Abstract
This chapter aims to throw light on the ways in which the concept of center of gravity interacted with some of the cosmological ideas conceived in antiquity and in particular with the idea of the figure of earth as presented in Aristotle’s De coelo. Developing earlier research, this study provides a better understanding of the scientific discussion that took place during the crucial first stage in the development of modern science. The origins and earliest stages in the development of the concept of center of gravity in Ancient Greece was briefly studied by Duhem, whose cursory analysis of a few texts by Pappus and Archimedes was undertaken with the specific purpose of showing the supposed faults inherent in Greek statics. The chapter will begin with a discussion of these Greek sources and attempt to follow the intellectual recovery of this key concept in the Renaissance and the first decades of the seventeenth century, relying in particular on a thorough study of textbooks used for teaching astronomy in Jesuit schools, for example, Commentaries on De sphera of Sacrobosco.
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07 February 2019
The book was inadvertently published with few errors in chapter 6 “A Journey to the Center of the Earth: Cosmology and the Centrobaric Theory from Antiquity to the Renaissance”.
Notes
- 1.
To the study of the centrobaric theory in the field of mechanics was later joined a more strictly mathematical study concerning the determination of rigorous methods for finding the center of gravity of solids. The lack of ancient specific texts made this study more complex and difficult, entailing a discontinuity between the two stages of development.
- 2.
“La Statique est la science de l’équilibre des forces […] L’équilibre résulte de la destruction de plusieurs forces qui se combattent et qui anéantissent réciproquement l’action qu’elles exercent les unes sur les autres; et le but de la Statique est de donner les lois suivant lesquelles cette destruction s’opère. Ces lois sont fondées sur des principes généraux qu’on peut réduire à trois ; celui du levier, celui de la composition des forces, et celui des vitesses virtuelles.” Lagrange (1811, 1–2).
- 3.
Volumes I–V, 1913–1917; volumes VI–X were published posthumously by his daughter, 1953–1959.
- 4.
The explanation of the functioning of the lever in the Mechanical Problems is vague. This fact was pointed out more than once by Renaissance scientists following the Archimedean tradition. The vagueness was explained as a first imperfect result of the earliest investigations on the lever, which were later to be fully developed only in Archimedes’ work and in the centrobaric theory. This point of view is clearly presented in Guidobaldo del Monte’s preface to his edition of Archimedes’ work In duos Archimedis aequiponderantium libros paraphrasis, Monte (1588, 4). The same point of view is present in Archimedes’ biography by Bernardino Baldi: “Since Archimedes (as it is probable and as Guidobaldo himself guessed in the preface to Book One of On the Equilibrium of Planes) regarded this Aristotelian work as being based on solid principles, but not being very clear in explaining them, he wanted to make it more explicit and more easily understandable by adding mathematical demonstrations to physical principles. Aristotle solved the problem of why the longer the lever, the easier it moves the weight, by saying that this happens because of the greater length on the side of the moving power; this was true according his principle, in which he supposed that the things that are at greater distance from the center move more easily and with greater force; the cause of which he saw in the greater speed with which the bigger circle moves compared to the smaller circle. This cause is indeed true, but lacks precision; for given a weight, a lever and a power, I do not know how I should divide the lever in the point where it turns, so that the given power balances the given weight. Archimedes accepted Aristotle’s principle, but went further; he was not satisfied with saying that the force would be greater on the longer side of the lever, but he determined how much longer it should be, that is, what proportion it should have with the shorter side, so that the given power would balance the given weight. [...] He established this with a brilliant demonstration in Book One of On the Equilibrium of Planes, which, as Guidobaldo pointed out, was the book of Elements of the whole field of mechanics. In the preface of his paraphrase of Archimedes’ work, Guidobaldo showed that Archimedes had followed Aristotle entirely, as far as the principles were concerned, but he had added his own exquisite demonstrations.” Baldi (1887, 54–55).
- 5.
