aristotle, philoponus, avempace, and galileo's pisan dynamics

Aristotle, Philoponus, Avempace, and Galileo’s Pisan Dynamics

by EDWARD GRANT*

In his brilliant monograph, “Galileo and Avempace, The Dynamics of the Leaning Tower Experiment,” Ernest Moody1 describes and analyzes with great thoroughness the dynamics of Galileo’s Pisan period. Two essential concepts dominate that dynamics and may be briefly summarized as follows:

1. The concept of specific gravity which determined the uniform speed of a falling body. Thus the speed of a falling body is equal to V = P - M, where “‘V’ represents the speed or velocity of the motion, ‘P’ the motive power measured by the specific gravity of the mobile body, and ‘M the resisting medium whose resistive power is measured by its spec&

2. The concept of a self-expending, or self-cormpting, impressed force (virtur impressa) employed in the causal explanation of projectile motion and natural downward acceleration.3

In this brief article, I propose to examine only the fist concept pertain- ing to Galileo’s early law of fall and involving the concept of specific gravity, or more precisely, specific weight. The latter expression “is not limited like the modem ‘specific gravity‘ to a comparison of the densities of substance and water,” and although in the fourteenth century it was often synonymous with density, the scholastics defined density “as the quantity of matter in relationship to a given volume, while specific weight was understood as the weight in relationship to a given volume.”4 Since our major concern is with “weight in relationship to a given volume,” the expression “specific weight” will be employed but on occasion, particularly in reference to Archimedes or in quotations, the expression “specific gravity” will appear.

gravity.”2

Department of History and Philosophy of Science, Indiana University, Bloomington. Indiana, U.S.A.

Ccnrnvrw 1965: voL 11: no. 2: pp. ’19-95

80 Edward Grunr

In his examination of the historical antecedents of Galileo’s Pisan law of velocities described in (1) above, Moody takes cognizance of the possibility that Galileo may have borrowed the law V = P - M from Giovanni Battista Benedetti who somewhat earlier had explicitly formulated the law that the speed of fall of a body is proportional to the difference between its specific weight and the specific weight of the medium through which it falls. Recognizing that Galileo was not the first to formulate this particular version of the law of fall, I shall none- theless, for convenience, confine my remarks to Galileo, for as Moody rightly observes, “even if Galileo borrowed from Benedetti, our whole problem recurs when we seek for the origin of Benedetti’s dynamics.”s

In his quest for an earlier and more ultimate source, Moody shows that Galileo’s law was enunciated many centuries before by Avempace (d. 1138), a Spanish Arab, whose views were transmitted to the Latin West in the commentary of Averroes (d. 1198) on Text 71 of Book IV of Aristotle’s Phy.sics.6 Avempace, in turn, was probably indirectly de- pendent for his ideas on John Philoponus, the 6th century Greek Neo- Platonist and Aristotelian comentator.7 It seems that not only was Galileo’s law derived ultimately from Avempace and Philoponus, but it was in fact identical with the law as expressed by his predecessors. Thus after quoting from Avenoes the crucial and sole passage containing Avempace’s theory, Moody comments: “The medium is not essential to natural motion at finite speed, as Aristotle held, because the speed is determined by the difference, and not by the ratio, between the densities of body and medium. For Avempace, as for Galileo in his Pisan dialogue, V = P - My so that when M = 0, V = P.”* Later we are told that “To Archimedes’ laws of equilibrium, Avempace’s theory adds a law of velocities as determined by specific gravities.”g Much earlier than Avem- pace, John Philoponus “rejected the assumption that motion essentially depends on a relation between the moved body and a corporeal medium, or that velocity is determined by the ratio of densities of the body and the medium. Like Avempace, he substituted the law of arithmetic difference, represented by the formula V = P - M, for Aristotle’s law of proportionality represented by the formula V = P/M.”lO It will be argued here that while Galileo was almost certainly influenced by the Philoponus- Avempace tradition,ll his theory, and Benedetti‘s before him, is yet different in a very significant manner. In order to show this, two aspects of Galileo’s theory of fall must be distinguished. First, there is the general

Aristotle. Philoponus, Avempace, and Galileo’s Pisan Dynamics 81

concept that velocity is determined by the difference - not the ratio - between the weight of a body and the resistance of the medium through which it falls. The resistance of a medium is something to be subtracted since it functions only as a retardation to motion. As a consequence, bodies falling freely in the void will always fall with a “natural,” finite velocity, proportional to their weights, since no medium acts to retard their fall. The second aspect declares the precise manner in which the difference between the weight of the falling body and resistant medium shall be measured in determining velocity. For Galileo this is measured by the difference in specific weight between the falling body and the resistant medium. It is the second aspect of Galileo’s theory that shall hold our attention, for while Galileo’s theory is in accord with the views of Avempace and Philoponus in the first aspect, it differs radically in the second, since neither of his predecessors utilized the concept of specific weight as a means of arriving at velocities of falling bodies. The introduction of the concept of specific weight was a significant addition to, and clarification of, “the difference theory” in its earlier vague and im- precise form.

