The Pendulum Swings again: A Mathematical Reassessment of

The Pendulum Swings again: A Mathematical

Reassessment of Galileo’s Experiments with Inclined Planes

Alexander Hahn

, Notre Dame

Abstract. After over 300 years of scrutiny, the subject of Galileo continues to be

pursued with unabating intensity. Dava Sobel’s Galileo’s Daughter points to the

popular interest in the man and his legacy. The Catholic Church, understandably

interested in dispelling the notion that its censure of Galileo centuries ago is proof

positive that religious faith and science as well as ecclesiastical authority and free

pursuit of scholarship are irreconcilable, continues to oﬀer explanations. New books,

articles and conferences probe both in breadth and in depth the magnetic ﬁeld charged

by Galileo, science, and the Church.

Galileo’s analysis of the physics of motion has also received considerable atten-

tion. In particular, a great deal has been written during the past thirty years about

the structure and objectives of three experiments with inclined planes. Galileo had

carried them out in Padua and recorded them in his working papers. The assessments

of the three experiments diﬀer widely in points of detail, but all regard them as so-

phisticated, ingenious, and remarkable. This article presents a new critical study of

these experiments. Its conclusion is that one of the experiments is indeed a success,

but that the other two fail and are abandoned because Galileo did not have a ﬁrm

enough grip on the underlying physical principles and mathematical relationships.

1. Galileo on Motion. Galileo’s journey of discovery of the laws of motion is

lengthy, twisted, and anything but smooth. We recall it only brieﬂy. It is described

in [10], [11] and, in compelling detail, in [38].

Galileo’s early eﬀorts in Pisa (1589-1592) to understand the phenomenon of motion

are the subject of the De motu manuscripts. Central to Galileo’s explanations are

two underlying concepts, a basic one of uniform speciﬁc speed of fall and the auxiliary

one of an impressed force, that account for the nonuniform motions actually observed.

The uniform speciﬁc speed of the object depends on the medium in which the object

moves and is determined - via Archimedean hydrostatics - by the diﬀerence in the

densities between object and medium. The impressed force is something that is put

into an object by an external mover. Once imparted, this impressed power decays

I wish to thank Ernan McMullin for much information about Galileo and his scientiﬁc pursuits,

and in particular for drawing my attention a number of years ago to the research of Stillman Drake

on Galileo’s working papers. Thanks also go to Noel Swerdlow for very useful insights, and to Neil

Delaney, David Kirkner, and Timothy O’Meara for many stimulating discussions about Galileo and

his work.

gradually and the speed of the body changes steadily until the impressed power is

completely dissipated and the object assumes its natural constant speed speciﬁc to

the medium in which the motion takes place. Galileo’s thinking during this time is

still heavily inﬂuenced by Aristotelian and medieval elements. Only towards the end

of the De motu does Galileo begin to accept the possibility that acceleration may be

a fundamental feature of fall.

During the Padua period (1597-1610) Galileo’s understanding of motion undergoes

transition. His 1604 letter to Fra Paolo Sarpi provides important testimony: ”the

spaces passed in natural motion are in proportion to the squares of the times taken,

and consequently that the spaces traversed in equal successive time intervals are to

the odd numbers ...” Diverted by his interests in astronomy and the well documented

conﬂict with the Church, Galileo does not present the ﬁnal synthesis of his theory

of motion until 1638. The Discorsi [2] is the exposition of a learned, four day long

conversation among Salviati who represents Galileo himself, Sagredo an open minded

supporter of the new science, and Simplicio, an adherent of the old Aristotelian

point of view. Days three and four of these discussions focus in vernacular Italian

on the treatise De motu locali - a book within a book in scholarly Latin - that

is Galileo’s own systematic treatment of motion. It is intended to be a clear and

rigorous presentation in ﬁnal form. Errors and missteps from earlier stages of Galileo’s

thinking are visible in the probing comments of the protagonists. The discussion

analyzes the motion of objects undergoing constant acceleration, both in the situation

of free fall and along inclined planes. Galileo derives his conclusions deductively,

often with geometric constructions. He expresses quantitative relationships between

time, distance, velocity, and acceleration, in terms of proportions; indeed, he uses

only proportions of magnitudes of the same kind, for example, distance to distance,

velocity to velocity, but not distance to time. In particular, he does not have the

real number system and decimal notation at his disposal and does not formulate his

conclusions in terms of equations involving variables and constants. (The systematic

development of all this was only in its infancy at this time.) Within his context,

Galileo arrives at the following basic insights.

1. All bodies falling in a vacuum do so with the same constant acceleration. For a

body falling from rest, the speed is proportional to the elapsed time. This is so both

in the situation of free fall and for balls rolling on an inclined plane.

2. The law of fall, namely, that the distance covered by a body moving from rest

(again, either in free fall or rolling on an inclined plane) is proportional to the square

of the time of the motion.

3. The trajectory of a projectile has parabolic shape.

Galileo regards two underlying principles as fundamental. One is a principle of

inertia, namely that a body moving with a certain constant velocity (speed and

direction) will continue to move with that same velocity unless an outside force acts

on it

. The second is a principle of superposition of motions, namely that projectile

motion can be conceptualized as an independent composite of a motion with constant

velocity and a vertical motion with constant acceleration. It must be emphasized that

Galileo’s basic concepts (for example, velocity and acceleration) lack the precision that

calculus would later give them and his principles of inertia and superposition never

reach a deﬁnitive and general state. See [5], [10] and [38] for instance.

2. Galileo’s Experiments. There is general agreement that Galileo arrived at his

conclusions by combining the thinking of predecessors, geometric deductions, thought

or ”arm chair” experiments, and actual ”hands on” experiments. However, the as-

sessments of the relative importance of these elements have changed over the years.

