bradwardine and galileo: equality of velocities in the void

Bradwardine and Galileo." Equali of I41ocities in the Paid

E D W A R D G R A N T

Communicated bY M. CLAGETT

In one of his m a n y a rguments against the existence of void space ARISTOTLE

insists t ha t one of the absurd consequences associated with e m p t y space is tha t

bodies of different weights would fall wi th equal velocit ies over the same distance,

when, in t ru th , t hey ought to fall wi th velocit ies t ha t are direct ly propor t ional

to thei r respect ive weights. Speaking first about bodies falling in a medium,

ARISTOTLE says (Physics IV.8. 216a .14- -20) :

We see that bodies which have a greater impulse either of weight or of lightness, if they are alike in other respects, move faster over an equal space, and in the ratio which their magnitudes bear to each other. Therefore they will also move through the void with this ratio of speed. But that is impossible; for why should one move faster ? (In moving through plena it must be so; for the greater divides them faster by its force. For a moving thing cleaves the medium either by its shape, or by the impulse which the body that is carried along or is projected possesses.) Therefore all will possess equal velocity. But this is impossible. 1

1 Translated by 1R. P. HARDIE & R. K. GAYE in Vol. II of the Works o] Aristotle edited, by W. D. Ross (Oxford. 1930). In tl~is paragraph how are we to interpret the equali ty of velocities - - are they finite or infinite ? M. COHEN & I. E. DRABKIN (A Source Book An Greek Science, Cambridge, Mass., ,t958, p. 2t7, n. t) maintain that in this passage equality of speed in the void is a consequence of the Aristotelian argument tha t all bodies of whatever size move instantaneoflsly in the void. Now it is true tha t in comparing the fall of .a body in medium and void, ARISTOTLE concluded tha t "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 ra t io" (Physics IV.8.215 b.20--23). But this was only one of a number of absurd consequences derived by ARISTOTLE concerning motion in a void. For immediately after concluding .*his comparison o f motion in medium and void, ARISTOTLE goes on to say tha t "these are the consequences tha t result from a difference in the media; the following depend upon an excess of one moving body over another" (ibid., IV.8.216a.t2--14). I t is precisely at this point that the passage quoted above in the body of this article commences. Here ARISTOTLE wishes to derive another absurd consequence of motion in the void. In this passage the bodies fall in one and the same medium, and ARISTOTLE argues that di/lerences in finite speed arise because a heavier body can cleave a medium :more readily than a less heavy body. But if the same two unequal bodies were to fall in a void where there is no medium to cleave, there would be no reason for ,a heavier body to traverse a given distance more quickly than a lighter body. He therefore concludes that ifi a void all bodies would fall with equal finite speeds, since there is no medium to overcome and no possible explanation for differences in finite speed. Thus only finite, not instantaneous, speeds are involved in this particular

Bradwardinc and Galileo 345

F o r ARISTOTLE, then, this i m p o r t a n t t r u t h was held to be an absu rd i ty , since i t confl ic ted wi th all t h a t he be l ieved t rue abou t local mot ion and the essent ia l role of a m e d i u m in such mot ion. When, in the s ix t een th century , GIOVANNI BATTISTA BENEDETTI and GALILEO, in his De Molu, r e p u d i a t e d ARISTOTLE'S exp lana t ion of na tu r a l mot ion, t h e y offered a r ad ica l ly different exp lana t ion founded upon hyd ros t a t i c principles. One consequence of the i r new approach was t ha t homogeneous bodies of different weight or size would fall in the vo id wi th equal f ini te velocit ies. This is, of course, well known. ~ Much less known, however , is the fact t h a t more than two centur ies earlier, THOMAS BRADWARDINE and others also a r r ived at , and accepted as true, t h e same ant i - Ar i s to te l ian conclusion. Thus, b y an his tor ical accident , w e have been p rov ided w i t h an o p p o r t u n i t y to make an i l lumina t ing compar ison be tween two u t t e r l y different conceptua l approaches t h a t led to the same conclusion. Indeed, i t is not an ex t r avagance to c l a i m tha t the difference in ra t iona le under ly ing these two approaches is r ep resen ta t ive of the difference be tween medieva l physics and the beginning of ear ly modern physics. In th is art icle, I propose to descr ibe and compare these fundamen ta l l y di f ferent pa thways .

I n his Tractatus de proportionibus (1328), Chap te r III, Theorem 12, THOMAS BRADWARDINE asser t s :

All mixed bodies of similar composition will move a t equal speeds in a vacuum, In all such cases the moving powers bear the same proport ion to their resistances. Therefore, by Theorem I of this chapter, all such bodies move at the same speed. 3 argument. - - I t is interesting to note tha t ARISTOTLE'S reasoning and his conclusion are found in EPIClJRtlS, who did not, however, think i t absurd but who had to make adjustments to explain how atoms in the void could collide if, regardless of size, they all fell with equal finite speeds. I t was to avoid this consequence tha t EPicuRus ascribed to his atoms the proper ty of "swerving. °' See COHEN & DRABKIN, Op. cir., p. 2t 5 and n. t , and PIERRE DUI-1EM, Le systkme du monde (10 vols. ; Paris 1913-- t 959), Vol. 8, p. t 04. - - One further question arises. Are the equal finite speeds of bodies falling in the void uniform, accelerated, or average ? ARISTOTLE does not say, al though he was well aware tha t freely falling bodies accelerated. Despite his awareness, no- where does he explicit ly reconcile this observation with his quasi-mathematical dis- cussions of natural fall which are presented as if the velocities were uniform. Whether this uniformity is absolute or merely an average 'of an accelerated motion is unclear. There is, of course, an enormous conceptual difference.

2 See A. KOYR~, t$.tudes Galildennes (Paris, t939), fasc. 1, pp. 52--53; I . E . DRAB- KIN, "G, B. Benedet t i ' and Galileo's De Motu," Acres du dixi~me congrks international d'histoire des sciences (Paris, 1964), Vol. I, pp. 627, 629; and E. J. DIJKSTERHUIS, The Mechanization o/ the World Picture (Oxford, 196t), p. 335.

3 Thomas o[ Bradwardine His Tractatus de proportionibus, edited and t ranslated by H. LAMAR CROSBY, Jr. (Madison, Wis., t955), p. 1 t 7. The Lat in t ex t reads (p. 1 t6) : "Omnia mixta compositionis consimilis aequali velocitate in vacuo movebuntur . Nam in omnibus talibus motores sunt proportionales suis resistentiis. Igi tur (primam huius) omnia ta l ia aequevelociter movebuntur ." For the significance of Theorem I, see below, n. 32. Although this proposition is discussed by CROSEY, who even mentions tha t BRADWARDINE clarified the ambiguities in the Aristotelian view "some two and a half centuries before GALILEO is said to have actual ly gone to the length of dropping things from the tower o f Pisa in order to prove tha t different quanti t ies of the same substance will fall a t the same rate" (p. 45), and is mentioned by Miss ANNELIESE MAIER (An der Grenze von Scholastik and Naturwissenschaft [Rome, t952], p. 241), no proper comparison and evaluation has ye t been made of the s~rikingly different paths tha t led BRADWARDINE and GALILEO to substant ial ly the same conclusion. - - BRADWARDINE was not the only scholastic who adopted this conchlsion. DUI-IEM,

346 EDWARD GRANT :

This brief, bu t remarkable, theorem raises impor t an t quest ions the answers to which reveal d ramat ic developments in medieval physics. W h y , for example, does BRADWARDINE speak of mot ion in the void as if it were commonplace ? W h y does he restrict equal i ty of fall in the void to "mixed bodies," thereby excluding from his proposit ion the fall of "pure e lemental bodies ."? If mixed bodies are capable of mot ion in the void, does any th ing resist or re tard their mot ion ? If so, what might serve such a funct ion in e mp t y space ? Has BRAD- WARDINE, in an ant i -Aris tote l ian mood, merely adopted as true the consequence deemed absurd by ARISTOTLE, namely tha t where a resis tant med ium is lacking, as in a void, bodies of unequa l weight would fall with equal speed ? As we shall see, this is unl ike ly for it is probable tha t another Aris totel ian consequence of mot ion in the void lies historically a t the root of BRADWARDINE'S surpris ing theorem. For ARISTOTLE also insisted tha t " b y so much as the med ium is more incorporeal and less resis tant and more easily divided, the faster will be the movemen t . " ( P h y s i c s I V . 8 . 2 1 5 b . 9 - - t t ) , so tha t a body "moves through the void with a speed beyond a n y r a t i o . ' ' ( P h y s i c s IV.8.215b.23) - - i.e., it would move ins tan taneous ly . I t was in response to this Aris totel ian claim tha t medieval scholastics formula ted answers to the quest ions raised above.

