“Galileo’s Machine”: Late Notes on Free Fall, Projectile Motion, and the Force of Percussion...

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Physics in Perspective ISSN 1422-6944Volume 16Number 4 Phys. Perspect. (2014) 16:440-460DOI 10.1007/s00016-014-0149-1

“Galileo’s Machine”: Late Notes on FreeFall, Projectile Motion, and the Force ofPercussion (ca. 1638–1639)

Stefano Salvia

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‘‘Galileo’s Machine’’: Late Notes on Free Fall,Projectile Motion, and the Force of Percussion(ca. 1638–1639)

Stefano Salvia*

My paper focuses upon the problem of determining the nature, establishing the propor-tionality, and measuring the intensity of the force of percussion of a projected or fallingbody, as treated in the Sixth Day of Galileo’s Discorsi e dimostrazioni matematiche intorno adue nuove scienze (1638). This fragment was written around 1638–1639 as part of twoadditional Days of the Discorsi, which Galileo never finished and remained unpublisheduntil 1718. Galileo’s last works on percussion show a significant step towards a general-ization of his own views on uniform and accelerated motion that will later lead to theNewtonian principle of inertia. The experiment with two equal weights hanging on a pulley,performed in Arcetri during the same period, is compared with the Paduan 1604–1608experience of the ‘‘water balance.’’ Both account for more than three decades of inquiriesinto what Galileo called forza della percossa, which marked the transition from preclassicalto classical mechanics.

Key words: Galileo Galilei; machine; percussion.

Measuring the Force of Percussion

Galileo’s interest in projectile motion and in the variable effects of percussion on

different materials should not be regarded as a consequence of his investigations

of the law of free fall, the inclined plane, the periodic motion of pendulums, and

that of bodies swinging along curved paths. On the contrary, the latter were led

and shaped by the former, at least after his move from Pisa to Padua in 1592. Here

the methodological reorientation from a physico-philosophical inquiry into ‘‘local

motion [de motu locali]’’ to a systematic quest for proportionality among physico-

mathematical quantities was made possible by a fertile interaction with the tech-

nical knowledge of the time related to applied mechanics, machine- and ship-

building, military architecture, and ballistics.1

However, to refrain from speculations on the ultimate nature of motion (motus)

and weight (gravitas) did not necessarily imply a merely kinematic approach to

* Stefano Salvia, PhD, is a research assistant in History of Science at the Department ofPhilosophy of the University of Pisa.

Phys. Perspect. 16 (2014) 440–460� 2014 Springer Basel

1422-6944/14/040440-21

DOI 10.1007/s00016-014-0149-1 Physics in Perspective

440

Author's personal copy

mechanics, but rather a reconsideration of dynamic explanations from a new and

broader perspective. Velocity and weight were in fact two inseparable quantities

involved in the study of shocks and impacts, respectively the kinematic and

dynamic components through which percussion must be analysed. As early as

1604–1608, by reappraising the Peripatetic (Aristotelian) treatment of the ‘‘extra

weight’’ acquired by a moving body striking another one, Galileo had come to the

conclusion that the force of percussion could be measured in terms of ‘‘equivalent

weight’’ proportional both to the intrinsic weight of a moving body and to its

velocity.2 His first experiment to measure the percossa, the balance described by

Aproino (named after a former student of Galileo’s) in the unfinished Sixth Day

and dating back to the same period, was actually based on such an equivalence.3

APROINO: [I]t was the Academician’s first idea to try to find out what part in the

effect and operation of impact belonged…to the weight…, and what [part

belonged to] the greater or lesser speed with which it was moved. He wanted if

possible to find one measure that would measure both of these, and would

assign the energy of each; and to arrive at this knowledge, he imagined what

seems to me to be an ingenious experiment.

He took a very sturdy rod, about three braccia* long, pivoted like the beam

of a balance, and he suspended at the ends of these balance-arms two equal

weights, very heavy. One of these consisted of copper containers; that is, of two

buckets, one of which hung at the said extremity of the beam and was filled with

water. From the handles of this bucket hung two rods, about two braccia each in

length, to which was attached by its handles another like bucket, but empty; this

hung plumb beneath the bucket already described as filled with water. At the

end of the other balance-arm he hung a counterweight of stone or some other

heavy material, which exactly balanced the weight of the whole assembly of

buckets, water, and ropes. The bottom of the upper bucket had been pierced by

a hole…, which…could be opened and closed.4

Since the very beginning of his inquiries on the force of percussion, Galileo

tried to express its measure as a ‘‘supplementary weight’’ acquired by a falling body

(whose ‘‘natural weight’’ at rest was already known) during its motion, until the

impact on another body or resisting surface. The balance (more precisely, the

equally armed balance) was by definition the suitable instrument to measure

weight and its possible variations over time and/or according to other factors such

as the inclination of a plane. Hence, to isolate and measure the effect (energia) of

percussion meant to quantify the momentary (impulsive) alteration of the equi-

librium between two equal weights.

* The Florentine braccio or braccio da panno (‘‘tailor’s arm’’), usually adopted by Galileo asthe standard measure of length, was equal to 58.4 cm.

Vol. 16 (2014) Galileo’s Machine 441


First Experiment: The ‘‘Water-Balance’’ (ca. 1608)

One possibility was to measure (by means of a needle and quadrant) the instan-

taneous ‘‘extra weight’’ acquired by a body previously weighted at rest and let fall

from a certain height on one of the two plates of a balance in equilibrium. The

system would reach a new equilibrium, displaying the actual difference of weight

at rest between the two plates. This was, however, a possibility that Galileo did not

consider around 1608. Moreover, he had no possibility to conceptualize the con-

servation of quantity of motion and mechanical energy in an isolated system as we

do nowadays. He went another way, which afterwards turned out to be a serious

failure but an important starting point at the same time. So the fictional character

Aproino describes the experiment on percussion performed by Galileo in Padua

around 1608 and its surprising outcome:

APR: Our first conjecture was that when the balance rested in equilibrium, the

whole apparatus having been prepared as described, and then the [hole in the]

upper bucket was unstoppered and the water allowed to flow, this would go

swiftly down to strike in the lower bucket; and we conceived that the adjoining

of this impact must add to the [static] moment on that side, so that in order to

restore the equilibrium it would be necessary to add more weight to that of the

counterpoise on the other arm. This addition would evidently restore and offset

the new force of impact of the water… The outcome was no less wonderful than

it was unexpected by us. For the hole being suddenly opened, and the water

commencing to run out, the balance did indeed tilt toward the side with the

counterweight; but the water had hardly begun to strike against the bottom of

the lower bucket when the counterweight ceased to descend, and commenced

to rise with very tranquil motion, restoring itself to equilibrium while water was

still flowing; and upon reaching equilibrium it balanced and came to rest.5

Falling bodies (also fluid ones) did not ‘‘weigh’’ while falling: weight (thus also

the possibility of its measurement) was always the result of an opposition, a

resistance (i.e., a reaction) to free fall. This was definitely not obvious to those who

still conceived weight as an intrinsic feature of ‘‘heavy bodies,’’ if not of any

material body, as Galileo already thought in his manuscripts De motu antiquiora

(ca. 1589–1592). Of course, Galileo never came to regard weight as a ‘‘force’’ in our

terms, let alone any force of attraction such as Newton’s later theory of gravita-

tion. Galileo always considered gravitas to be a natural property of matter, a

tendency of material bodies to move straight forward with uniform acceleration

towards a common center of gravity.

Sagredo, who represents the point of view of the younger Galileo, is asked for a

conclusion that can be derived from the experience of the ‘‘water balance’’:

SAGREDO: This result certainly comes as a surprise to me. The outcome differed

from what I had expected, and from which I hoped to learn the amount of the

force of impact. Nevertheless it seems to me that we can obtain most of the

442 S. Salvia Phys. Perspect.


desired information. Let us say that the force and moment of impact is

equivalent to the moment and weight of whatever amount of falling water is

found to be suspended in the air between the upper and lower buckets, which

quantity of water does not weight at all against either upper or lower bucket….

Hence it follows that all the water contained in the jet is as if it were not in the

balance.6

The unexpected behavior of the apparatus built by Galileo in Padua did not allow

him to isolate and visualize the effect of the water column impacting on the

bottom of the lower bucket: the balance just showed a temporary ‘‘loss of weight’’

at the very first instants, when water started to fall from the upper bucket, until it

gradually regained its state of equilibrium. Actually, the system is much more

complicated to describe according to modern mechanics and fluid dynamics. There

is indeed a ‘‘loss’’ of mechanical energy in the system, while water falls from the

upper to the lower bucket (constantly but not uniformly). At the end of this

process, the same water has a lower gravitational potential than before: part of its

kinetic energy is converted into the oscillatory motion of the balance, usually

dissipated by friction among its different components. Galileo could not notice this

secondary phenomenon, because his apparatus was not enough sensitive to

appreciate it (figure 1).7

Impact Theory and ‘‘Galilean Relativity’’

The problems that emerged during his first, unsuccessful attempt to measure the

force of percussion led Galileo to a fundamental insight: the intensity (momento)

of the percossa was due to a combined effect of ‘‘intrinsic weight’’ and the relative

velocity of the two colliding bodies, specifically (in modern terms) the (vectorial)

sum of their ‘‘absolute’’ velocities, according to their directions of motion

(according to what we presently call ‘‘Galilean transformations’’). Salviati, who

represents the standpoint of the older Galileo, expresses the idea in the following

words:

SALVIATI: It is evident that the property [faculta] of force in the mover and [that]

of resistance in the moved is not single and simple, but is compounded from two

actions, by which their energy must be measured. One of these is the weight, of

the mover as well as of the resistant; the other is the speed with which the one

must move and the other be moved.8

Galileo’s ‘‘research program’’ to isolate and quantify the different effects of

impacting bodies by analyzing them in terms of weight and relative velocity did

not simply derive from his studies on free fall and projectile motion, as if it were

merely a mere by-product. It was, on the contrary, the core of these inquiries,

together with the outlines of a new ‘‘science of materials,’’ which was also insep-

arable from his concern with hydrostatics and hydro- and aerodynamics. The



conceptual roots of ‘‘Galilean relativity’’ itself are to be found in the problem of

measuring the force of percussion because of the crucial role played by the relative

velocities of falling/projected bodies in order to account for their behavior during

impact. By the way, what we nowadays refer to as the ‘‘Galilean’’ (i.e. classical)

principle of relativity of motion was never formulated as such by Galileo but was a

post-Newtonian coinage of the late nineteenth century, at the point when ‘‘Gali-

leo’s transformations’’ needed to be distinguished from Lorentz’s non-classical

ones. Let us read how Galileo deals with the relative motion of colliding bodies as

expressed by his character Salviati:

SALV: Imagine striking with your hand against a ball that comes falling from

above, and tell me: if, when this arrived upon your hand, you were to have your

hand sinking along the same line and with the same speed as the ball, what

shock would you feel? Surely none. But if, upon the arrival of the ball, you

yielded only in part, by dropping your hand with less speed than that of the ball,

you would indeed receive an impact—not as with the whole speed of the ball,

but only as with the excess of its speed over that of the dropping of your hand.

Thus if the ball should descend with ten degrees of speed, and your hand

yielded with eight, the blow would be made as by two degrees of speed of the

ball. The hand yielding with four [degrees], the blow would be as six; and the

Fig. 1. A modern reconstruction of Galileo’s balance (2005). Credit: Roberto Vergara Caffarelli

(Fondazione Galileo Galilei, Pisa).



yielding being as one, the blow would be as nine; the entire impact of the speed

of ten degrees would be [only] that which struck the hand that did not yield.9

In his Mecanique analytique (1788), Joseph-Louis de Lagrange actually was the

first to maintain that the Discorsi (and all the materials related to them) must be

regarded as Galileo’s most important legacy in the history and development of

modern mathematical physics (including astronomy).10 We can go a little bit

further: after the general reorientation of his approach to motus and gravitas

through a closer commitment to mechanical arts, craftsmanship, and applied sci-

ences, Galileo put the forza della percossa in relation to projectile motion and free

fall at the top of his research agenda.