In particular, the question - central to an understanding of Galileo the physicist -

as to the quality and purpose of Galileo’s hands on experimentation has received a

diversity of answers.

Galileo’s contemporaries Descartes and Mersenne exercised skepticism about his

experiments. See [5, p. 107] and [8, p. 20]. For the most part, however, historians

have taken Galileo’s word for it (as expressed on the third day of the Discorsi ) that

he used carefully constructed experiments as an important tool of both discovery and

veriﬁcation of his fundamental insights into motion. This includes Kant [3, preface]

and Mach [4]. This point of view prevailed until the late 1930’s when the inﬂuen-

tial historian Koyr´e proclaimed that Galileo’s real experiments were most certainly

inadequate and that he relied on thought experimentation. See [5, pp. 106-107]

and [6, p. 224]. The pendulum begins to swing back in 1961 when Settle [8] recon-

structed Galileo’s basic apparatus and demonstrated that Galileo could have carried

out with satisfactory precision the experiments with inclined planes he described.

While emphasizing the point that the progressive evolution of Galileo’s understand-

ing of motion must surely have had an experimental aspect, essays written as recently

as 1967 were still circumspect and tentative about the precise nature and impact of

these experiments. See [10, pp. 11-12] and [11, pp. 328-329]. The backward swing

of the pendulum gained speed instantly in the early 1970s when Drake [12] perused

Galileo’s unpublished working papers on motion in the Biblioteca Nazionale Centrale

It is important to note that in this article ”inertia” will be used to refer simply to a principle

of uniform motion and not to Newton’s deﬁnitive version of the concept. For Newton, inertia is

an inherent property of the mass of a body that causes it to resist any change in its state of rest

or motion along a straight line. This article parallels Galileo understanding of inertia as only a

kinematic principle or an assertion about the motion of an object under ideal conditions.

in Florence and found written records of Galileo’s experiments with inclined planes.

They are the ”smoking gun” that refutes the position of Koyr´e. These working papers

- there are 160 sheets or folios - are now bound as Volume 72, or Codex 72, of the

Galileo manuscripts. The websites

http://www.imss.ﬁ.it/ms72/INDEX.HTM

http://www.mpiwg-berlin.mpg.de/Galileo

−

Prototype/index.htm

provide wonderful and very useful electronic renditions of the folios of the codex.

Many of the folios are drafts of theoretical discussions that would later appear in the

Discorsi. (The electronic version provides precise connections with cross-references.)

Many folios are ﬁlled with computations. A number of folios, 80r, 81r, 86a r, 87, 90r,

91v, 102, 107, 111, 113r, 114v, 115r, 116v, 117, 152r, and 175v among them, contain

diagrams and data that suggest studies of motion. The abbreviations r and v stand

for ”recto” and ”verso”, the ”front” and ”back” of the sheet in question. (In the

listing just given, if neither r nor v appears, then both sides of the sheet are relevant.)

Some of these folios are geometric explorations of the parabola and some are records

of experiments. The historians who have studied them consider it ”well substantiated

by the evidence” (watermarks, for example) that they stem from the later Paduan

period 1604-1610. For example, see Naylor [20, p. 366].

The present article will focus on 81r, 114v and 116v. Each of these folios gives

evidence of an experiment in which Galileo has placed an inclined plane on a table,

lets a ball roll down the plane, and records quantitative data about the ball’s ﬂight

from the table’s edge to the ground. Salviati informs us on the third day of the

Discorsi that Galieo repeated some of his experiments ”a full hundred times.” Thus

it would seem that each recorded measurement represents a cluster of trials. The

general conclusions of Drake [12, 27, 32, 36, 37], Drake-MacLachlan [16], Naylor [13,

18, 19, 20, 25, 26, 28], and Hill [33, 35] - these are the historians who have studied

them most thoroughly - are in agreement:

Drake [32, p. 4] uses folios 81r and 114v to conclude that Galileo is a ”skilled

experimentalist capable of holding his results within a variance of four units ...” The

unit referred to here is Galileo’s punto, or ”point”, a unit of length slightly less than

one millimeter.

Naylor [18, pp. 168-169], reﬂecting about 81r, speaks of ”indications that Galileo

carried out meticulous, thorough-going studies of the form of projectile motion” and

suggests that ”Galileo had a striking talent for combining a mathematical approach

to nature with a considerable mathematical technique. The simplicity and power of

this particular form of experiment is quite remarkable.”

Hill [35, p. 666] comments that ”worksheets 81r, 114v, and 116v reveal an impres-

sive experimental program, ingenious in structure, ambitious in concept, eminently

successful in execution. This series of procedures enabled Galileo to provide powerful,

perhaps empirically decisive, evidence for both the new speed law and the parabolic

trajectory.”

It is a fact that Galileo’s record of the experiments on these folios omits impor-

tant details, in reference to both the descriptive and numerical elements. Thus, an

important ingredient in the studies of these folios has been the careful reconstruction

of the experiments from the information that Galileo does supply. These reconstruc-

tions - both actual and mathematical - become an important part of the evidence.

The numerical data that they generate is carefully compared with the analogous data

from Galileo’s record. These comparisons are used to inform the authors’ comments

about the plausibility of their reconstructions and the validity of their analyses of

the experiments. Unfortunately, in terms of particulars (for example, the inclination

of the inclined plane and release heights of the balls), these reconstructions as well

as the conclusions drawn from them - speciﬁcally the purpose and precision of the

experiments - diﬀer widely.

