλαφυρωγεωμετρίαν, και λογικής και γραμμικής αποδείξεως ουκ εύποροΰντα προβλήματα δΓαισθητών και οργανικών παραδειγμάτων ύπερείδοντες, ως το περί δύο μέσας ανά λόγονπρόβλημα και στοιχεϊον επί πολλά των γραφομένων άναγκαΐον εις όργανικάς έξήγοναμφότεροι κατασκευάς, .. Brought to you by | University of Haifa

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Plutarch criticizes the use of instruments even for the problem of theduplication of the cube (= finding two mean proportionals) which cannotbe solved through traditional arguments and diagrams. Tellingly, how-ever, Plutarch's terms of abuse are the same as Hero uses to describe hismachines and methods. In Hero's work a term such as 'various' or'embellished' (ποκίλος) expresses qualities of mechanical inventions,and in the Automaton Construction 'the most subtle (γλαφυρωτάτη) ar-rangement' is presented as something to strive for.49

Living just a generation after Hero, it could well be Hero andlikeminded authors at whom Plutarch lashes out with his contempt forinstrumental methods. Plutarch may be taken as evidence of the successof Hero's project since he denotes the methods 'admired and famous'.Furthermore Plutarch's virulent attempt to disassociate Archimedesfrom mechanical and instrumental methods indicates that Hero or otherswere successful in giving instrumental methods the Archimedean stampof approval.

Geometrized Devices

We have now shown how Hero places instrumental solutions on anequal footing with Archimedean-style geometric proofs and makes prac-tical methods and instruments an integral part of a complete geometry.Now we consider how Hero incorporates geometry into the descriptionof mechanical devices.

In his Dioptra Hero does not distinguish between physical and geo-metric space. The problems are presented in a similar vein to Metrica, asproblems pertaining to specific numerical examples, but Hero remindsthe reader of their physical provenance by giving the measures in actualunits such as feet or ells. Although the problems considered clearly dealwith a physical landscape that includes growth, harbours, rivers andtunnels, they are presented as geometrical propositions. Problems areintroduced with standard formulae such as 'Let the given points be Aand B', but while the 'given' is usually a point, circle segment or the like,it might also be a trench. When the lettered points have a physical

49 'Varied', ποκίλος, Pneumatics 2.18, 28.14, and Automaton Construction 338.4, 342.6,404.15; 'subtle', γλαφυρός, Automaton Construction 410 21-2.Brought to you by | University of Haifa

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Hero of Alexandria's Mechanical Geometry 47

condition attached, for instance that they cannot be accessed or observed,this is simply included in the geometrical presentation.50 The dioptraitself is also presented as a geometrical object with phrases such as 'letthe dioptra be constructed', or 'stood up' or 'set up'51 and Hero refers to'demonstrations' (άπόδειξις and δεΐξις) involving and concerning instru-ments.52

The space in which Hero solves problems with the dioptra is thussimultaneously a geometrical and a physical space and at times theproblems seem to be more concerned with covering all the geometricalpossibilities than with practical application. Hero, for instance, showshow the outline of a harbour can be drawn not only in the shape of acircular segment but also in an elliptical, parabolic, hyperbolic or anyother shape we may choose!53 These possibilities seem to be motivatedby a desire to give a geometrically complete account in the same vein asdiscussed in the case of Metrica, but without losing the appeal to practicalconsequence. Hero aims to satisfy both practical and geometrical re-quirements.

Similarly Hero translates a practical motivation into a geometricalproblem when he explains the importance of finding a straight line earlyon in Dioptra. To 'escape cost' he will demonstrate how to find thestraight line between two points 'for this is the shortest of all lines thathave the same end-point.'54 The latter part of this statement is almostword for word the definition that Archimedes gives of the straight linein his On the Sphere and the Cylinder.55 Thus the pragmatics of finding thecheapest construction is directly linked to a central definition of Ar-

50 Dioptra 214.18-19,218.20-2

51 Dioptra 214.21-2 (.. και κατεσκευάσθω ή διόπτρα ...), 222.21-2 (και καθεστάσθω ήδιόπτρα ...), 234.25 (κείσθω δη ή διόπτρα).

