The Principle of the Impenetrability of Bodies in the History of Concepts of Separate Space from the...

The Principle of the Impenetrability of Bodies in the History of Concepts of Separate Spacefrom the Middle Ages to the Seventeenth CenturyAuthor(s): Edward GrantSource: Isis, Vol. 69, No. 4 (Dec., 1978), pp. 551-571Published by: The University of Chicago Press on behalf of The History of Science SocietyStable URL: http://www.jstor.org/stable/231092 .

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The Prnciple of the Impenetrability of

Bodies in the History of Concepts of

Separate Space from the Middle Ages

to the Seventeenth Century

By Edward Grant*

IN THE TREATMENT of place and vacuum in the fourth book of his Physics, Aristotle rejected the existence of a separate, three-dimensional void space. He

devised numerous arguments to convince his audience that such a space could not possibly function as a place where material bodies could move or rest. Two of these are relevant to the theme of this paper.

The definition of void as "place bereft of body"' implied for Aristotle that place is something distinct from the bodies which occupy it. Indeed, since most people believed that "everything is somewhere and in place," it would seem that, like Hesiod's chaos, place (or space) would necessarily precede the things that must move and rest in it. "If this is its nature," Aristotle declares, "the potency of place must be a marvellous thing, and take precedence of all other things. For that without which nothing else can exist, while it can exist without the others, must needs be first; for place does not pass out of existence when the things in it are annihilated."2

If Hesiod and the atomists could find justification for the existence of such a separate, empty space, Aristotle found it an unintelligible conception from which only absurdities and impossibilities followed. Leaving aside his numerous arguments against the possibility of motion in a separate space,3 let us now describe the two arguments which concern this paper.

The first declares that if place or space were three dimensional, it would be a body. "But the place cannot be body," Aristotle declares, "for if it were there would be two bodies in the same place."4 As a subsequent illustration of this absurdity, Aristotle compared the immersion of a wooden cube in a material medium, such as water or air, with its "immersion" in a dimensional empty space. In water or air, the cube would, of course, displace an equal volume of water or air respectively. When placed in empty space, however, the void space would not be displaced, but, on the contrary, penetrate the wooden cube throughout its extent. As Aristotle put it, it would be "just

Received July 1977; revised/accepted August 1977. *Department of History and Philosophy of Science, Indiana University, Bloomington, Indiana 47401. 'Physics IV 1, 208b 25; see also IV 7, 213b 32 and 8, 214b 18-19. Quotations from Aristotle's Physics are

from the translation by R. P. Hardie and R. K. Gaye in The Works of Aristotle Translated into English, ed. W. D. Ross, 12 vols. (Oxford: Clarendon Press, 1908-1952).

2Physics IV 1, 208b 34-209a 1. 3For these, see ibid., IV 7-8. 4Ibid., 1, 209a 5-7.

ISIS, 1978, 69 (No. 249) 551



552 EDWARD GRANT

as if the water or air had not been displaced by the wooden cube, but had penetrated right through it."5 By interpreting a three-dimensional void space as if it were a material, extended body, Aristotle made no distinction between the penetration of void space throughout the dimensions of the material cube and the penetration of water or air throughout those same dimensions. Since the interpenetration of material medium and material cube is impossible, so also is the interpenetration of the dimensions of an alleged void space with that material cube impossible. Thus, the simultaneous interpenetration and coexistence of separate dimensional entities is impossible; or, to express it in its most concise form, two bodies cannot occupy the same place. For Aristotle it was now obvious that not only is alleged void space not dimensional, but it is nothing at all. The analogy, and indeed the basic likeness, which he saw between the conception of a three-dimensional void and a material body was, however, destined to play a significant role in the controversy over the possible existence of a separate space in the period from the Middle Ages to the seventeenth century. The equation of dimensional void with body had an interesting history. Not only was it used to reject void space, as Aristotle had intended, but, as we shall see below, it would be turned against Aristotle and used to justify motion in an extended vacuum.

But if the idea of extended void space was deemed ridiculous by virtue of the simultaneous occupation of one and the same place by two "bodies," it was also to be rejected because it was utterly superfluous. To demonstrate this Aristotle asks that we imagine the impossible and conceive of the dimension, or volume, of that same wooden cube as if it existed independently from all its other inseparable properties, such as hot or cold, heavy or light, and so forth. Under these circumstances, the abstracted volume of the cube would "occupy an equal amount of void, and fill the same place, as the part of place or of void equal to itself." "How then," inquires Aristotle, "will the body of the cube differ from the void or place that is equal to it? And if there can be two such things, why cannot there be any number coinciding?"6 The consequence is obvious: either a separate void space is superfluous, or any number of such spaces could coincide simultaneously, which is absurd. Thus if a void space differs in no way from the abstracted dimensions of the body which occupies it, is it not plausible to infer, argues Aristotle, that void place is simply superfluous? After all, if the dimension of a body is one of its fundamental attributes-an attribute that it retains wherever it may be-why assume that it requires yet another void dimension "in" which to be located?

In the Aristotelian tradition of the fourteenth to seventeenth centuries, the existence of a separate, extended, void space, whether always filled or occasionally, or even permanently, devoid of body,7 was almost always rejected on the basis of the arguments just described. A typical defense of Aristotle's position was offered by the Averroist John of Jandun (d. 1328), who argued that if place were a space or corporeal dimension, it would follow that two bodies, or dimensions, could coincide simultaneously in the same place. And if two bodies could coexist in the same place, so could an infinite number. To illustrate, Jandun imagines the heavens, or world, to be subdivided into an equal number of particles each the size of a millet seed. Since any two bodies could occupy the same place, and therefore any number whatever, the absurdity would follow that all the particles into which the world is divided could

5For the entire argument, see ibid., 8, 216a 26-216b 2. 6 For the whole argument, see ibid., 216b 3-1 1. 7As will be seen below, Latin scholastics, who frequently distinguished between these two possible types

of space, derived the distinction from Averroes.



THE IMPENETRABILITY OF BODIES 553

occupy the same place, and thus the world could be contained within a single millet seed.8 While scholastics would readily have conceded that God could, if He wished, create two or more bodies in the same place simultaneously,9 they were agreed that it was naturally impossible.

The superfluous nature of a separate space was also frequently emphasized. Albert of Saxony (c. 1316-1390) argued that if a body required a separate space as its place, this would only be so because of the body's dimensions rather than from its matter, form, or qualities, all of which were assumed to inhere in those dimensions. But if the dimensions of a natural body are located in a separate three-dimensional space, that dimensional space would itself require a separate dimensional place, and so on ad infinitum.10 Extending the argument to the cosmos itself, Jean Buridan (c. 1300-c. 1358) insisted that the world needed no separate space or place because it was itself a magnitude with a dimension, as were all its parts. Thus God did not require a pre- existing dimensional space in which to create the world. For if such a space did exist, God could by His absolute power destroy it and subsequently re-create it. But in re- creating it, God would surely not require yet another space in which to place it. To argue that God needed a space in which to create a space would be to limit His infinite power to create a space without something there to receive it. In the same manner, then, in which God does not require a pre-existing space in which to create a possible space for the world, so also does He not require a separate dimensional space or magnitude in which to create the world itself."I Since what applies to the world, applies to any part of it, Buridan concludes that a separate space is superfluous.12

Acceptance of the idea of a three-dimensional, separate void space was thus

8 Questions on the Physics, Bk. IV, ques. 4, in Questiones Joannis de Janduno De Physico auditu noviter emendate. Helie Hebrei Cretensis questiones: De primo motore; De efficientia mundi; De esse essentia et uno; annotationes in plurima dicta commentatoris (Venice, 1519), fol. 48r, col. 2. To conserve space, I have, with a few exceptions, omitted citation of supporting Latin texts. In the context of God's absolute power, Albert of Saxony also imagined a situation in which God could place the whole world within a millet seed and achieve this without any condensation, rarefaction, or penetration of bodies. As a supernatural act, this was not deemed absurd. See my article, "Place and Space in Medieval Physical Thought," in Peter K. Machamer and Robert G. Turnbull, eds., Motion and Time, Space and Matter, Interrelations in the History of Philosophy and Science (Columbus, Ohio: Ohio State University Press, 1976), p. 154.

9E.g., see William Ockham in Guillelmus de Occam, Ouotlibeta septem; Tractatus de sacramento altaris (Strasbourg, 1491; reprinted in facsimile at Louvain: Editions de la Bibliotheque S.J., 1962), Quotlibet primum, ques. 4, sig. a4v, col. 1 (the pages are unfoliated). By conceding to God the power to locate two or more bodies in the same place, Ockham probably had in mind Art. 141 of the Condemnation of 1277, which denounced the opinion that God could not make more than several dimensions exist simultaneously ("Quod Deus non potest facere accidens esse sine subjecto, nec plures dimensiones simul esse," H. Denifle and E. Chatelain, eds., Chartularium Universitatis Parisiensis, Vol. I, Paris, 1889, p. 551.)