Thus, we should reverse the traditional and still common point of view

according to which the principle of ‘‘Galilean relativity’’ found its birthplace and

main treatment in the Dialogo sopra i due massimi sistemi del mondo (1632), while

in the Discorsi it was considered in a very specific way, limited to the problem of

measuring the force of percussion. In my opinion, exactly the opposite is the case:

almost forty years of Galileo’s research into percossa are condensed in the dif-

ferent texts we read today as his Discorsi, including the additional Fifth and Sixth

Days and other notes. This inquiry into percossa might be viewed as the long-term

thread of his scientific activity, at least since 1608. The extension of his relativity

principle to astronomy in the Dialogo should rather be seen as a process of

knowledge-transfer and generalization from one field to another.11

Velocity, Gravity, Momento

Galileo supposed that the force of percussion was proportional to both weight (at

rest) and to velocity. The easiest way was to assume a direct and linear propor-

tionality. To better appreciate how close Galileo came to the later notion of

quantity of motion as mechanical constant by interpreting the percossa as

impulsive force, we can make an anachronistic but still useful translation of his

1638–1639 argument into modern terms. This will help us to ‘‘visualize’’ it through

the formalism of classical mechanics, of course keeping aware of the fact that it

was not available at the time when Galileo wrote the Discorsi.*

According to that modern argument, the percossa is directly proportional to the

relative velocity between ball and hand, as Galileo states in his text. When the

hand is at rest, both before and after the impact, the intensity of the percossa only

depends on the (final) velocity of the ball. Descartes would later define his notion

* Let mb be the mass of the ball, mh that of the hand, vb the velocity of the ball, vh that ofthe hand, and Dp the shock fully absorbed by the hand, supposing that the hand stillmaintains its original velocity after catching the ball. The relation between their quantitiesof motion, before and after the impact, is given by the following expressions:mbvb ? mhvh = (mb ? mh)vh ? Dp and, Dp = mbvb ? mhvh – mbvh – mhvh = mbvb – mbvh =mb(vb – vh).



of quantitas motus as the (scalar) product of mass and velocity, defining that of

mechanical work (actual or only virtual) as the (scalar) product of weight and

distance. Apart from the absence of any clear distinction between mass and

weight, as well as between scalar and vectorial quantities, the reason why Galileo

never managed to formulate an explicit equation like Descartes’s expression

q = mv is the same that prevented him from providing a full account of how the

force of percussion of falling bodies increases during the accelerated motion

proportionally to the height, hence proportional to what we call their final velocity

v (more accurately, to v2). Galileo and his contemporaries were simply unable to

conceive (average) velocity as we do nowadays.*

Classical mathematics previous to algebraic geometry and differential calculus

was based upon the traditional theory of proportions coming from Euclid and

Archimedes. According to it, ratios were admissible only between two homoge-

neous quantities, two quantities of the same kind, such as two distances or two

speeds. In the case of heterogeneous quantities, one could only compare their

ratios as pure numbers. Therefore, Galileo’s pre-classical notion of velocitas is not

defined as a ratio between space and time, but as the space covered in a given time

or alternatively the time elapsed in covering a given space: something that might

appear very close to the modern notion of average speed, which however pre-

supposes the possibility of writing the equation v = Ds/Dt (in the limit of vanishing

time Dt leading to the instantaneous velocity vt = ds/dt).12 Galileo’s conceptual

difficulties are evident when he speaks about equal momenti di gravita (degrees of

gravity) or momenti di velocita (degrees of velocity) gained by a falling body in

equal times, while he tries to explain in ‘‘quasi-infinitesimal’’ terms (so to speak)

the effects of different weights falling from different heights (for instance, by using

a battipalo or berta, a pile-driver) or being projected from different distances and

angles on the same resistant materials.

I shall quote the second of seven fragments, which were collected and published

at the end of the Discorsi by the editors of the 1718 edition of Galileo’s works.13

As Stillman Drake remarks in his notes, these fragments probably come from no

longer extant manuscripts and their authenticity remains controversial today.

Fragment 2 in particular seems spurious, because (citing Drake himself) ‘‘the word

momento is used in both senses of ‘static moment’ and ‘momentum’…, not

alternatively, but in a mixed way uncharacteristic of Galileo.’’14 It fits much better

with Torricelli’s later modification of Galileo’s thought than with his own writings,

and it may be apocryphal. The idea here is that moments, like degrees of speed,

are uniformly added with time, so that the momentum of a body on impact

resembles its terminal speed in free fall as being a finite aggregate of infinitely

many unquantifiable parts.’’15 At any rate, here is the key passage in Fragment 2

about ‘‘momentum’’ and percussion:

* Namely as the ratio between the displacement Ds and the time interval Dt, which becomesthe instantaneous velocity v(t0) at a given time t0 as Dt ? 0.



[2] The momentum of a body in the act of impact is nothing but a composite

and aggregate of infinitely many momenta, each of them equal only to a single

moment [al solo momento], either internal and natural per se, as is that

[moment] of its own absolute weight which it eternally exercises when placed

on any resistant body, or else extrinsic and violent, as is that [momentum] of the

moving power [forza]. Such momenta go accumulating during the time of

[naturally accelerated] motion of the heavy body from instant to instant with

equal increments, and are stored therein, in exactly the way that the speed of a

falling body goes increasing; for as in the infinitely many instants of time,

however short, a heavy body goes ever passing through new and equal degrees

of speed, always retaining those acquired in the previously elapsed time, so also

in the moveable those momenta (either natural or violent, conferred on it by

nature or by art) go conserving [themselves] and compounding from instant to

instant, etc.16

Galileo’s need to distinguish the contribution of gravity from that of velocity in

determining the intensity (momento) of the percossa during the final impact of a

projected/falling body might be translated into the language of Newtonian

mechanics as the difficulty of distinguishing between quantity of motion, impulse

(i.e. its variation through time), mechanical work, and (the change of) potential/

kinetic energy, as well as to conceive an instantaneous velocity, such as one body

would give another on collision.