Now although Philoponus and Avempace subscribed to a law of fall represented by V = P - M and Aristotle to a law usually represented as V = P/M (strictly speaking, this is not an accurate reflection of Aristotle’s scattered remarks since natural motion was for him not only acceleratedlz but assisted13 as well as retarded by the air; it is retained here, however, because our interest shall center on the relationship between the weights of falling bodies and the densities of the media through which they fall, and because in the Middle Ages it was one of the standard interpretations of Aristotle’s rules of motion), Moody insists that all three treated both the falling body and the resisting medium in terins of densities or specific weights.14 In fact, as we hope to show, while these three compared, and frequently spoke of, the densities of media, the bodies falling or rising through those media were conceived in terms of absolute weight or lightness, and not in terms of density or specific weight. It is for this reason that their laws of fall were vague and con- ceptually confused. For they were trying to formulate relationships between the densities of media involving quantity of matter and volume and the gross weights of bodies where volume is not a factor. The Aris- totelian doctrine of absolute heavy and light posed great difficulties here, for until this troublesome notion was abandoned the concept of relative

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density - or relative light and beavy - could not be employed to formulate or express velocity in terms of a difference between the specific weights of a falling body and the medium through which it falls. And without this, the most fundamental conclusion of Galileo’s Pisan dynamics - that bodies of unequal size but of the same material or specific weight fall with equal velocities in a uniform medium or in a void15 - was not likely to have been formulated (even Thomas Bradwardine, who, in the fourteenth century, enunciated a similar proposition for homogeneous mixed - but not pure elemental - bodies falling in a void, employed a kindred concept involving intensive versus extensive factors similar to the distinc- tion between specific and gross weight).’6 Indeed, this very consequence itself follows so naturally from Galileo’s law that differences in specific weight determine velocity, that its absence in Philoponus and Avempace constitutes almost prima facie evidence that in their formulations of the V = P - M law, they were not thinking of differences in specific weights. But let us now offer specific evidence and arguments that Aristotle, Philoponus, and Avempace were not concerned with the specific weights of falling bodies, if indeed they had such a concept.

Taking Aristotle first, we see immediately that while he did speak of density, he could have had no clear comprehension of specific weight. The major obstacle here is connected with his insistence that the element fire must be weightless (see De caelo IV.4.311b.27). Now Aristotle’s notion of density could be made compatible with a weightless element, since the density of fire could be thought of as quantity of matter per unit volume and in this sense one might conceive of a ratio of densities relating a unit volume containing a quantity of fire and a unit volume containing any one of the elements having weight (air, water, and earth). But no relationship of specific weights can be rendered intelligible with a weightless element. A ratio of the specific weights of air to fire is un- intelligible involving as it does a certain weight of air per unit volume to a zero weight of fire for the same unit volume. Indeed, the situation becomes even murkier if the unit volume is increased, for this involves a greater quantity of fire and the greater the quantity of weightless fire, the lighter it becomes although its state of weightlessness remains un- changed (this seems a consequence of De caelo IV.4.309b. 13-16 and IV.4.311a 20-22). The concept of specific weight could hardly have played a role in a physics that embraced a weightless element. While Aristotle utilized the notion of quantity of matter per unit volume in

Aristotle, Philoponus, Avcmpace, and Galileo’s Pisan Dynamics 83

his discussions of media (as when he tells us that dense and rare differ “in that the former contains a greater quantity in an equal bulk” [De caelo III.1.299b. 8-91), he tended to ignore the weight relationships between unit volumes.

Although Aristotle was certainly aware that earth was denser than air and water, he would have denied that t h i s in any way explained the fall of a stone through air or water. A stone falls only because it is absolutely heavy. Fire does not rise because it is less dense than earth, water, and air, but because it is absolutely light.17 Indeed, f i e does not even possess weight in its own natural place, so that if it were in its own place and the air below were removed it would not fall or move downward.l* Thus where utilization of a notion of relative density would have required that fire fall if the denser medium below were removed, Aristotle denies the consequence for it was incompatible with fire’s most basic property of absolute lightness. That “P”, in the formula V = P/M, must not be construed as density or specific weight receives further confirmation in Aristotle’s observation that “. . . a larger quantity of gold or lead moves downwards faster than a smaller, and so with all heavy bodies.”lg Thus we see that the greater the quantity of a heavy element possessed by a body, the more absolute heaviness does it have, and the greater will be its downward speed. Indeed the concept of absolute heaviness and lightness must be understood to take precedence even when, in one and the same passage, Aristotle speaks of the relative densities of bodies. In the Physics, he says :