A first stage in the transformation of the term ῥοπή can be seen in a passage from De justo ascribed to Plato, where the verb ῥέπειν usually refers to the motion of a heavy body downwards, seems to be used also with reference to the motion of a light body upwards. But the thing seems rather dubious. This brief text attempts to define what is just through a Socratic discussion, where the solution of the problem is found through a series of short questions and answers concerning less problematic fields of knowledge than the one discussed here. Among these is the doctrine dealing with the concepts of heavy and light. The question then is: How can we judge whether a body is heavy or light? By its weight. How can the weight of a body be assessed? By the art of weighing. Now the art of weighing makes use of the scale, and therefore “what on the scale inclines downwards is heavy, what inclines upwards is light” (τὸ μὲν κάτω ῥέπον ἐν τοις ζυγοις, βαρύ· τὸ μὲν ἄνω, κουϕον). In the second part of this sentence, the verb is missing, but it seems correct to assume that the term ῥέπον should be repeated. But this not a question of a natural tendency of the light body to move upwards but rather of the simple observation of what happens on the scale: the equilibrium has been disrupted, one body goes downwards, the other goes upwards. We should not think of a scale for weighing the lightness, but of a determination of heavy and light as relative terms.
- 6.
The original reads: “trium quorum [elementa] quodlibet terram orbiculariter undique circumdat, nisi quantum siccitas terre humori aque obsistit ad vitam animantium tuendam.” Thorndike (1949, 78–79).
- 7.
“Omnia etiam preter terram mobilia existunt, que ut centrum mundi ponderositate sui magnum extremorum motum undique equaliter fugiens rotunda spere medium possidet.” Thorndike (1949, 79).
- 8.
Sacrobosco might have deliberately avoided a discussion of this type of problem in his treatise. It would seem that he abstained from referring to Aristotle’s De coelo. At the time that Sacrobosco wrote his De sphera, Aristotle’s work was already available both through Gerardo of Cremona’s translation, and through the Pseudo-Avicenna’s Liber celi et mundi, which was much more common. This work was a paraphrase of some sections of De coelo and at the time of Sacrobosco, generally regarded as an original work by Aristotle, Gutman (1997, 121–8). The tenth chapter of the Liber celi et mundi which is titled Quod figura terre sperica est, contains an interesting discussion of the problem I have been considering here. I will discuss the content of this chapter on another occasion. Gutman (2003, 181–183).
- 9.
It is interesting to observe that even in this case Sacrobosco never refers to De coelo II, 4 which would have given him a good argument to prove that water takes on a spherical shape. In the passage concerned, Aristotle presents a geometrical demonstration of the type of reductio ad absurdum, in which the conclusion is not in contradiction with the assumed geometrical principle, but rather it conflicts with the evidence of sense experience: water runs by nature from higher places to lower places. It would have been impossible for Sacrobosco to refer to the proposition at the beginning of Archimedes’ On Floating Bodies, which was later translated by William of Moerbeke. Only during the Renaissance were references to Aristotle and Archimedes made by those authors who wrote commentaries on the Sphere, thus showing the need for more rigorous demonstrations on this important point.
- 10.
This must be obviously related to Michael Scot’s activity as a translator of Aristotelian works (with Averroës’ Commentaries): he had translated into Latin an Arabic translation of De coelo before 1230. Another, perhaps not complete, translation was made by Robert Grosseteste soon after 1230. After these works became available, it was unlikely that those who wrote on cosmography could ignore Aristotle’s work.
- 11.
The original reads: “ut ex pugillo terre per rarefactionem et subtiliationem fiunt pugille aque decem etc. Et econverso ex decem pugillis ignis per condensationem et inspissationem fit unus pugillus aeris et deinceps.” Thorndike (1949, 263–264).
- 12.
Further on in the text just quoted (Thorndike 1949, 296), Michael Scot discussed the same question from the point of view of concept of place by investigating the reason for the placement of the earth within the sphere of the immediately higher element. The internal surface of the sphere of water, as it was not in contact with all parts of the sphere of the earth, could not be considered as the place of the earth, whereas in some passages in Aristotle’s works it appeared that the internal part of the sphere of any element was the place of the element immediately lower. The solution given by Michael Scot was similar to that briefly mentioned by Sacrobosco: according to the form proper to each element, the earth should have been contained by water, but the world was not perfect and the great majority of the animals and of the plants could not live in water, therefore part of the earth had been cleared of water. No reasons are given to explain how this situation came about.
- 13.