Last Experiment: ‘‘Galileo’s Machine’’ (ca. 1638)

These fundamental problems remain in the background of Galileo’s later attempts

to isolate and measure the force of percussion, which he must have performed

around 1638 in Arcetri together with his disciple Vincenzo Viviani (1622–1703),

before he became completely blind a few months thereafter. I will focus in par-

ticular on the crucial experiment proposed and thoroughly described by Salviati as

follows:

SALV.: I want you to imagine a solid weight of, say, 1,000 pounds, placed on a

plane that sustains it. Next, I want you to think of a rope tied to this weight and

led over a pulley fixed high above. Here it is evident that when force is applied

by pulling down on the end of the rope, it will always meet with quite equal

resistance in raising the weight; that is, the opposition of 1,000 pounds of

weight. For if from the end of the rope there were suspended another weight,

equal to the first, equilibrium would be established; and being raised up without

support from anything below, they would remain still; nor would this second

weight descend and raise the first unless given some excess of weight. And if we

rest the first weight on the said plane that sustains it, we can use other weights

of varying heaviness (though each of them less the weight sustained at rest) to

test what the forces of different impacts are. [This would be done] by tying such



weights to the end of the rope and then letting them fall from a given height,

observing what happens at the other end to that great solid that feels the pull of

the falling weight, which pull will be to that large weight as a blow that would

drive it upward. Here, in the first place, it seems to me to follow that however

small the falling weight, it should undoubtedly overcome the resistance of the

heavy weight and lift it up. This consequence seems to me to be conclusively

drawn from our certainty that a smaller weight will prevail over another,

however much greater, whenever the speed of the lesser shall have, to the speed

of the greater, a greater ratio than the weight of the greater has to the weight of

the smaller…; hence the force of this percussant will overcome the resistance of

that on which it makes its impact.17

The issue remains always the same: to quantify percussion indirectly through a

mechanical equivalent to which the force of percussion itself can be reduced. In

Padua, more than thirty years before, such equivalent was a momentary ‘‘extra

weight’’ that his balance was expected to measure. In this case, Galileo tries to

express the forza della percossa in terms of equivalent (final and constant) velocity.

His idea here is to use another classical elementary machine: the pulley. Like the

lever (which the balance is based on), the pulley was known since remote antiquity

as a multiplier of mechanical work allowing to pull/lift a heavy body by means of a

smaller ‘‘dead weight’’ or ‘‘living force.’’

In the passage quoted above (reformulated in modern terms), Galileo actually

maintains that if two weights gm1 and gm2 (with g = 9.81 meters per second

squared) are at the two opposite ends of a rope hanging on a pulley and m1 [ m2,

then m2 will prevail on m1 (i.e. m2 will pull/lift m1) if gm1v1 \ gm2v2. This state-

ment does not seem to be the result of a mere thought experiment. On the

contrary, it seems to come from direct experimental evidence: Galileo describes

his device starting from the case of two different weights hanging on a pulley

mounted on the top of an inclined plane (figures 2–3); then he considers the

situation of two different weights vertically suspended on a pulley; finally he treats

the case of two equal weights.

One cannot exclude that Galileo already explored such an alternative to inves-

tigate the force of percussion around 1608, while still in Padua. Actually, we do not

know if Aproino’s and Sagredo’s suggestions reflect partial—and unsatisfying—

results dating back to the same period; certainly they reflect Galileo’s original ideas

on the problem. He was already well acquainted with an apparatus such as that

represented in figures 2–3, since it had allowed him to measure the variation of (the

acceleration of) gravity of a body along an inclined plane proportionally to sin a(using our terminology), where a is the inclination angle of the plane.

Adapting his ‘‘machine’’ to the new goal by removing the inclined plane,

Galileo aims at going beyond a purely qualitative evaluation to achieve a precise

measurement of the forza della percossa. He wants to measure the velocity v1

instantaneously acquired by m1 when m2 is left falling down from a certain height



h above its resting point and gives an impulsive jerk (strappata) to the rope with its

final velocity v2 = v (supposing that m1 is at rest). As was already noted, velocitas

must be intended here as space covered in a given time or the time elapsed in

traversing a given space. Galileo formulates a first hypothesis through Salviati’s

words:

SALV.: Next we shall seek to find out how great is the space through which the

impact received will raise it, and whether perhaps this [distance] will corre-

spond to that of other mechanical instruments. Thus it is seen in the steelyard,

for example, that the rise of the heavy weight will be that part of the fall of the

counterweight, which the weight of the counterweight is of the greater weight.

So in our case we should have to see, supposing the weight of the big resting

solid to be 1,000 times that of the falling weight—which falls, let us say, from a

Fig. 2. Pulley and inclined plane in the Third Day of Galileo’s Discorsi (1638). Source: Galileo

Galilei, Opere (ref. 4), 8:215. Credit: Library of the Galileo Museum, Florence.

Fig. 3. Pulley and inclined plane in the Sixth Day of Galileo’s Discorsi (1638). Source: Galileo

Galilei, Opere (ref. 4), 8:339. Credit: Library of the Galileo Museum, Florence.



height of one braccio—whether this raises the other [weight] one one-hun-

dredth of a braccio; if so, it would appear to be following the rule for the other

mechanical instruments.18

Salviati (alias Galileo) decides to consider the easiest situation, when the two

weights are equal and asks both Aproino and Sagredo what the result should be in

this case:

SALV.: Let us imagine making the first experiment by dropping from some

height, say one braccio, a weight equal to the other, which we have placed on a

[supporting] plane, these weights being tied to the opposite ends of the same

rope. What shall we believe to be the effect of the pull of the falling weight, with

regard to the moving and raising of the other, which was at rest? I should be

glad to hear your opinion.19

Aproino answers in these terms:

APR.: [I]t appears to me that the two weights being equally heavy, and the one

which falls having in addition the impetus of its speed, the other one must be

raised by it far beyond equilibration, inasmuch as the mere weight of the other

was sufficient to hold it in balance. Hence, in my opinion, it will rise through

much more than a space of one braccio, which is the measure of the descent of

the falling weight.20

Sagredo is much more cautious in his conclusions. Like his alter ego, the young

Galileo in Padua, he learned from past errors that intuition is often misleading.