A l t h o u g h JOHN PHILOPONUS (6 th cen tu ry A.D.) 4 and AVEMPACE (d. t138) 5 had argued against ARISTOTLE'S conten t ion tha t mot ion in the void would be

who devoted a special section to a discussion of this very conclusion ("Tousles corps tombent-ils dans le vide avec la m~me vitesse ? - - Les rGponses donnGes a cette question au moyen-Age" in Le syst~me du monde, Vol. 8, pp. 104-- t t2) , believed that those who accepted motion in the void were committed to this conclusion because of their acceptance of a certain proposition in ARISTOTLE'S P h ys i c s (I believe DUHEM is mistaken, bu t this will be discussed below), Despite his generalization, DUHEM could single out only ALBERT OF SAXONy as an avowed supporter of this view (p. 110). Here is what ALBERT says (DUHEM gives only a French translation, omitting the Latin text) : "Eigh th Proposi t ionl Mixed bodies of homogeneous composition are moved equally in a vacuum, but not in a plenum. The first part of this is obvious for this reason: if they are of homogeneous composition the ratio of the force over the total resistance would be the same in the one body as in the other, since there is no resistance except an internal one." (Octava conclusio. Mixta consimilis compositionis equaliter moventur in vacuo, sed non equaliter in plen.o. Pr imum patet ex quo: essent consimilis mixtionis consimilis esset proportio motive potentie super totam resistentiam in uno sicut in alio, ex quo non haberent resistentiam nisi intrinsecam. Questions on the Phys ics , Book IV, qu. 12 in Questiones et decisiones physicales i n s i g n i u m v irorum : Albert i de S a x o n i a in Octo libros phys i corum . . . Th imon i s in Quatuor libros Meteororum ... B u r i d a n i in lib. de sensu et sensato .. . Aris totel is . . . recognitae rursus et emendatae s u m m a accuratione .. . Magi s t r i Georgii Loker t . . . (Paris, t5t8), f:51 r.~ c.2,

COHEN • DRABKIN, op. cir., p. 2t8. 5 Knowledge of AVEMPACE'S argument was known in the Middle Ages only through

a report given by AVERROES in the latter 's Commentary on Aristot le 's Physics: To the Latin scholastics this famous passage was known as AVERROES' commentary on Text 71 and can be found in the J imta edition Opera Aristotel is cure Averrois com- mentari is (Venice, 1572), fol. 160r., c.1--160 v., c.2. A complete translation of it was made by ERNEST MooDY in his long article "Galileo and Avempace,"' J o u r n a l o / t h e His tory o / I d e a s , 12 (1951), pp. 184--t86. The Arabic text of AVEMPACE'S commentary on the Physics - - from which AVERROES' quotation was taken - - is extant in a Bodleian manuscript and is discussed by SALOMON PIN~S, ~'La Dynamique d ' Ibn ]3ajja," Mdlanges A lexandre Koyrd, Vol. I: L'aventure de la science (Paris, 1964), pp. 442--468.

Bradwardine and Galileo 347

instantaneous, it is with St. THOMAS AQtlINAS that our story may properly begin. Without specifically mentioning AVEMPACE, THOMAS agrees that motion in a hypothetical void space would be finite, rather than instantaneous. He insists that ARISTOTLE was wrong to assume "that if a motion occurs in a void it would bear no ratio in speed to a motion made in a plenum. Indeed, any motion has a definite velocity Earising~ from a ratio of motive power to mobile - - even if there should be no resistance. This is obvious by example and reason. An example is that of the celestial bodies whose motions are not impeded by anything, and yet they have a definite speed in a definite time. An appeal to reason is this: just as there is a prior and posterior part in a magnitude traversed by a motion, so also we understand that in the motion Eitself~ there is prior and posterior. From this it f011o~s that motion takes place in a definite time."6 Thus motion in the void is necessarily successive and, therefore, finite in duration because void space, like any full medium, is an extensive magnitude. To move from one distinct point to another a body must traverse the parts of the intervening magnitude, or distance, in order of succession, thus necessitating that the parts nearer the starting point be gone through before those further re- moved. 7

But if, as almost universally held, all local motion was the resultant e f fec t of a conjoint action between two entities, one identifiable as a motive force and the other as a resistance, what could serve as resistance in the void after a body has been set in motion ? That is, what could produce motion that was successive and of finite duration, the requirements that any resistance had to fulfill. THOMAS solved this by identifying the dimension, or magnitude, or the body itself - - he called it corpus quantum ~ as resistance, s Others, however, chose to identify the void itself as the resistance. As an extended dimension, the void had parts that could only be traversed successively and in time, provided that the motive force was itself finitel Thus motion did not require a resistant medium to be successive and of finite duration as ARISTOTLE had insisted. Ex- tension was the key to successive motion, not a resistant medium. In its essentials, this view was adopted by PETER JOHN OLIVI, WILLIAM of WARE, DUNS SCOTUS, and others. 9

I t was soon objected, however, that successiveness or succession (successio) alone did not guarantee t h a t motions in the void would be of finite velocity. For it was possible that without a resistant medium an infinite velocity might occur where the distance between two distinct termini would be traversed in an infinitely small

Commentaria in octo libros Physicorum Aristotelis, which is Vol. 2 of Sancti Thomae Aquinatis ... opera omnia iussu impensaque Leonis XlII, P. M. edita (Rome, t884), p. t82, c.2. The translation is mine and is quoted from my article "Motion in the Void and the Principle of Inertia in the Middle Ages," Isis 55 (1964), 269. Another translation can be found in MOODY, "Galileo and Avempace," p. 381.

THOMAS was thought by many scholastics to Oe the Originator of this position which came to be designated as incompossibilitas terminorum or distantia terminorum. See ERNEST MOODY, "Ockham and Aegidius of Rome," Franciscan Studies 9 (1949), 424--425.

8 See my IsSs article, op. cir., p. 270. 9 See MooDY, "Galileo and Avempace," pp. 385--388 and MAIER, ,zl~ tier Greme,

pp. 228--23t.

348 EDWARD GRANT:

t i m e - - rather than in a durationless instant.I° From this standpoint, motion between two distinct termini in the void would be successive, since in an infinitely small time a body would traverse those parts of the intervening void nearer the starting point before it would move through the more remote parts . But an infinite velocity was an impossibility. Thus even if motion over an extended distance in the void could be conceived as successive, it would be an absurdity if it were also of infinite speed. Some means had to be found to guarantee that hypothetical motions in the void would not only be successive but also of finite speed. Further- more, the argument that motion in the void would necessarily be successive and finite was a kinematic one expressed in terms of distance and time. But how could such motion be justified dynamically and yet meet the Aristotelian requirements that every violent motion be the effect of a conjoint action between a mot ive force and a resistive power, and that every natural • motion be made against a resistant medium. I t appears that in responding to these problems, the concept of internal resistance n was formulated and applied to mixed bodies falling naturally in a void. 1. Let us examine first the medieval • notion of a mixed body.

A mixed body was one composed of various proportions of two or more of the four Aristotelian elements, is in such a combination, ARISTOTLE had held that one of the elements would dominate and thereby determine the natural motion of the whole body m i.e., whether it would naturally rise or fall. lsa The scholastics, however, altered this i •No longer did the predominant element determine the upward or downward direction of motion, 14 but the light and heavy elements in the mixed body were assigned degrees or parts which upon summation would reveal whether the heavy or light predominated and thus determine the direction Of natural motion. To use Miss MAIER'S example, 15 let us assume that a mixed body consists of three parts 'or degrees of fire, two of air, two of water, and two of earth. Although fire predominates in the mixture, this tells us nothing about the behavior of the body until we know its location. If it is found in air, for example, fire will be a light element and water and earth heavy elements. Because an element in its natural place lacks all inclination for mot ion and is

• lo MAI.ER, A n der Grenze, pp. 249---250. n Where, when, and by whom this concept was first introduced has not yet been

determined. MAIER observes (An d.er Grenze, p. 240) that BRADWARDXNE already presupposes it and utilizes it in a number of propositions of his Tractatus de proportionibus.

1, In violent motion in the hypothetical void, impetus, or an impressed incorporeal force, was invoked by some to account for the continuance of the motion when the projectile lost contact with the projector. The weight of the body served as the • resistance. See my article in Isis, p. :274.

is If only two or three elements were combined, the mixed body was called a mixture imperfectum. (MAIER, A n der Grenze, p. 244.) For ARISTOTLE all mixed or compound bodies are composed of all four simple elemental bodies (De Generatione et Corruptione, II.8.334b.3t--335a.24).

18a De Caelo 1.2.268b.30~-269a.6. 14 As Miss MAIER explains (An der Grenze, p. 237), the scholastics were disturbed

by the fact that one and the same piece of wood fell in air but rose in water. If the wood had a predominant element, it clearly did not determine the natural motion of the body, for no single element could rise naturally, in water and fall naturally in air.

