Abstract
The Mechanical Problems traditionally attributed to Aristotle is a short problem collection that also contains an ambitious project of reduction, which traces various mechanical devices back to the lever, the balance and the radii of a circle. This work is thus not just a collection of problems, but also the first theoretical mechanical treatise that has come down to us: Basic concepts of technical mechanics such as force, load, fulcrum are abstracted from an analysis of simple technology, and the workings of this technology are explained by arguments cast in syllogistic form. This chapter traces the origins of mechanical theory in this work and analyzes the form and structure of its argument. The key steps in the concept formation of basic mechanics carried out in this treatise are analyzed in detail. We focus on the special role of the balance with unequal arms in the early development of mechanics, on the interaction of various forms of explanatory practice and on their integration into systems of knowledge in the Peripatetic school.
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Notes
- 1.
- 2.
Galilei (1638, 11).
- 3.
There is no accepted critical edition of the Mechanical Problems. We use the Greek text given in Hett 1936, cited, as is customary, according to page and column of the Bekker edition of 1831. All translations from the Greek are our own. They are intended to be as literal and interpretatively open as possible.
- 4.
- 5.
- 6.
There is no consensus on authorship or dating. Based on the way letters are used to locate points and figures, the work predates the Euclidean reform. On such formal questions see Heiberg (1904) and Netz (1999). Euclid’s Elements are generally taken to have been compiled shortly after 300 BCE; Aristotle died in 322 BCE. If we don’t want to resolve the question of Aristotle’s possible authorship by stipulation, we have to date the work at some time between 330 and 270 BCE. Recent commentators tend to favor the later date.
- 7.
- 8.
- 9.
For the manuscript tradition see van Leeuwen (2012, 2013, 2016); for the Arabic translation see Abattouy (2001). On the ancient lists see Flashar (2004, 189–191) and Hein (1985, 304). On Athenaeus, see Whitehead and Blyth (2004, 44) and Bodnár (2011); Vitruvius (1931–1934, Bk. 10,3); Hero of Alexandria (1900 , Bk. 2.8; 2.33, pp. 114, 170).
- 10.
Clagett (1959, 71, n.5 and 75–76, n. 6).
- 11.
- 12.
The reduction program was already noted by Duhem (1905, 8).
- 13.
Problem 4: The oars of a ship are identified as levers.
Problem 6: The mast of a ship is identified as a lever.
Problem 9: The wheels of pulleys are identified as levers.
Problem 13: Handles of spindles and windlasses are identified as levers.
Problem 14: A piece of wood broken over the knee is identified as a lever.
Problem 15: Pebbles at the beach rotated and worn down by water are identified as levers.
Problem 16: Wooden planks raised are identified as levers.
Problem 17: Wedges are identified as consisting of levers.
Problem 18: A pulley is identified as a lever.
Problem 20: An asymmetric balance is identified as a lever.
Problem 21: The forceps of a dentist are identified as a pair of levers.
Problem 22: A nutcracker is identified as a lever.
Problem 26: Wooden planks carried on the shoulder are identified as radii.
Problem 27: Wooden planks raised up to the shoulder are identified as radii.
Problem 29: A plank carried by two men is identified as a lever.
- 14.
- 15.
There has been some disagreement in the literature on the meaning of ischus (force) here, Duhem (1905 and 1913), De Gandt (1982), and Krafft (1970). See the discussion in Schiefsky (2009, 59–61). For our purposes it suffices to note that motion presupposes a force and that wherever the force is applied to the rotating radius or beam: outside points move faster than inside points.
- 16.
Since Krafft’s analysis in Dynamische und statische Betrachtungsweise (1970); this argument about the geometry of a moving point has been the focus of attention of scholars dealing with the Mechanical Problems. See also Mark Schiefsky, (2009) and De Groot (2009, 2014). No convincing argument has so far been advanced as to why the later problems should need this proof.
- 17.
Reading phalanx (beam) for plastinx (pan). Not only does the sense of this text argue for this reading, but also the next problem, which deals with a material beam balance (without actually calling it a beam), would be much easier to understand if the material beam had already been introduced. No modern translation takes the author to mean ‘pan’. Forster (1995, 1302) and Hett (1936, 347), (both take the author to mean “balance” and not “scale pan.” The Arabic translation (Abbatouy 2001, 114–115) renders whatever was in the original Greek text all three times as beam or pole. We thank Sonja Brentjes for advice on the Arabic.
- 18.
The term spartion (the diminutive of sparton: cord) becomes a technical term meaning suspension point after its identification with the center of a circle here. The two later occurrences of the non-diminutive sparton (in Prob. 3 and 20) may be copying mistakes, since in each case the text refers back to a cord previously mentioned in the diminutive form. In Problem 1, on the other hand, we are dealing with larger balances, where the larger cord might actually be meant.
- 19.
The phrase menei gar touto otherwise occurs only in Problem 27, where it plays exactly the same role as in 1, 3, and 4. The verb ginetai (it becomes) is used to express the identification of a part of a device with the fulcrum or center in Problems 1, 3, 4, 5, 6, 12, 15, 16, 19, 20, 26, 29. In Problems 9, 13, 14, 17, 21, 22, and 27 only the verb to be or no verb at all is used.