This state of aﬀairs calls for a sober re-examination of these folios. What aspects

of his insights about motion did Galileo put to the test? How precise were his experi-

ments? What conclusions can legitimately and compellingly be drawn from Galileo’s

record of them? Is there indeed convincing evidence that they were successful? The

answer to these questions is the purpose of the discussion that follows. The focus

will be on the folios themselves (rather than the reconstruction of the experiments)

and on related aspects of the Discorsi. The folios 116v, 81r, and 114v and all the

information on them are reproduced below. The originals can be studied at either of

the websites listed above. The organization of the calculations on 116v and 114v into

rectangular ”frames” follows the practice of the websites.

3. The Experiment of Folio 116v. The statement punti 828 altezza della tavola

tells us that Galileo recorded distances in units he calls punti (that is to say ”points”)

and that he had a table 828 punti high. There is agreement among the historians

already mentioned (based on evidence from folio 166r) that one punto is equal to

approximately 0.94 millimeters. The diagram together with the computations on

the folio conﬁrm that he placed an inclined plane on the table, ﬁxed an angle of

inclination, and released a ball (likely of bronze) from the respective heights h of

300, 600, 800, 828, and 1000

punti above the horizontal table top. Galileo might have made use of a curved de-

ﬂector to provide a smooth transition for the ball from the inclined plane to the

horizontal table. His sketch on folio 175v shows that he considered such deﬂectors.

After a short run on the table, the ball ﬂew oﬀ to land on a horizontal ﬂoor. Galileo

measured the respective distances from the point of impact of the ball to the base of

300

600

800

828

1000

300 . 800

800

2 1 3 3

6 4 0 0 0 0

2 1 3 3

800

1 3 0 6

1 7 0 6 4 0 0

2 3

260

300

2 2 0 8

300 . 828

800

2 2 0 8

6 6 2 4 0 0

800

1 3 2 9

1 7 6 6 4 0 0

2 3

300 . 1000

800

2 6 6 6

8 0 0 0 0 0

300

2 4 0

8001172

1328

1340

1500

doveria 1460

dria 41

doveria esser 1306

doveria per rispondere al primo esser 1131

punti 828 altezza della tavola

doveria 1330

2 6 6 7

8 0 0

1 4 6 0

2133600

1 1 3 1

1600

800

1280000

2 1

2 2 3

731

C06

C07

C09

C01

1 4 3 0 6 2

6.2

2.2

1640

820

1344800

1 1 5

32800

1312

30 . 60

160 0

480 0

30 . 80

300 640000

2 1 3

1394

1603

1866

236

247

2 1

2 2 5

286

1 3 7

262

264

2920

C08

C10

C11

C12

1 6 1 6 0 4

2 8 4 9 0 0

C03

C02

C05

C04

dria 10

dria 22

dria 40

Folio 116v (size of original: 306 by 207 mm)

the table (the point directly below the start of the ball’s ﬂight) and recorded these

on the folio as

(a) 800, 1172, 1328, 1340, and 1500

punti. These are the experimental values that correspond to the various heights of

release listed above.

3A. Understanding the Folio. We now turn to the analysis of the experiment

as well as the computations that Galileo carried out. Consider the ball in its initial

position on the inclined plane. Let

h = the height of the ball above the table, and

d = the distance from the ball to the bottom of the inclined plane.

Now release the ball and let

t = the time it takes for the ball to descend to the bottom of the plane,

v = the speed of the ball at time t. This is also the speed of the ball at

the beginning of its fall from the table. Finally, let

R = the distance from the point of impact of the ball to the point on the

ground precisely below the starting point of the ball’s ﬂight.

At the time of the experiment - before the end of the Paduan period in 1610 -

Galileo had discovered, or at least wrestled with, all essential aspects of his program

on motion as outlined in Section 1 above. In particular, he was in a position to put

to the test the proportion

(i) v ∝ t

as well as the square law

(ii) d ∝ t

(deduced from (i) in Proposition II. Theorem II of the Discorsi). From his principle

of inertia he could assume that the horizontal component of the velocity is constant

throughout the ball’s ﬂight and hence equal to v. (Given the relatively small velocities,

distances and times, Galileo could safely assume that air resistance would not play a

signiﬁcant role. See [19, p. 408].) In reference to the vertical component of the ball’s

ﬂight, Galileo knew that the time of fall of the horizontally projected ball from the

table to the ground is independent of its starting velocity v. So this time is equal to

the time t

that it takes for a ball to fall vertically from rest through the height of the

table. Notice that these observations rely on the principle of superposition. Galileo

can conclude that

(iii) R ∝ v

with t

the constant of proportionality. By similar triangles (the angle of inclination

of the inclined plane is ﬁxed) h ∝ d. After putting the above proportions together,

Galileo has

(iv) h ∝ d ∝ t

∝ v

∝ R

Therefore, R

∝ h. So, if releases of the ball at the heights of h

and h above the

table result in points of impact at the respective distances of R

and R from the foot

of the table, then

(v)

It is this relationship that the experiment recorded on folio 116v is designed to con-

ﬁrm. Galileo’s next step is to insert the values h

= 300 and R

= 800 from the

experiment. By doing so, he in eﬀect determines, or at least approximates, the con-

stants of proportionality that link R

and h, or equivalently, R and

√

h. The equation

(vi) R =

800

√

300

√

captures what he does. It remains for Galileo to compute R for h successively equal to

600, 800, 828, and 1000, and to compare the resulting values with the measurements

for R that were provided - see (a) - by the experiment. The successive values for R

that Galileo computes are (in punti)

(b) — , 1131, 1306, 1330, and 1460.