15 Ibid:, p. 238; for a thorough discussion of mixed: bodies see pp. 236--246.


therefore deemed neither heavy nor light, the elemental air in the mixture is inactive and neutral in its own natural place. Our mixed body will, consequently, m o v e downward~ s~nce the sum of the degrees of the heavy elements is four, while that of the light element, fire, is only three degrees. If, however, this same mixed body were found in water, the body would rise, since the sum of fire and air is five degrees or parts, while that of earth is only two degrees. In i ts own natural place, the water would be inactive. On the basis of Miss MAIER'S example, it appears that as long as this particular body retained its identi ty, it would move perpetually upward from water to air and then down again to water as it became alternately relatively light .and heavy. Thus, as Miss MAIER observes, the location of the body determines the relative heaviness or lightness of a mixed body.1, s

From this it was an easy step to the concept of internal resistance. Since the heavy and light elements must, by their Very natures, move in contrary directions, the greater number of degrees of one or the other came to be designated as a motive force and the lesser number as a resistance. The inclination to do this probably arose from the practice of arbitrarily assigning numbers of degrees to the heavy and light elements and subsequently summing each category. If, now, two mixed bodies are compared such that in one of them heaviness predominates over lightness by eight to three, and in the second body by eight to five, it seems reasonable to assume that in the same external medium, the body with the fewer degrees of lightness will fall with the greater velocity. I t would seem natural to account for this by assuming that the quicker moving body has less lightness, or-internal resistance. On the other hand, two mixed bodies with equal degrees of lightness might vary in their velocities of fall because one had more heaviness than the other. Here then we have an intrinsic or internal force. Thus in a falling body, heaviness would be construed as motive force, and lightness as resistance; in a rising body, lightness would be motive force and heaviness resistance. All this was also compatible with the widely held opinion that a greater ratio of motive force to resistance produced a greater speed. Thus in our example, if F~/R2= { and FI/RI-~- a, then V~>V 1, where F is the force of the heavy elements, R the internal resistance of the light elements~ and V is velocity. Any alteration in the ratio of heavy and light would, of course, correspondingly alter the velocity, and perhaps, even the direction of motion.

The ideas and developments just described must serve as the basis, for our historical understanding of BRADWARDINE'S use of the concept of internal resistance in a mixed body, since he himself apparent ly thought it superfluous to justify the use of what must already have been a sufficiently well known physical doctrine. The introduction of internal resistance was prompted by a desire to explain causally finite motion in the void while remaining faithful to the basic Aristotelian physical principle that locomotion occurs only when there is resistance: to the motion of a bodyY But it was one thing to explain how motion

is Ibid., p. 238. 17 Some of the most dramatic departures from ARISTOTLE that occurred within

the framework of medieval Aristotelian physics derived, first from the internalization of motive force (impetus) and then resistance. Paradoxically, both of these anti- Aristotelian moves were made for the purpose of preserving the fundamental Aristo- telian principle that motion is only maintained and continued by the conjoint action af a motive power and a resistance.

350 EDWARD GRANT :

in the void could be f inite and qui te ano the r to conclude t ha t two homogeneous bodies of di f ferent size and weight would fall in t h a t same vo id wi th equal speed. How d id BRADWARDINE arr ive at th is ProPosi t ion ?18

The answer seems to lie in BRADWARDINE'S app l i ca t ion o f a wide ly accep ted d is t inc t ion be tween in tens ive and ex tens ive measure. F o r example , b y BRAD- WARDII~E'S d a y the d is t inc t ion h a d been made in med ieva l science be tween i n t ens i t y of hea t (or t empera tu re ) and q u a n t i t y of heat , and also be tween specific weight (an in tens ive factor) and gross weight (an ex tens ive factor). 19 I t was th is t y p e of d i s t inc t ion t h a t BRADWARDINE app l i ed to ra t ios of force and re- s is tance in mixed bodies. Concerning res is tance he insists t h a t :

... i t is not inconsistent for the same [agent] to have an identical proport ion qualitatively, i.e. in its power of acting, with both the whole and the par t [of its resistance 3, but not so quantitatively. For although the whole and the par t are unequal in quanti ty, they can however, be equal in the qual i ty of resisting. And therefore just as they do not differ in their qual i ty of resisting, so their movements through media do not differ in qual i ty of motion (which is swiftness and slowness), but ra ther in quan t i ty of motion, which is Einl length or brevi ty of time. 2°

Similar ly , mo t ive forces

. . . can be proport ional to their resistances qualitatively, t ha t is, in their power of acting. From such propor t ional i ty arises equal i ty of motions qualitatively, tha t is, in swiftness and slowness. 21

To arr ive a t equa l i t y of speed for two unequa l bu t homogeneous bodies fall ing in the void, BRADWARDil~E has t r e a t ed force (heaviness) and in te rna l res is tance (lightness) in tens ive ly or qua l i t a t i ve ly r a the r t han quan t i t a t ive ly . Qua l i t a t ive ly , a ra t io of force to in te rna l resis tance, "or h e a v y to l ight , is the same for a n y uni t or q u a n t i t y of homogeneous ma t t e r . W h a t e v e r t h e size of the two unequa l homogeneous m i x e d bodies , in each of t hem the ra t io of force to res is tance per un i t of m a t t e r is equal . Now since "from such p ropo r t i ona l i t y arises equa l i t y O f mot ions qualitativdy, t h a t is, i n swiftness and s lowness," i t follows - - in accordance wi th the un ive r sa l ly accep ted ax iom of med ieva l physics t h a t equal ra t ios of force to res is tance p roduce equal speeds - - t h a t these two unequa l homogeneous bodies would fall w i th equal speeds in the void. aS

18 To most of his predecessors who accepted motion in the void - - and this in-. c ludes PHILOPoNUS and AVEMPACE - - i t was probably self-evident t ha t a heavier homogeneous body would .fall .with propor t ionate ly greater speed than a l ighter body (I exclude EPicuRus, who, as we have seen, accepted equal i ty of speed for unequal homogeneous atoms bu t introduced a swerve in order to produce collisions between a toms and avoid recti l inear fall.

x9 See C L A G E T T , Science of Mechanics, p. 2t 2. 20 CLAGETT'S t ransla t ion (the italics are lxis), ibid., p. 232. The Lat in t ex t is on

p. t | 8 o f CROSBY'S e d i t i o n . 21 CLAGETT'S translation, ibid. In CROSBY'S edition, the Latin. tex t is on p . t t8 . 22 We must now consider DUHEM'S explanat ion of the fOrmulation Qf this con-

clusion. He insists (Le syst~me, Vol. 8, p. t 08) tha t this conclusion was forced upon those who accepted motion in the void because they also accepted a rule enunciated by ARISTOTLE in Book v n (presumably Ch. 5) of the Physics, where 1~ says: " I f a' given force move a given weight a certain distance in a certain t ime and half t h e distance in half the time, half the motive power will move half the weight the same distance in the same time. Let E represent half the motive power A and F half the weight B: then the rat io between the motive power a~ld the weight in the one case


Al though BRADWARDINE operates wi thin the general framework of 'Aristotel ian physics and GALILEO, as we shall see, abandons tha t physics, the m a n n e r in which each concluded tha t unequa l bodies of similar composit ion would fall wi th equal speed in the void is s t r ikingly similar. Each saw fit, for wholly different reasons, to make velocity dependent u p o n intensive r a the r t han extensive" factors. Where BRADWARDINE made a ratio of force to resistance p e r un i t of ma t t e r the de te rminan t of speed, GALILEO was to account for speed of fall in terms of weight per un i t volume. The crucial m o v e common ±o both was to make veloci ty of fall dependent upon an in tensive factor.

Mixed bodies were representat ive of real physical bodies. Bu t the scholastics also considered the fall of pure or simple elemental bodies which were conceived as hypothet ica l abstract ions no t actual ly found in nature . The fall of pure, unmixed elemental bodies such as air, water, a n d ear th (as an absolutely light element, fire could no t possibly fall th rough a ny medium) could occur only in media rarer t han themselves. Free fall in void space was, however, impossible since no dynamic explanatiol i in terms of mot ive force and resistance could account for their fall. 2a While the weight or g rav i ty of a simple heavy body might be taken as a mot ive force, no other ent i ty , ei ther in te rna l or external ,

is similar and proportionate to the ratio in the other, so that each force will cause the same distance to be traversed in the same time." (Physics VII.5.250a.4--9; trans, cir.) Although DUHEM quotes a number of medieval authors who discussed and accepted this rule, he fails to connect it with the single author, ALBERT OF SAXONY, who enunciated the conclusion that unequal homogeneous mixed bodies fall with equal speed in the void. Indeed, as we saw in note 3, ALBERT cites this conclusioll in commenting on Book IV of the Physics - - not Book VII ; furthermore, he makes no mention of ARISTOTLE'S rule. BRADWARDINE, who did not enunciate his conclusion in a commentary on the Physics, also fails to mention ARISTOTLE'S rule - - in fact ARISTOTLE is not mentioned in any context - - in connection with his pllonouncement of the theorem. - - I t is possible, of course, that ARISTOTLE'S rule played a role in all this, bu t it is noteworthy t h a t that rule is concerned with two separate bodies, one a motive force and the other a resistance, whereas the theorem about equality of fall in the void located the motive force and resistance in one and the same mixed body. DOHEM does not hesitate to bridge this gap, however, for he says (p. t09): " . . . mais si l 'on admet que le grave se peut mouvoir avec une vitesse finie dans le vide, en l'absence de route r6sistance extrins~que et parce qu'il poss~de de lui-m@me une r6sistance intrins~que; si l 'on admet qu 'en coupant un grave en deux parties @gales on divise la r6sistance intrins@que, tout comme le poids, en deux parties @gales, on arrive forc@ment ~ cette conclusion : chacune des deux moiti@s d 'un grave homog~ne tombera, dans le vide, avec la m@me vitesse que le grave tout entier." But there is no evidence - - and DUHEM really offers none - - to link ARISTOTLE'S rule with the conclusion about •equality of fall in the void. But in virtue of the two quotations above from BRADWARDINE'S Traaatus de proportionibus, there is good reason to believe that he arrived at, or at least justified, this important theorem on the basis of the widely accepted distinction between intensive and extensive factors.