I0Albert of Saxony, Questions on the Eight Books of the Physics, Bk. 4, ques. 1, in Questiones et decisiones physicales insignium virorum: Alberti de Saxonia in octo libros Physicorum; tres libros De celo et mundo; duos libros De generatione et corruptione; Thimonis in quatuor libros Meteororum . Buridani in librum De sensu et sensato . . . Aristotelis . . . recognitae rursus et emendatae summa accuratione . . . Magistri Georgii Lokert . . . (Paris, 1518), fol. 43r, col. 2. Albert probably drew the substance of this argument from Jean Buridan, who gives it in his Questionts on the Physics, Bk. 4, ques. 2, in Acutissimi philosophi reverendi Magistri Johannis Buridani subtilissime questiones super octo Phisi- corum libros Aristotelis diligenter recognite et revise a Magistro Johanne Dullaert de Gandavo antea nusquam impresse (Paris, 1509; reprinted in facsimile under the title Johannes Buridanus, Kommentar zur Aristotelischen Physik, Frankfurt a.M.: Minerva, 1964), fol. 68r, col. 1. In the 16th century, Franciscus Toletus (1532-1593) offered a similar argument in his commentary on Aristotle's Physics. See D. Francisci Toleti Societatis Iesu Commentaria una cum Quaestionibus in octo libros Aristotelis Dephysica auscultatione, nunc secundo in lucem edita (Venice: Apud luntas, 1580), Bk. 4, Ch. 5, ques. 3, fol. 117v, col. 1.

"Jean Buridan, Questions on the Physics, ed. cit., Bk. 4, ques. 2, fol. 68r, cols. 1-2. 12The superfluous character of space was manifested in yet another way. John of Jandun argued that as

a mere empty, separate, homogeneous dimensionality, space would lack all capacity to influence or affect the bodies that might occupy it. Questiones Joannis de Janduno De physico auditu noviter emendate, Bk. 4, ques. 4, fol. 48v, col. 1.



554 EDWARD GRANT

confronted with formidable obstacles. Either it was conceived as a corporeal dimension and thus incapable of receiving material bodies because of the obvious violation of the self-evident principle that two bodies cannot occupy the same place simultaneously; or it was simply superfluous. Unless plausible and acceptable solutions to these problems could be formulated, the idea of a separate, void space appeared doomed. The dilemmas posed by Aristotle had, somehow, to be answered. Would a three- dimensional vacuum be truly incapable of receiving a three-dimensional material body because of the impossibility of the interpenetration of bodily dimensions? Were other options and interpretations available that might plausibly, and perhaps even compellingly, avoid interpenetration? The various responses to the problems posed by Aristotle's criticisms against a separate space represent a significant aspect of the history of spatial doctrines up to approximately 1700. In what follows, I shall attempt to describe some of the solutions that were proposed.

I. INTERNAL SPACE

One significant reaction was to avoid altogether the assumption of an external space and to internalize it instead by equating it with material extension. In the conflation of space with material extension, the problem of interpenetration was completely avoided by reducing to one the number of extensions involved. This position could have been derived as a consequence of Aristotle's argument described above, namely his insistence on the superfluousness of external space since all the dimensions of such a space are already found in the extension of any material body. From this interpretation, one could easily deny the existence of external space on grounds that all the space that a body needed was already within it, in the form of its own extension or dimension. It is also possible that John Philoponus played a significant role in the development of the concept of an internal space. In his De aeternitate mundi contra Proclum of 529, a work unknown in the Latin Middle Ages but which was translated into Arabic and then from Greek into Latin in the sixteenth century, when it was widely read, Philoponus argued that the substance of a corporeal entity is its three- dimensional extension.'3 Rejecting the traditionaf interpretation of Aristotle's concept of prime matter as something without definite properties, Philoponus assigned three-dimensional extension to it. 14 Although Philoponus seems not to have conceived of this three-dimensionality as the internal space of bodies15-indeed, as we shall see below, he believed in the existence of a separate void space-those who

'3See Michael Wolff, Fallgesetz und Massbegriff. Zwei wissenschafthistorische Untersuchungen zur Kosmologie des Johannes Philoponus. Quellen und Studien zur Philosophie, ed. Gunther Patzig, Erhard Scheibe, Wolfgang Wieland, Vol. II (Berlin: Walter de Gruyter, 1971), p. 109; for German translations of relevant passages from the Greek text, see pp. 141-143.

'4Giacomo Zabarella (1533-1589) interpreted Philoponus as one who identified prime matter with three-dimensional extension, an interpretation with which he agreed, and which he says had also been adopted by the Stoics. Jacobi Zabarellae Patavini, De rebus naturalibus libri XXX (Frankfurt, 1607; first published 1590 and reprinted, Frankfurt: Minerva, 1966), De prima rerum materia, liber secundus, col. 211. Following in the tradition of Plotinus, Simplicius-in his commentary on Aristotle's Physics-notes difficulties and even a contradiction in Aristotle's conception of matter and then denies extension to prime matter, which acquires tridimensionality only upon receipt of a distinct corporeal form. See Harry A. Wolfson, Crescas' Critique of Aristotle, Problems of Aristotle's 'Physics' in Jewish and Arabic Philosophy (Cambridge, Mass.: Harvard University Press, 1929), p. 582. For Wolfson's valuable discussion on the history of the problem of "corporeal form," see Pt. II, n. 18, pp. 579-590.

15 ow close one could come to describing internal space without actually proclaiming it is made evident by Zabarella, who, in citing Plotinus with approval, declared that "if we mentally contemplate matter abstracted from all forms, we can conceive nothing other than a certain extended and indistinct body and a




rejected any kind of external space could easily have adapted his ideas on extended substance to the concept of internal space.

Among those who adopted the idea of internal space, we may mention Jean Buridan in the fourteenth century, Franciscus Toletus (1532-1596) and Franciscus Suarez (1548-1617) in the sixteenth, and Rene Descartes (1596-1650) in the seventeenth. As a typical expression of internal space, the Jesuit commentator Toletus described it as the extended quantity of a body's matter. The existence of internal space was undeniable because body and space "imply each other as a mutual consequence. For if there is a body, there is a space; and if there is a true space, there is a body in it."1 6 A compelling illustration of the identification of internal space and material extension was offered by Buridan and Toletus. Thus Buridan, who believed that space is nothing but the dimension of a body, asks that we imagine a man's body, or part of his body, to be located in a void beyond the last sphere or heaven. Should that man now raise his arm beyond the last celestial sphere, Buridan explains that "before you raise your arm outside this [last sphere] nothing would be there; but after your arm has been raised, a space would be there, namely the dimension of your arm. "17 Offering much the same example of internal space, Toletus observes that it matters little whether this internal space is really (realiter) and truly distinct from quantity, or only formally (formaliter) so. Despite his declared indifference, however, Toletus concludes that it is not a separate space, but only a proper accident inhering in a body'8 and, therefore, presumably distinguishable only formally, or by reason alone.

Internal space, or the dimension of a body, was easily linked with mathematical magnitude. Duns Scotus observed that the volume of a body, or its quantum as it was called in Latin, is naturally prior to its qualities and, although it does not exist apart from the body, is the primary concern of mathematics when it is considered per se. That Scotus' quantum is akin to, if not identical with, internal space may be seen from his further claim that it is the quantum which causes separation of the sides of a body. 19

The concept of internal space, whether medieval or early modern, is the physical counterpart of the purely geometric space of Euclid's Elements. Where the internal space or dimension of a material body is inseparable from that body, so that

certain void mass ("si materiam mente contemplemur abiunctam a formis, nil aliud concipere possumus nisi corpus quoddam vastum, et indistinctum, et molem quandam vacuam, ut dicebat Plotinus." De rebus naturalibus libri XXX, col. 217).

16"Esse autem hoc intrinsecum cuiusque corporis spatium negari non potest, ut videtur, quia invicem sese mutua consequentia inferant. Nam si corpus est, spatium est; et si verum spatium est, in eo corpus est." Toletus, Commentaria una cum Quaestionibus in octo libros Aristotelis De physica auscultatione, fol. 123r, col. 2.

17"Dico enim quod spacium non est nisi dimensio corporis et spacium tuum dimensio corporis tui et antequam elevares brachium ultra illam speram, nichil esset ibi; sed brachio elevato esset ibi spacium, scilicet dimensio brachii tui." Jean Buridan, Questions on the Physics, ed. cit., Bk. 4, ques. 10, fol. 77v, col. 1. See also my article, "Place and Space in Medieval Physical Thought," p. 152. For Toletus' similar example, see his Commentaria una cum Quaestionibus in octo libros Aristotelis Dephysica auscultatione, fol. 123r, col. 2.

18Toletus, Commentaria, fol. 123r, col. 2. That Toletus believed Philoponus held a doctrine of internal space is made evident a few paragraphs later (fols. 123r, col. 2-123v, col. 1) when Toletus cites Philoponus as one who believed that internal space (spatium intrinsecum) is not a separate space but is something that inheres in bodies as a proper accident.

19Johannes Duns Scotus, Opera omnia (Hildesheim: Georg Olms, 1968; facsimile reprint of the Luke Wadding edition, Lyon, 1639), Vol. VI, Pt. 1: Quaestiones in Lib. II Sententiarum, Distinction 2, ques. 6, p. 192, where Scotus discusses the place of a natural body.



