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ABSTRACT
Modern commentators have doubts about the authenticity and cogency of the early propositions of Archimedes’ On Equilibrium of Planes Book 1. Ernst Mach famously said that the proof of Prop. 6, the so-called law of the lever, assumes what is to be proven. Comparing the initial text in Heiberg’s modern edition (1881, 1913) to the first propositions in Eutocius’ commentary on EP 1, J. L. Berggren ([1976]. ‘Spurious Theorems in Archimedes’ Equilibrium of Planes: Book I’, Archive for History of Exact Sciences 16.2 (1976), 87–103.) claimed that the propositions up through Proposition 3 of the standard modern edition are schoolbook additions written by an ancient author inferior to Archimedes. The present paper argues for the logical connectedness of Postulates 1–5 to Props. 1–6, by means of a detailed examination of the course of the argument and a re-examination of Eutocius’ remarks. The paper reinterprets the role of the empirical in the early propositions and offers a reading of the contribution of Archimedes’ mechanics to the method of EP 1.
Acknowledgements
The author thanks the anonymous referees of this paper for their very valuable comments, which helped to improve the paper.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Correction Statement
This article has been corrected with minor changes. These changes do not impact the academic content of the article.
Notes
1 References to Archimedes' texts in this paper (e.g., Heiberg, Citation1881) are to Heiberg’s 1st edition of Archimedis Opera omnia, 3 vols. (1880–81) (reprint Cambridge University Press, 2013). Any significant difference from the 2nd edition (1913, Heiberg and Stamatis, 1910–1915, reprint Teubner Citation1972) is noted.
2 All extant manuscripts and early editions are traced back to two lost exemplars, A and B. Manuscript C, the Archimedes palimpsest, does not include any of Book 1 of EP. For the manuscript tradition, see Dijksterhuis Citation1987 [Citation1956]: 33–49, and Knorr’s additional notes in Dijksterhuis Citation1987: 424–28, Berggren’s summary (Citation1976: 90–91), and Netz Citation2004: 10–18. Knorr (Citation1978a) argues against Berggren’s dismissal of EP 1 as not by Archimedes (185–86) but concedes that EP 1 could be a compilation of more than one Archimedean writing (224). Reviel Netz expresses doubt about Berggren’s thesis (Citation2004: 12) due to the character of Archimedes’ reasoning in other treatises. In volume 3 of his Works of Archimedes (forthcoming), Netz concludes that the two books of EP were written by Archimedes and belong together (Introduction, §3).
3 A note on chronology is in order. Berggren and Heiberg agree that the two books of EP were not composed by Archimedes as a single treatise. As Heiberg says, EP 1 is presupposed by the treatise Quadrature of the Parabola, which is itself presupposed by EP 2 (Heiberg Citation1907: 299). This view is widely accepted and not disputed here. Heiberg points to the unoriginality of the title, Equilibrium of Planes, as an indication it was not chosen by Archimedes. More compelling is that Archimedes referred to the content of Book 1 as Elements of Mechanics (Στοιχϵῖα τῶν μηκανικῶν) in On Floating Bodies and as Mechanics (μηκανικά) in Quadrature of the Parabola (Heiberg Citation1907: 299).
4 1–0 reasoning overlaps with proof by contradiction, reductio ad absurdum or ad impossibile, but is not the same as it. On per impossible reasoning (hē eis to adunaton apagōgē), see Proclus’s commentary on Book I of Euclid’s Elements (Proclus Citation1873 [repr. 1992]), 255.8–12: ‘Every reduction to impossibility takes the contradictory of what it intends to prove and from this as a hypothesis proceeds until it encounters something admitted to be absurd and, by thus destroying its hypothesis, confirms the proposition it set out to establish’ (translation by Morrow Citation1970, 198).
5 Postulate 6 in Heiberg’s edition is Postulate 8 in the early modern reworkings of EP 1 used by J. M. Child (Citation1921). Child’s sources are Guido Ubaldi del Monte and Guldinus.
6 Netz, Works of Archimedes vol. 3 (forthcoming), § Archimedes’ First <Book> of ‘Balancing Planes’.
7 For an ancient account of the different terms for starting points or first principles in geometry, see Proclus (Citation1873, 180.23–182.20). Proclus addresses the first postulate of Archimedes’ ‘aitoumetha’ directly at 181.16-21. He says that some prefer to call it an axiōma and indicates the terminology has come to be used loosely (Proclus Citation1873, 181.2 –182.4).