23 KinematicaUy i t mattered not at all whether Bodies moving through void space were mixed or pure. Finite, successive motion would be characteristic of both types. Indeed, those who accepted the incompossibilitas or distantia terminorum (see note 7) as justification for finite, successive motion in the void found it unnecessary to specify the type of body undergoing such motion. But, as we shall see, this indifference was no longer possible when a dynamic explanation was sought. Indeed a curious situation developed for those scholastics who accepted motion in a hypo- theticaLvoid - - the motion of a pure elemental body could be justified kinematically, bu t was impossible dynamically.

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could be i so la ted and ident i f ied as a resis tance. W i t h o u t resis tance to the i r mo t ion in the void , pure e lementa l bodies would move ins t an taneous ly jus t as ARISTOTLE h a d said or wi th an inf in i te veloci ty , which is absurd . ~4 As a consequence, compar isons be tween the veloci t ies of a h e a v y m i x e d b o d y and a pure e lementa l b o d y could be made on ly in the following c i rcumstances : (t) when b o t h fall t h rough ma te r i a l media , and (2) when the h e a v y m i x e d b o d y falls in the vo id and the pure e lementa l b o d y falls in a m a t e r i a l medium.

B o t h of these possible compar i sons were made b y BRADWARDINE in his Tractatus de proportionibus3 s Let A represent a h e a v y mixed b o d y and B a pure e lementa l b o d y as smal l as you please; fur thermore , le t //r be t h e ra t io of heaviness to l ightness o r in te rna l res is tance in b o d y A, and assume t h a t R is the ex te rna l medium. BRADWARDI~E now shows t h a t A, the h e a v y m i x e d body , m a y fall in the same m e d i u m slower or fas ter t h a n B , the pure e lementa l b o d y ; or, indeed, b o t h m a y fall w i th equal speeds. Now if m e d i u m R is raref ied so t ha t B/R>/ /r , t hen when A a c t u a l l y falls in R i t follows t h a t B / R > H r + R so t ha t VB> VA, where V is the speed. Therefore A falls more s lowly t h a n B for the condi- t ions given. ~s B u t now if t h a t same m e d i u m R were condensed to the po in t where

24 The difference in behavior between a heavy mixed body and a simple or pure elemental body is expressed as follows b y ALBERT OF SAXONY: "Seventh Proposition. Although a lead weight would weigh more on a scale than simple [or elemental] water, nevertheless if both were placed in a vacuum the water would be moved, or would descend, infinitely quickly; the lead, however, would descend successively. This is obvious, for al though the lead would have no external resistance, i t would have an internal resistance because i t is a mixed ~ody. Water , however would have neither an internal nor an external resistance." (septima condusio. Quavis massa plumbea ponderaret plus in equilibra quam aliqua aqua simplex, tamen si ambo ponerentur in vacuo, aqua in infinitum velociter moveretur vel descenderet; p lumbum autem descenderet successive. Pa te r hoc quia p lumbum quavis nullam haberet resistent iam extr insecam tamen qum esset mixtum habe re t l e s i s t en t i am intrinsecam [the t ex t has "extr insecam"]. Aqua tamen non habere t resistentiam, nec intrinsecam nec extrinsecam. Op. cir., f.51r., c.2.)

25 Ch. I I I , Theorem XI in CROSBV'S edition, p. t t 4 (Latin) and p. t t5 (Engiish). ,s I quote CROSBV'S translat ion (p. t 1 5) of this .par t icular case. "Let , fo r example,

A represent a heavy mixed body composed of heavy and l ight and having a certain weight, and let B represent some pure heavy body, as small as you please. Now let a given medium be rarified to the point a t which B bears to it a proport ion equal to, or greater than, t ha t of the heaviness t o the lightness in A. Then let both bodies be placed in the same medium. The heaviness of A win now be in a lesser proport ion to i ts to ta l intrinsic and extrinsic resistance than B to i ts resistance. Therefore, by Theorem I, Chapter I I ! , A moves more slowly than B." - - CROSBY'S interpretat ion of this case is quite confused. He observes properly (p. 4t) t ha t "if a mixed body may be represented by lit, then the velocit3r of such a body through a medium

possessing the resistance R, is to be represented as V = / and not V = (~)~' P$

However, in representing this he says (p. 4 t ) :

Let F represent a mixed body (l/r), and

let F ' represent a pure body, and

R ~ F ' F F" let ~-F' = r / . . . . . Then < ~ - , and - ~ - < ~ - and V < V ' "

But if t/r is subst i tu ted for F in FIR, we get (//r)/R which; as CROSBY himself tells us, would violate BRADWARDINE'8 intent. There are, indeed, other deficiencies in his symbolic representat ion of the various cases of this theorem.

]3radwardine and Galileo 353

B/R<//r so tha t when A falls in R we obtain B/R<//r+R, it follows tha t VB<VA SO tha t A falls more quickly than B in medium R. 27 Finally, let medium R be adjusted so tha t once again B/R<]/r.2S This time, however, the difference between the ratios B/R and ]/r is sufficiently less than in the preceding case so tha t they will be equalized when A actually, falls through R, at which time we get B/R----//r+ R, so tha t V B : V A . In this third comparison, then, the mixed and pure bodies fall with equal speed in the medium R.

The same results would be obtained if, initially; A fell in a vacuum (although BRADWARDINE makes no mention of it, the pure elemental body, B, could not possibly fall in the void with a finite speed) and then in material medium R. 29 BRADWARDINE assumes tha t mixed body A falls in the void with speed C. Now let some med ium R be rarified to the point where B, the pure elemental body, will fall in it with a speed equal to, or greater ' than, C. In symbols, B/R >= [/r, so tha t VB=>C. Should A then fall in m e d i u m R, it must necessarily fall more slowly than B, since B/R> l/r+ R. Next, let R be condensed to the point where A ' s velocity in the void, namely C, is sufficiently greater than B's velocity in R (i.e. C>V~) so tha t when A is let fall in medium R it will still have a greater velocity than B 1 despite a certain loss of velocity by A in R. Thus when both B and A fall in R, we have B/R<//r~R, so tha t VB<V a. And, finally, let the medium be condensed somewhat less than in the previous case, where once again C > VB, so tha t when A falls in R its speed is equal to tha t of B - - i . e . B/R= //r + R.

By appropriate adjustments of the external medium, BRADWARDINE was able to produce three different results in his comparison of speeds for a given mixed body and a pure elemental body. Basically, such results were at tainable only because in heavy mixed bodies heavy and light functioned as moti.ve power and internal resistance respectively. Odd as BRADWARDINE'S results ma y seem, it should be noted tha t he varied only the external medium and left constant the ra t io of heavy to light in the mixed body. Such restraint was not exercised by some later authors. For example, ALBERT OF SAXONY was to conjure up situations in which the ratio of force to internal resistance was altered as a heavy mixed body fell t h r o u g h different media. Indeed, by such alterations he was abl6 to demonstrate tha t the natural motion of heavy mixed bodies was

27 I have added a step to BRADWARDINE'S description. He does not specify, as I have, that initially B/R<//r, but this seems implied by the steps followed in the first case. BRADWARDINE says only this: "Conversely, let the medium be condensed to the point at which the propOrtion of B to it is less than the proportion of the heaviness of A to its entire intrinsic and extrinsic resistance. Then by Theorem I of this chapter, A moves faster than B." CROSBY'S translation, p. t.1 5- Clearly, BRAD-

WARDINE assumes that the medium R is condensed sufficiently so that B/R< / and VB<V a. r+R

2s I have again added this step which is not made explicit by BRADWARDINE. 29 The three cases are very briefly summarized by BRADWARDINEL: "Alternately,

let A be supposed to have a determinate speed, C, in a vacuum, and let some medium be rarefied until B falls'in it with spee d C or faster; then A, placed in the Same medium, will fall more slowly than B. Conversely, let the medium now be condensed as required, and the remaining two consequences will follow." CROSBY, ibid. p. 11 5. In representing these three cases I have retained the letter designations used in the previous three c a s e s .