556 EDWARD GRANT

whatever its location, a body has its own space, so also does a geometric figure in Euclid's Elements possess its own "internal" space, namely the space of its own configuration, which accompanies it wherever it might be located or moved.20 There is nothing in Euclid's geometry to suggest that he assumed an independent, infinite, three-dimensional homogeneous space "in" which the figures of his geometry were located. In a purely geometric sense, such a space would have been superfluous, since every geometric figure has its own "internal" space. To assume that it also lies in a separate and independent three-dimensional space would have served no purpose whatever. Moreover, if the space of the geometric figure and the independent space it is alleged to occupy are conceived as indistinguishable, an infinity of spaces could be postulated in one and the same place. Indeed, a separate dimensional space might also have confronted Euclid with the Aristotelian problem of penetrability: could the extension of a geometric figure "penetrate" the dimensionality of the separate space? Thus if Euclid had assumed the existence of a separate, three-dimensional homogeneous infinite void space in which his geometric figures are assumed to be located, he would have confronted the mathematical analogy to the problems formulated by Aristotle with respect to the relationship between material bodies and separate dimensional space. Perhaps the dilemmas posed by Aristotle were known to Euclid, who, as a consequence, found it prudent to refrain from the postulation of a separate space in his geometry, a space he in no way needed.21 It is a mistake, then, to insist, with Shlomo Pines that "Euclidean geometry appears to require a three-dimensional infinite space, whose subsistence . . . is wholly independent of any relation it may have to bodies,"22 and to infer from this, with Alexandre Koyre, the equally mistaken notion that in the seventeenth century the "differentiated set of innerworldly places" of Aristotelian cosmology was replaced by the "essentially infinite and homogeneous extension" of Euclidean geometry, which came to be identified with the real space of the physical world, a process encapsulated in the expression "the geometrization of space."23 No plausible evidence can be adduced for the claim that space was con- sciously geometrized by application of the "infinite space" of geometry to the physical world. Prior to the eighteenth century, it is doubtful whether the space of Euclid's geometry was even conceived as a separate three-dimensional, infinite, homogeneous space. If anything, as we saw, Euclidean geometrical space was the space of bounded geometric figures, which when applied to material bodies was conceived as an internal space.

20Hermann Weyl recognized that Euclid's geometry was not "a doctrine of space itself," but rather "like almost everything else that has been done under the name of geometry," was "a doctrine of the configura- tions that are possible in space." Hermann Weyl, Space- Time-Matter, trans. Henry L. Brose (1 st American printing of the 4th German ed. [1922]; New York: Dover, n.d.), p. 102. I am indebted to my colleague Prof. J. Alberto Coffa, who drew my attention to Weyl's judgment and also informed me of its significance.

2IConsidering the close conformity and agreement between Euclid's Elements and Aristotle's conceptions of definitions, axioms, postulates, and other geometrical matters, it is not implausible to suppose that Euclid may have been familiar with the works of Aristotle and perhaps also with the arguments on void space described above. See Thomas L. Heath, The Thirteen Books of Euclid's Elements translated from the text of Heiberg with Introduction and Commentary, 3 vols. (2d ed.; New York: Dover, 1956; reprinted from the Cambridge edition of 1926), Vol. 1, pp. 117-124, 143-151. To my knowledge, Euclid did not use a term for "space" in his geometry.

22Shlomo Pines, "Philosophy, Mathematics, and Concepts of Space in the Middle Ages," in Yehuda Elkana, ed., The Interaction Between Science and Philosophy (Atlantic Highlands, N.J.: Humanities Press, 1974), p. 84.

23Alexandre Koyre, From the Closed World to the Infinite Universe (Baltimore: Johns Hopkins Press, 1957), p. viii. Although I shall elaborate on this in the future, the concept of infinite space adopted in the 17th century resulted primarily from the divinization of space, a process begun in the 14th century, and perhaps certain requirements of physics and cosmology. But it did not arise from any straightforward application of Euclidian geometric space to the physical world.




Perhaps it was the intrinsically mathematical nature of the concept of internal space which appealed to that great geometer Rene Descartes, whose name is most conspicuously associated with it. More explicitly than his predecessors, Descartes identified internal place with space24 and assumed that "the same extension in length, breadth, and depth, which constitutes space, constitutes body."25 It would appear, then, that like Buridan and Toletus before him, Descartes had avoided the Aristotel- ian problem of interpenetration of separate extensions by a total identification of matter with its internal place or space, a move that implied rejection of a separate, three-dimensional space divorced from materiality.

And yet Descartes occasionally speaks as if a separate, empty extended space might exist, as when he explains that "in body we consider extension as particular and conceive it to change just as body changes; in space, on the contrary, we attribute to extension a generic unity, so that after a body which fills a space has been changed, we do not suppose that we have also changed the extension of that space, because it appears to us that the same extension remains so long as it is of the same magnitude and figure, and preserves the same position in relation to certain other bodies, whereby we determine this space;"26 or when, upon imagining all attributes stripped from body, we find that "there is nothing remaining in the idea of body excepting that it is extended in length, breadth, and depth; and this is comprised in our idea of space, not only of that which is full of body, but also of that which is called a vacuum."27 Thus it would appear that Descartes might have allowed for the possible existence of a three-dimensional vacuum, which could then be equated with body, a move that was indeed made in the Middle Ages, as will be seen below. Such an interpretation must not, however, be attributed to Descartes, who insisted that every extension is associated with substance, that is, with material body, "because it is absolutely inconceivable that nothing should possess extension."28 Thus any alleged void space assumed to have extension must also be assumed to have substance,29 a conception which enabled Descartes, and all who adopted the doctrine of internal space, to avoid the Aristotelian dilemma wherein two separate extensions might occupy one and the same place.

II. EXTERNAL SPACE

A. Void space always occupied by body

The Latin Middle Ages learned from Averroes that the external, separate, three- dimensional void space which Aristotle had rejected could be, and had been, distinguished in two basic ways. In his commentary on Aristotle's Physics, at a point where he had occasion to inject the opinion of John Philoponus, or John the Grammarian,

24"Locum autem aliquando consideramus ut rei, quae in loco est, internum, & aliquando ut ipsi externum. Et quidem internus idem plane est quod spatium; . . ." Principiaphilosophiae, Pt. 2, Principle 15 in Oeuvres de Descartes, ed. Charles Adam and Paul Tannery, Vol. VIII, Pt. 1 (Paris: Librairie Philosophique J. Vrin, 1964), p. 48.

25"Revera enim extensio in longum, latum, & profundum, quae spatium constituit, eadem plane est cum illa quae constituit corpus." Ibid., Principle 10, p. 45. The translation is from The Philosophical Works of Descartes rendered into English by Elizabeth S. Haldane and G. R. T. Ross, 2 vols. (New York: Dover, 1955; reprinted from the 1st ed. in 1911 as corrected in 1931 and published by the Cambridge University Press), Vol. I, p. 259. See also Principle 11 (ibid., p. 46), where Descartes reiterates the identification of space and body.

26Principles of Philosophy, Pt. 2, Principle 10 on p. 259 of the Haldane and Ross translation, which I have altered. For the Latin, see Oeuvres de Descartes, Vol. VIII, Pt. 1, p. 45.

27Principles of Philosophy, Pt. 2, Principle 11 on p. 259. 28Ibid., Principle 16, p. 262. 29 Ibid.



558 EDWARD GRANT

Averroes declared that void or separate space was conceived to exist either as something always occupied by body, or as something distinct from body and unoc- cupied by it, though capable of receiving it. As a supporter of the first approach, Averroes cites Philoponus,30 but presents no supporting arguments or discussion, a circumstance that may explain the little attention that was paid to it in the Middle Ages. On the false assumption that Philoponus conceived void space to be nothing in itself, one thirteenth-century commentator, pseudo-Siger of Brabant, argued that Philoponus' position avoided the Aristotelian dilemma of interpenetration of dimensions. For if the void space in which bodies exist and move is itself nothing at all, then the bodies in those empty spaces will not interpenetrate with "anything." To all this pseudo-Siger responded by arguing that "if such dimensions are assumed to be a place, and since such dimensions are nothing, it follows that place is nothing, which is impossible."31

Philoponus' conception of a void space as the container of all bodies, and indeed of the whole material world, gained little support in the Middle Ages; nor did it seem to fare much better in the sixteenth century when Philoponus' works became available in the original Greek as well as in Latin translations.32 Since Philoponus believed,

30 Here is the brief, but complete, relevant text: "loannes vero propter hoc obedit huic, scilicet locum esse et dimensionem et vacuum, non finem continentem, ut dicit Aristoteles. Licet apud ipsum non possit inane separari a corpore quoniam dicentes vacuum esse sunt bipartiti: alii enim dicunt ipsum separari a corporibus, et alii non, quorum est loannes Grammaticus." Commentary on the Physics, Bk. 4, Comment 43 in Aristotelis opera cum Averrois commentariis (Venice, 1562-1574; reprinted in facsimile, Frankfurt a.M.: Minerva, 1962), Vol. IV, fol. 141r, vol. 2. Averroes' description was derived ultimately from Philoponus' Commentary on Aristotle's Physics, Bk. 4, in a section titled "Corollary on Place." Here, after arguing that the place of a body is a void extension distinct from the body occupying it, Philoponus concluded that the void itself is never without an occupying body, just as matter, though it differs from forms, is never without them. For the Greek text, see H. Vitelli, ed., Joannis Philoponi in Aristotelis Physicorum libros quinque posteriores Commentaria in Commentaria in Aristotelem Graeca, Vol. XVII (Berlin, 1888), pp. 568-569. The first Greek edition of Philoponus' Physies commentary, published in 1535 by Victor Trincavelli, was followed by Latin translations published at Venice in 1554, 1558, and 1569. In the translation of 1569 by Johannes Rasario (Joannes Grammatici, cognomento Philoponi, in Aristotelis Physicorum libros quattuor explanatio, lo. Baptista Rasario Novariensis, interprete), see cols. 339-340 for the relevant discussion. A German translation of the Greek pitssage, and much else from the works of Philoponus, appears in Johannes Philoponus, Grammatikos von Alexandrien (6 Jh. n. Chr.), Christliche Naturwissenschaft im Ausklang der Antike, Vorldufer der modernen Physik, Wissenschaft und Bibel, Ausgewdhlte Schriften ubersetzt, eingeleitet und kommentiert von Walter Bohm (Munich/ Paderborn/ Vienna: Verlag Ferdinand Schoningh, 1967), pp. 92-93. For a brief description of Philoponus' theory of space, see S. Sambursky, The Physical World of Late Antiquity (New York: Basic Books, 1962), pp. 6-7. Sambursky observes that Philoponus' views are virtually identical with those of Strato of Lampsacus (c. 300 B.C.), whose opinions are described on p. 3. Valuable information on the early history of Greek spatial concepts is preserved by Simplicius in the discussion of place in his commentary on the fourth book of Aristotle's Physics (for the Greek edition, see H. Diels, ed., In Aristotelis physicorum libros quattuor priores commentaria in Commentaria in Aristotelem Graeca, Vol. IX, Berlin: Berlin Academy, 1882, pp. 601-645). Since the physics commentaries of Philoponus and Simplicius were not translated into Latin in the Middle Ages, they were not directly influential until the 16th century, when they became highly significant following publication of the Greek texts and Latin translations.