Bartholomaeus Anglicus (1505, sign. [s 3v]) “Terra, ut dicet Philosophus, est propriis equilibrata ponderibus. Quelibet enim suarum partium suo pondere nititur ad mundi medium, quo nisu et inclinatione singularium partium, tota circa centrum equilibrata suspenditur, et equaliter immobilis retinetur.” The first part of this passage seems to be in relation with Publius Ovidius Naso (Metamorphoseon, I, 1, vv. 12–13) “nec circumfuso pendebat in aere tellus ponderibus librata suis…”
- 14.
De caelo et Mundo, Book. 2, lectio, 27, de Aquino (1866, 142–143). Author’s italics.
- 15.
“Quod terra est in medio mundi quo ad centrum sue gravitatis. Probatur: nam omnes partes terre tendunt ad medium per suam gravitatem, sicut dicit Aristoteles in littera; et verum est. Modo pars que esset gravior depelleret aliam tam diu quod medium gravitatis totalis terre esset in medio mundi; e tunc starent due partes eque graves; licet una maior et alia minor quantum ad magnitudinem contra se invicem; sicut duo pondera in equilibra.” Albert of Saxony (1520, 40r, lib. 2, quaestio 25).
- 16.
We could even think that Albert refers to the law of lever as presented by Archimedes in Proposition 6 of Book one, but from the Latin passage quoted in the note above it would seem that Albert only points out that in this case the heavier weight is nearer the center of the world, and the lighter weight is further away from it.
- 17.
By referring to what Aristotle says in chapter XIV of Book Two of Meteorologica, Albert probably thought that the continuous changing location of lands, rivers and seas involved a continuous changing arrangement of the weighing parts of the earth.
- 18.
Here, I deliberately disregard the tradition of the medieval Latin Scientia de ponderibus, which greatly influenced discussions of the problem of the equilibrium of balances during the Renaissance. This is not because I assume there is no relation between the question I discuss and those treated in the works ascribed to Jordanus Nemorarius. On the contrary, the premises of this work treat explicitly the relation between gravitas and rectitudo of the path of descent of heavy bodies (with reference to the line that ends in the center), Moody and Clagett (1960, 128–129). I chose not to deal with this tradition because the problems of the centrobaric theory are not mentioned. For an extensive treatment of the dispute between followers and opponents of the Scientia de ponderibus in the sixteenth century, see Renn and Damerow (2012). In the Pseudo-Aristotelian Mechanical Problems there is also no reference to the concept of center of gravity, but in this case there is a tendency to adopt Archimedes’ method, on the assumption of a continuity between Archimedes’ work and the mechanical tradition of Pseudo-Aristotle.
- 19.
Maurolycus’ peculiar way of phrasing the question must be noted. For him, the problem was the need to explain “ut hanc terrae marisque congeriem conglobatam esse.” Maurolico (1543, 7r/v). This seemed to go back to the tradition of discussions on the causes of why the earth was not completely covered by water. Sacrobosco instead discussed separately, one after the other, the question of the spherical shape of the earth (“Quod terra etiam sit rotunda sic patet,” Thorndike (1949, 81)) and of water (“Quod autem aqua habeat tumorem et accedat ad rotunditatem sic patet,” Thorndike (1949, 83)).
- 20.
“Nimirum, quod utrique nos erectos arbitramur, verum est; quod vero nos illos, illique nos in caput versos putamus, falsum. Siquidem utrique recti stamus; ipsumque terrae centrum locus est infimus utrisque communis; ad quem sane duo pondera utrinque suspensa pendent, et dimissa concurrerent.” Maurolico (1543, 16v).
- 21.
This argument was also derived from Albert of Saxony (1520, 41v.), Quaestiones subtilissime, in the 4th corollary to Question XXVII (already mentioned) of Book Two of De coelo.
- 22.
“Punctum videlicet, quod utcunque ac quotiescunque suspenso corpore, semper versus universale centrum pendet ad perpendiculum; quodque, dimisso corpore, modo absint obstacula, ipsi universali centro connitur.” Maurolico (1543, 18v).
- 23.