“For just as a cube displaces its own bulk of water if immersed in it, so does it also if the medium be air, only that the displacement is not perceptible to the senses. So, whatever the yielding medium may be, it must (unless it condenses) yield in whatever direction is determined by the natural movement of the intrusive body, always making way underneath it if it be earth, above it if it be fire, or below it or above it according as the medium may be lighter or heavier than the moving body.”20

Even in this passage, where relative densities and volumes of bodies are the paramount consideration, it is the “natural movement of the intrusive body” that determines the direction of motion - and “natural movement” is a function of absolute heaviness and ligbtness, although, to be sure, it is also incidentally compatible with the motion of one elemental body through another in terms of their relative densities - except that, as we have already noted, a fiery body would not move downward even if the

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air below were removed. But the passage just quoted does not represent Aristotle’s conception of natural motion. The displacement of volumes is introduced only for the purpose of setting up a specific argument against the existence of void. The aim of the argument is to show that in void the volume of a wooden cube, or any other body, taken as pure dimension would occupy a volume of void equal to itself. Aristotle concludes that two equal dimensions, namely of body and void, would then occupy the same space and “this is one paradoxical and absurd implication of the void.”21 Thus hydrostatic principles are employed in this passage for special and limited purposes and do not represent Aristotle’s usual way of considering the natural motions of elemental bodies. We have already seen more than enough evidence to show that no reasonably consistent use of hydrostatic principles was possible in Aristotle’s scheme where reliance on the notion of absolute heaviness and lightness, and gross weight, were central and fundamental physical notions. It seems an unavoidable conclusion that “P” must be interpreted as total weight, since velocity is determined, for the most part, by the ability of the gross weight of a body to cleave through a medium.= As already noted, a greater quantity, or gross weight, of gold would cleave a medium more readily and produce a greater velocity than would a smaller quantity or weight of gold.

Aristotle’s insistence that finite velocities in nature are impossible without the presence of a medium was rejected by John Philoponus, who regarded the medium as a hindrance to motion and quite unessential to it. The sole cause of finite downward motion is the natural downward tendency of heavy bodies - a downward tendency that is proportional to the total weight of the body, but not to its specific weight. Since the medium is conceived only as a retarding factor, quite unessential to the motion itself, Philoponus concluded that true natural motion could occur only in a void. The time in which a given distance would be traversed in the void is the original, or true, time for that particular body, since that same body traversing the same distance in a medium would require an additional time proportional to the density of the medium. The emphasis is on the relationship between gross weight - not specific weight - and downward tendency, as Philoponus stresses in this passage:

“And so, if a body cuts through a medium better by reason of its greater downward tendency, then, even if there is nothing to be cut, the body will none the Iess

Aristotle, Philoponus, Avempace, and Galileo's Pisan Dynamics 85

retain its greater downward tendency. . . . And if bodies possess a greater or a lesser downward tendency in and of themselves, clearly they will possess this difference in themselves even if they move through a void. The same space will consequently be traversed by the heavier body in shorter time and by the lighter body in longer time, even though the space be void. The result will be due not to greater or lesser inter- ference with the motion but to the greater or lesser downward tendency, in proportion to the natural weight of the bodies in question. . . ."23

It might now be argued that Philoponus' celebrated observation that two unequal weights dropped from a given height strike the ground at almost the same time may have involved the concept of specific weight. Even if the bodies involved were of the same material - and there is no evidence whatever to indicate this -, it is almost certain that specific weight or gravity played no role.24 In view of the importance of this observation, let us summarize the two interpretations of Aristotle's law of fall that Philoponus gives just prior to the invocation of the observation itself. In his initial representation, Philoponus says that ". . . Aristotle wrongly assumes that the ratio of the times required for motion through various media is equal to the ratio of the densities of the media . . ."25 Philoponus, then, seeks to repudiate the relationship represented by T2 D2 - = -, where Tis time and D is the density of a medium. In an example T1 D1 representing his own interpretations, Philoponus tells us26 that if a stone takes one hour to fall a given distance in the void, it would take a longer time to fall the same distance in water. Assuming that in water the stone falls the same distance in two hours, the additional time - one hour in this case - would thereafter be reduced proportionally as the medium is rarefied. Thus if the water is changed to air and the latter is half the density of water, the total time of fall would be one hour and a half. Should the density of air be successively halved, the additional times would be correspondingly halved. In this example, the successive times of fall, beginning with the time of fall in the water and approaching the time of fall in the void as a limiting case, are representable as 2: l#: I t : 1 i