31 Philippe Delhaye, ed., Siger de Brabant Questions sur la Physique. Les Philosophes Belges, Textes et Etudes, Vol. XV (Louvain, 1941), pp. 153-154 (Bk. 4, ques. 7). Although pseudo-Siger of Brabant (the work is incorrectly ascribed to Siger of Brabant) names Averroes as his source, the latter did not attribute to Philoponus the belief that void is "nothing in reality." Thus pseudo-Siger either learned of this elsewhere, or simply assumed it.

32An exception was Pico della Mirandola. In his Examen Vanitatis, first published in 1520, Pico repeats, with approval, Philoponus' argument by declaring that "place is space, vacant (vacuum) assuredly of any body, but still never existing as a vacuum alone of itself. It is like the case of matter, which is something other than form; but, nevertheless, never without form." In the 1520 edition, published at Mirandola, see Bk. VI, Ch. 4, p. 768. The Examen Vanitatis also appears in the Opera Omnia Ioannis Francisci Pici Mirandulae . . . (Basel: Henricus Petrus, 1573; reprinted in facsimile as Giovanni Pico della Mirandola Gian Francesco Pico Opera Omnia [1557-1573], 2 vols., Hildesheim: Georg Olms, 1969), where the




with Aristotle, that the world was a material plenum without actual void spaces, and since he also denied the existence of extracosmic void space,33 his conviction that the world and all its parts are contained in a three-dimensional void which is never actualized seemed to serve no purpose.

B. Material and immaterial dimensions distinguished

But if Philoponus' version of a separate void space won few adherents, his arguments against Aristotle's charge of interpenetration between body and extended space are of interest and may have played a significant role in winning support for the idea of a separately existing space.34 In rejecting Aristotle's concept of place as the two- dimensional concave surface of the containing body in contact with the contained body, Philoponus had declared in favor of the interval between the containing surfaces as the true place of the bodies which came to occupy it; that is, he assumed that the place of all bodies is a three-dimensional void space. To meet Aristotle's criticism that such a dimensional space would, by virtue of its tridimensionality be a body and therefore unable to receive any other body, Philoponus simply denied that three-dimensional void space is a body. In effect he distinguished between material body and immaterial three-dimensional extension; the former is a substance, the latter is not. Although two or more material bodies cannot occupy one and the same place, a material body can occupy an equal empty space with which it coincides. Indeed, Philoponus insists that the void space which a body occupies serves as its volumetric measure. That such a volumetric measure is independent of the things that occupy it, and is therefore a separate empty space, follows for Philoponus from an analysis of the successive occupations of a pitcher by different bodies. For if a pitcher contains air, the dimensions between the pitcher's inner surfaces must not only equal the volume of air but serve as its exact measure. That the dimensions lying between the pitcher's inner surfaces are independent of the air is evident when the air is displaced by water. Now it is the three dimensions of the water that are exactly measured by the volumetric dimensions within the pitcher. Philoponus thus assumes that the fixed volume within the pitcher measures first the air and then the water which displaces it. Since the pitcher's inner volumetric dimensions constitute the measure of successive and different material and extended bodies, the pitcher's dimensions cannot themselves be corporeal, for then two corporeal entities namely, the dimensions of the pitcher and the dimensions of the occupying body-would coincide and occupy the same exact place, which is impossible. Philoponus, therefore, concludes that the place of successive occupants of the interior of a pitcher for example, air and water-is a three-dimensional incorporeal void, which is but part of an absolute, three-dimensional void space that not only contains the entire cosmos, but is coterminous with it. This finite, void space is thus the place and measure of all bodies in the universe. It is obvious, then, that place cannot be a two-dimensional

passage appears in Vol. II, p. 1189. The translation cited above is by Charles B. Schmitt, Gianfrancesco Pico della Mirandola (1469-1533) and His Critique of Aristotle (The Hague: Martinus Nijhoff, 1967), pp. 140-141. Since the Greek edition of Philoponus' Physics Commentary was not published until 1535 and Latin translations even later (see above, n. 30), Pico, who knew Greek well (see Schmitt, p. 9), probably consulted Greek manuscripts directly.

33 For Philoponus' arguments against extracosmic void, see H. Vitelli, ed., Ioannis Philoponi in Aristotelis Physicorum libros quinque, pp. 582-583; Rasario's translation of 1569, col. 348; and Bohm's German translation, p. 95.

34What follows on Philoponus has been drawn from B6hm, Johannes Philoponus, pp. 86-89, 92-93, where the page references to Vitelli's Greek edition are also given.



560 EDWARD GRANT

surface, as Aristotle contended, since a two-dimensional entity cannot function as the measure of three-dimensional bodies.

In Philoponus' cosmos, bodies move in an absolutely immobile, three-dimensional void space. When a body moves, it leaves behind successive parts of that void equal to itself and occupies other parts equal to itself. Although bodies occupy and then depart from successive parts of an absolute void space, the latter remains immobile. By virtue of its absolute immobility, then, no part of void space can be transported elsewhere to occupy another part of void space. But even if one part of cosmic void were admitted into another part, this coincidence of equal voids would no more cause an increase or decrease of the original void than would the superposition of any number of equal lines or any number of equal mathematical surfaces. Thus did Philoponus reply to another of Aristotle's major criticisms against extended void space, namely Aristotle's claim that an infinity of places could coincide if the space between the inner surfaces of a vessel were conceived as the place of the bodies which occupy it.35 For Philoponus, however, it matters not at all whether one conceives of absolute void space as a single, three-dimensional volume or as a series of two or more superposed equal volumes. Since the latter conception must ultimately reduce to the former, the two modes are, in the final analysis, identical. An infinity of superposed equal void extensions differs in no way from any one of them taken singly. Thus Philoponus not only resolved the dilemma of interpenetration involving corporeal and incorporeal extensions, but he also explained how two or more incorporeal dimensions could coincide, thereby countering another of Aristotle's criticisms.

C. The distinction between material and immaterial dimensions applied to void space capable of existence per se

Although Philoponus' arguments on interpenetration were probably influential in the sixteenth century, his conception of vacuum as a three-dimensional extension that is always filled with body and never existent per se was not destined for popularity. It was the second of the voids described by Averroes that emerged as a significant rival to Aristotle's plenum, namely the void which, though capable of receiving body, could also exist independently. The Greek atomists were, of course, the major representatives of this interpretation, which few were prepared to adopt during the Middle Ages. One who did was a Spanish Jew, Hasdai Crescas (c. 1340-1412), who, in a Hebrew treatise titled Or Adonai (The Light of the Lord), believed with Philoponus (whose work he does not appear to have known directly) that our cosmos is a material plenum located in a three-dimensional vacuum. But unlike Philoponus, he assumed that the three-dimensional vacuum extended infinitely beyond our world in every direction.36 Cognizant of Aristotle's argument about the impenetrability of

35See Aristotle, Physics IV 4, 21 lb 19-29 and also Aristotle's report of Zeno's argument in IV 1, 209a 24-26.