The — refers to the value R = 800 that was used along with the corresponding

h = 300 to obtain (vi).

Galileo records these numbers on the folio with the phrase doveria esser (or simply

doveria) meaning ”ought to be.” He also includes his calculations. For example,

the calculation for h = 600 is carried out in frame C01. Galileo ﬁrst computes

800·800·600

300

= 1600 · 800 = 1280000 and calculates R =

√

1280000 = 1131.

For h = 800, this is done in C06. For h = 1000, the computation can be seen in

frames C09 and C08. In C09, Galileo computes 1000 ×800 = 800000 and divides this

result by 300 to get 2666. In C08, he multiplies the more accurate value 2667 of this

computation (the actual value is 2666

) by 800 to get 2133600. This is R

.TogetR,

he calculates

√

2133600 = 1460. The computation in frame C10 is analogous to that

of C01 and suggests that Galileo also considered a table height of 820 punti. Note

that some of the computations are only approximations and that the computation

√

1344800 = 115 in frame C10 is incomplete. In the course of computing the square

root of a number, Galileo crosses the digits of the number out. In the rendition of

the folio above these numbers are entered in a lighter shade.

Galileo compares his experimental values (a) to his theoretical values (b) and

records the respective diﬀerences of 41, 22, 10, and 40 punti using the abbreviation

dria for diﬀerentia. The fact that the theoretical values fall short of the experimental

values (from about 1 to 4 centimeters) seems contrary to expectation. After all,

the experimental values are subject to the retarding eﬀects of the imperfections in

Galileo’s experimental setup, whereas the theoretical values are not. The explanation

is provided by the fact that Galileo’s theory, as captured by equation (vi), depends

on one data point from the experiment. We will see, in particular, that the measured

distance of 800 punti (corresponding to the height of 300 punti) falls short of the

predicted mark. So the constant

800

√

300

is too small, and thus all of Galileo’s computed

values are too small as well.

We turn next to the question of the precision of the experiment of folio 116v.

We will test the accuracy of the experimental values (a) against the predictions of

elementary mechanics. (Galileo’s theory can’t be used because it depends on his

experiment.) We will only outline these matters here. The details are available in

many texts, for example, in Chapter 9.3 of the basic calculus text [42]

. Note that

the analysis that follows goes far beyond what Galileo was familiar with.

3B. The Underlying Mathematics. Return to the ball on the inclined plane and

assume that the ball is homogeneous. Let t = 0 be the instant at which it is released.

For any time t ≥ 0, let f(t) be the frictional force on the rolling ball (a priori it

depends on t). This is the force that rotates the ball. Assume that there is neither

slippage (as the ball would experience on a frictionless surface) nor any additional

retardation of the motion down the plane (as would be the case if the surface were

”bumpy” or ”sticky”). The connection between the torque produced by the frictional

force, the resulting angular acceleration of the ball, and the ball’s index of inertia (this

connection is provided by the rotational analogue of force = mass × acceleration),

My interest in the experiments of Galileo had its beginning in my eﬀorts to develop applications

of calculus with interesting historical connections for this book.

leads to the equation

f(t)=

ma(t)

where m is the mass of the ball and a(t) is its linear acceleration down the plane. By

Newton’s second law and the fact that the component of gravity down the plane is

F = mg sin β, where β is the angle of inclination of the plane, we get

ma(t)=F − f(t)=mg sin β −

ma(t) ,

and therefore,

a(t)=

sin β.

This informs us in turn that the velocity of the ball at the bottom of the plane is

v =



gh. (Alternatively, this equation can be established by using the law of

conservation of energy. See [17, pp. 398-399].) Combining this with one of the basic

equations of projectile motion and letting y

be the height of the table, provides the

connection

R =2



√

between the starting height h and the distance R from the point of impact of the ball

to the foot of the table. With the substitution y

= 828 this equation becomes

(vii) R =2



828

√

Plugging Galileo’s starting heights of 300, 600, 800, 828, and 1000 into equation

(vii) for h, we get the values (again in punti)

for the corresponding distances R.

This model applies to the ideal situation: a perfectly round and homogeneous

ball; a path that is perfectly smooth and ﬂat with no tilts other than the inclination

of the plane; a force of friction that rotates the ball without slippage but provides

no additional impedance; and a deﬂector that provides a perfectly smooth transition

from the plane to the table. In addition, to conform to the situation of the model, the

table as well as the ﬂoor on which the ball impacts need to be perfectly horizontal.

There is, of course, no such perfection in the context of Galileo’s experimental setup.

In sum, the expectation is that the ball will land short of its theoretical target. A

comparison of the lists of numbers (a) and (c) conﬁrms this. We know, of course,

from the discussion on the third day of the Discorsi, that Galileo is fully aware that his

fundamental laws of motion apply only in idealized situations and that any experiment

or ”real” situation will encounter ”impediments.” Notice that the ”bottom lines” of

the analyses of Sections 3A and 3B, namely the equations (vi) and (vii), diﬀer only

in the value of the constant, and that

800

√

300

≈ 46.19 falls short of the correct value



· 828 ≈ 48.64.

So far we have said nothing about the groove that guides the ball down the plane.

The description of an inclined plane experiment in the Discorsi [2, Crew-Salvio p. 171,

compare Drake p. 169] informs us that there was a channel ”a little more than one

ﬁnger in breadth” cut into the inclined plane, and that ”having made this groove very

straight, smooth, and polished, and having lined it with parchment, also as smooth

and polished as possible, we rolled along it a hard, smooth, bronze ball ...” The

fact that Galileo says nothing speciﬁc about the groove presents a problem, because

diﬀerent conﬁgurations of the cross-section require diﬀerent theoretical explanations.