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slower i'n the end than at the beginning which was contrary to the universal opinion, based ,on. experience, that the natural motion of heavy bodies was quicker at the eild than in the beginning, 3° namely that natural downward motion was accelerated motion. Such a result could occur, hypothetically, if the components of a heavy mixed body were as follows: three degrees each of earth and water, and two degrees each of air and fire. Assuming next that the heavy mixed body begins its fall in the sphere of fire, which is arbitraril# assigned a resistance of l, m ALBERT shows that in fire the ratio of force to total resistance is 8 to t, where 8 is the sum of all the degrees of the active elemental components (fire is excluded since, as we have seen, it would have no inclination for motion in its natural place)and I is the resistance of the external medium - - fire in this case. On descending to the natural place of air, the ratio of force to resistance is diminished to 6 t o 3. Here the total resistance consists of 2 degrees of fire as an internal resistance and t degree of external resistance. The motive force is compounded of 3 degrees of water and 3 degrees of earth (air is neutral in its natural place). Since the ratio of force to resistance has diminished, so .also has the downward speed of the body. Obviously, the speed of the body will be less than a t the beginning of its fall. Indeed, ALBERT tells us that the spee(1 in air will be four times slower than in fire; 32

That ALBERT should 'give as the ratio of speeds V~r/Vnre = ~ is itself quite revealing and significant, for it exemplifies another major difficulty that plagued medieval attempts to formulate laws of motion while, at the same time, it offers a significant contrast between ALBERT and BRADWARDINE. In computing the speed relationships, ALBERT is clearly applying to natural downward motior/ the function VocF/R, where V is velocity, F is motive force and R resistance, a3 Thus, in fire F,/R~= ~ a n d in air F~/R~= {, so that V~/V~= ~. I n considering violent motion, however, ALBERT related speeds by another function, an ex- ponential one, fomlulated by BRADWARDINE himself and called by later scholastics a "ratio of ratios" (proportio proportionum) and labeled by modern scholars as

8o "Sed dices quid igitur dicetur de communi dicto scilicet quod motus naturalis est velocior in fine quam. in principio ? Potest dici quod hoc universaliter est verum de motu gravium et levium simplicium; non tamen de motu gravium et levium mixtorum." This appears_ in ALBERT'S Questions on the Physics, edit. cit:, f.~0v., c.t:

st Apparently the resistance of any external medium in which a body fell was assigned a value of t. See MAIER, An der Grenze, p. 9.5t. ALBERT'S example is summarized by MAIER on pp. 250---25t and actually appears in ALBERT'S Questioues de celo et mundb, Book I, qu. I art. 4, fol 86 r, c.2 of the same edition containing his Questions on the Physics, which has been cited in note 3.

3, In another example (summarized b_y MAIER, An der Greme, pp. 25t--252), ALBERT shows that under certain conditions a heavy mixed body could descend more quickly in a plenum than in a. vacut~m. The text appears in his Questiones de celo et mkndo, edit. cit., ft. 86r., c.2-7-86v., c.t ; a similar example is given by ALBERT in his Questions on the Physics, edit. cit,, f. 50v, c.t. (This issummarized in E. GRANT, "Motion in the Void," Isis 85, p.~284.) By the use of internal resistance, such results were easily manufactured.

s~ Most scholastics ascribed this position to ARISTOTLE for.both violent and natural motion. In natural motion, the gross weight of the falling body functioned as the downward motive force, while R would represent the external medium..With the introduction of internal resistance, the total resistance was"the sum of internal and external resistances.


"BRADWARm~'S function," which they often represent as ~ / R ~ : (F1/R1)V*lV~. ~ Had ALBERT applied "BRADWARDI~E'S •function," the ratio of velocities would have been V1/V~= ~ since { = (~-)~. Thus ALBERT used two different "laws of motion" - - BRADWARDI~E'S function for violent motion and the relation V e o F / R for natural • motion. In sharp contrast, BRADWARDINE was consistent and thorough- going, for he applied his function to both violent and natural m o t i o n i n void and plenum, s5 In this he may have been unique. The concept of internal resistance made such a move easy, for with its introduction a motive force and a resistance were conjointly operative in all natural and violent terrestrial motions, whether in a plenum or hypothetical void.

We have now summarized the background that must be presupposed • in order to make sense of BRADWARDINE'S conclusion about "equality of fall;" and we have examined that conclusion itself. Finally, we saw how ALBERT OF SAXOny, exercising little of the restraint show.n by BRADWARDINE,. extended th e concept • of internal resistance to derive consequences contrary to experience. Let us now examine GALILEO'S argument in his essay De Motu ,~ which may possibly represent his earliest extant attempt at sCientific expression. In Ch. 8, GALILEO argues that homogeneous bodies of unequal size and, therefore, unequal weight will fall in a uniform medium with equal speeds. The essence • of his argument emerges in a refutation of the Aristotelian contention that the doubling of a weight results in a doubling of its speed. GALILEO argues that "if we suppose that bodies a and b are equal and are very close to each other, all will agree that they will move with equal speed. And if we imagine that they are joined together while moving, why, I ask, will,they double the speed of their motion, as ARISTOTLE held, or increase their speed at all ? Let us then consider it sufficiently corroborated that there is no reason p e r se why bodies of the same material• should move [in natural motion] with unequal velocities, but every

34 See E. GRANT, "Nicole Oresme and his De proportionibus proportionurn," Isis 51 (t960), pp. 293, 295, and C~-AGETT, Science o~ Mechanics, p. 438. ALBERT employs "BRADWARDINE'S function" in his Questions on the Physics, Book VII, qu. 7, edit. cir., f. 76r., c.1, where he says: "Quantum ad primum sit ista prima suppositio" quod velocitas motus sequitur proportionem potentie moventis ad suam resistentiam ita quod proportio velocitatum est sicut proportio proportionum potentiarum moventium ad suas resistentias ..."

3~ Theorem I, Ch. I I I of BRADWARmNE'S Tractatus de proportionibus is a statement of his function (edit. cir., p. 112 [Latiff] and ttB [English]), In Theorems I I - - X of Ch. Ill., he seems to apply it to violent motion~ while in Theorem Xi (p, t t4 [_Latin] and 115 [English~), cited above, lie invokes Theorem I (where the function is ex- plicitly stated)in comparing the natural fall of heavy mixed bodies and heavy simple bodies, and cites it in support of his theorem that unequal homogeneous mixed bodies fall with equal speeds in the void. Strictly speaking, we should represent their equality

[ -F1 ~ , where --V~V* --1 of fall in the void as ~ = k-~l] . V1 "

3s My quotations will be from I. E. DRABKIN'S translation of De Motu in DRABKIN & STILLMAN DRAXE,. Galileo Galilei "On Motion" and "On Mechanics" (Madison, Wis., t 960)~ Hereafter all references to DRABKIN'S translation -,vill take the for~: DRABKI~, De Motu. Since DRABKIN has supplied in the margins the page numbers from FAVARO'S edition of the Latin text, it will be unnecessary to add references to the National Edition of the works of GALILEO.

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reason why they should move with equal velocity."3~ This conclusion, first formula ted for fall in a medium, is later ex tended to cover fall in the void when GALILEO says tha t "we can show tha t bodies of the same mater ia l bu t of diffe- ren t size move with the same speed in a void." ~s

GALILEO was undoub ted ly led to this impor t an t conclusion by his adopt ion Of the c o n c e p t of effective weight, ra ther t han gross weight, as the u l t imate de t e rminan t of velocity. Bu t the effective weight, or heaviness, of a body in a med ium was, for GALILEO, dependent upon the difference in the specific weights of the body and the med ium th rough which it fell. Hence, it is actual ly a difference in specific weights tha t determines velocities. The free fall of a body m a y be represented as Vo~ specific weight of body - - specific weight o / m e d i u m , where V is speed; the veloci ty of a rising body would be V o~ specific weight of medium - - specific weight of body. 39 Clearly, then, if there is equal i ty between the

87 DRABKIN, De Motu, p. 30. Recognizing that one of the two homogeneous bodies falling in a plenum might be so small tha t its fall would be impeded by the medium, GALILEO offers the following qualification: "Our conclusion must therefore be under- stood to apply to [two~ bodies when the weight and volume of the smaller of them are large enough not to be impeded by the small viscosity of the medium, . " (Ibid. C[, I)RABKIN, "G. B. Benedetti and Galileo's De Motu," op. cir., p. 628.) I)RABKIN observes (p. 30, n. 9) tha t BENEDETTI'S proof of the same proposition differs from GALILEO'S since it "involves the severing of connected weights . . . , while GALILEO'S involves the joining of separate weights." The Latin text with French translation of BENEDETTI'S proof appears in KOYRg, Etudes Gatildennes, fasc. t, pp. 52--53.

88 DRABKIN, De Motu, pp. 48--49. GALILEO'S final and correct version of this important proposition appears in the First Day of the Discourses. There, all bodies of whatever composition and specific weight fall with equal uniformly accelerated speeds in the void, bu t not in a plenum. See Discorsi e Dimostrazioni matematiche intorno a due Nuove Scienze in Le Opere di Galileo Galilei (Florence, 1898), Vol. 8, pp. 117, 1 t9. For an English translation see Two New Sciences by Galileo Galilei, translated by HENRY CREW • ALFONSO DE SALVIO (NewYork, t9t4), pp. 72, 74.

39 "Therefore, the body will move in a void in the same way as in a plenum. For in a plenum the speed of motion of a body depends on the difference between its weight and the weight of the medium through which it moves. And likewise in a void [the speed off its motion will depend on the difference between its own weight and that of the medium. But since the latter is zero, the difference between the weight of the body and the weight of the void will be t.he whole weight of the body. And therefore the speed of its motion [in the void~ will depend on its own total weight. But in no plenum will it be able to move so quickly, since the excess of the weight of the body over the weight of the medium is less than the whole weight of the body. Therefore its speed will be l e s s than if it moved according to its own total weight." DRABKiN, De Motu, pp. 45--46. Throughout this passage "weight" is to be taken as "specific weight." In the very next paragraph (p. 46), GALILEO explains effective specific weight when he remarks " tha t in a plenum, such as tha t which surrounds us, things, do not weigh their proper and natural weight, bu t they will always be lighter to the extent that they are in a heavier medium. Indeed, a body will be lighter by an amount equal to the weight, in a void, Of a volume of the medium equal t o the volume of the body." - - Even in the l a t e r Discourses, GALILEO retained the law that the velocity of a body is determined by the difference in its specific weight and that of the medium. Although GALILEO fully recognized that natural downward motion is uniformly accelerated and takes this into account when he introduces the concept of terminal velocity (Le Opere, V01.8, p. t t9 ; Two NewSciences, tr. cir., p. 74), all of his numerical examples in which velocities are compared by differences i.n specific weights of body and medium assume uniform downward speed and are almost identical in form with those given many years earlier in De Motu.


speCific weights of two bodies of unequal size, they must fall with equal speeds in the same medium or in the void.