- 20.
The author discusses the horizontal motion of a suspended beam balance in Prob. 10.
- 21.
Stevin (1586, 65, 509) condemned Aristotle for this internally contradictory formulation. The strict proportionality of lengths and weights holds only in equilibrium, that is, in that case in which there is neither a moving weight nor a moved weight. When one weight moves the other, they are not in equilibrium. If we assume the author was aware of this fact, we have an explanation for his avoidance of the language of proportions.
- 22.
For instance: Problems 16, 20, 29. As we shall see in Problem 20: With a given lever and a given force, the closer to the load the fulcrum is put, the greater the load that can be moved, but the increase in effect is not proportional to the change in distance from the fulcrum.
- 23.
There are two uses of logos in a different sense in Prob. 19 and 23.
- 24.
- 25.
In this particular case the rower in the middle does not in fact effect more, he just has an easier time of it. Moreover, if the arm-length of the rowers is the same, the rower on the longer lever moves it the same length as the rowers on the shorter levers (oars) and thus actually effects less. Problem 3 asserted: the farther, the more easily; Problem 4 asserts: the farther, the more effective. Whereas Problem 3 held that the same effect is achieved with less force, Problem 4 wants to assert that a greater effect is achieved with the same force.
- 26.
The verb apereidein means to fix or support and is related to the term peisma for the ship’s cable used to tie down the ship to land. This sentence may be a later insertion since the text then continues with a renewed identification of the sea as the load and a repetition of the explanation of why the lever (now of the first kind again) is longer in the middle. There is clearly some corruption in the text. Renaissance authors often pointed out that Aristotle should have used a lever of the second kind in his analysis. See Galileo’s letter to Giacomo Contarini, March 22, 1593, in Galilei (1968, vol. 10, 55–57); Biancani (1615, 159); Baldi (1621, 41).
- 27.
The three sentences quoted make up the entire Problem 6, 851a38–b6. Note that the principle invoked is that the same force moves the same load more easily and thus more swiftly—and thus has a greater effect.
- 28.
hêmizugiou. This is the only occurrence of the term in the classical Greek corpus: it could either be an adjective (hêmizugios) meaning “forming a half-balance,” Liddell et al. (1996) or (as Markus Asper has suggested to us) a diminutive noun (hêmizugion) meaning “small half-balance.”
- 29.
On the history of the balance in general, see Robens et al. (2014).
- 30.
From the end of the Anglo-Hanseatic War (1474) until 1598, Hanseatic merchants (who apparently used such devices) had a compound on the north bank of the Thames near London Bridge called the “Stalhof” (steel yard).
- 31.
Aristophanes (1998, lines 1245–49, pp. 58, 304).
- 32.
See Damerow et al. (2002).
- 33.
sphairôma: literally “round thing”
- 34.
The term weight can be used in two quite different senses: Like length it names an abstract physical quantity, but it also can name a concrete material thing such as a measuring weight. If a piece of silver weighs three talents, the silver has a weight, the talent is a weight. Greek of Aristotle’s time sometimes distinguished the two senses by the gender of the noun: ho stathmos (masculine) could refer to the abstract quantity measured and to stathmon (neuter) to the standardized measuring weight placed in the pan [see Liddell et al. (1996)]. The author of the Constitution of Athens (presumably Aristotle) uses the terms in this manner (Aristotle 1995, pp. 2346, 2373; ch. 10.1–2 and 51.3). However, Aristotle at one prominent place (Metaphysics N, 1087b37) uses the masculine form for the measuring unit. The standard published versions of Problem 20 use both forms of the word: masculine twice and neuter once. Of the relevant manuscripts, all but the one on which the first print edition was based have only the masculine form. We thank Joyce Van Leeuwen for checking the manuscripts for us.
- 35.
Reading (with Cappelle) phalanx (beam) for plastinx (pan).
- 36.
In the text just quoted, the appeal to the lever principle of Problem 3 “the greater the length … the more easily (rhaon) it moves” is marked by the tantum-quantum formula hosô … tosoutô. On the other hand, the state of affairs that this principle is supposed to explain “the nearer the cord … the greater the load it draws” is not so marked.
- 37.
Biancani (1615, 183); cf. Piccolomini (1547, 42v); de Monantheuil (1599, 147): “… aequipondium, Graecis dictum sphairôma, nostris Marcum vel Romanum….” Even Baldi (1621, 134), who mentions that “we could use the steelyard in a different manner” and then describes a bismar, does not suggest that this might have been intended by Aristotle himself.
- 38.
On the possibility of comparable developments in China in the same period, see Renn and Schemmel (2006).
- 39.
See the already cited passage from the Constitution of Athens, ch. 10, in Aristotle 1995, p. 2346.
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Renn, J., McLaughlin, P. (2018). The Balance, the Lever and the Aristotelian Origins of Mechanics. In: Feldhay, R., Renn, J., Schemmel, M., Valleriani, M. (eds) Emergence and Expansion of Preclassical Mechanics. Boston Studies in the Philosophy and History of Science, vol 270. Springer, Cham. https://doi.org/10.1007/978-3-319-90345-3_5
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