We now let d be the diameter of the ball and consider the most likely possibilities.

If the cross-section of the groove is a circular arc of radius greater than the radius

of the ball, then in the ideal situation, the ball will roll on the bottom of the groove

throughout its descent. This is a situation to which the mathematical model already

described applies. Asssume next that the groove has rectangular cross-section and let

w>0 be its width. If d ≤ w, then the ball is supported by the bottom of the groove

and rolls entirely within the groove. Again, the model already described applies. But

if d>wand the groove is deep enough, then the rolling ball does not touch the

bottom of the groove and is instead supported by its two edges. In this case, the

dynamics are diﬀerent. The mathematical model of this situation (obtained by an

analysis similar to that above) provides the relationship

(viii) R =2



−w

√

This equation also applies to a groove with a cross-section in the shape of an isosceles

triangle, if w is taken to be the distance between the two points of contact of the ball

with the groove. Let y

= 828 punti be the height of the table. Because

−w

> 1,

the value of equation (viii) is less than the value of equation (vii) for any h>0. In

particular, the values for R that equation (viii) supplies for the respective starting

heights h equal to 300, 600, 800, 828, and 1000 are less than the values (c) supplied by

equation (vii). Hence the values provided by equation (viii) will be closer to Galileo’s

experimental values (a).

Now to the comparison of Galileo’s experimental data against the predictions

of the theory. It follows from the analysis of the cross-section of the groove that

the respective diﬀerences between the experimental data (a) and the predictions

experiment, these diﬀerences provide the worse case scenario. The diﬀerences are

−42 = 800 − 842, −19 = 1172 − 1191, −48 = 1328 − 1376, −60 = 1340 − 1400 and

−38 = 1500 − 1538 punti. In terms of percentages, this amounts to −5.0%, −1.6%,

−3.5%, −4.3%, and −2.5%, respectively. What can be said about this discrepancy?

While the inclined planes used by Galileo seem no longer to exist, we do know - see

[34] for example - that the apparatus that Galileo used in other investigations was well

crafted. The physicists Shea and Wolf [17], considering the many sources of possible

experimental error in the folio 116v experiment, regard the data generated by Galileo

to fall ”within acceptable limits of experimental error.” All indications are that this

assessment is correct. For example, Naylor [13, pp. 109-111] reconstructed the folio

116v experiment with considerable care (the cross-section of the groove was a circular

arc of radius greater than

) and obtained distance data very close to Galileo’s.

4. The Experiment of Folio 81r. There is a consensus among historians - see [18],

[35], and [37, Chapter 8] - that folio 81r focusses its attention on the trajectories of

balls that are propelled obliquely into space after having descended down an inclined

plane placed on a table. In important contrast to folio 116v, the balls drop directly

from the inclined plane and there is no horizontal deﬂection. Each of the three

curves on the folio corresponds to a certain ﬁxed angle of inclination of the plane

and ﬁxed starting height of the ball. In repeated trials Galileo intercepts the ﬂight

of the ball with horizontal planes placed at diﬀerent heights and marks the points

of impact. Evidently, he starts by placing the intercepting plane at a distance of

53+53+77

+146 = 329

punti below the plane of the table and ”calibrates” the three

trajectories so that the the points of impact are at the respective horizontal distances

of 250, 250 + 250 = 500, and 250 + 250 + 250 = 750 punti from the table. He then

successively raises the intercepting plane as indicated and measures the horizontal

distances of the points of impact of the ball for each of the three trajectories. The

Latin phrase on the folio informs us that these vertical and horizontal distances are

given in the same scale. Galileo, interested in the shape of the three curves, has in

eﬀect provided ﬁve sets of coordinates for each of them.

Observe that other than the diagram, the folio contains only three small clusters of

numbers (some of which are related to those of the diagram). One of these considers

a group of four numbers and divides each by 4. However, there is nothing to suggest a

successfully completed experiment. Nor does Codex 72 appear to contain folios with

computations that inform 81r. Before jumping to conclusions, however, it is necessary

to look at 81r in the light of the relevant context. We know that Galileo regarded the

veriﬁcation of the parabolic law as a primary goal of his studies on motion. During

Altitudines in ab mensuratae sunt

cum scala maiorum transversalium bc

250 - 250

170 - 177

121 - 1301/2

81 -

121

1/2

1301/2

250

146

1/2

178

131

250

170

177

1/2

185

131

200 . 86

50 21 23

250

141

Folio 81r (size of original: 304 by 205 mm)

the Paduan period he invested considerable eﬀort towards this end. See [14, p. 336]

and [38, pp. 205-213] for instance. Given this, one would have to believe that Galileo

designed the experiment of folio 81r - at least in signiﬁcant part - to test the parabolic

hypothesis, and that he would have regarded the experiment a success only if it

provided solid evidence for or against it. Let’s consider what Galileo was up against.

The diagram below abstracts the essence of what is happening. It shows the path of

the ball down the inclined plane from right to left to the edge of the inclined plane

at the point O. Thereafter, the ball is in ﬂight and is shown in typical position B.

The plane determined by the incline and the horizontal plane of the table top are

denoted (both in cross section) by P and H. The line V is the vertical through O.

The distance from B to the line V is denoted by x and that from B to H by y. The

question facing Galileo for each of the three curves is this. Are the pairs x and y that

he singles out in the diagram of the folio in a relationship that is parabolic, or at least

close to parabolic?