SAGR.: The reasoning seems conclusive to me at first glance; but, as I said a

while ago, many experiences have taught me how easily one may be deceived,

and accordingly how necessary it is to go circumspectly before boldly pro-

nouncing and affirming anything. Hence I shall say, still dubiously, that it is true

that the weight of 100 pounds of the falling heavy body will suffice to raise the

other, which also weighs 100 pounds, as far as to equilibrium, even without its

being endowed and supplied with speed; [to do this] the excess of a mere half-

ounce will suffice. But I also think that that equilibration will be made very

slowly, and hence that when the falling body acts with great speed, it will

necessarily raise its companion on high with like speed. Now, there seems to me

no doubt that greater force is needed to drive a heavy body upward with great

speed than to push it very slowly; so it might happen that the advantage of the

speed acquired by the falling body in free fall through one braccio would be

consumed, and so to speak spent, in driving the other with equal speed to a like

height. Hence I am inclined to believe that these two movements, upward and

downward, would end in rest immediately after the rising weight had gone up

one braccio, which would mean two braccia of fall for the other, counting the

first braccio of free fall as executed by that one alone.21



Sagredo supposes a continuous process: the rope regains its maximum elon-

gation when m2 reaches its maximum velocity v by falling down from the height h,

then m2 gives the same velocity v to m1 through the tension of the rope. According

to him, m1 and m2 should decelerate uniformly from their initial velocity v to rest

by covering upwards and downwards a space equal to h, respectively. Sagredo

explicitly refers to the equivalence between the final velocity v reached by a body

falling from a given height h and the initial velocity v that a body must acquire to

reach the same height h when projected along the vertical. Salviati agrees only

with the first part of his argument:

SALV.: I truly lean toward the same belief. For though the falling weight is an

aggregate of heaviness and speed, the operation of its heaviness in raising the

other [weight] is nil, this being opposed by the resistance of equal heaviness in

that other, which clearly would not be moved without the addition of some

small weight. Therefore the operation is entirely that of the speed, which can

confer nothing but speed. Being unable to confer other [speed] than what it has,

and having nothing other than that which it acquired in the descent of one

braccio after leaving from rest, it will drive the other upward through a like

space and with a like speed, in agreement with what can be discerned in various

experiences; namely that the falling weight, leaving from rest, is everywhere

found to have that impetus which suffices to restore it to the original height.22

Towards a Generalization of ‘‘Inertial Motion’’?

When Sagredo suggests a possible analogy between the motion of a swinging

pendulum and that of two equal weights vertically hanging on a pulley, Salviati

underlines the fundamental difference between the two situations:

SALV.: I find no trifling discrepancy between these two operations; I mean

between that of the heavy solid hanging from the thread, which released from a

height and descending along the circumference of a circle, acquires impetus to

transport itself to another equal height, and this other operation of the falling

body tied to the end of a rope in order to lift another one equal to itself in

weight. For that which descends along the circle continues to acquire speed as

far as the vertical [position], favored by its own weight, which impedes its ascent

as soon as the vertical is passed, ascent being a motion contrary to its heavi-

ness.… But in the other case, the falling weight comes upon its equal placed at

rest, not only with its acquired speed but with its heaviness as well; and this

[heaviness], being maintained, by itself alone removes all resistance on the part

of its companion [weight] to being lifted. Hence the [previously] acquired speed

meets with no opposition from any weight that resists rising; and just as impetus

conferred downward on a heavy body would encounter no cause in that [body]

for annihilation or retardation [of that impetus], so none is encountered in that



rising weight whose [effective] heaviness remains nil, being counterpoised by

the other, descending, weight.23

At this point, Salviati/Galileo introduces a conceptual novelty by considering

the case of two equal weights vertically suspended at the ends of a rope hanging

from a pulley. The gravitational accelerations of m1 and m2 cancel each other out,

so the state of the system is actually comparable to that of a ball resting on a

horizontal plane:

SALV.: Here, it seems to me, precisely the same thing takes place which happens

to a heavy and perfectly round moveable placed on a very smooth plane…. If

the plane were not inclined, but horizontal, then this round solid placed on it

would do whatever we wish; that is, if we place it at rest, it will remain at rest,

and given an impetus in any direction, it will move in that direction, maintaining

always the same speed that it shall have received from our hand and having no

action [by which] to increase or diminish it, there being neither rise nor drop in

that plane. And in this same way the two equal weights, hanging from the ends

of the rope, will be at rest when placed in balance, and if impetus downward

shall be given to one, it will always conserve this equably. Here it is to be noted

that all these things would follow if there were removed all external and acci-

dental impediments, as of roughness and heaviness of rope or pulleys, of

friction in the turning of these about the axle, and whatever others there may be

of these.24

Here we are dealing with a crucial shift in Galileo’s thoughts about uniform

motion, which of course he never called ‘‘inertial,’’ for he never used the word

inertia, which entered the vocabulary of natural philosophy only at the end of the

seventeenth century. Historians of science usually refer to Galileo’s ‘‘circular

principle of inertia’’ as derived from uniform motion along the horizontal (but

ideally circular) plane, which Galileo regarded as ‘‘neutral’’ and perpetual, being

neither natural (accelerated) nor violent (decelerated). In his mature works

(including the Dialogo), Galileo identifies this kind of (inertial) motion with

uniform motion along a circle, which is supposed to pertain mainly to celestial

bodies within a finite universe, while rectilinear motion is generally identified with

uniformly accelerated motion and is supposed to be characteristic of terrestrial

bodies.

In my opinion, Galileo’s 1638 attempt to measure the force of percussion in

terms of uniform velocity by means of his ‘‘machine’’ marks a crucial point both in

the theoretical and experimental development of classical dynamics. The fact that

it was one of his last experiments, probably performed under precarious condi-

tions, and the fact that Galileo never published his two additional Days of the

Discorsi might explain why its relevance and potential implications remained

neglected for so long.