52 Dioptra 214.11 (proof of how the straight line is found using the dioptra), 286.21 and23 (uses of the dioptra have been proved), 290.13 (demonstrates the working of the'star'), 298.28 (the working of the road measurer has been shown) and 308.19-20(reference to the proofs of the simple powers i.e the pulley, screw, windlass, leverand wedge).

53 Dioptra 246.10-14

54 Dioptra 214.12-14

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48 Karin Tybjerg

chimedean geometry. Hero thus blurs notions of geometrical and physi-cal space, mechanical and geometrical objects, practical and geometricalconcerns.

In general purely geometrical demonstrations are moved to the back-ground in the Dioptra. The treatise contains just three geometrical proofs,including a version of the proof of 'Hero's formula' that is also found inMetrical But where the proofs took the centre stage in Metrica they arepresented as an aside in Dioptra. Hero first solves the problem of meas-uring land with the dioptra and offers demonstrations of the geometricalrelations he used only afterwards. The geometrical proofs are treated likelemmas to the practical problems of land measurement. In this way, Heropresents Dioptra as geometrically based, but makes pure geometry aux-iliary to the geometrical work done with the dioptra.

Hero also uses the ambiguity of geometrical language to geometrizemechanics. The term 'to construct' (κατασκευάζω), for instance, is com-mon in geometrical language, where it usually refers to the constructionof the diagram. It can, however, also mean 'to furnish' or 'make'. InCatoptrics — a treatise concerning reflection in mirrors — Hero begins witha geometrical section where he proves the path of reflection for mirrors ofdifferent shapes. He begins these propositions in standard geometricalstyle with 'Let there be', for example, 'a plane mirror ab'.57 When hemoves onto more complex mirrors — which he also describes how tomanufacture — he changes to a terminology of construction, for instance:To construct a mirror that shows the right on the right'.58 Here, Hero usesthe range of meaning of the term 'construct' to move inconspicuously fromgeometrical demonstrations to the construction of mirrors.

In Catoptrics Hero again combines a Euclidean-Archimedean traditionwith a project centred on mechanics. Many of the proofs and types ofmirrors that Hero describes are also included in the Pseudo-EuclideanCatoptrics, which offers an axiomatic-deductive treatment of reflectionin mirrors.59 But if we compare the example of the mirror that shows the

56 Dioptra 276.5, 274.14,268.10

57 Catoptrics 326.3: Sit enim speculum planum ab, ... . 'Sit' is here equivalent to theGreek έστω.

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Hero of Alexandria's Mechanical Geometry 49

right on the right, Pseudo-Euclid does not, like Hero, describe the actualconstruction of the mirror. Where Hero advises his reader to use Corin-thian bronze, Pseudo-Euclid merely affirms the possibility that such amirror can be constructed in the opening of the proposition: 'It is possibleto construct ..Λ60 These subtle differences in the cases where bothtreatises describe the same example are underscored by Hero's inclusionof an account of how mirrors are manufactured. By describing in detailhow to polish up a surface in order to make it reflective he makes thematerial foundation of catoptrics evident. By the same token he excludessome of the geometrical proofs and thereby shifts the focus from geome-try towards physical devices.

When Hero uses the phrases 'let there be' and 'to construct...' aboutdifferent cases he does not draw on a standard distinction in geometricalvocabulary. Bringing geometrical entities into existence and construct-ing geometrical objects are both common ways of proceeding in ageometrical proposition. Hero uses the term 'construction' to mergegeometry and mechanics. He simultaneously constructs a diagram thatis the site of the geometrical proof and a working device that can producecertain effects. He thus again combines the rigour of geometrical dem-onstration with practical expertise.