Let’s analyze the matter in Galileo’s terms. The principle of superposition of

motion tells us that the motion of the ball from O to its typical position B is the

composite of an inertial component along P and a uniformly accelerated component

down to B. These components result in the vertical displacements D

and D

re-

spectively. The distance D

can be obtained by measuring up to the inclined plane

from the table top at a distance of x units from O. Notice that D

= c

x where c

is the slope of the inclined plane. What about D

? Let t be the elapsed time of the

motion of the ball from O to B. The proportion x ∝ t is a consequence of the inertial

motion. So t

∝ x

. By considering the downward accelerated motion, D

∝ x

Galileo could have approximated the corresponding constant of proportionality c

la 116v by using D

= y − D

and the ﬁrst data point (x =81,y = 53 for the inner

curve). Since he knew the basics about parabolas, it would remain for him to test

whether the pairs of distances x and y that he has recorded satisfy the equation

y = D

+ D

= c

+ c

in an approximate way.

Why doesn’t Galileo carry this out on folio 81r? Unlike Fermat and his famous

margin, he had more than room enough (on the bottom half of the page and the

empty reverse side). Or on a separate folio? Or elsewhere? The short answer is that

he was not able to do so. Why not?

We have just seen that a meaningful assessment of the data of 81r relies on the

principles of superposition and inertia. The principle of superposition isolates from

the ball’s ﬂight a component of motion along the non-horizontal line P and it is this

non-horizontal motion to which the principle of inertia is applied. The fact is, however,

that Galileo neither formulated nor used the principles of inertia and superposition

at this level of generality. The studies [5, pp. 186-187], [10, pp. 27-31], and [38, pp.

225-226] support this point. For example, on the ﬁrst day of the Dialogo [1] - its

publication in 1632 occurs more than twenty years after the experiment of folio 81r -

Sagredo tells us:

”In the horizontal plane no velocity whatever would ever be naturally

acquired, sice the body in this position will never move. But motion in

a horizontal line which is tilted neither up nor down is circular motion

about the center; therefore circular motion is never acquired naturally

without straight motion to precede it; but being once acquired, it will

continue perpetually with uniform velocity.”

Early on the fourth day of the Discorsi (published in 1638), the principles of inertia

and superposition are discussed only in the context of a motion that is the composite

of a horizontal motion of constant speed and a vertical motion that is accelerated.

Galileo then uses them to provide a proof of the parabolic law in the case of horizontal

projection. But he presents no such proof in the oblique case. There is convincing

evidence that Galileo’s protracted eﬀorts to come to grips with the obliquely projected

trajectory - the gunner’s problem - were unsuccessful. The analysis in [38, pp. 205-

213, 225-226, 241-243, 251-258, 262-264] draws this conclusion from a careful study

of both the Discorsi and the relevant theoretical studies in the folios of Codex 72.

Indeed, when the proof in the oblique case is supplied by Cavalieri in 1632, Galileo

reacts with dismay in a letter (see [7, p. 75] or [10, p. 23]) written in the same year

to his frequent correspondent Cesare Marsili:

”I cannot hide from your most excellent Lordship that the information

scarcely pleased me, seeing that the ﬁrst fruits of more that forty years’

study, of which I had revealed a large part in close conﬁdence to the

said Father, were to be taken away from me, and I was to be deprived

of that glory that I desired so ardently and that I was promising myself

after such long eﬀorts; for really my ﬁrst intention, that which incited

me to meditate upon motion, had been to ﬁnd that line, and though

I succeeded in demonstrating it, I know how much trouble I had in

arriving at that conclusion.”

The proof of the parabolic ”line” that ”said Father” Cavalieri and later Torricelli in

1644 provide are based on the more general principles of inertia and superposition.

See [5, pp. 236-244] and [38, pp. 274-276]. They are the same as those used in the

analysis of 81r presented above. In spite of Galileo’s assertion that he succeeded, the

demonstration he refers to does not appear in any of his published works, letters, or

manuscripts. The empty bottom half of folio 81r (as well as the empty reverse side) is

a concrete example of this gap. Galileo was unable to exploit the promising diagram

and abandoned this early eﬀort to analyze the trajectory of an obliquely projected

ball.

However, the diagram does provide information about the accuracy with which

Galileo observed and measured the trajectories. Return to the previous ﬁgure and

take the lines H and V to be the x and y axes of a standard coordinate system

with the punto as the unit of length. Because the horizontal and vertical distances

travelled by the balls were small, their speeds were small as well. So we will assume

that air resistance was negligible, and that the trajectories that Galileo observed were

precise parabolas. In terms of the coordinate system, each parabola is given by an

equation of the form y = −ax

+ bx, where a is a positive constant, b is non-negative,

and x ≤ 0. The line of the sloping plane P has equation y = bx and ax

measures the

vertical deviation of the motion of the ball from this line. (This parallels the earlier

analysis of folio 81r.) Let’s turn to the inner trajectory. In reference to the coordinate

system, Galileo’s data points are

(−81, −53), (−121, −106), (−170, −183.5), (−250, −329.5) .

The parabola of the form y = −ax

+ bx that ﬁts these data points best - in the sense

of least squares - is

(∗) y = −0.00351906 x

+0.446797 x.