Following Salviati’s argument just quoted above, we can see that the properties

of uniform and ‘‘neutral’’ motion along a horizontal plane (i.e. along a circular

pathway ideally coinciding with a terrestrial meridian) are transposed into an

analogous motion along the vertical (i.e. along a straight line passing through the

centre of the Earth): to use modern terms, this hints at a fundamental general-

ization of Galilean ‘‘circular inertia’’ towards Newton’s ‘‘rectilinear inertia.’’ By the

way, such a generalization can already be traced in Galileo’s account of the uni-

form motion (moto equabile) of raindrops falling from great heights, a problem

dating back to Aristotle’s Meteorologica. Their constant speed is in fact due to the

compensation between acceleration of gravity and aerodynamic attrition, when

they reach a critical velocity.25

Salviati/Galileo’s proposal is to measure the ‘‘momentum of percussion’’ of a

falling body by measuring the ‘‘momentum of velocity’’ it acquires and maintains

when its uniformly accelerated motion is converted into a uniform motion (along

the vertical). To say that in modern terms, if one of the two equal weights

m1 = m2 = m of the ‘‘machine’’ is left falling down from a given height h above its

resting point, then its final ‘‘momentum of velocity’’ (m)v—the strappata trans-

mitted to the system by the tension of the rope—should be equal to the constant

velocity (instantaneously) acquired by both masses in opposite directions when the

effects of gravity are removed and those of friction reduced to a minimum.

Knowing the period of a pendulum, the supposed velocity of the two masses can be

determined by measuring the space covered by them during a given number of

oscillations. Alternatively (the easiest way in this case), knowing the space covered

during their uniform motion, the supposed velocity of the two masses can be

measured by means of Galileo’s water-clock.26

SALV.: But since we are considering the speed acquired by one of these weights

in descent from some height while the other remains at rest, it will be good to

determine what and how much must be the speed with which both would be

moved after the [initial] fall of the one, this descending and the other ascending.

From what is already demonstrated, we know that a heavy body which falls

freely on departing from rest perpetually acquires a greater and greater degree

of speed; hence in our case, the greatest degree of speed of the heavy body,

while it descends freely, is that which it is found to have at the point at which it

commences to lift its companion. Now it is evident that this degree of speed will

not go on increasing when its cause of increase is taken away, this being the

weight of the descending body itself; for its weight no longer acts when its

propensity to descend is taken away by repugnance to rising of its companion of

equal weight. Hence the maximum degree of speed will be conserved, and the

motion will be converted from one of acceleration to uniform motion.27

Furthermore, Galileo has already demonstrated an important theorem in the

Third Day of the Discorsi, which might help him to control his experimental

results: the time t elapsed to cover a given space s with a uniformly accelerated



motion is equal to the time t elapsed to cover the same space s with a uniform

motion whose constant velocity v0 is equal to �v, where v is the final velocity

reached by the uniformly accelerated motion.*

SALV.: What the future speed will then be is manifest from the things demon-

strated seen in the [discussions of the] past days. That is, the future speed will be

such that, in another time equal to that of the [initial free] descent, double the

space of [free] fall would be passed.28

The possibility of converting uniformly accelerated motion to some equivalent

uniform motion is essential in Galileo’s preclassical treatment of acceleration,

before the development of infinitesimal calculus, precisely because of his non-

differential notion of velocitas within the framework of the classical theory of

proportions. For him, this is the only way to understand/formulate something

otherwise very difficult to conceive and even more to measure: the idea of an

instantaneous velocity.

As a matter of fact, his assumption about the final velocity of m1 and m2 is

wrong (the result obtained should be �v), because Galileo does not consider them

as forming an integrated system whose total mass is m1 ? m2, as we know today.

After all, it was not easy for him either to provide a measure of the final velocity of

the two equal masses, from an experimental point of view, or to figure out the

actual situation, from a theoretical perspective. As in the case of the experience of

the ‘‘water balance’’ performed in Padua, intuitions based on common sense were

difficult to avoid; the whole dialogue among Aproino, Sagredo, and Salviati is

definitely written to show to the reader the typical difficulties of a work in pro-

gress. Ultimately, Galileo’s confinement in Arcetri and his blindness from 1639

onwards prevented him from carrying on his studies (figure 4).

As noted above, the Fifth and Sixth Day of the Discorsi were published only in

the Florentine edition of Galileo’s works (1718). In 1638, Lodewijk Elzevier

(1604–1670) in Leyden was impatient to have Galileo’s manuscript of the Discorsi

ready for publication. He decided to print the first four Days and the Appendix on

the centers of gravity of solid bodies without waiting for the two additional Days,

which Galileo was dictating to his friend and collaborator Marco Ambrosetti.

They were not included in the first edition of Galileo’s work (Bologna, 1656), so

they remained unpublished and unknown to most of Galileo’s pupils.29 Viviani

himself, who had inherited all of his master’s papers, just mentioned these addi-

tional Days in his Quinto libro degli Elementi d’Euclide (1674) as dealing

respectively with the Euclidean theory of proportions (reappraised in Viviani’s

book) and the forza della percossa, without giving more detailed information

about their content.30

* An equivalent formulation is: a body moving with a constant velocity v will cover a space2s in a given time t, where s is the space covered in the same time by a body moving with auniformly accelerated motion from v0 = 0 to vt = v.



Challenging Experiments, Challenged Solutions

I have focused my attention on the 1638 experiment with the ‘‘machine,’’ in par-

ticular on the special case of two equal weights hanging on a pulley, because this

last achievement of the Discorsi is explicitly presented in the Sixth Day as an

evolution of the experience performed more than thirty years before, thus showing

a striking continuity of method and issues in Galileo’s lifelong attempts to cope

with the force of percussion. Both in the case of the balance built in Padua and of

the apparatus used in Arcetri, the final outcome of the experiment was in con-

tradiction with its initial assumptions and with the heuristic strategies they implied.

In his 1604–1608 experiments, Galileo did not consider the water column falling

from the upper to the lower bucket hung on one arm of the balance to be literally

weightless, until some resistance occurred against its accelerated motion: some-

thing that was expected to contribute to the behavior of the system must be

excluded. In his last experiment on free fall and percussion, the situation was

Fig. 4. A modern reinterpretation of ‘‘Galileo’s machine’’ (2007). Credit: Roberto Vergara

Caffarelli (Fondazione Galileo Galilei, Pisa).



exactly the opposite: m1 should only compensate the gravity (i.e. the accelerated

motion) of m2, so that both are expected to move with the final velocity acquired

by m2, but they actually behave as a unique body with total mass

m1 ? m2 = 2m. Thus, something supposed not to affect the dynamics of the sys-

tem must be taken in account.