36 The Hebrew text with accompanying translation, introduction, and notes has been published by Harry Austryn Wolfson, Crescas' Critique of Aristotle, Problems of A ristotle's "Physics" in Jewish and Arabic Philosophy (Cambridge, Mass.: Harvard University Press, 1929). With an infinite void surrounding our finite world, which is a plenum, Crescas' cosmology is more akin to that of the Stoics than the atomists, although in contrast to the Stoics, Crescas not only allowed for the possibility of the existence of other worlds beyond ours, but eventually conceded that "we are unable by means of mere speculation to ascertain the true nature of what is outside the world . . ." (p. 217). For an explicit statement by Wolfson that for Crescas the vacuum does not exist within the universe, but only outside, see his The Philosophy of Spinoza (Cambridge, Mass: Harvard University Press, 1934), p. 275.




body and dimensional void, Crescas formulated a clear and significant response, one that bears a striking resemblance to that of Philoponus. Crescas insisted that only material dimensions are mutually impenetrable, but that an immaterial dimension, such as a vacuum, could receive and accommodate a material dimension.37 A three- dimensional vacuum could, therefore, receive a material, extended body and function as its place. The bogey man of impenetrability was thus destroyed. To the further Aristotelian argument that if a dimensional vacuum were a place, then, insofar as it is also like a body, it would require a three-dimensional void place, and so on ad infinitum, Crescas argues that only material dimensions require places. Immaterial dimensions are not in anything else and have no need of separate places.38

Thus did Crescas, possibly following in the path of Philoponus, deny that three- dimensional empty space was a body. The problem of the impenetrability of bodies was not applicable to the relationship between a dimensional vacuum and the material bodies it could receive. Crescas, and before him Philoponus, had thus anticipated Pierre Gassendi, John Locke, and others in the seventeenth century who made an infinite three-dimensional void space the basis of a new cosmology. Like his predecessors, Gassendi insisted that incorporeal dimensions, such as volume or space, could exist independently of corporeal dimensions, the only kind acknowl- edged by Aristotle. As representative of the ancient tradition opposed to Aristotle, Gassendi quotes Nemesius who declared that "Every body is endowed with three dimensions. But not everything endowed with three dimensions is a body. For of this sort are Place and Quality, which are incorporeal entities."39 For Gassendi, not only are incorporeal extended spaces infinite and immobile, but, by virtue of their incorporeality, "have no resistance, or can be penetrated by bodies, or as it is even commonly said, can coexist with them; so that wherever there is a body either permanently or transiently, it accordingly occupies an equal part of space."40

Even more explicitly than Gassendi, did John Locke distinguish between corporeal and incorporeal extension. Reacting against the Cartesian identification of space and body, Locke, relying on the concept of clear and distinct ideas, insisted that the idea of space was clearly distinct from that of solidity. Indeed the very controversy about the existence of vacuum was evidence of this. For "it is not necessary to prove the real existence of a vacuum, but the idea of it; which it is plain men have when they inquire and dispute whether there be a vacuum or no; for if they had not the idea of space without body, they could not make a question about its existence; and if their idea of body did not include in it something more than the bare idea of space, they could have no doubt about the plenitude of the world."'4' Assuming the clear difference between void space and body after all, no one would dare deny that God could

37 As Crescas put it, "the impenetrability of bodies is due not to dimensions existing apart from matter, but rather to dimensions in so far as they are possessed of matter." Wolfson, Crescas' Critique, p. 187.

38 Ibid. 39 Milic Capek, ed., The Concepts of Space and Time, Their Structure and Their Development (Dor-

drecht/ Boston: D. Reidel, 1976), p. 91. The section from Gassendi in Capek's volume (pp. 91-95) is drawn from Gassendi's Syntagmaphilosophicum, Physica, Sectio I, Liber 2: "De Loco et Duratione Rerum," in Opera Omnia (Florence, 1727), Vol. I, pp. 162-163, 170. The translation is by Capek and Walter Emge. In the recent facsimile reprint of Gassendi's Opera omnia (Lyon, 1658; Stuttgart/ Bad Cannstatt: Friedrich Frommann Verlag, 1964), see Vol. I, p. 182. Nemesius was the bishop of Emesa and lived around 400 A.D.

His treatise in Greek, "On the Nature of Man," from which Gassendi has quoted, was translated into La- tin in the 12th century and was widely read.

4OCapek, Concepts of Space and Time, p. 93. 41An Essay Concerning Human Understanding, Bk. 2, Ch. 13, par. 24 in J. A. St. John, ed., The

Philosophical Works of John Locke, 2 vols. (London: George Bell and Sons, 1905, 1903), Vol. I (1905), Bk. 2, Ch. 13, par. 24, p. 296. See also par. 22, p. 295 for a similar statement.



562 EDWARD GRANT

annihilate a body at rest and thereby leave behind a total vacuum42-Locke laments the exclusive application of the term "extension" to body. Those who conceive body as "pure extension without solidity" must then designate a vacuum as an extension "without extension," "for vacuum, whether we affirm or deny its existence, signifies space without body."43 Such absurdities might have been avoided, Locke observes, if only separate terms had been employed for matter and space, for example, if "the name extension were applied only to matter, or the distance of the extremities of particular bodies, and the term expansion to space in general, with or without solid matter possessing it, so as to say space is expanded and body extended."44

Despite arguments of the kind just described, Crescas, operating within a Hebrew tradition, was one of the very few, if not the only one, in the European Middle Ages to reject Aristotle's identification of void with body, an identification which, left unchallenged, allowed Aristotle's supporters to argue that a body could not occupy a void space because the former would be unable to penetrate the latter. The doctrine of the impossibility of the interpenetration of bodies was thus ready at hand to render acceptance of a separate, three-dimensional void space untenable.

D. Is motion possible in a void space equated with body?

And yet, by a curious irony, Aristotle's identification of extended void with body would be turned against one of his major arguments denying the existence of void, namely that motion in a vacuum would be instantaneous.45 The basis of the medieval Latin reaction against Aristotle was Avempace's argument, as transmitted by Aver- roes, that motion in a resistanceless medium, such as the celestial aether, must be finite and successive, as evidenced by the planetary motions themselves. Averroes, who disagreed with Avempace's conclusion, equated motion through a resistanceless medium with motion in a vacuum and therefore attributed to Avempace the opinion that "as every motion involves time, that which is moved in a void is also necessarily moved in time and with a divisible motion."46

In the description of Avempace's argument, Averroes had not included any justification for the finitude and successiveness of motion in a void. It remained for Latin scholastics to supply the deficiency. Two of the earliest discussants were Roger Bacon and Thomas Aquinas, who formulated an argument that came to be called the distantia terminorum or incompossibilitas terminorum. Adopting Aristotle's assumption that a three-dimensional vacuum is like a body, or material medium, with separate and distinct termini, motion through a vacuum was conceived as if it were in a plenum. Thus if motion in a plenum is temporal and successive, so also would it be

42 In effecting this result, God is first assumed to have caused all motion of the material universe to cease. See Essay, Bk. 2, Ch. 13, par. 22, pp. 294-295. To demonstrate the existence of vacuum, Locke also appealed to our experience of motion (par. 23, pp. 295-296).

43Ibid., par. 22, p. 295. 44Ibid., par. 27, p. 298. 45After arguing that zero bears no ratio to a number, Aristotle insists that "Similarly the void can bear no

ratio to the full, and therefore neither can movement through the one to movement through the other, but if a thing moves through the thickest medium such and such a distance in such and such a time, it moves through the void with a speed beyond any ratio." Physics IV 8, 215b 20-23 in the Oxford English translation by Hardie and Gaye.

46Averro6s cites Avempace's arguments in his Commentary on the Physics, Bk. IV, Comment 71 in Aristotelis opera cum Averrois commentariis, Vol. 4, fols. 160r, col. 2-160v, col. 1. Averroes' Comment 71 has been translated in its entirety by John E. Murdoch and Edward Grant in Edward Grant, ed., A Source Book in Medieval Science (Cambridge, Mass.: Harvard University Press, 1974), pp. 253-262. For Avempace's argument and Averro-s' remark, see pp. 256-257.




temporal and successive in a vacuum, rather than instantaneous as Aristotle had argued. The rationale for motion in a void is found in the idea that a dimension is divisible into parts that must, of necessity, be traversed in sequence. Since a dimensional vacuum is like a body, Bacon, for example, argued that motion through a vacuum would be finite because "every body is divisible and has a distance [or separation] between its boundaries; therefore it has a prior part and another posterior part, and will traverse one part and then another. For this reason, before and after [or prior and posterior] occur within the parts of the magnitude and there will be a before and after in the motion and, therefore, in the time."47

For Bacon, then, as for many other subsequent scholastics, if the existence of a three-dimensional vacuum were assumed, finite and successive motion ought necessarily to occur in it, since a body could not simultaneously occupy any two of the distinct termini of an extended vacuum. The very traversal of the prior and posterior parts of a void dimension necessarily implied a measurable temporal interval. In order to save Aristotle, and reduce to absurdity the possibility of finite motion in a vacuum, one must, Bacon argues, deny dimensionality to the vacuum-that is, insist that vacuum is nothing whatever, neither accident nor incorporeal substance. For if the vacuum is nothing at all, motion would indeed be instantaneous, since no ratio could obtain between a dimensionless nothing and a plenum. If, however, vacuum is assumed to be an extended, separate space, it would be a corporeal substance and motion through it would be finite.48

Convinced that Bacon holds the opinion that a body could move with finite speed in a separate, three-dimensional vacuum, readers of his next question ought to have been shocked to learn that he abandoned this position upon confronting the issue of interpenetration of dimensions. Now he insists, with Aristotle, that if a vacuum had