. . . 1 + -, where q is 3, 4, 5, 6, and so on to infinity.27 Now whether

the body dropped in both media was thought of in terms of specific or gross weight is irrelevant here, but there is certainly no indication that specific weight was intended. As for the media, while Philoponus relates them as a ratio of densities, he says nothing about any relationship be-

1 29

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tween their weights.28 Had he possessed a notion of specific weight, he might have suggested weighing unit volumes of each medium instead of admitting that when densities are employed in representing Aristotle’s law of fall “. . . the fallacy is not easy to detect because it is impossible to find the ratio which air bears to water, in its composition, that is, to find how much denser water is than air, or one specimen of air than another.”29 For this reason, Philoponus found it necessary to resort to another formulation expressed in terms of two unequal bodies falling through one and the same medium (thus eliminating density consider- ations) in a ratio of times inversely proportional to the ratio of weights

- 1.e. . - w2 = - where W is the weight of a body and Tits time of fall. Wi T2

This was verifiable, for if W2 should be twice W1, it follows that W2 should reach the ground in half the time required for W1 to fall through the same height. Since we have already seen that for Aristotle it is the gross weight of a body that is considered when it falls in a medium, there is good reason to assume that Philoponus’ argument also involves gross weight. That is, it seems reasonable to assume that in order to refute Aristotle, Philoponus would adopt the basic conditions inherent in the Aristotelian position - and gross weight is one of those basic conditions.

But even more important is the fact that this devastating observation is introduced in a completely ad hoc manner for the sole purpose of destroying and discrediting the Aristotelian position. It could have played no part in leading Philoponus to formulate his own position as represented by V = P - M. On the contrary, had Philoponus examined the implica- tions of his own important observation, he would have seen that its destructive power made him as vulnerable as Aristotle. This becomes apparent when Philoponus insists that “the same space will consequently be traversed by the heavier body in shorter time and by the lighter body in longer time, even though the space be void.”30 Thus the Aristotelian position that he repudiates as a proper representation of natural downward motion in a medium, is the very one he adopts to represent the

W2 Ti Wi Ta’

For Philoponus, a five pound body will take only one-Mth as long as a one pound body to fall a given distance in the void. Should these same two bodies fall, subsequently, in a uniform medium, some additional

downward motion of two unequal bodies in the void - i.e. - _ - -

Aristotle, Philoponus, Avempace, and Galileo’s Pisan Dynamics 87

time must be added in proportion to the density of the medium. But whatever the time added, by virtue of the retardation caused by the medium the times of fall for these unequal weights could not be equalized. Indeed, the disparity in the times should become even greater, since Philoponus, following Aristotle, believed that a heavier body could more readily cleave a medium - “for that which has a greater downward tendency divides a medium better.”31 In order to reconcile more nearly his own law of fall with the observation that two unequal heavy bodies fall a given distance in almost equal times, Philoponus would have had to assume not only that the additional times are directly proportional to the densities of the media but also to the weights of the bodies. In- stead he assumed an inverse relationship between time and weight since the greater weight “divides a medium better.”

Thus even if Philoponus had specific weight in mind to explain the almost simultaneous fall of two unequal weights, the observation itself could have played no role in formulating his own theory, since that theory is also incompatible with the observational evidence used to refute Aristotle. I have tried to argue, however, that it is implausible to suppose that the concept of specific gravity or specific weight played any part in the law of fall proposed by Philoponus. But should my arguments be unconvincing, I add a final quotation from Philoponus that reveals the incompatibility of the concept of specific weight with the peculiar results that he believed to arise from the addition and subtraction of weights.

“Qualities become more powerful when p@ of the same form act together. This is obvious and can be seen more clearly in the case of weights. When you join together two weights of a pound, the combined weight will be heavier than the sum of the two, for it will not be two pounds, but more. Similarly, when you divide a weight of two pounds into two equal parts, each will not be a pound, but less. Thus things of the same form coming together become more powerful, and being divided they become weaker.”32

Although the view expressed by Philoponus in this passage does not affect his representation of motion in the void - despite the lack of precise additivity of any two weights, the ratio of their times of fall will remain inversely proportional to the weights - it does have the peculiar consequence that when exactly half of a given body is taken its weight is not half of the whole. This would seem to exclude specific weight as a factor, since the addition of two homogeneous unit volumes will always

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result in a total weight more than double the weight of each of the com- ponent volumes. In effect, the specific weight of a unit volume would be less than the specific weight of that same unit volume after it had been added to another equal unit volume.