Thus GALILEO was able to reach the same conclusion as did BRAD'~VARDINE long before him. This is not mere coincidence but stems from the fact that both men used intensive, rather than gross or quantitative factors. Where BRAD- WARDINE employed a ratio of force to resistance per unit .of matter, GALILEO used weight per unit volume. Equality of the respective intensive factors in bodies of similar composition but unequal in size would produce equality of speed. But leaving aside this quite significant similarity, however, one is struck even more by the basic dissimilarity in the conceptual apparatus ~. Despite the formulation and adoption of many non-Aristotelian positions, BRADWARDINE is essentially an Aristotelian on the fundamental principles of physics. With GALILEO we have moved outside of the framework of medieval Aristotelian physics. I n fact, we have moved into early modern physics.

Where BRADWARDINE adhered to the notion of heaviness and lightness as absolute properties of bodies and employed them in mixed bodies as contraries operating as motive force and internal resistance, GALILEO has completely abandoned absolute heaviness and lightness. He insists that everything has weight and that things are only less heavy, equal to, or heavier than other things; ~/oid alone is weightless - - and it is not a substance. 4° Adopting the notion of relative density, GALILEO utilized it in its more precise form of specific weight. While the tree weights of bodies could onlY be determined in the void, *1 their effective weights in a given medium could be determined by their specific weights,

With the abandonment of absolute heaviness and lightness and the adoption of relative density and specific weight, GALILEO could avoid many of the strange consequences, distinctions, and explanations that were unavoidably associated with the use of th~s pair of contraries. Where BRADWARDINE and medieval- Aristotelians explained direction of motion in terms of absolute lightness and heaviness functioning as motive qualities, GALILEO relied on the relation between the specific weight of body and medium. No longer was it necessary to distinguish the behavior of simple or pure elemental bodies from that of mixed bodies. The distinction forced upon most medieval physical speculators that only certain bodies (mixed bodies) could fall with finite speed in hypothetical void space

4o GALILEO devotes Chapter i2 to argue "in opposition to Aristotle ... that the absolutely light and the absolutely heavy should not be posited; and that even if they existed, they would not be earth and fire as he believes." DRABKIN, De ,~Iotu, p. 55. If there could be a heaviest thing it would not be called heaviest "except in comparison with other things .which-are less heavy, since the heaviest cannot be defined or conceived except insofar as it lies below the less heavy; and in the same way . . . . it is impossible for anything-to be called lightest, except in comparison with things which are less light and above which it rises; and ... the lightest substance is not that which lacks all weight - - for this is void, not some substance - - but that which is less heavy than all other substances that have weight." Ibid., p. 60. In fact, even on the basis of relative weight, GALILEO. denied that there could be a "heaviest" or "lightest" body. He concluded that "it is clearly absurd to posit an [absolutely]. greatest lightness or heaviness. For just as, when any speed, howeVer great, is assumed, another speed greater than it can b e assigned; so, when any heaviness or lightness, however great, is assumed, another greater than it can be assigned." Ibid., pp. 117--118.

41 DRABKIN, De ~rotu, p. 62.

358 EDWARD GRANT"

and tha t all others (pure bodies) could not became u t te r ly meaningless in the physics of De Motu. For GALILEO, whose most fundamenta l concept in De Motu is specific weight, all bodies are t reated alike, regardless of composition, and all bodies would fall in void and p l e n u m . Indeed with GALILEO, homogeneous Archimedean magni tudes replaced the simple and heavy mixed bodies of the later Middle Ages. The emptiness of this d i cho tomy in the analysis of mot ion and the rejection of absolute heaviness and lightness rendered the concept of internal resistance meaningless. In ternal resistance, which in the Middle Ages had been invoked to permi t an explanat ion in dynamic terms for finite mot ion .in the void, depended upon the con t ra ry tendencies of elements distinguishable as light and heavy in a mixed body. In downward motion, heavy and light funct ioned as mot ive force and internal resistance respectively; in upward mot ion their roles would be reversed. GALILEO, however, required neither internal nor external resistance for the product ion o f finite speed in a void. In void space, the speed of a falling b o d y would be directly proport ional to its specific weight.

Thus where specific weight provided GALILEO with a consistent s tandard of comparison for the motions of bodies in void and plenum, those who a d o p t e d internal resistance in mixed bodies were led to peculiar and often bizarre con- clusions. In compar ing the fall of heavy and pure bodies BRADWARIOINE ar r ived at the odd results a lready described. And al though BRADWARDINE did not himself consider the case of two homogeneous, bu t unequal, mixed bodies falling in a plenum, ALBERT OF SAXONY did make the comparison concluding tha t in one and the same medium they would not fall with equal speeds (they would in the void), but , depending upon the part icular numbers assigned to their equal ratios of force to internal resistance, either the greater or the lesser homogeneous mixed b o d y might fall with a greater speed. 4~ And as m e n t i o n e d earlier, the same

42 "Eighth Proposition. Mixed bodies of homogeneous composition are moved equally in a vacuum, but not in a plenum. The first part of this is obvious [see note 31 .-. The second part is also obvious. For if there is a mixed body, say A, which has a heaviness as 8 and a lightness as 4, and if there is another mixed body, B, whose heaviness is as 4 and lightness as 2, then A and B are homogeneous. Let the medium be assumed to have a resistance of t. I [now] demonstrate tha t A descends more quickly than B. This is obvious in terms of the greater ratio of motive power to resistance. For by computing both the internal and external resistances, the total resistance to A will be as 5 and the motive power as 8; in B, however, the total resistance is as 3 and the motive power as 4. Now the ratio of 8 to 5 is greater than the ratio of 4 to 3." (Octava conclusio. Mixta consimilis compositionis equaliter moventur in vacuo, sed non equaliter in pleno. Primum pater ex quo ... Secundum pater, ham si unum mixtum cuius gravitas sit sicut octo et levitas ut quatuor; et sit illud A. Dei~ade sit aliud mixture B cuius gravitas sit sicut quatuor et levitas sit sicut duo, tunc A et B sunt consimilis mixtionis. Ponantur ergo ad medium resistens sicut unum'. Probo quod A descendit velocius quam B. Pater quia ex maiori proportione pot~ntie moventis ad resistentiam. Computando enim resistentiam intrinsecam et extrinsecam simul, tota resistentia A est sicut quinque et potentia mofiva sicut octo; in B veto, tota resistentia est sicut tria et potentia motiva sicut quatuor. Modo maior est proportio octo ad qninque quam sit proportio qu~tuor ad tria. Questions on the Physics, Book IV, qu. t2, edit. cir., f. 5tr., c .2- -5tv . , c.t.) VVhether A is greater than, equal to, or less than B is a matter of indifference. Indeed, had ALBERT inifiMly assigned ratio ~- to B and ~ to A, body B not A, would fall more quickly in ~ plenum with resistance 1. Thus regardless of size, one homogeneous body would always fall more quickly than another if its equal ratio is assigned numbers whose difference is

Bradwardine and Galileo. 359

ALBERT OF SAXONY could show tha t a heavy mixed body might fall more slowly at the end of its natural mot ion than at the beginning (see note 30), as well as "demons t r a t e " tha t the very same heavy mixed b o d y could fall more quickly in a p lenum than in a void (see note 32).

We have now seen how BRADWARDINE and GALILEO, despite radically different physical assumptions, arrived at the same basic conclusion about the equal speed of fall for homogeneous bodies of different size. Now it is overwhelmingly likely ~hat GALILEO was unacquain ted with BRADWARDINE'S treatise. But even if he knew it, or had heard of the part icular conclusion we have discussed, it is not likely tha t he would have ment ioned it or tha t he would have cited BRADWARDINE or anyone else by name, for GALILEO rarely ment ioned his megieval predecessors, except, on occasion, to criticize them. Nevertheless, GALILEO was not averse to utilizing laws and e~tplanations tha t had formed an integral par t of the medieval physical tradition. Indeed, his law of fail in which velocity is determined by the difference in specific weight between the falling body and the medium through which it fell was actual ly drawn from a t radi t ion tha t is traceable to AVEMPACE '(d. t t38) and beyond him to the 6 *h century Aristotelian commenta to r JOHN PHILOI'ONUS. 43 Those who followed AVEMPACE in the Lat in West conceived of veloci ty or speed as something determined by the diHerence - - not the ratio, as ARISTOTLE would have it - - between the weight of a b o d y and the resistance of the medium through which it falls. The resistance of a medium is something tha t mus t be sub t rac ted since it serves only to re tard motion. In the absence of a resistant medium they concluded tha t bodies falling freely in the void would have a natural finite velocity.** To m y knowledge, however, no one in the Middle Ages - - not even AVEMPACE - - specified how this subtract ion was to be made. 45 In fact, the b o d y was usually taken in terms of gross weight and the resistance

greater than that which is found between thenumber s assigned to the equal ratio of the other homogeneous body. Obviously, a very small mixed body could be made to fall more quickly in a plenum than a very large mixed body of homogeneous composition. One has only.to assign to the smaller body the equal ratio whose numbers have the greatest difference. I t is painfully obvious how bewildering and devastating were the consequences tha t could be derived from the use of internal resistance based on heaviness and lightness.