47As justification for this conclusion, Bacon cites Aristotle, Physics VIII, to the effect that "before and after" in space are the cause of the translation over a space, and that "before and after" in space cause "before and after" in time and motion. Opera hactenus inedita Rogeri Baconi, Fasc. 13: Questiones supra libros octo Physicorum Aristotelis, ed. Ferdinand M. Delorme, O.F.M. with the collaboration of Robert Steele (Oxford: Clarendon Press, 1935), Bk. 4, p. 234. It was more customary to leave the crucial analogy between a three-dimensional vacuum and body only implicit, as Thomas Aquinas did. For my translation of the relevant passage from Aquinas' Commentary on Aristotle's Physics, see Grant, Source Book, p. 334 and also Grant, "Motion in the Void and the Principle of Inertia in the Middle Ages," Isis, 1964, 55:268-271. Although Bacon cites the eighth book of Aristotle's Physics as a justification for relating space and time (perhaps VIII 1, 251b 10-12 is intended), a more relevant passage in the Physics from which the distantia terminorum argument may have been derived is IV 11, 219a 14-19, where, in considering the nature of time, Aristotle declares that "since 'before' and 'after' hold in magnitude, they must hold also in movement, these corresponding to those. But also in time the distinction of 'before' and 'after' must hold, for time and movement always correspond with each other" (Hardie and Gaye). In his discussion of the distantia terminorum, Walter Burley cited this very passage to justify motion in a vacuum, declaring that as "the Philosopher says in his treatise on time that before and after in time are taken from before and after in magnitude...." Burleus super octo libros Phisicorum (Venice, 1501; reprinted as Walter Burley, In Physicam Aristotelis expositio et quaestiones, Hildesheim: Georg Olms, 1972, fol. 116r, col. 2.). It is obvious that where Aristotle links magnitude, or distance, and time he intended to apply this relationship only to motion in a plenum and not to any dimensional vacuum, the possible existence of which he emphatically denied. By treating a hypothetical three-dimensional vacuum as if it were a body, medieval scholastics such as Bacon and Burley quite naturally extended the argument to empty space. For other possible origins of the distantia terminorum argument see Grant, Source Book, p. 334, n. 4. A clear description of the incompossibilitas terminorum in Albert of Saxony's Questions on the Physics appears in Grant, Source Book, p. 273, n. 6.

48Bacon, Questiones supra libros octo Physicorum Aristotelis, pp. 234-235. Like most scholastics, Bacon rejected the actual existence of a three-dimensional void and assumed that vacuum was an absolute "nothing," lacking dimensions and attributes or properties. But the assumption of a hypothetical, separate, extended vacuum demanded, for all the reasons cited above, that one concede the possibility of finite motion.



564 EDWARD GRANT

dimensions and a cubic body were assumed within it, motion would be impossible.49 Two dimensions, or bodies, could not possibly occupy the same place. In contrast to some other scholastics, who, while they believed it possible that finite motion could occur in a dimensional vacuum, ignored the problem of interpenetration,50 Bacon realized that the validity of the distantia terminorum argument depended on the possibility of a body being received into an empty dimension. In the end, he appears to have sided with Aristotle and denied that a body and a hypothetical extended vacuum could interpenetrate.

E. The mutual exclusivity of body and space

Apart from distinguishing between corporeal and incorporeal dimensions, thereby denying that an extended vacuum is a body, as Philoponus, Crescas, and others would do, was there perhaps another solution to this perplexing problem? Such a solution was indeed suggested by John Duns Scotus and more fully described, though not subscribed to, by Walter Burley. Ironically, the solution for preserving the possibility of finite motion in an extended void depended on a more extensive and systematic exploitation of the analogy and even identification between body and extended vacuum. For if an extended vacuum is really like a body, or material medium, with respect to divisibility, then just as a material medium yields to a body moving through it, so also ought an extended vacuum to yield to a body moving through it. On this assumption, as well as another wherein the matter surrounding a hypothetical dimensional vacuum would not collapse inward to fill the void which nature so violently abhorred, Duns Scotus allowed that "the motion of a heavy body in a vacuum would be successive because a prior part of the vacuum would yield [cederet] first, and the whole heavy body would traverse that part of space and then this [part]."'51

The concept of yielding, mentioned so briefly here by Duns Scotus, would prove crucial to the proposed solution. But it was Walter Burley, whose works were composed in the generation immediately following Scotus, who actually described, and then rejected, the implications of the latter's approach. In the course of a lengthy commentary on Aristotle's discussion of void space, Burley considers the opinions of those who conceive vacuum as a positive quantity, namely "a quantity separated from every subject and every sensible quantity."52 With respect to such a quantitative vacuum, Burley distinguishes two opinions, which, in effect, defined two different types of separate, immobile vacua. The first kind, which he attributes to certain ancients, was capable of receiving bodies and being coextensive with them, from which it followed that two quantities could occupy the same place simultaneously. The second kind of vacuum, however, was assumed incapable of receiving a body, although it could yield to one, from which it followed that two quantities could not occupy the same place simultaneously.

Although the first kind of vacuum is, by assumption, able to receive bodies, finite

49"Aristoteles arguit sic, si vacuum esset hujusmodi spatium habens dimensionem, tunc erunt ibi plures dimensiones, si ibi ponatur corpus cubicum et debeat moveri, quia illud corpus habet suas dimensiones naturales: quod est impossible; quare nullam translationem habebit in vacuo. Quod concedo. Set negan- dum est quod aliqua translatio ibi posset fieri, imo stant semper dimensiones vacui cum dimensionibus corporis, quare nec subita nec successiva potest ibi fieri translatio." Bacon, Questiones supra libros octo Physicorum Aristotelis, p. 235.

50E.g., Thomas Aquinas; for relevant passages, see n. 47. 5IJohannes Duns Scotus, Opera omnia, ed. cit., Vol. VI, Pt. 1: Quaestiones in Lib. II Sententiarum,

Distinction 2, ques. 9 ("Utrum Angelus possit moveri de loco ad locum motu continuo?"), p. 300. 52Burleus super octo libros Phisicorum, fol. 1 16v, col. 2. Unless otherwise specified, the summary of

Burley's views given below is dra'wn from this same folio.




motion would not be possible between any two distinct points within it. Lacking the capacity to yield to entering bodies, no resistance to bodies would be possible. Consequently, a body placed in such a void would move instantaneously between any two separated points; that is, it would move from one extremity to the other without passing through the middle or any other part of the space.

What Burley reports here reflects the standard medieval interpretation of Aristotel- ian physics: that resistance to a body is essential if finite motion is to occur in either plenum or vacuum. In the plenum of the Aristotelian cosmos, air or water functions as a material resistance to the motion of bodies. But what could serve this function in a vacuum? To resolve this potential dilemma, many, if not most, of those who accepted the distantia terminorum as justification for finite motion in a vacuum identified the dimensions of the void itself as an external "resistance" to the motion of bodies in empty space. For them the very dimensionality of the vacuum served as a resistance to motion and was, indeed, the cause of finite motion in a vacuum, since a body had to traverse the prior parts of its path before the posterior parts, a process which required a measurable quantity of time. Since a major function of any resistance was to produce a finite, te-mporal motion, the dimensions of a vacuum could be conceived as a resistance.53

But Burley, who rigorously adhered to Aristotle's dictum that a resistance to the motion of bodies is essential to finite, successive motion, rejects the distantia terminorum as a resistance that can be attributed to a medium (exparte medii) because, in order to resist, a medium must yield. A distance in and of itself does not possess the property of yielding. Since that property has not been assigned to the first kind of vacuum, Burley insists that a body placed in such a vacuum would move instantaneously.

In turning to the second type of vacuum, Burley admits the greater difficulty in formulating an adequate response. For although supporters of this second opinion also accept the distantia terminorum, the fact that their vacuum yields and does not allow a body to occupy and fill a portion of vacuum equal to itself puts them in agreement with Aristotle's claim that a separate dimensional space would resist the entry of another quantity or body.54 By virtue of yielding, it allegedly offers resistance to bodies that attempt to occupy it. Thus the second type of vacuum can be conceived to function as a material medium, although, as with the first kind of vacuum, its resistance arises, as we shall see, from the concept of the distantia terminorum, or its extension. Because it can be conceived as a medium, heavy or light bodies ought to be able to move successively and finitely through it.55

53In the passage cited above in n. 51, Duns Scotus declares that the successiveness of motion is the extension of the space itself. Thus while Scotus speaks of the "yielding" of the vacuum, he did not attribute the successiveness of motion to it, but assigned that role to the extension of the vacuum. In addition to Scotus, Anneliese Maier, An der Grenze von Scholastik und Naturwissenschaft (Rome: Edizioni di storia e letteratura, 1952), pp. 228-234, also identifies Peter John Olivi, William of Ware, and William of Ockham as among those who conceived of void space as a resistance by virtue of the distantia terminorum; see also Ernest A. Moody, "Galileo and Avempace," Journal of the History of Ideas, 1951, 12: 385-388. Although Albert of Saxony classified the incompossibilitas, or distantia, terminorum as one of seven kinds of external resistance, he subsequently rejected it (see my article "Motion in the Void," p. 282).

54 Aristotle's purpose, of course, had been to show the impossibility of the existence of void, whereas those who supported the view described here were apparently seeking to save the concept of a dimensional void by replacing the idea of the coexistence of body and void with that of the resistance of void to body.

55"Mihi tamen videtur secundum intentionem Philosophi quod talis quantitas separata resisteret alteri quantitati et non permitteret secum aliquam aliam quantitatem corpoream. Et secundum hoc esset dicendum quod grave vel leve posset moveri motu successivo in tali spatio repleto quantitate separata resistenti mobili quia ad motum successivum sufficit resistentia ex parte medii, et quantitas, secundum istam opinionem, resistit quia sola quantitas facit distare, secundum Philosophum." Burleus super octo libros Phisicorum, fol. 116v, col. 2.