Turning now to Avempace, we find that while his opinions, as summarized for us by Averroes, are not as explicit as those of Philoponus, they seem, nevertheless, to be closely related to those of his Greek pre- decessor. Avempace insists, says Averroes, “that it does not follow that the proportion of the motion of one and the same stone in water to its motion in air is as the proportion of the density of water to the density of air, except on the assumption that the motion of the stone takes time only because it is moved in a rnedium.”33 And a few lines below, in a direct quotation from Avempace, we read: “For the proportion of water to air in density is not as the proportion of the motion of the stone in water to its motion in air; but the proportion of the cohesive power of water to that of air is as the proportion of the retardation occurring to the moved body by reason of the medium in which it is moved, namely water, to the retardation occurring to it when it is moved in air.”34 If by “motion” (motus) and “retardation” (tarditas accidens) Avempace meant to convey total speed or velocity and loss of speed or velocity respectively, then we could properly represent Avempace’s verbal de-

scription as - v2 = - D1; but if these terms signrfy respectively total time

and additional time required for bodies falling through a medium, then

Ti Di pression proposed by Philoponus. As a formulation for representing a ratio of total velocities or total times Avempace deems it false, but adopts it as a means of indicating correctly a ratio of retardations in media, whether these retardations involve losses of velocity or increases of time.

But if Averroes tells us how Avempace wished to represent additional time, or loss of velocity, for bodies falling in a medium, he fails to report Avempace’s opinion on the manner in which the true time of fall, or true velocity, is to be conceived in the void itself. Instead, he tells us how Avempace explained the difference between the uniform velocities of the different celestial bodies moving through a resistanceless aether (this is as close to the void as we come). This difference “is caused only by the difference in perfection between the mover and the moved. When there-

Vi D2

- _ - Tz - D2 would be appropriate and in agreement with the initial ex-

Aristotle, Philoponus, Avempace, and Galileo’s Pisan Dynamics 89

fore the mover is of greater perfection, that which is moved by it will be more rapid; and when the mover is of lesser perfection, it will be nearer (in perfection) to that which is moved, and the motion wdl be slower.” 35 Although differences in the perfection of celestial intelligences account for the multiplicity of different uniform celestial velocities, no explanation is offered for differences in velocities of terrestrial bodies falling in a hypothetical void. Is velocity directly proportional to weight, or to dimension, or to some indwelling motive power? Whatever Avem- pace’s thoughts on the matter, there is no evidence for asserting that in the void “the essential or ‘natural’ velocities of bodies of diverse nature or density are held to be proportional to the ‘perfection’ of their nature, as to the degrees of their densities.”36 There is here an unwarranted assumption that Avempace considered the motion of a terrestrial body “as an absolute indwelling power of self-motion animating the body like a soul” by analogy with “the movements of the celestial spheres, which were thought to be caused by incorporeal substances called Intelligences, distinct from the spheres moved by them, yet operative on them from the inside, like Idea “motivating” desire. It is this conception of indwelling power, or of operative Ideas giving motion to a magnitude, that Avempace brings down from the Greco-Arab heavens and extends to the dynamics of terrestrial bodies. He thus conceives a single universal dynamics, breaking down the barrier between the heavens and the earth which is so distinctive of Aristotle’s cosmology. Yet this universal dynamics is modelled on the theory of Aristotle’s theological astronomy, of celestial rotations caused by immaterial and intellectual “motives”. Gravity is conceived as mover of a stone toward the center of the world, in the manner that the separated Intelligences were conceived to be movers of the spheres.”37 No part of the passage in Averroes suggests a universal dynamics of the kind assumed by Moody for Avempace.38 But if there is little jusacation for seeing an analogy between Avempace’s terrestrial dynamics and the motion of the celestial spheres, there is even less reason to link this “indwelling power of self-motion”, or degree of “perfection”, of a terrestrial body with its density. Talk of the density of media was common enough, but it would be striking if Avempace interpreted the bodies falling in those media in terms of density or specific weight. Neither Averroes, who rejected Avempace’s position, nor the medieval Latin authors39 who adopted Avempace’s criticism of Aristotle ever understood him to have held that velocities in the void

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were proportional to densities, or that velocity in a plenum is determined by a difference in specific weights between moving body and resistant medium.

But at some point in the later Middle Ages someone - perhaps Be- nedetti - did so interpret the Philoponus-Avempace tradition and thereby added a dramatic new dimension to this anti-Aristotelian position. It was Galileo, however, who, in his De Motu, offered the most developed exposition of the new interpretation. Although Galileo agrees with the Avempacean tradition in regarding the medium as a factor retarding motion and unessential to it, he, as did Benedetti before him, introduces Archimedean hydrostatic concepts as theoretically objective means of determining velocities. But, as Moody observes,4* where Archimedes utilized hydrostatic theorems to determine the direction of movement of a body and its conditions of equilibrium in a fluid medium, Galileo employed such theorems to determine not only the direction but also the speed with which a body moves toward equilibrium. For Galiieo, the difference between the specific weights of a falling (or rising) body and the medium determines the speed. Although it is true to say that some earlier authors had related velocity and specific weight (e.g., in the fourteenth century, Albert of Saxony did almost the converse of Galileo by utilizing speed of descent in the same medium as a criterion for deciding whether two bodies of equal volume had the same specific weight),41 it was not until the Philoponus-Avempace law of fall was interpreted in terms of precise hydrostatic concepts that velocities were determined by any direct relationship involving the specific weights of both fuZZing body and medium. This seems not to have occurred until the sixteenth century.