(Any of several software packages - in the present case Mathematica - can be used to

supply this equation.) If Galileo’s cluster of data points were to fall on a parabola

precisely, then (∗) would be that parabola. Therefore, an assessment of the extent

to which they do not gives insight into the accuracy of Galileo’s measurements. The

x-coordinates of the points on parabola (∗) whose y-coordinates correspond to

−53, −106, −183.5, and − 329.5

are −74.7, −121.4, −173.5, and −249.0, respectively. When comparing these numbers

with the horizontal distances that the folio records for the inner curve, we see that

the diﬀerence 81 − 74.7=6.3 is the largest. The indication is that at least some

of the measurements deviate from the reality that Galileo observes by more than

6 punti. Similar analyses of the middle and outer curves provide inaccuracies in

excess of 9 and 12 punti. Inaccuracies of this size are hardly surprising considering

what is involved. Galileo has to set the intercepting planes perfectly horizontally

at the vertical distances of 53, 106=53+53, 183.5 = 53+53+77.5 and 329.5=

53+53+77.5 + 146 punti below the plane of the table, locate with precision the

points of impact of the ball, and measure the distances from these points to the table.

This analysis applies only to distance measurements involving the trajectory. In

the context of folio 116v, there will be additional inaccuracies having to do with the

inclined plane, the groove, and the ball. In particular, we can safely conclude that

the experimental inaccuracies will be much greater than 6 punti in each of the trials

of the folio 116v experiment. The claim by Drake that Galileo experimented ”within

a variance of four units” is therefore not sustainable.

5. The Experiment of Folio 114v. There is general agreement that the sketch

on folio 114v also depicts the trajectories of a ball that had rolled down an inclined

plane placed on a table. The angle of inclination is ﬁxed, the ball rolls down from

various heights, and the numbers

253, 337, 395, 451, 495, 534, and 573

are the recorded distances (in punti) of the corresponding points of impact of the ball

from the foot of the table. In contrast to 116v, there is no information about the

116

928

200

580

6728

8 2

141

116

200 .

846

141

1 6 3 5 6

8 1

472

413

4602

141

259

111

1369

1089

784

3242

5 6

9 9

1 0 8 9

784

18868

1 5 5

1 3 7

24135

621

404

4661

4900

4812

3925

6 2

9 1

135

193

253 337 395 451 495 534 573

1 4 1 . 5 9

2 0 0 . 1 4 1

7 8 . 2 6

20000

1 4 1

1 6 4

2 4

2 8

. . . . .

700

624

. . . . .

122

8417

335

181

. . . . .

267

19 9

1 0 8

305

1 6

1 10

C02

C03

C04

C05

C06

C07

C08

C09

C10

C12

C11

C01

Folio 114v (size of original: 285 by 202 mm)

starting heights of the ball. Nor is there any information about the initial direction

of the motion (because the angle of inclination of the plane is not supplied).

What information do Galileo’s calculations provide about the experiment? Frame

C01 computes 116 · 58 = 6728 and then

√

6728 = 82. Frame C02 calculates

141·116

200

81. Frame C03 shows only a fragment of a multiplication. (The top right corner of

the folio is missing). Frame C04 carries out

√

20000 = 141. Frame C05 computes

59·78

141

= 32 and lists two pairs of numbers. Frame C07 shows that the squares of 33

and 28 are 1089 and 784 respectively. Frame C06 shows the square of 37 to be 1369,

shows that the sum of 1369, 1089 and 784 is 3242, and calculates the square root of

this number as 56. Frames C08 and C10 compute

√

24135 = 155 and

√

18868 = 137.

Frame C09 computes the squares of 70 and 69 to be 4900 and 4661 respectively and

lists the number 4812. Frame C11 calculates

√

3925 as 62 and lists the more accurate

value 63. Frame C12, ﬁnally, computes

√

8417 = 91 and seems to point out that

135

193

approximately

. Note that some of these computations provide only approximations

and that 69

= 4761 (and not 4661). The study of these computations does not reveal

any apparent connections with the diagram on the folio and the sequence of distances

that are recorded on it. (Nor are such connections provided in previous studies of

folio 114v.) In particular, there appears to be no evidence on the folio that suggests

a successful experiment.

What might the purpose of 114v have been? The diagram on the folio shows that

the trajectories are oblique. Because the three folios under study were crafted at

about the same time, it seems reasonable to suppose that Galileo intended to pursue

for oblique projection what he had accomplished on 116v for horizontal projection.

So he might have attempted (perhaps among other things) to establish a relationship

between the distances R = 253, 337, 395, 451, 495, 534, and 573 of the points of impact

of the ball from the foot of the table and the other relevant parameters: the height

of the table, the angle of inclination φ

of the plane, and the heights h from which

the ball was released.

Let’s have a look at the relationship between these parameters that a current

mathematical model provides. Inserting the speed v =



gh that the ball has at

the bottom of the plane (see Section 3B) into the standard equation for the range of

a projectile, shows that

R = −

h sin(2φ



sin

(2φ

cos

The most simple special case of this equation (other than the case φ

= 0 that 116v

considers) is φ

= sin

−1

(

)=30

◦

which Galileo could have achieved simply by raising

the end of the plane a distance of one half its length above the table top. In this case,

sin 2φ

= cos φ

√

and the equation simpliﬁes to

R = −

√

3 h +

√



= −

√

3 h +

√



h +4y

Finally, taking y

= 53+53+77.5 + 146 = 329.5 from folio 81r (as [35] plausibly

suggests), we get

R =

√

3 h +

√



h + 1318 .

It is most unlikely that Galileo - working with proportions - could have made much

headway in the direction of understanding the relationship between h and R even in

this simple situation. Only algebraic equations arising from analytic geometry could

have provided the necessary insights.

Therefore, as the examination of the folio itself already suggests, Galileo’s diagram

is the residue of an inconclusive and abandoned experiment and his calculations served

a later and diﬀerent purpose. Naylor [19, p. 403] observed long ago that the diagram

on the folio is unrelated to the calculations, but concluded instead that the latter

must be lost.