566 EDWARD GRANT

To imagine motion in such a vacuum, we must analyze the manner in which bodies move through a material medium. When a body moves through air, for example, the air simultaneously yields and resists it. And since two bodies cannot occupy the same place simultaneously, a portion of air equal in volume to the body is continually displaced. Where air exists, there is no moving body; and where the moving body is located, no air is found. A separate void space that resists and yields simultaneously will operate in much the same manner. Wherever body is located, an equal volume of void space is imagined to have been displaced, so that where body exists, void does not; and where void exists, body is absent.56 Therefore void and any body in the void will never coincide, and two bodies will not occupy the same place simultaneously.57

In this way Aristotle's assumption that space is a quantity, or body, so that space and body, as two quantities, could not occupy the same place,58 was used to win acceptance for an idea that Aristotle had vigorously rejected, namely that motion would be finite and successive in an extended vacuum.

But if supporters of the opinion just described had sidestepped the Aristotelian dilemma of simultaneous occupation of the same place by equal volumes of space and body, their conception entailed a strange physical consequence. If void space yields to a material body, did this not imply that empty space is somehow capable of rarefaction and condensation, that is, of expansion and contraction, in a manner analogous to a plenum? Since every quantity is necessarily rare or dense,59 and since void space is a quantity, presumably possessed of all the properties of quantity, it would seem to follow that extended empty space is capable of condensation and rarefaction!

But how is it possible to assign qualities such as rarity and density to an empty space that was assumed to be separated from all qualities, including rarity and

56What has been described here appears to have much in common with the ancient atomists, who, according to G. S. Kirk and J. E. Raven (The Presocratic Philosophers, A Critical History with a Selection of Texts, Cambridge: Cambridge University Press, 1957, p. 408), "had no conception of bodies occupying space" but "for them the void only exists where atoms are not, that is, it forms gaps between them." The concept of void space yielding to body and being displaced by it was described, and rejected, by William of Auvergne. In his De universo, written between 1231 and 1236 , William declares that a vacuum must be either divisible or not. If not, motion through it would be impossible, just as it would be if air and water were assumed indivisible, "since every body which is moved through another [body] makes a path for itself by division in it or through it." Thus if a void were truly indivisible, it would be impenetrable and be the strongest and most solid of all things, in which event we might well ask: if it has such a nature, in what sense is it a vacuum? But if void is divisible, and a body moves through it, the body will displace, or expel, parts of the vacuum equal to itself. At this point, however, William departs from those who supported the opinion described by Burley. The displacement of vacuum by a body moving through it signifies for William that the displaced void was itself in a void space now occupied by the body which displaced it. William uses this argument in support of Aristotle's claim that if a dimensional void exists, it would need a three-dimensional place, and so ad infinitum. Guilelmi Alverni ... Opera omnia (Paris, 1674; reprinted in facsimile, Frankfurt a.M.: Minerva, 1963), Vol. I, p. 607.

57 In his Questions on the Physics, Bk. IV, ques. 12, John of Jandun denied the possibility that the parts of a vacuum had the capacity to "yield" to a body moving through them. Jandun argued that if a dimensional vacuum existed, which he denied, it would, by virtue of its quantity, be divisible into prior and posterior parts. But a body could move through those parts only if the parts yielded to its motion; otherwise, two three-dimensional entities would occupy the same place simultaneously. Denying the possibility that parts of a vacuum could yield to a body, Jandun rejected the concept of successive motion in a void on grounds that void and body would have to interpenetrate, which is impossible. Questiones Joannis de Janduno de Physico auditu noviter emendate . . . (Venice, 1519), fol. 57r, col. 1.

58To illustrate the bodily nature of empty space, Burley had earlier cited Aristotle's example of a cube space separated from every substance and sensible quality, which, upon being immersed in water, causes the latter to separate and yield to it. Thus the water will be separated by distances equal to the dimensions of the cube. Burleus, fol. 116v, col. 1.

59" ... omnis quantitas est necessario rara vel densa . Ibid., fol. 117r, col. 1.




density? And if extended space cannot possess the qualities rarity and density, it could bear no ratio to a full, material medium, from which it followed that motion made in a void space could bear no ratio to motion made in a plenum. Consequently, motion in such an extended void would be instantaneous.60

Although Burley will eventually reject the claim that finite motion can occur in an extended void which yields to but does not receive bodies, he first attempts to offer some justification for assigning rarity and density, and therefore, resistance, to void space. It was this justification, drawn either from supporters of the second type of vacuum, or formulated by Burley himself, which may underlie his earlier remark that the second opinion was more difficult to respond to than the first. Burley distinguishes two kinds of rarity and density: qualitative and quantitative.6'

If rarity and density are conceived as tangible qualities, then separate, dimensional space will be neither rare nor dense, since void space is assumed devoid of qualities.62 But if rarity and density are taken quantitatively, that is, conceived as parts of a quantity in greater or lesser proximity to one another, then, since every quantity is necessarily rare or dense, it would follow that even a quantity separated from every substance and quality must also be rare and dense. The analogy, nay identification, between extended space and material body guarantees this. It follows, then, that a "ratio of medium to medium in rarity is as the ratio of motion to motion in speed."63 These comparisons are possible because quantity, as an active principle, has the capacity to resist,64 a capacity that arises from the distantia terminorum, that is, from the impossibility that in an extended space a body can be in two distinct places simultaneously.

But how could the contraction and expansion of an extended homogeneous void space be conceived analogously to that of a material medium such as air? Once again supporters of the position under discussion, or Burley himself, responded by distinguishing two kinds of resistances, positive and privative. A positive resistance always involves a reaction of the medium to the body in motion, as when a body penetrates earth or moves against flowing water or air that flows against it.65 Obviously this resistance cannot be applied to a vacuum. But air, or any medium, need not always offer violent, or positive, resistance to a body. For we can imagine that a stone occupies a quantity of air equal to itself so that stone and air coincide in the same place simultaneously. Under these circumstances, the air would, by assumption, offer no positive resistance to the stone. And yet we can say that the air does indeed resist the stone because the stone cannot be in the different parts of the air simultaneously. And this constitutes a second kind of resistance, which was called privative, and is nothing other than the distantia terminorum. Since privative resistance prevents a

6OIbid., fol. 116v, col. 2. 61"Dicendum quod raritas et densitas, seu spissitudo et subtilitas, uno modo sunt qualitates conse-

quentes calidum et frigidum. Alio modo accipiuntur pro approximatione partium quantitatis adinvicem, vel pro elongatione partium adinvicem." Ibid.

62Ibid., fols. 116v, col. 2-117r, col. 1. 63Ibid., fol. 117r, col. 1. 64"Et ideo dico quod quantitas est sic activa quia est resistiva, quamvis non sit activa sic quod sit

alterativa." Ibid. 65After declaring that some believe that a heavy body could fall in a vacuum because of the distantia

terminorum, Burley explains that these same individuals then reply to the charge that such a motion would be impossible because of a lack of resistance in a void by distinguishing two kinds of resistance, "scilicet quedam positiva, et illa est semper cum resistentia et reactione. Et talis resistentia nunquam est sine violentia resistentis, et hoc si resistens vincatur sicut patet quando aliquod corpus penetrat terram, vel movetur contra aquam currentem, vel contra motum aeris." Ibid., col. 2.



568 EDWARD GRANT

body from occupying all places in a void simultaneously, it is concluded that motion would be successive and finite.66

Despite the unfortunate example to convey the notion of privative resistance in a vacuum, that is, locating a stone and an equal volume of air in the same place simultaneously, it is obvious that those who assumed the possibility of finite motion in the second kind of vacuum did not believe that a body moving through that kind of void occupied a volume of void equal to itself. Since they identified extended void with body, this would have been contrary to their assumption that two bodies cannot occupy the same place simultaneously. But it is obvious that the analogy between body and void has broken down. A dimensional void does not yield to bodies as does a material medium; nor does- it expand or contract. For if it did, positive resistance would apply in void as well as plenum. It was only by calling the distantia, or incompossibilitas, terminorum a privative resistance that any sense, however feeble, could be assigned to the concept of resistance in a void.

Although Burley would muster a number of arguments against the idea that a vacuum could offer any resistance to a body moving through it,67 his summary account of the opinion he would reject is of interest as illustrative of a significant attempt to make sense of motion in a void by accepting Aristotle's identification of void as body, as well as the inevitable consequence therefrom that body could not exist simultaneously with void. It was undoubtedly with all this in mind that the property of yielding to body was assigned to the vacuum, while at the same time the capacity to receive bodies was denied it. Thus a body could, in principle, move through a vacuum and not occupy any part of that vacuum equal to itself in volume. Thus it would not coincide with the vacuum anywhere, and two equal dimensions would not occupy the same place simultaneously. Of course, the persistent effort to extend the analogy between void and body resulted in the assignation of properties to vacuum that made its behavior incomprehensible. While associating density and rarity with the vacuum, they failed to relate it properly to the distantia terminorum, which was the sole concept of resistance to which they appealed, a concept which relied on the impossibility of a body being in two distinct and separate places simultaneously.68

66"Resistentia privativa est incompossibilitas aliquorum ad tertium vel alicuius unius ad aliqua diversa. Verbi gratia, posito quod lapis posset esse simul cum aere, sic quod lapis non pelleret aerem extra locum suum secundum quod aer et lapis possent esse simul in eodem loco adequato, tunc aer non resistit lapidi positive quia non oportet aerem violenter moveri ad motum lapidis. Tamen aer secundum diversas partes eius resisteret lapidi privative et illud non est aliud dicere quam quod lapis non posset esse simul in diversis partibus aeris. Et ista resistentia requiritur necessario in omni motu locali recto; et talis resistentia sufficit ad motum localem." Ibid.