8 Proclus says the theorems are poikilōtera, more varied or subtle, than their simpler starting points (ProclusCitation1873, 57.25). Along similar lines, Aristotle says that arriving at a demonstration is seeing that something to be proven falls under another more general and already known mathematical truth. His example is proving the angle in a semi-circle is a right angle, by recognizing the angle makes a triangle with the endpoints of the diameter of the circle (Po An 1.1, 71a19–29).
9 Netz believes the commentator on EP 1 is not the same as the Eutocius who commented on other works of Archimedes (Works of Archimedes vol. 3 [forthcoming], ‘Introduction’ §3). This ps.-Eutocius is not to be relied upon for interpretation to the same extent as the Eutocius of other treatises, he says. For the sake of simplicity in this paper, I will still call this commentator ‘Eutocius’.
10 Volume II of Heiberg’s 1881 edition is hereafter cited as ‘H II, 1881’. Volumes 1 (1880) and 3 (1881) are cited similarly.
11 Archimedes defines ‘concave’ in his Sphere and Cylinder in axiōmata 1–2 (H I, 1880: 6.14–23; Netz Citation2004: 35).
12 Berggren (Citation1976: 100–103) argues that even this latter account is not by Archimedes, but this is a separate topic involving different concepts of centres of gravity and an argument by Drachmann that an Archimedean proof of the concept is reproduced in Heron’s Mechanics (Drachmann Citation1963).
13 Καλῶς δἑ δοκϵῖ ὁ Γϵμῖνος ϵἰπϵῖν πϵρὶ τοὺ Ἀρχιμήδους, ὅτι τὰ ἀξιώματα αἰτήματα λέγϵι. τὰ γὰρ ἴσα βάρη ἀπὸ ἴσων μηκῶν ἰσορροπϵῖν ἀξίωμά ἐστι καὶ τὰ ἑξῆς, καί ἐστιν πάντα σαφῆ τοῖς μϵτρίως αὐτὰ ἐπισκϵπτομένοις (H III, 1881: 308.3–7).
14 If the first sentence of the treatise alone were read as a postulate, P1.2 and P1.3 could be taken as comprising a separate postulate parallel in formulation to 2.1, 2.2 and 3.1, 3.2.
15 έσσϵῖται: Doric future. The sentence could be translated as, ‘If the weights were to be unequal, when the excess of the greater weight over the lesser were taken away, the remaining weights would not be in equilibrium.’ The counterfactual character of the claim is significant.
16 Brackets are added to the Greek in H II, 1913.
17 On this latter point, see below §7 Discussion.
18 Mach writes, ‘Stellt sich der Beschauer selbst in die Symmetrieebene der betreffenden Vorrichtung, so zeigt sich der Satz 1 [P1] auch als eine sehr zwingende instinctive Einsicht, was durch die Symmetrie unsers eigenen Körpers bedingt istʹ (Mach Citation1897, 11).
19 See for instance Aristotle, Physics 1.1; Posterior Analytics 1.1–2; pseudo-Aristotle, Mechanics 1, 847a22–b17; 4, 850b10–11; 5, 850b30–34. Hero of Alexandria starts with the simplest starting points of mechanics (Mechanica [On the lifting of heavy bodies], ed. Nix et al, vol. II [Citation1976] Bk I, §2–15, and at the beginning of Bk II, the so-called simple machines. For an early forerunner, see Archytas’ reference to wholes and parts in his commendation of the success of harmonics and the mathēma of ‘risings and settings’ of his day (5th century BCE) (D-K B1).
20 Knorr held that Archimedes does not make this assumption: ‘ … [Archimedes] never articulates the concept of ‘moment,’ the product of magnitudes times distance’ (Citation1978b: 106). See also Knorr’s review of scholarship in Dijksterhuis Citation1987: 435–38.
21 H II, 1881: 148.8–12. The sentence is enclosed in brackets in the 1913 edition (H II, 1913: 128.22–23). An anonymous referee points out that prodedeiktai may be a later insertion and not Archimedes’ words. Eutocius comments, ‘it was said earlier (anōterō) that there is a [point the] centre of two magnitudes, from which a beam suspended has its parts remaining parallel to the horizon’ (H III, 1881: 312.21–25).