The original reads: “Suspendatur proposita res utcunque, ut libere pendeat; mox ab ipso suspensionis signo ad horizontis planum perpendicularis recta ducatur, quemadmodum Euclidis in 11 undecimi docet. Rursum ab alio signo similiter res ipsa appendatur; et a signo rursum perpendicularis agatur ad horizontem. Oportebit nanque utranque perpendicularium per centrum incedere gravitatis, quandoquidem tale centrum in ipsa semper ad horizontem perpendiculari, utcunque res pendeat, invenitur. Punctum igitur, in quo se vicissim perpendiculares intersecane, erit proculdubio quaesitum gravitatis centrum.” Maurolico (1543, 19r/v).
- 24.
- 25.
Knobloch (1999, 57). In the 1581 edition published in Rome, the definition of the center of gravity taken from Maurolycus is followed by the one taken from Pappus’ Mathematical Collections, Book Eight. Since any reference to the motion of the body let loose from the point at which it was hanging is missing, Pappus’ definition would have been more suitable for Clavius’ argument; but its inclusion in this argument is in no way justified. Any comparison between the two texts is missing, and the new definition seems to be an unnecessary addition.
- 26.
- 27.
- 28.
Blancanus (1615, 155–157). In this work the Jesuit scholar had also discussed the passage on the aporia in Chapter XIV of Book Two of De coelo. On the basis of the concept of center of gravity, the motion of a heavy body towards the center of the world and its resting at this center at the end of the motion were immediately understandable: as soon as the center of gravity of the body and the center of the world coincided, the body would stop moving. But Aristotle could not have conceived of this center in this way since the concept of center of gravity was first used by Archimedes. Therefore Biancani thought that Aristotle meant the center of magnitude and hence was wrong. “Iuxta mathematicos duplex esse medium, sive centrum cuiusvis magnitudinis: aliud enim est centrum molis, aliud est centrum gravitatis. [...] Quando igitur Aristoteles ait, grave descensurum, donec ipsius medium, sive centrum, mundi centrum attingat, bene dicit, si de medio gravitatis intelligat, male autem si de medio molis, quia gravia omnia ratione centri gravitatis ponderant, neque manent, nisi ipsum maneat; quare nisi ipsum attingant centrum mundi, semper gravitabunt, et movebuntur. Verum enim vero ex antiquorum monumentis manifestum est, Archimedem, qui multo post Aristotelem floruit, primum omnium de centro gravitatis esse philosophatum, qua ratione dicendum esset, Aristotelem de centro, molis loquutum esse, et perinde non usquequaque vere.” Blancanus (1615, 81). The explanation of the phenomenon described in the aporia was simplified and transformed by Biancani: the greater part that pushes the smaller one is regarded as being inside the solid body. “Sensus Aristotelis est, debere nos existimare, quod si quaepiam gravis magnitudo descendat ad centrum mundi, eam non permansuram, statim ac ipsius extremum centrum mundi attigent; sed eo usque descensuram, quosque ipsius medium, mundi medium, sive centrum assequutum sit; maior enim ipsius pars, in qua scilicet medium est, minorem partem propellit, donec utrinque a centro mundi aeque emineat. Omne enim grave hucusque habet propensionem, sive hucusque gravitat, v.g. si lapis illuc descenderet, non quiesceret statim ac prima ipsius pars ad mundi centrum pertingere, sed reliquae ipsius partes adhuc gravitarent, sicque ulterius primam partem impellerent, donec lapidis medium mundi medio congrueret.” Blancanus (1615, 80–81).
- 29.
“Terrae moles ita circa mundi centrum constituta est, ut in aequilibrio sita sit, idest, partes eius circa mundi centrum aeque ponderent, ac propterea immota consistat; quae vero in aequilibrio manent, quovis minimo ex una parte addito, vel ablato pondere, ab aequilibrii situ dimoventur, ut experientia quotidiana in lancibus, ac stateris ostendit, et rationes Mechanicorum evincunt. Cum igitur perpetuo circa Terram, res variae modo illi addantur, modo demantur (ut eum lapis in altum proiicitur, vel cum aves ab ea avolant, et ad eandem advolant, aut cum aliquid super eam saltat) necessarium esse videtur ipsa in perpetua quadam trepidatione insensibili tamen, titubare, ac vacillare.” Blancanus (1620, 76).
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Nenci, E. (2018). A Journey to the Center of the Earth: Cosmology and the Centrobaric Theory from Antiquity to the Renaissance. 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_6
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