From a historico-epistemological point of view, Galileo’s late notes on free fall

and the force of percussion would deserve to be systematically compared with

cross-references to his correspondence, especially with Viviani, Giovanni Battista

Baliani (1582–1666), and Evangelista Torricelli (1608–1647).31 No less important

are other materials related to the Discorsi, some of them dating back to the

Paduan period, in order to highlight the elements of conceptual innovation implied

by ‘‘Galileo’s machine,’’ as well as the theoretical, experimental, and contingent

limitations which affected the 1638 experiment.

It would also be very tempting to regard that mechanical system—and its

variants with single or double inclined planes—as a direct ancestor of the more

famous device built and patented in 1784 by George Atwood (1745–1807). Despite

their material identity, however, the two machines definitely have different

functions: the former is a work-in-progress invention designed to measure the

Galilean ‘‘momentum of percussion’’ by trying to reduce it to the sole ‘‘momentum

of velocity,’’ suppressing the ‘‘momentum of gravity’’ as an otherwise inseparable

cofactor in determining the impact force of a falling/projected body; the latter is a

demonstrative apparatus (also useful for didactic purposes) designed to show the

validity of Newton’s principles of dynamics. Therefore, Galileo’s and Atwood’s

devices embody very different models of mathematical physics, as well as two

distant stages in the long and complicated transition from early modern to classical

mechanics (figures 5–6).32

Logical difficulties, unstable concepts, and related terminological oscillations

are particularly significant in Galileo’s account of the forza della percossa, since

they reflect the epistemic uncertainty, the questionable status of this problematic

and elusive physical entity. At the same time, they testify to the fundamental,

leading role played by the issue of measuring the force of percussion in the con-

struction of the Galilean ‘‘two new sciences’’ [due nuove scienze] at the intersection

of kinematics and dynamics, free fall and projectile motion, the study of the

resistance of materials, and the foundations of a general theory of shocks.

The main theoretical challenge faced by Galileo was also the main reason why

he had to give up on his project of an additional Day of the Discorsi entirely

devoted to percussion. On the one hand, he needed to isolate either the contri-

bution of gravity or that of velocity to treat the momento della percossa through a

‘‘quasi-infinitesimal’’ approach (by means of Cavalieri’s method of indivisibles33)

as an ‘‘aggregate’’ of infinite number of momenti di gravita or momenti di velocita

uniformly acquired through time.34 On the other hand, he was not able to go

beyond the traditional Euclidean theory of proportions in order to conceive

velocity (and acceleration) in differential terms as derived from a direct ratio



between space and time: so that in Galileo’s language the word forza—sometimes

replaced by energia as a synonym of effect(iveness)—essentially remained unde-

fined and ambiguous. The introduction of a notion like that of ‘‘quantity of

Fig. 5. The pulley in Galileo’s Meccaniche (ca. 1594–1598). Source: Galileo Galilei, Opere (ref.

4), 2:172. Credit: Library of the Galileo Museum, Florence.

Fig. 6. The pulley in Galileo’s Meccaniche (ca. 1594–1598). Source: Galileo Galilei, Opere (ref.

4), 2:173. Credit: Library of the Galileo Museum, Florence.



motion’’ [quantitas motus] and later that of kinetic energy, indispensable for a

systematic treatment of impact dynamics, would necessarily require this concep-

tual shift from proportions to derivatives, from indivisibles to infinitesimals.35

Acknowledgements

Special thanks to Roberto Vergara Caffarelli (Department of Physics, University

of Pisa) for his inspiring reconstructions of Galileo’s experiments, as well as for the

coinage of the name ‘‘Galileo’s machine’’ itself. I am also very grateful to

Massimiliano Badino (Centre d’Historia de la Ciencia, Universitat Autonoma de

Barcelona) for his helpful remarks during the writing of my paper.

References1 Jurgen Renn, ‘‘Galileo’s Theorem of Equivalence: The Missing Keystone of His Theory of

Motion,’’ in Trevor H. Levere and W. R. Shea, eds., Nature, Experiments, and the Sciences: Essays

on Galileo and the History of Science in Honour of Stillman Drake (Dordrecht: Kluwer Publi-

cations, 1990), 77–103; Renn, ‘‘Proof and Paradoxes: Free Fall and Projectile Motion in Galileo’s

Physics,’’ in Peter Damerow, Gideon Freudenthal, Peter McLaughlin, and Jurgen Renn, eds.,

Exploring the Limits of Preclassical Mechanics (Berlin: Springer, 1992), 126–268; Renn, Peter

Damerow, Simone Rieger, and Domenico Giulini, ‘‘Hunting the White Elephant: When and How

did Galileo discover the Law of Fall?’’ Science in Context 13 (2000), 299–419; Renn, ed., Galileo in

Context (Cambridge: Cambridge University Press, 2001); Matteo Valleriani, Galileo Engineer

(Berlin: Springer, 2010).2 Gianni Micheli, Le origini del concetto di macchina (Florence: La Nuova Italia, 1995).3 Thomas B. Settle, ‘‘An Experiment in the History of Science,’’ Science 133 (1961), 19–23.4 Galileo Galilei, Discorsi e dimostrazioni matematiche intorno a due nuove scienze, attenenti la

meccanica e i moti locali (Amsterdam: Elzevier, 1638); Galilei, Opera, ed. A. Favaro, I. Del

Lungo, U. Marchesini (Florence: Giunti-Barbera, 1890–1909; second edition 1939, third edition

1964), 8:323–324; Galilei, Two New Sciences, tr. S. Drake (Madison, WI: University of Wisconsin

Press, 1974), 283–284.5 Galilei, Opera (ref. 4), 8:324–325; Two New Sciences (ref. 4), 284–285.6 Galilei, Opera (ref. 4), 8:325; Two New Sciences (ref. 4), 2857 Ernst Mach, Die Mechanik in ihrer Entwicklung, historisch-kritisch dargestellt (Leipzig: Brock-

haus, 1883; second ed. 1908); The Science of Mechanics: A Critical and Historical Account of Its

Development, tr. T. J. McCormack (Chicago: The Open Court Publishing Co., 1919); La mecca-

nica nel suo sviluppo storico-critico, tr. A. D’Elia (Turin: Bollati Boringhieri, 1977), 149–178.8 Galilei, Opera (ref. 4), 8:329; Two New Sciences (ref. 4), 289.9 Galilei, Opera (ref. 4), 8:331–332; Two New Sciences (ref. 4), 292.10 Joseph-Louis Lagrange, Mecanique Analytique (Paris: Desaint, 1788; fourth edition 1811),

1:158–159: ‘‘Galilee a fait le premier ce pas important [i.e. the law of free fall and the parabolic

trajectory of projectiles], & a ouvert par-la une carriere nouvelle & immense a l’avancement de la

Mechanique. Ces decouvertes sont exposees et developpees dans l’ouvrage intitule: Dialoghi delle

Scienze nuove, &c., lequel parut pour la premiere fois a Leyde en 1637: elles ne procurerent pas a

Galilee, de son vivant, autant de celebrite que celles qu’il avoit faites sur le systeme du monde,

mais elles font aujourd’hui la partie la plus solide & la plus reelle de la gloire de ce grand homme.’’