The diagrams in the best manuscript edition of Catoptrics supportthe geometrical style of the propositions, even where Hero is dealingwith mechanical devices such as a window mirror. They resemble thelettered diagrams of geometrical treatises and represent mirrors andvisual rays simply as lines. In the modern editions of Hero's work thisaspect of the diagrams is underplayed and the reproductions of thediagrams have tended to picture the device rather than just the geo-metrical situation.61

59 On the similarity of propositions and interests between Hero's and Pseudo-Euclid'sCatoptrics, see Heiberg, 1925, 78 n 2; Lejeune, 1957, 137-42; and Knorr, 1994, 70-8.Lejeune argued that the Pseudo-Euclidean Catoptrics is a compilation made afterHero's Catoptrics, with the subtext that Euclidean formalisation constitutes 'im-provement', Knorr rejected this view and showed how Hero clarified and added tohis source.

60 Pseudo-Euclid Catoptrics 338.7-11

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50 Karin Tybjerg

The majority of Hero's diagrams occupy this uncertain position be-tween geometrical diagram and illustration. Hero follows the conven-tions of the geometrical lettered diagram, but the surfaces of devices areoften drawn with thicker lines and ropes on lifting devices are includedin the diagrams. They do not however depict the machines as they wouldappear. The diagrams focus on elements that are important for theworking of the devices and enlarge the parts that are relevant to theirfunction.62 In this way mechanical diagrams are analogous to geometri-cal diagrams, which do not depict quantitative relations either, but rather'qualitative' geometrical relations. Hero's diagrams simultaneously actas a geometrical diagram where the diagram is the object manipulatedand a technical illustration that represents the object. In this way Hero'sdiagrams bear out the same ambiguity as his geometrical mechanics.

The tendency to geometrize devices is also evident in the treatises wheregeometry plays a lesser role, such as Pneumatics and Automaton Construc-tion. The descriptions of mechanical devices in Pneumatics resemblegeometrical propositions both in language and in structure. Each de-scription consists of a presentation of the problem, the construction ofthe device aided by a lettered diagram, and an account of the functioningof the device in lieu of the actual geometrical demonstration. Moreover,parts of the devices are often described as geometrical objects such asspheres, cylinders or parallellopipeds.63

Lastly, the development of devices in the course of the treatise can beseen as parallel to the development of geometrical propositions. In thepreamble to the description of the clepsydra, which is the first device inPneumatics, Hero writes that he will begin with the smaller devicesbecause these are 'elemental'.64 The term used here is the same as is usedin the title of Euclid's Elements and when Apollonius denotes the firstbooks of Conies as elementary.65 Hero thus indicates that his descriptionsof mechanical devices constitute a geometry of machines where simplemachines are combined systematically to produce more complex ones.

62 Shickelberger, 1994

63 E.g., Pneumatics 70.13 and 120 4-5.

64 Pneumatics 56.12-16

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Hero of Alexandria's Mechanical Geometry 51

By the same token Hero writes in the introduction to Pneumatics howdifferent arrangements can be made by 'combining three or four ele-ments',66 thus linking his account to systematic works of geometry orphysics.

Geometry and Expert Knowledge

I have now discussed how Hero's demonstrative practices constitute anamalgam of the rigour and demonstrability of geometry with the skilland expertise of mechanics. In this last section I want to briefly considerthe link between the claims that Hero makes in his introductions that hiswork is 'useful' and the geometrization of mechanics.

In the introduction to Dioptra Hero gives a long list of applications ofthe device, which range from aqueduct-construction to measuring areasoccupied by enemies or featuring natural obstacles such as currents that'can suck you down.' Lastly, and rather surprisingly, Hero gives anexample of how the dioptra can be used in a military attack on a city.Hero writes that

... many who attempt a siege construct ladders or siege-machines,which are smaller than is needed, and, when they attack the walls, bringthemselves under the control of their enemies, having miscalculatedthe measurements of the wall because they were not acquainted withthe study of the dioptra. For one must always measure the intervalmentioned above carefully while being outside shooting range.67

This is not an obvious use of a surveying instrument, but it does allowHero to connect the dioptra's measurements powerfully to issues of