43 A thorough study of AVEMPACE'S anti-Aristotelian position is given.by MOODY in "Galileo and Avempace," Journal of the History of Ideas, Vol. 12, pp. t83-- t93. The passage in which PHILOPONUS expresses the law is given in COHEN & DRABKIN, op. cir., pp. 2 t7- -22t .

• 44 Although it is to criticize them, GALILEO does mention some of his predecessors w h o accepted f ini temotion. in a void. In a memorandum, he remarks: "Philoponus, Avempace, Avicenna, Saint Thomas, 'Scotus and others who t ry to maintain tha t motion takes place in time [i.e., not instantaneouslyJ in the void, are m~staken when they assert a twofold resistance in the moving body, viz, one accidental and due to the medium, the' other intrinsic and due to the body's own weight. These two resistances are clearly one, for ~the medium, insofar as it is heavier, both offers more resistance and [by t h a t very fact~ renders the, body lighter." DRXBKIN, De Mogu, p. 50, n. 24. See also MooDY, "Galileo and Avempace," pp. 384--389. : 45 Although MooDY claims that AVEMI'ACE determined velocities by a subtraction of specific gravities (,Galileo and Avempace," pp. t 86, 4t 7), this is unsupported by t h e quotation of AVEMI'ACE'S views in AVERROE~ commentary on ARISTgTLE'S Physics (see note 5). The medieval Latin authors who agreed with AV~MPAC~- did not interpret b;s law in.terms of.specific gravltms. . . . . ~,,':/

360 EDWARD GRANT:

of the medium was characterized by its density. By introducing specific weight as the criterion for measuring the difference between body and medium, GALILEO, and BENEDETTI before him, made this vague law precise and intelligible.

But GALILEO did not stop here. His hydrostat ic law of fall was initially applicable only to bodies falling uniformly in medium or void, since body and medium were assumed t o remain constant. But it was well known th'at free fall was accelerated motion. Therefore, GALILEO - - b u t not BENEDETTI - - extended the concept of specific weight and Arichimedean hydrostatics to account for the forced u p w a r d motion of a body and its subsequent downward acceleration. Here, once again, GALILEO utilizes an explanation that was widely known in the Middle Ages and' which wasbased on the account of HIPPARCHUS asrepor ted in SIMPLICIUS' commentary on ARISTOTLE'S De Caelo. 48 In his interpretation of HIPPARCHUS' explanation of accelerated fall, GALILEO employed the popular medieval physical doctrine o f ' a self-expending, incorporeal impressed force or impetus. 47 In GALILEO'S exposition, the notion of a residual force plays th,~ leading role. Initially, the mover imparts an impressed force to a stone tha t is hurled aloft. As- the force diminishes, the body gradually decreases its upward speed unti l the impressed force is counterbalanced by the weight of the stone at which moment the stone commences to fall, slowly at first and then more quickly as the impressed force diminishes and gradual ly dissipates itself. The acceleration arises from the continual increase of the difference between the weight of the stone and the diminishing impressed force. 4s Other than his ex- plicitness abou t a self-expending impressed force, GALILEO'S account seems not to differ from tha t offered by HIPPARCHUS. But a closer examination will reveal t ha t not only does it differ from HIPPARCHIJS, but it also marks a rather unique interpretation of the incorporeal self-expending impressed force that was employed in the Middle Ages. For GALILEO:S impressed force vd.U be interpreted hydrostatically in an apparent effort to apply the same basic concept tha t was employed so effectively in his explanation Of uniform fall.

GALILEO characterizes his impressed force as if it were a kind of internal hydrostat ic medium possessfng a specific weight greater than tha t of the stone. Indeed, he calls this impressed force "lightness ''49 because it enables a stone,

4~ SIMPLiClUS' commentary on De Caelo was translated into Latin in 127t by WILLIAM OF MOERBEKE. In De Motu, GALILEO claims that he had arrived at HIP- PARCHUS' explanation of acceleration two months before he read about it in SIMPLI- ClUS' commentary (see DRABKIN, De Motu, pp. 89--90). On the basis=of GALILEO'S interpretation, DRABKIN questions (p. 90, n.7) whether GALILEO actually knew, at first hamd, the relevant sections in $IMPLICIUS.

iT Although HIPPARCHUS' explanation was well known, it does not seem that a self expending impetus was actually applied to the case of accelerated free fall in medieval Latin physics. MOODY, however, believes that T~OMAS AQUINAS "in one of his arguments against AVERROES, offers very much the same explanation of de facto acceleration of falling bodies:' ("~alileo and Avempace", p. 384) as did GALILEO with his impressed force theory. Whatever the merit of this comparison, it should be noted that THoMAs rejected the notion of an impressed force (see CLAGETT, Science

o f Mechanics, pp. 516---517.) 4, For GALILEO'S complete description see DRABKIN, D~ Motu, p. 89. a9 Ibid., p. 93-


for example , to become "acc iden ta l l y " l ight 50 and, con t r a ry to i ts na tu r a l down- w a r d mot ion , to rise in a m e d i u m : 1 I t seems t h a t b y ana logy wi th the hydro - s t a t i c re la t ions ob ta in ing be tween a b o d y and i ts ex te rna l medium, GALILEO conceived of his impressed force as a quas i - in te rna l m e d i u m wi th a specific g r a v i t y g rea te r t h a n t h a t of the b o d y and the ex te rna l medium. This seems borne ou t b y his asser t ion t h a t " in the case of th is s tone i ts na tu r a l and intr insic weight is lost in the same w a y as when i t is p laced in med ia heavier t han itself . . . Wood , too, becomes so l ight in wate r t h a t i t cannot be kep t down excep t b y force. A n d yet , ne i the r the s tone nor the wood loses i ts n a t u r a l weight , bu t , on being t aken f rom those heavie r media , t h e y bo th resume the i r p roper weight. I n the same way, a project i le , when freed f rom the pro jec t ing force, manifests , b y descending, i ts t rue and in t r ins ic weight . ''52 Thus, a s tone th rown aloft is e x t r u d e d and pushed u p w a r d b y i ts quas iAnterna l medium, or impressed force, whose specific weight m u s t be assumed grea ter t h a n t h a t of the s t o n e : 3 GALILEO t r ea t s the upwardl3~ p ro jec t ed s tone and the quas i - in ternal med ium as an i so la ted sys tem ac t ing who l ly i ndependen t ly of the ex te rna l medium. This isola t ion endures un t i l the comple te d iss ipat ion of the in te rna l medium, a t which i n s t a n t t h e normal re la t ion be tween s tone and ex te rna l m e d i u m is resumed, and any fur ther descent of the s tone will be wi th a uni form ve loc i ty s4 p ropor t iona l to the difference be tween the specific weights of s tone and ex te rna l medium. Pr io r

50 "But , what is more, they say tha t they cannot Conceive tha t a heavy stone should be able to become light by receiving a motive force from a projector. But this force, since i t is lightness, will indeed render the body in motion light by inhering in it. Yet these same people say tha t it is u t ter ly ridiculous to suppose tha t a stone has become light after its upward motion and weighs less than before. But their judgment of things is not based on a sober and reasonable consideration. For I too would not say tha t a stone, after its [upwArd] motion, has become [permanently~ light. I would s a y ra ther tha t i t retains its natural weight, jus t as the hot glowing iron is devoid of coldness but, after the heat [is used up], i t resumes the same coldness t h a t is its own. And there is no reason for us to be surprise6 tha t the stone, so long as i t is moving [Upward], is l ight." Ibid., pp. 80--81; see also pp. 88--89. In the same way as GALILEO used the notion of "accidental ly" light in his explanat ion of acceleration, so did BURIDAN and others speak of "acc iden ta l ly" heavy in their account of acceleration (see note 57). In my judgment, GALILEO'S use of "accidental ly" l ight is rooted analogically in his use of hydrostat ics to deal with the problem of upward motion.

sx In opposition to ARISTOTLE, GALILEO finally adopted the view tha t all Upward motion was forced or unnatural . Ibid., p . t 17.

sz Ibid., p. 81. sa In GALILEO'S physics of the De Motu upward :mot ion can occur, under two

different conditions. One t y p e of upward motion occurs when the specific weight of a fluid medium is greater than tha t of the body immersed inli t . Here the cause of the upward motion is external resulting from an extruding o~pushing action of the medium on the body (see DRABKIN, De Motu, pp. 22--23, 3S, 39, and t20). A second type of upward motion can occur when the specific weight of a fluid m e d i u m is less than tha t of the body itself. I n this case, however, the cause of upward motion is internal and requires an incorporeal impressed force. Of great interest is the f ac t t ha t GALILEO treats this impressed force as i / i t were an internalized uniform fluid medium whose specific weight, s tar t ing from a certain initial maximum, varies gradual ly to zero. Otherwise, the quasi-internal medium functions precisely as a n external medium.