22 Berggren notes this characteristic of Prop. 4 also and Eutocius’ interpretation of it (Citation1976: 101).
23 On Prop. 7, see also Hayashi and Saito Citation2009 (in Japanese) reviewed by Nathan Sidoli Citation2012, 222.
24 Netz, Works of Archimedes, vol. 3 forthcoming, ‘EP 1, Prop. 4, Comments’. Netz’s Prop. 4 is Heiberg’s Prop. 6.
25 Knorr gives a fairly literal translation of the proof of Prop. 6, indicating sentences or phrases he regards as interpolated because of their being explications (Citation1978a, 222). For the sake of simplicity, I follow Knorr in omitting most of his suspected interpolations. I include one set, which make more definite the progress of the proof in ways Archimedes likely intended and which may not be interpolations. The sentences I have included are lettered proportions. I place them in brackets to indicate their exclusion by Knorr.
26 Terms used by Archimedes: ὁσαπλάσιος, τοσαυταπλάσιος, ἴσα τῇ πλήθϵι
27 Knorr says the ‘let … be’ sentence (l. 11–12) does not ‘set the ground for proving a proportionality’ (Citation1978a, 223). It does not define ‘equimultiples.’ That is, it is simply part of the specification. Knorr says that the ‘equimultiples are equivalent to the proportionality LH:N = A:Z’ (1978, 223).
28 Knorr did not flag this sentence as a possible interpolation (despite its odd τὰ κέντρα τῶν μέσων μϵγϵθέων) though he removed the immediately preceding sentence containing a reference to Prop. 5, as well as lines 156.6–8 of similar import (Citation1978a, 223). I mention this, because one criticism of the proof of Prop. 6 is that it relies illicitly on Prop. 5, since the application of Prop. 5 to the equilibrium of EC, CD respectively depends on the whole ECD being already in balance with the weights A, B at E and D. I will return to the objection below and in §7 Discussion.
29 The assumption that A>B is to distinguish H from C for cases where A≠B.
30 This could be viewed as the reason the proportion is inverse, since the greater multitude of Zs is correlate to the longer length, but those Zs are matched with the shorter length in equilibrium.
31 Hertz Citation1899, xxi–xiv, 1–6; Hölder 2013 [1899]: 47, n. 66.
32 Child held that Cavalieri was the first to articulate and see the significance of a notion of moment (Citation1921: 493–94).
33 Mach uses the expression . Greek proportion theory did not, however, multiply or divide heterogeneous magnitudes like weight and distance by one another, nor did the Greeks have the notion of a function. Using Mach’s terms, Child (Citation1921) says the supposedly hidden assumption,
, is really the very general and openly embraced
. Equilibrium is a function of weight and distance combined. Both Child and Dijksterhuis (Citation1987: 293–94) reject the function language in relation to Archimedes.
34 Vailati, trans. Palmieri Citation2009: 319, col. 2, 320, col. 2; Vailati Citation1911: 497, 499.
35 Pappus of Alexandria Citation1878, Sunagōgē Book 8, ¶5–7. Pappus credits Archimedes with discovery of the general barycentric theory.
36 For instance, Prop. 6 ultimately could include P1 and Prop. 1 as limiting cases within the wider scope of an inverse ratio principle.
37 As mentioned earlier, in this sequence of propositions Archimedes never begins with a sole weight ‘centered’ on a beam as both Vailati and Hölder do.
38 Vailati notes Hölder’s account but prefers his own possibly more elegant reductio argument based on the notion of centre of gravity (Vailati, trans. Palmieri Citation2009, 320, col. 1–2).
39 Netz points out that this theorem (his Prop. 7) does not require the law of the lever (Heiberg’s Prop. 6 and Netz’s Prop. 4) and could have been placed after Heiberg’s Prop. 5 and its corollaries (Works of Archimedes vol. 3 forthcoming).
40 One ratio may still be greater than or less than the other.
41 The mechanical concepts of EP 1 may derive from some other work of Archimedes’ own, now lost to us, which focused on principles of balance alone. Α work Peri zugōn,‘On Balances,’ is attributed to Archimedes by both Hero of Alexandria and Pappus (Hero, Mechanica (Kitāb fī raf’ īlashyā‘ althaqīla, ‘Book on the Lifting of Heavy Bodies’) ed. Nix, Bk I, 71, l. 8; Pappus, Sunagōgē VIII, ed. Hultsch, 1068, l. 20).