Previously, problems involving a connection between speed of fall and specific gravity or specific weight were expressed in such a way that either (1) the same body was assumed to fall in two distinct media of different densities, or (2) two bodies of different specific weight were assumed to fall in one and the same medium. Where the densities of the media differ, as in (l), the status of the body is left vague and unspecified. That is, whether its weight is to be taken as gross or specific is often left indeterminate since the ratio of velocities is here partially or wholly related inversely to the ratio of the densities of the media (for Philoponus and Avempace the ratio of velocities is only partially determined by the densities of the media because the "original" or "natural" velocity would occur only in the void). For Aristotle, Philoponus, and Avempace, I

Aristotle, Philoponm, A vempace, and Calileo’s Pisan Dynnmics 91

have argued that gross weight is very likely intended. In instances of (2) - as in the case of Albert of Saxony cited above - velocity is associated only with the specific weight of the falling bodies, but not with the medium. Since the medium is here treated as a resistance common to both bodies, it is simply eliminated from any further consideration. In the absence of any reasonable evidence, however, it would be highly improper to ascribe to Albert of Saxony the view that velocity of fall is determined by some ratio of the specific weight of the body to the specific weight of the medium through which it falls.

For the many reasons cited in this article, it would be even more improper to attribute such a view to Aristotle, Philoponus and Avempace. In determining speed of fall, no direct relationship between the specific weights of body and medium - whether by division or subtraction - seems to have been enunciated until the sixteenth century.

As long as the relationship between falling body and medium was thought of in terms of gross weight and density respectively, the sub- tractive law of fall associated with Philoponus and Avempace, while important, would remain vague and unilluminating. No objective, definable criteria were proposed - or could be proposed - to calculate the Merence between the total, or absolute weight of a falling body and the density of its resistant medium. Neither Philoponus nor Avempace and his Latin followers offer a single example in which the medium is actually subtracted from the weight of the body falling through it. Such examples were not proposed because the total or gross weight of a body and the density of a medium were not reducible to a common factor. This helps explain why the additional times of fall in media, or the loss of velocity in media, were calculated, as we have seen above, in accordance with the Aristotelian rules for bodies falling in media. But Galileo, who did have such a common factor in the concept of specific weight, was able to offer many examples by assigning arbitrary numerical values for the specific weights of bodies and media and subtracting to obtain the velocities.42

There seems little doubt that Galileo, at least, was applying to the problem of natural fall traditional principles and theorems in hydrostatics and statics. Undoubtedly impressed by the success of mathematical physics in the realm of hydrostatics and statics, Galileo sought to treat the problem of natural motion or free fall as a special case of hydrostatics and to show that bodies in free fall would behave as did weights

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on a balance. Thus after noting that “there are many media through which motions take place, e.g., fire, air, water, etc. and in all of them the same principle applies,” Galileo takes the case of a body falling in water and demonstrates in Chapters 4 and 5 of De Motu “that bodies equally heavy with the water itself, if let down into the water, are completely submerged, but then move neither downward nor upward. Secondly, . . , that bodies lighter than water not only do not sink in the water, but are not even completely submerged. Thirdly, . . . that bodies heavier than water necessarily move downward.”43 But if the natural motion of bodies could be properly explained hydrostatically as motion through a fluid medium, it was also clear to Galileo that “the motion of bodies moving naturally can be suitably reduced to the motion of weights on a balance. That is, the body moving naturally 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. So that, if a volume of the medium equal to the volume of the moving body is heavier than the moving body, and the moving body lighter, then the latter, being the lighter weight, will move up. But if the moving body is heavier than the same volume of the medium, then, being the heavier weight, it will move down. And if, finally, the said volume of the medium has a weight equal to that of the moving body, the latter will move neither up nor down, just as the weights in the balance, when they are equal to each other, neither fall nor rise.

“And since the comparison of bodies in natural motion and weights on a balance is a very appropriate one, we shall demonstrate this parallel- ism throughout the whole ensuing discussion of natural motion. Surely this will contribute not a little to the understanding of the matter.”&

Galileo’s purpose in all this seems clear. After treating natural motion or free fall in air as a particular exemplification of hydrostatic principles, he tries to render this plausible by reducing the hydrostatics of motion to the familiar behavior of weights on a balance, where the weights represent equal volumes of the falling body and of the medium. In this way, the complex problem of free fall was reduced to, and explained in terms of, physical principles and laws that, along with optics, had con- stituted the most highly developed branches of mathematical physics.