66 Pneumatics 2.14-16

67 Dioptra 190.10-21: πολλάκις γαρ έμποδών ϊσταταί τι είργον ήμας της προθέσεως, ήτοιδια πολεμίων προκατάληψιν ή δια το άπρόσιτον και άβατον είναι τον τόπον παρε-πομένου τινός ιδιώματος φυσικού ή ρεύματος οξέα ύποσύροντος. πολλοί γοΰν πολιορ-κεϊν έπιχειοϋντες κλίμακας ή μηχανήματα κατασκευασάμενοι ελάσσονα ων χρή καιπροσα<γα>γόμενοι τοις τείχεσιν υποχείριους εαυτούς παρέσχον τοις άντιπάλοιςπαραλογισθέντες τη αναμετρήσει των τειχών δια το απείρους είναι της διοπτρικηςπραγματείας, αΐεΐ γαρ εκτός οντάς βέλους άναμετρεϊν δει τα προειρημένα διαστήματα.Brought to you by | University of Haifa

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52 Karin Tybjerg

military control and security. If no proper measurements are taken andthe siege-ladders are too short the enemy might gain the upper hand.

If we now consider the Dioptra as a whole, it becomes clear that Heroappeals to a generalised demand for control over external conditions,which he links to making measurements with the dioprra. Measurementis the key to avoiding mistakes and to remaining in control of the projectsundertaken. In one problem, Hero, for instance, shows how to replacethe boundary stones that mark property after a flood.68 Although Herogives no reason for including this example, it is perhaps significant thatit relates to reinstating property-markers and thereby maintaining socialorder. Later in the treatise Hero deals with the problem of how to dig atunnel in a straight line through a mountain.69 He gives directions fordetermining the starting point at either end, such that the workers willmeet in the middle. From other sources we know that failing to meet inthe middle was a real danger. An inscription from the second centurytells us about a tunnel project where the working teams missed eachother because T^oth the tunnels deviated from the straight line'.70

Both Hero and the inscription formulate the problem of constructinga tunnel in terms of determining a straight line, and, in general, straightlines are presented in Dioptra as a means to control the environment withmechanical and geometrical expertise. Finding the straight line betweentwo points is one of the first problems that Hero treats in Dioptra, but heremarks that it is not always easy to draw the shortest line between twopoints as hindrances such as mountains or unhealthy swamps must benegotiated.71 Hero describes these hindrances in a military vocabulary.We hear for instance how the hindrances 'fall upon' the straight line andcost 'must be escaped'.

This control of both the physical and political environment by build-ing tunnels, replacing property markers and drawing straight lines foraqueducts is closely related to the geomerrisation of the landscape thatI considered earlier. To regard the landscape as a geometrical space is

68 Dioptra 268.17-272.15

69 Dioptra 238.3-240 27

70 Corpus Inscriptionum Latinarum VIII.2728. See translation in White, 1984, 215.

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Hero of Alexandria's Mechanical Geometry 53

not only a convenient way to measure it, it is also a way to control it andto deal with military and environmental dangers.

Turning back to Metrica, Hero describes, in the introduction to the thirdbook, how geometry improves current methods of measuring and divid-ing land. Land is, according to Hero, normally divided such that greaterpeoples get more land and smaller peoples get smaller parts. Also peoplewith a talent for leadership get big cities while smaller minds are leftwith tiny villages. Hero, however, rejects this method of distribution andhe suggests that division of land is better done geometrically. He writes,

But in these cases the proportions were estimated in a relatively roughand lazy manner. If someone really wants to divide areas according toa given proportion, so that not a single grain, so to speak, exceeds or isleft over from the given ratio, then only geometry is required. Ingeometry the fit is fair, justice lies in proportion and the proof (άπόδει-ξις) concerning these things is indisputable; this no other art or sciencecan promise.72

Here, we see how Hero combines the practical relevance of measurementwith the status, precision and especially the demonstrative powers ofgeometry. It is the indisputability of geometry that makes it superior formeasuring and dividing land. And the fact that we are dealing with thehighly political and social issue of dividing land means that Hero canassociate geometry with ethical values such as justice and fairness. Inthis way, Hero's extended concept of demonstration allows for a broaderclaim to expertise.