[Galileo made first this important step, through which he opened a new and fruitful pathway for

the advancement of mechanics. These discoveries are exposed and developed in the work entitled:

Dialoghi delle scienze nuove, &c., which appeared for the first time in Leiden in 1637: they did not

give Galileo the same celebrity as those he made about the system of the world, during his life, but

nowadays they represent the most solid and significant part of the glory of this great man].11 Renn, ‘‘Galileo’s Theorem’’ (ref. 1).12 Pierre Souffrin, ‘‘Du mouvement uniforme au mouvement accelere: une nouvelle lecture du

theoreme du plan incline dans les Discorsi de Galilee,’’ Bollettino di Storia delle Scienze Mate-

matiche 6 (1986), 135; Souffrin, ‘‘Sur l’histoire du concept de vitesse d’Aristote a Galilee,’’ Revue

d’histoire des sciences 45 (1992), 231–267; Enrico Giusti, ‘‘Il problema della velocita in Galilei,’’ in

Enrico Giusti and Enrico Bellone, ed., Argomenti di storia della scienza: Matematica e Fisica

(Pavia: University of Pavia, 1986), 33–64.13 Galileo Galilei, Opere di Galileo Galilei (Florence: alla Condotta, 1715–1718).14 Cf. Paolo Galluzzi, Momento: Studi galileiani (Rome: Edizioni dell’Ateneo & Bizzarri, 1979).15 Galilei, Two New Sciences (ref. 4), 304.16 Galilei, Opera (ref. 4), 8:344; Two New Sciences (ref. 4), 304.17 Galilei, Opera (ref. 4), 8:332–333; Two New Sciences (ref. 4), 293–294.18 Galilei, Opera (ref. 4), 8:333; Two New Sciences (ref. 4), 294.19 Galilei, Opera (ref. 4), 8:333–334; Two New Sciences (ref. 4), 294.20 Galilei, Opera (ref. 4), 8:334; Two New Sciences (ref. 4), 294.21 Galilei, Opera (ref. 4), 8:334–335; Two New Sciences (ref. 4), 294–295.22 Galilei, Opera (ref. 4), 8:335; Two New Sciences (ref. 4), 295.23 Galilei, Opera (ref. 4), 8:335–336; Two New Sciences (ref. 4), 296.24 Galilei, Opera (ref. 4), 8:336–337; Two New Sciences (ref. 4), 296–297.25 Roberto Vergara Caffarelli, ‘‘Il principio di inerzia negli ultimi scritti di Galilei,’’ Cuadernos

Valencianos de Historia de la Medicina y de la Ciencia 10 (2007), 63–85.26 Thomas B. Settle, ‘‘Galilean Science: Essays in the Mechanics and Dynamics of the Discorsi,

PhD diss., Cornell University, 1966; Settle, ‘‘Galileo’s Use of Experiment as a Tool of Investi-

gation,’’ in Ernan McMullin, ed., Galileo, Man of Science (New York: Basic Books, 1968),

315–337; Settle, ‘‘Galileo’s Experimental Research, preprint (Berlin: Max Planck Institute for the

History of Science, 1996).27 Galilei, Opera (ref. 4), 8:337; Two New Sciences (ref. 4), 297–298.28 Galilei, Opera (ref. 4), 8:337; Two New Sciences (ref. 4), 298.29 Galileo Galilei, Opere (Bologna: Evangelista Dozza, 1665–1656).30 Vincenzo Viviani, Quinto libro degli Elementi d’Euclide (Florence: alla Condotta, 1674–1676).31 G. B. Baliani, De motu naturali gravium, fluidorum et solidorum (Genoa: Typographia Johannis

Mariae Farroni, Nicolai Pesagnij et Petri Francisci Barherij, 1638); Evangelista Torricelli, De motu

gravium naturaliter descendentium, et projectorum (1644), in Opere di Evangelista Torricelli, ed.

Gino Loria and Giuseppe Vassura (Faenza: Stabilimento Tipo-Litografico Montanari, 1919–1944),

vol. 2.32 Roberto Vergara Caffarelli, Galileo Galilei and Motion: A Reconstruction of 50 Years of

Experiments and Discoveries (Berlin: Springer, 2009). See also https://www.youtube.com/

watch?v=SGrO4X1Newo&list=UUVfq6BX8XmjCoJJdxRMCpgg, accessed November 20, 2014.



https://www.youtube.com/watch?v=SGrO4X1Newo%26list=UUVfq6BX8XmjCoJJdxRMCpgg

https://www.youtube.com/watch?v=SGrO4X1Newo%26list=UUVfq6BX8XmjCoJJdxRMCpgg

33 Bonaventura Cavalieri, Geometria indivisibilibus continuorum nova quadam ratione promota

(Bologna: ex Typographia de Ducijs, 1635).34 Francois De Gandt, ‘‘Cavalieri’s Indivisibles and Euclid’s Canons,’’ in Peter Barker and Roger

Ariew, eds., Revolution and Continuity: Essays in the History and Philosophy of Early Modern

Science (Washington DC: The Catholic University of America Press, 1992).35 Antoni Malet, From Indivisibles to Infinitesimals: Studies On Seventeenth-Century Mathemati-

zations of Infinitely Small Quantities (Barcelona: Universitat Autonoma de Barcelona, 1996).

Via Gaspare Morardo 3/10

18100 Imperia, Italy

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