Perhaps for the first time in the history of physics, theoretically objective and measurable criteria were offered for determining the velocities of freely falling bodies. That Galileo’s theory was false is far less important

Aristotle, Philoponus, A vempace, and Galileo’s Pisun Dynumics 93

than what it signified. An old and vague anti-Aristotelian law of velocities was mathematized and rationalized. Already in the De Motu (both essay and dialogue), Galileo’s earliest extant attempt at scientific expression, we see him seeking to mathematize a dynamic law of falling bodies. Thus the desire to discover mathematical laws for the motion of real physical bodies was already a prominent and characteristic feature of Galileo’s earliest period of scientiiic activity. By employing specific weight as the basis of a law of motion, Benedetti and Galileo ushered in a new era in the history of physics. By expressing laws of motion in terms of clearly definable concepts and objective criteria, they established a new ideal and a new direction to aid in the solution of one of the most vexing problems in the long historical evolution of physics.

NOTES A N D R E F E R E N C E S

1. This appeared in two parts in the Journal of the History of Zdeas, Vol. 12 (1951),

2. “Galileo and Avempace,” p. 173. 3. Ibid., pp. 390-396. 4. M . Clagett, The Science of Mechanics in the Middle Ages (Madison, Wisc.. 1959),

5. “Galileo and Avempace,” p. 413. For a discussion of some important quaUcations that Benedetti imposed upon his law see I. E. Drabkin, “Two Versions of G. B. Benedetti’s Demonstratio Proportionum Motuum Localium,” Zsis, Vol. 54 (1963), pp. 259-262. Benedetti’s law and his impetus theory are summarized in A. KoyrC, gtudes GaliZ6ennes (Paris, 1939). fasc. 1, pp. 45-54. The laws of fall of Benedetti and Galileo arc compared brieffy by Drabkin, “G. B. Benedetti and Galileo’s De MOZU,” Acres du dixiPrne congrb international d‘histoire des sciences (Paris, 1964), Vol. I,

6. Moody supplies a complete translation of the Latin text on pp. 184-186 of “Galileo and Avempace.” For biographical information on Avempace see pp. 187-188. The text of Averroes from which Moody made his translation is Opera Aristotelis . . . cum Averois commentariis (Venetiis. 1560). Vol. IV, fol. 13 1 verso. A later edition of this volume, Aristotelis opera cum Averrois commentariis (Venctiis, 1562). was re- printed by Minerva, Frankfurt am Main, 1962. The passage on Avempace appears on fol. 160r. c.l-l6Ov, c.1.

163-193, 375-422.

pp. 94-95.

pp. 627-630.

7. “Galileo and Avempace,” p. 192. 8. “Galifeo and Avempace,” p. 186. Galileo’s Pisan dynamia is extant, for the most

part, in a dialogue and an essay containing substantially the same ideas but whose order of composition is unknown. These two versions, which were never published by Galileo, are titled De Motu in Le Opere di Galileo Galilei, Edizione Nazionali,

94 Edward Grant

Vol. I (Florence, 1890). Moody relied exclusively upon the dialogue (“Galileo and Avempace,” p. 167, n. 7). The essay has been fully translated into English in Galilco Galilei on Motion and On Mechanics comprising De Motu (ca. 1590) translated with Introduction and Notes by I. E. Drabkin and Lc Meccanichc (ca. 1600) translated with Introduction and Notes by Stillman Drake (Madison, Wisc., 1960).

9. “Galileo and Avempace,” p. 417. 10. Ibid.. p. 192. 11. In a memorandum (see p. 50, n. 24 of Drabkin’s translation of De Mot& Galileo

12. See, for example, De caela 1.8.277a. 27-33 and Physics V.6.230b. 24-25. 13. De caelo III.2.301b. 30. 14. Quotations relevant to Philoponus and Avempace have been cited above. For Ari-

stotle, see “Galileo and Avempace.” p. 172. Although Moody states (p. 172, n. 14) that Aristotle differentiated between mobile bodies by “weight” or “nature”, he maintains throughout that relationships between falling bodies and media were conceived in terms of densities or specific weights.

15. See p. 30 and pp. 48-49 of Drabkin’s translation of Galileo’s De Mom. Although this conclusion was later modified in the Discourses where the concept of terminal velocity was introduced, Galileo retained the law that the velocity of a body is determined by the difference between its specific weight and that of the medium. See Discorsi e Dimostrazioni matematiche intorno a due Nuovc Scienze in Lc Operc di Galileo Galilei (Florence 1898). Vol. 8, pp. 117-121 (for an English translation see Two New Scfcnccs by Galileo Galilei, translated by Henry Crew and Alfonso de Salvio [New York, 19141,

16. See Thomas of Bradwardine His Tractatus de proportionibus. edited and translated by H. Lamar Crosby, Jr. (Madison, Wisc.. 1955). pp. 44-45 and 117. I compare Brad- wardine’s theorem to Galileo’s law of fall in an article, “Bradwardine and Galileo: Equality of Velocities in the Void,” to be published soon in Archive for History of Exact Sciences.