72 Metrica 140.16-142.2: άλλα τα μεν παχυμερεστέραν πως καΐ άργοτέραν εϊληφε τηνάναλογίαν · ει δε τις βούλοιτο κατά τον δοθέντα λόγον διαιρεΐν τα χωρία, ώστε μηδέ ωςειπείν κέγχρον μίαν της αναλογίας ΰπερβάλλειν ή έλλείπειν του δοθέντος λόγου, μόνηςπροσδεήσεται γεωμετρίας· εν ή εφαρμογή μεν ίση, τη δε αναλόγιο: δικαιοσύνη, ή δεπερί τούτων άπόδειξις αναμφισβήτητος, όπερ των αλλων τεχνών ή επιστημών ουδεμίαΰπισχνεΐται. Brought to you by | University of Haifa

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54 Karin Tybjerg

Conclusion

Hero adapts Euclidean-Archimedean demonstrations and methods toproduce a more mechanical and practical geometry. Machines and meas-urement are integrated into geometry and Hero presents his materialwith seamless transitions from geometrical to mechanical tools and fromgeometrical to mechanical objects. Neither diagram nor formal presen-tation allows the reader to set the geometrical apart from the mechanical.This is the aim of Hero's Mechanical Geometry.

Hero creates an authoritative foundation for his geometry by castinghis mechanics in a geometrical form and extending the concept ofdemonstration to include instrumental proofs. He associates his demon-strations with the incontrovertibility of the Archimedean proof and isthereby able to present mechanics as a theoretical discipline based ondemonstration. But Hero's mechanical geometry has a larger area ofvalidity than traditional Euclidean-Archimedean geometry as he alsoincludes areas such as irregular figures.

Moreover Hero vastly extends the power of geometry in social andpractical contexts and he blurs the boundary between professionalmathematics and geometry that is often used to degrade practical skillrelative to theory. By presenting problems of land measurement andsiege war as susceptible to geometrical methods, Hero's geometry ofmachines combines the authority of geometrical demonstration with thepower of practical consequence.

References

Primary

Apollonius. Conies in Apollonius Pergaeus, ed J.L. Heiberg, Vols 1-2(1881/1974). Stuttgart.Teubner.

Archimedes. De Sphaera et Cylindro (On the Sphere and the Cylinder), in Archimedis OperaOmnia, ed. J.L. Heiberg and E.S. Stamatis, Vol. 1 (1910/1972) Leipzig: Teubner.

Archimedes. Ad Eratosthenem Methodus (Method), in Archimedis Opera Omnia, ed. J.L.Heiberg and E.S. Stamatis, Vol 3(1915/1972) Leipzig: Teubner.

Cicero. Tusculan Disputations, Engl. trans. J.E. King (1945/1971). London: Hememann(Loeb).

Euclid. Elementa, ed. J.L. Heiberg, 5 vols., (1883-1888). Leipzig Teubner.[Euclid]. Catoptnca, ed. J.L. Heiberg (1895). Leipzig: Teubner.Eutocius, Eutocn Commentarium in Libruni I de Sphaera et Cylindro, in Archimedis Opera Omnia

ed. J.L Heiberg and E.S Stamatis, Vol. 3 (1915/1972). Leipzig: Teubner.Brought to you by | University of Haifa

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Hero of Alexandria's Mechanical Geometry 55

Diodorus Siculus, Bibliolhcca, Engl trans. C.H. Oldfather el al. (1933-1967) London: Heine-mann (Loeb).

Hero of Alexandria Pneumatics, ed. and German trans. W. Schmidt, in Heroins AlexaiidnniOpera, Vol. 1 (1899). Leipzig. Teubner.

Hero of Alexandria Antoinntopoetica (Automatic Construction), ed. and German trans. W.Schmidt, in Heroins Ale\andriin Opera, Vol 1 (1899). Leipzig· Teubner.

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