67Because he believed that a vacuum could not function as a resistance when it served as a medium (ex parte medii), Burley was convinced that neither of the two vacua he distinguished could offer resistance to motion. But finite and successive motion was yet possible in a vacuum with respect to the body itself (ex parte mobilis), provided the body in motion is a corpus mixtum, or compound-that is, a body in which are mixed contrary light and heavy elements, one serving as motive force, the other as internal resistance. In a further qualification, Burley insists that such is only possible if the elements in the compound retain their respective identities so they might operate as contraries. Ibid., fols. I 17r, col. 2-11 7v, col. 1. For more on mixed bodies and internal resistance, see my article "Bradwardine and Galileo: Equality of Velocities in the Void," Archive for History of Exact Sciences, 1965, 2, 4: 348-351. Herman Shapiro ("Walter Burley and Text 71," Traditio, 1960, 16: 395-404) attributes to Burley the very opinion which he rejected, and thus mistakenly argues that Burley believed that an extended vacuum could resist the motion of a body (pp. 402-403). Shapiro not only fails to distinguish between vacua that yield and those that do not, but makes no mention of Burley's crucial discussion of mixed bodies.

681f proponents of this opinion had abandoned the distantia terminorum argument and insisted instead that a body could not occupy two separate places simultaneously because of resistance generated by density and rarity, they would have had to explain how a homogeneous vacuum could possibly be rare and




Burley's report about the property of yielding and its connection with the problem of the interpenetration of bodies seems to have implied a discontinuous void space, so that where body is, void is not; and where void exists, body does not. The analogy in this conception is that of a body displacing another body, space in this case, much as a stone might displace the air it moved through. Moreover, in yielding, space is said to offer resistance to bodies, not, however, by virtue of the density and rarity attributed to extended space by analogy with a material medium, but only because of the distantia terminorum, by which it is impossible that a body be simultaneously in any two distinct and separate points of an extended vacuum.

F. The nonequivalence of void space and body

In the sixteenth century, a second kind of yielding can be discerned in the significant spatial discussions of Francesco Patrizi.69 Unlike so many medieval scholastics, Patrizi refused to equate extended void space with body;70 and since he insisted that only bodies could offer resistance, it followed that space could not. Since three- dimensional void space offered no resistance to the bodies moving through it, Patrizi assumed that space simultaneously yielded to and penetrated those bodies,7' a clear indication that he considered space to be permanently continuous and homogeneous. For in yielding to a body without offering resistance, space necessarily "penetrated" that body and came to coincide with it rather than being "displaced" by it.

Despite Patrizi's description of space as a place, or locus, when filled with body (in contrast to its designation as void, or inane, when empty of body), space continues to exist in that locus, since it coincides with the body occupying that locus. For Patrizi, then, yielding of space did not signify its displacement by body in the sense that space and body are mutually exclusive, a condition that would imply a discontinuous space. To the contrary, yielding (cessio) implied continuity of space, since in yielding to a body without resistance the space simultaneously "penetrated" the body and coincided with it at every position or place in its path. In Patrizi's conception, the interpenetration of body and space was an essential feature of the cosmos. It posed no problem because space, though dimensional, was not a body, offered no resistance to body, and could therefore coincide and coexist with it.

dense. Since this could not be done, the distantia terminorum, as the obviously more intelligible of the two options, was retained.

691n Ferrara, 1587, Francesco Patrizi published De rerum natura libri IIpriores: Alter de spaciophysico, alter de spacio mathematico. The treatises on physical and mathematical space were republished, apparently with alterations, in Patrizi's Nova de universis philosophia (Ferrara, 1591; Venice, 1593) where they form part of the Pancosmia, the fourth part of the work, which follows Panaugia, Panarchia, and Pamsychia, in that order. From the Venice (1593) edition, Benjamin Brickman translated De spacio physico and part of De spacio mathematica in "On Physical Space, Francesco Patrizi," Journal of the History of Ideas, 1943, 4: 224-245, which I shall cite here. For my earlier discussion of Patrizi, see my article "Place and Space in Medieval Physical Thought," pp. 159-161.

70Patrizi explains that "When, however, it is said that locus is different from the locatum, this is to be taken to mean that every locatum is a body, while locus is not a body, otherwise two bodies will interpenetrate. Hence, locus not being a body, will of necessity be a Space (spacium) provided with three dimensions-length, breadth, and depth-with which it receives into itself and holds the length, breadth, and depth of the enclosed body." Brickman, "On Physical Space," p. 231. On each page of his translation, Brickman specifies the folio and column designations from the 1593 edition.

7lAlthough space is an infinite whole, Patrizi distinguished the space which contains the world and the space beyond the world. He insists, however, that "Neither of these two kinds of Space is a body. Each is three-dimensional. Each can penetrate the dimensions of bodies. Neither offers any resistance to bodies and each cedes and leaves a locus for bodies in motion. And just as resistance (resistentia, renitentia, and antitypia) is the property of a body which makes it a natural body, so is a yielding offered to bodies and their motion the property of each kind of space." Brickman, "On Physical Space," p. 238.



570 EDWARD GRANT

G. The Divine solution

The dilemma posed by Aristotle concerning the penetration or nonpenetration of extended void space and body received at least one other solution, one which involved the Deity and concepts about His mode of extension. I refer here to the assumption of God's omnipresence in an infinite void space beyond the world, a space that was often identified with God's immensity (immensitas). Although that identification had already been made in the fourteenth century by Thomas Bradwar- dine, Jean de Ripa, and Nicole Oresme,72 the issues that are of interest here were either left implicit or simply ignored until the sixteenth and seventeenth centuries when both scholastic and nonscholastic authors found occasion to consider them. Among those who assumed that God's immensity or substance was an infinite void space-and here we must include Francisco Suarez, Pierre Gassendi, Henry More, Otto von Guericke, Benedict Spinoza, Isaac Newton, Samuel Clarke, and Joseph Raphson-some believed that the infinite void was nondimensional, others that it was dimensional; some held that God was nondimensional though omnipresent, while others believed that He was necessarily dimensional if the space which was His immensity was dimensional. Although controversy on these points was significant,73 the association of God with space provided theological solutions for the old problems that were acceptable to all. Whether God and/or space was dimensional or not, no one would have wished or dared to argue that a space that was God's immensity could possibly offer resistance to bodies moving and existing within it, bodies that He Himself had created. Indeed, as God's omnipresent immensity, space could not vanish when occupied by a body, a consequence which prompted the Jesuit commentator on Aristotle, Bartholomaeus Amicus (1562-1649), to insist that space could not be a vacuum, since by definition a vacuum ceased to exist when filled with body. But space remains whether occupied or not.74 Although Berkeley and Leibniz began the assault that would drive God from space by the latter part of the eighteenth century,75 Isaac Newton, in the General Scholium of the second edition (1713) of the Principia, was still a representative spokesman of all who believed that space was a property, quality, or attribute of God when he declared that "In him are all things contained and moved; yet neither affects the other: God suffers nothing from the motion of

72For their opinions, see my articles: "Medieval and Seventeenth-Century Conceptions of an Infinite Void Space beyond the Cosmos," Isis, 1969, 60: 39-60, and "Place and Space in Medieval Physical Thought," pp. 137-167.

73The details of the controversies will be described in a volume I am now preparing on medieval and early modern concepts of space.

74Although Amicus calls both vacuum and space "negations," the former is the negation of the body which fills it-i.e., it does not exist when filled with body-while the latter is a negation that is not destroyed by the reception of a body. In Aristotelis libros De physico auditu dilucida textus explicatio et disputationes in quibus illustrium scholarum Averrois, D. Thomae, Scoti, et Nominalium sententiae expenduntur earumque tuendarum probabiliores modi afferuntur; auctore P. Bartolommeo Amico, Societatis Jesu theologo, Vol. II (Naples, 1629), p. 763, col. 1 (B-C); also p. 761, col. 2 (C), where Amicus declares that "space exists simultaneously with the body filling it" ("spatium existit simul cum corpore replente illud ... ;") and p. 764, col. 1 (B) where he reiterates that in contrast to a vacuum, a space exists whether or not it is filled with body. On p. 766, col. 1 (C), Amicus equates infinite space with the divine immensity ("Ad 4, respondeo spatium illud esse infinitum, quia adequari debet immensitati Divinae"). Although Amicus is not in the group which equated God's immensity with an infinite void space, he did equate the divine immensity with infinite space, which he described as an independently existing negation without positive attributes.

75See Koyre, From the Closed World, pp. 275-276.




bodies; bodies find no resistance from the omnipresence of God."76 With this supernatural solution to the old problem of interpenetration, we may fittingly conclude, since all potential physical difficulties were transcended and resolved.

76Sir Isaac Newton's Mathematical Principles of Natural Philosophy and His System of the World translated into English by Andrew Motte in 1729. The translations revised, and supplied with an historical and explanatory appendix, by Florian Cajori (Berkeley: University of California Press, 1947), p. 545. That Newton understood God to be actually extended in space is evident from the preceding sentence where he declares that "He is omnipresent not virtually only, but also substantially; for virtue cannot subsist without substance." For the Latin text, see Isaac Newton's Philosophiae Naturalis Principia Mathematica, 3rd ed. (1726) with variant readings assembled by Alexandre Koyre and I. Bernard Cohen with the assistance of Anne Whitman (2 vols.; Cambridge: Cambridge University Press, 1972), Vol. II, p. 762. The passage is quoted and discussed by Koyre, From the Closed World, p. 227.



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