17. For a definition of absolute heavy and light see Dc caclo IV.4.311a. 16-24. Air and water are each relatively light and heavy (De caclo IV.4.31 la. 24-30).

18. De caelo IV.4.312b. 13-16. 19. Zbid. IV.2.309b. 13-16. Immediately preceding these lines, Aristotle says: “A large

and small quantity of fire will contain more void and solid in the same proportions, yet the large quantity moves upwards more quickly than the small.”

20. IV.8.216a. 28-33. The translation is by Philip H. Wicksteed and Francis M. Cornford in the Loeb Classical Library.

21. Physics IV.8.216b. 6-13. 22. Ibid., IV.8.216a. 13-21. 23. This passage, from Philoponus’ Commentary on Aristotle’s Physics, is translated by

I. E. Drabkin and M. Cohen in A Source Book in Greek Science (Cambridge, Mass., 1958), p. 21 8.

24. Clagett (Science of Mechanics, p. 435) denies generally any attempt on the part of Philoponus to claim “that the speed of fall in the case of natural movement is proportional to specific gravity rather than to gross weight? and thinks it was in no way implied in the observation on the fall of unequal bodies in the same time.

mentions both as proponents of finite motion in the void.

pp. 72-76).

Aristotle, Philoponus, Avrmpacr, and Galileo’s Pisan Dynamics 95

25. Cohen and Drabkin, op. cit., p. 219. 26. Zbid. 27. See E. Wohlwill, “En Vorganger Galileis im 6. Jahrhundert,” Physikalische Zeit-

28. Cohen and Drabkin, op. cit., p. 219, n.3, believe that Philoponus. and the ancients

29. Zbid., p. 219. 30. Zbid., p. 218. 31. Zbid., p. 217. 32. From Philoponus’s Commentary on the Physics as quoted and translated by S. Sam-

33. Translated by Moody in “Galileo and Avempace,” pp. 184-185 (the italics are his). 34. Zbid., p. 185 (the italics are Moody’s). 35. Zbid. 36. Zbid., p. 186. 37. Zbid., pp. 186187. 38. In this connection, it should be mentioned that Salomon Pinks, who reports the exist-

ence of Avempace’s commentary on Aristotle’s Physics in an Arabic manuscript at the Bodleian Library, Oxford, conjectures - and in this he agrees with Moody - that Avempace may have treated terrestrial dynamics by analogy with celestial dynamics. “Mais tout cela n’tst pas trQ sQr. Les Ctudes qui restent B faire et qui devront porter sur le texte intcgral du Commentairr qu’a fait Ibn Bajja des Naturalia d‘Aristote Cclaircront, peut-ttre, les points obscun qui nom empkhent <&re certains d’avoir saisi le rapport subsistant entre la physique celeste et la physique sublunaire du philosophe arabe.” “La dynamique d’Ibn Bgjja,” in L’aventure de la science, Mdlanges AJexandre Koyd (Paris. 1964), pp. 460-461. Rather than “an absolute indwelling power of self-motion animating the body like a soul,” Avempace may have followed Philo- ponus and conceived of heavy (and light) as an indwelling physical force that was simultaneously a quality of the body (see E. Wohlwill, “Ein Vorganger Galileis im 6. Jahrhundert,” Physikalische Zeitschrvt, Vol. 7, p. 30, col. 2.

39. For example, Thomas Aquinas, Peter John Olivi, William of Ware, and Duns Scotus (see “Galileo and Avempace,” pp. 380-389).

40. “Galileo and Avempace,” p. 170. 41. See Clagett, Science of Mechunics, pp. 137 and 141 ; and ibid., p. 141, for similar ideas

expressed in the much earlier Liber de pondrroso et levi. 42. Such examples may be found on pp. 34, 35.36.40, and 43-45 of Drabkin’s translation

of De Mom. 43. Zbid., p. 17. 44. Zbid., p. 23. The comparison between natural motion and weights on a balance is

reemphasized in Chapter 9 of De Motu (pp. 38-41) titled: “In which all that was demonstrated above is considered in physical terms, and bodies moving naturally are reduced to the weights of a balance.”

schrift, Vol. 7 (1906). p. 29, col. 1.

generally, had no quantitative notion of density as mass per unit volume.

bursky, The Physical World of Lute Antiquity (New York, 1962). p. 67.

aristotle, philoponus, avempace, and galileo's pisan dynamics

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