54 DRABK!N, De Motu, pp. 100--..-101,

3_62 EDWARD GRANT :

to this instant, however, the stone will undergo first an upward deceleration and then a downward acceleration. This results from the continual diminution of the impressed force throughout the upward motion and through part or all of the downward motion. As the impressed force diminishes, the effective weight of the stone gradually increases until it equals' the upward buoyant force of the impressed force acting as an internal fluid medium. At this point the body ceases to decelerate 5s and commences to accelerate downward. As the impressed force continues to diminish the stone will accelerate. Explained hydrostatically, the downward acceleration is determined by the continually increasing difference between the constant specific weight of the stone and the gradlaally diminishing specific weight of the quasi-internal medium. When the impressed force vanishes completely, any remaining portion of the descent will be with uniform speed since the relation will now be one involving stone and external medium, not stone and internal medium.

My 'interpretation of the incorporeal impressed force as a quasi-!nternal medium wi{h a specific weight is wholly compatible with GALILEO'S insistence that the impressed force reduces the effective weight of a body projected up- wards and compatible with the assumed behavior of the incorporeal impressed force which is capable of variation in the sense that it diminishes from some initial hypothetical maximum. Thus GALILEO took a common medieval conception of a self-expending incorporeal impressed force, or impetus, and interpreted it hydrostatically. I t w a s no more a strain on the imagination to conceive an incorporeal impressed force as an entity that could vary quantitatively~ than to conceive of that same entity as a weightless fluid having a hypothetical specific weight that could vary.

In addition to its function of lightening the weight of the moving body, the internal medium also served as a kind Of internal resistance that prevented a

ss When we interpret in hydrostatic terms the relation between the impressed force and the upwardly projected body, we see why it was necessary for GALILEO

• to treat the relation in isolation from the external medium. I1 the external medium were involved, the stone would never rise but rather fall to the ground after projection since the specific weight of the internal medium must of necessity, be greater than that of the exteynal medium in order that the stone be pushed upward. - - Many years later, when GALILEO came to compose his Discourses and completely abandoned his impressed force theory of downward acceleration, his conception of the role of the external medium was drastically revised, for by that time he had arrived at the concept of terminal velocity. The resistive capacity of the external medium now assumed a crucial importance in accelerated fall. As the speed of a downward ac- celerating body increases, the resistance o~ the external medium also increases until "the resistance of the medium becomes so great that, balancing each other, they prevent any further acceleration and reduce the motion o1 the body to one which is uniform and which will thereafter maintain a constant value. There is, therefore, an increase in the resistance of the medium, not on account of any change in its essential properties, but on account of the change in rapidity with which it must yield and give way laterally to the passage of the falling body which is being con- stantly accelerated." Two New Sciences, tr. cir., p. 74 (in Le Opere, Vol. 8, p. t t 9). Where in the De Motu the external medium was ignored while there yet remained any quantity of impressed force - - i.e. while the body accelerated - - in the Discourses impressed force is abapdoned, and the external medium is conceived as actively resistive throughout the acceleration ultimately producing a constant terminal velocity.


body from attaining its normal effective weight in the external medium through which it falls. For GALILEO only a medium could he called a resistance in the proper sense. Referring to an external medium, he says that "the medium insofar as it is heavier both offers more resistance and Eby that very fact~ renders the body lighter. ''~6 Now an impressed force functioning as an internal medium plays the same role as does an external medium - - i.e. it renders a body lighter. On the upward leg of a forced motion, the internal medium must be assumed "heavier" or "denser" than the body it moves. Therefore, not only does it enable a given body to rise hydrostatically, but i t also serves to resist and retard its upward motion. Even on the downward leg of its fall, when the body is denser than its residual internal medium, the internal medium must still be construed as an internal resistance since it continues to make the body lighter, although diminishingly so. In this way, GALILEO'S physics of fall also relied upon an internal resistance. But it was a very different kind than its medieval counter- part. Indeed, one can say that iust as the medieval internal resistance was a consequence of absolute heaviness and lightness, so GALILEO'S was a direct consequence of his hydrostatic interpretation of the problem of fall.

I n truth, however, GALILEO'S explanation of the dynamics of fall was no more successful than were the efforts of his medieval predecessors, some of whom utilized an impressed force, or impetus, theory. 57 Long after the De Motu, t ry as he would, GALILI~O could produce no satisfactory dynamic explanation for bodies undergoing free fall or projectile motion. In the Discourses, published almost at the end of his life, he had to confess that seeking after the cause of accelerated motion was a fruitless endeavor. After describing some explanations that had been proposed by others, his spokesman, Salviati, remarks that "all these fantasies, and others too, ought to be examined; but it is really not worth- while." ~ GALILEO decided that he would rather describe the kinematic properties of real motions than seek in vain after their causes. Like his predecessors who had applied their ingenuity to this enormous problem, GALILEO failed to expla in the cause of dynamic fall; unlike them, however, he knew it.

GALILEO'S contribution to the dynamics of fall, then, is not to be found in specific and positive causal explanations, but rather in his rejection of certain ingrained medieval assumptions which, as long a s they were accepted, were destined to keep dynamics on a fruitless path. His glory must lie in the out- right rejection~ or avoidance, of such traditional ideas and entities as absolute heavy and light, internal resistance based on absolute lightness and heaviness,

®

56 DRABKIN, De Motu, p. 50, n. 24. 57 JEAN BURIDAN and his followers assumed that at every instant the gravity

or heaviness of a body, which initiated its downward fall, also produced successive and cumulative increments of impetus, or "accidental gravity" as it was sometimes called, in the failing body. These successive increments of impetus generated successive and cumulative increments of velocity, thus producing a continuously accelerated motion. See CLAGETT, Science of Mechanics, pp. 561--562. This is, Of course, a radically different use of impressed force than "is found, in GALILEO'S De Morn, where, instead of a continuous accumulation of impressed force, the acceleration arises from a continuous diminution of the impressed force (see MooDY, "Galileo and Avempace," p. 407).

58 Third ~Day of The Two New Sciences, tr. cir.;p, 166 (in Le Opere, Vol. 8, p. 202).

36~ EDWARD GRANT: Bradwardine and Galileo

pure and mixed bodies, the r equ i rement t ha t a mot ive force mus t a c c o m p a n y the b o d y it moves, 59 and t h a t eve ry mot ive force mus t exceed i ts res is tance in order to produce motion. 6° These a c h i e v e m e n t s are b y no means the unique p roduc t of GALILEO'S genius, b u t he emphas ized t hem as never before, and t hey s t and as a t r i bu t e to his g rea t intel lect . In germ or full blown, all of these ant i - med ieva l pos i t ions can be found in the D e Motu.

But GALILEO'S con t r ibu t ions to the dynamics of fall were not solely nega t ive in character . B y founding his law of mot ion on differences in specific w e i g h t s , GALILEO, even more than BENEDETTI before him, in t roduced objec t ive cr i ter ia and theore t i ca l ly measurab le quant i t i es in to the ancient p rob lem of fall ing bodies. He had replaced the vague, i l l -defined, and often Undefined force (virtus or potentia) and resis tance (resistentia) of med ieva l physics wi th .the more posi t ive q u a n t i t a t i v e concept of specific weight . B y ex t end ing and app ly ing hydros t a t i c s and s tat ics , 61 the most h igh ly m a t h e m a t i c a l branches of ea r ly dynamics , to the dynamics of fall, GALILEO gave a new direct ion to an old problem. IdeMly, and where possible, causal exp lana t ions of mot ion should be expressed in the language of m a t h e m a t i c s or in t e rms of ob jec t ive ly quan t i f i ab le cri teria.

59 In De Motu, only the case of horizontal motion does not require an accompanying motive force since it is neither natural nor forced motion (see DRABKIN, De Motu, p. 66). For upward or downward motion GALILEO assumes either an impressed ]~orce or an external medium as the accompanying motive force.

so Ibid. 61 After accounting for the free motion of bodies hydrostat ical ly, GALILEO believed

tha t " the motion of bodies moving natura l ly can be sui tably reduced to the motion of Weights in a balance. That is, the body moving natural ly plays the role of one weight in the balance, and a volume of the medium equal to the volume of the moving body represents" the other weight in the balance . . . " - - "And since the comparisonof bodies in natural motion and weights on a balance is a very appropr ia te one, "we shall demonstra te this parallelism throughout the whole ensuing discussion of natural motionl Surely this will contr ibute not a lit t le to the understanding of the mat te r . " (Ibid., p. 23). The heading of Ch. 9 of De Motu reads (p. 38) : "In which all tha t was demonstrated above is considered in physical terms, and bodies moving natura l ly are reduced to the weights of a balance."

Depar tment of the History and Philosophy of Science

Indiana Univer.~ity "

(Received 2VIarah 30, 1965)

bradwardine and galileo: equality of velocities in the void

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