6. Previous Assessments of the Folios. It was argued in Sections 4 and 5 that

the experiments whose footprints are visible on folios 81r and 114v failed and that

Galileo abandoned them. The experiment of 81r failed because Galileo did not have

a broad enough grasp on the principles of inertia and superposition, and that of

114v was not successful because the relationships between the numerical parameters

of the experiment were beyond his understanding. But how is it that accomplished

historians are able to conclude that these experiments were signiﬁcant achievements?

And what are their arguments?

Drake agrees that the experiments recorded on the two folios did not succeed, but

does not leave it at that and says ([37, p. 124]):

”Without analytic geometry, he lacked the mathematical tools for suc-

cess, which is why he never mentioned the investigations. It is precisely

because they did not succeed that they are of great interest, for they

show how Galileo went about attacking a physical problem when it lay

beyond his powers of solution.”

He then goes on to provide very speciﬁc details about Galileo’s experimental setup

about which Galileo himself was completely silent. See [32, p. 4] and [37, pp. 124-

127]:

”Analysis of Galileo’s measurements of downward oblique projections,

on ﬀ. 114v and 81r, made it possible to describe in fair detail the

apparatus and procedures he had used. His inclined plane was grooved

to guide the ball, as decribed in Two New Sciences [i.e., the Discorsi],

width of groove 8

punti being such as to reduce the linear acceleration

by 5.75%, the ball used being of radius 10 punti. For f. 114v the angle

of the plane was set so that horizontal advance was double the vertical

descent; that is, at tan

−1

or 26

◦

.565. For f. 81r, the angles where

sin

−1

, and

. Galileo’s measurements of projections on both doc-

uments were accurate within 4 punti.”

How does Drake arrive at this? He reconstructs the experiments of the folios from

the fragments that Galileo provides. When a reconstruction provides data that are

in tight agreement with the recorded measurements of the folio, Drake regards both

the reconstruction and the conclusions drawn from it as the likely explanations of

Galileo’s procedures. In light of his analysis of folios 114v and 81r, Drake revises his

earlier assessment of folio 116v. See [32, pp. 4-5] and [37, pp. 110-113]:

”This information, gleaned over a period of years, made it no longer

tenable to suppose that on f. 116v Galileo had measured incorrectly

by as much as 40 punti, as seemed to be implied by calculations of

his own on that page. A skilled experimentalist capable of holding his

results within a variance of four units would not err by ten times that

amount in two out of ﬁve recorded measurements. Accordingly, I have

re-examined f. 116v ... ”

Whereas his earlier conclusions rely on actual reconstructions of the experiments,

his re-examination of 116v turns to theory. In particular, Drake selects the mathemat-

ical model of equation (viii) of Section 3B. His choices d = 20 punti for the diameter

of the ball and w =8.5 punti for the width of the rectangular groove ”slow down” the

theory, but not enough to bring the predictions to within 4 punti of Galileo’s experi-

ment. To accomplish this, Drake argues that the height of Galileo’s table must have

been y

= 800 for some trials of the experiment and y

= 820 for the others, instead

of the 828 punti that the folio speciﬁes. Refer to [32, pp. 5-13] and [37, Chapter 7].

That Drake’s strategy of reconstruction is fundamentally ﬂawed is demonstrated

by the other analyses of folios 81r and 114v. Their authors produce diﬀerent recon-

structions that also generate data that is in close agreement with Galileo’s fragmen-

tary record. However, the insights about the experiments that are derived from them

deviate substantially from those of Drake.

Even though a comparison of the diagrams of 114v and 116v suggests the contrary,

Naylor infers from his reconstruction that folio 114v is an experiment about horizontal

projection rather than oblique projection. He thinks that the experiment covers the

same territory as 116v and considers it a success. In reference to Drake’s analysis he

says ([13, p. 115]):

”The view that Galileo would roll spheres down an incline, compile a

list of observations, and then realize his total inability to interpret the

information, certainly seems a little out of character.”

Naylor regards the experiment of 81r to have been a successful test of the parabolic

trajectory. In [18, p. 167] he argues that

”... there are good grounds for believing that Galileo suspected that

the curve of the projectile trajectory could well be a parabola. If this

was the case, then his knowledge of the parabola would enable him to

design his experiment in such a way as to test this hypothesis. His

experiment suggests that it was designed on the basis of knowledge of

certain relatively simple yet unique properties of parabolas having a

common axis.”

In Naylor’s version of Galileo’s experiment the three parabolic trajectories are com-

pared not only under the assumption that all three parabolas have the same axis

but that they also have the same apex. But is this assumption consistent with the

data on the folio? The same least squares approximation that produced the equation

y =0.446797 x − 0.00351906 x

for the inner trajectory (see Section 4) provides the

equations y =0.177481 x − 0.00097007 x

for the middle trajectory and the equation

y =0.10604 x − 0.000447278 x

for the outer trajectory. Elementary calculus tells us

that the axes of these parabolas cross the x-axis near x =63,x =91, and x = 119,

respectively. Therefore, the axis of the parabola that ﬁts the inner data best is 56

punti from the axis of the parabola that ﬁts the outer data best. In the context of the

distances recorded on folio 81r, this is excessive. In other words, there is no indication

that the experiment of folio 81r deals with parabolas that have a common axis.

Hill [35] also makes use of reconstructions. He considers 114v and 81r to be

connected and argues in [35, p. 661] that 114v provisionally, but eﬀectively, tests

whether the speed of the ball in its descent (from rest) down the inclined plane is

proportional to the square root of the distance that it covers. He ﬁnds his data to

”broadly approximate” this interpretation of the folio. Would broad approximation

have been good enough for Galileo? Note that the 116v experiment tests the speed

law in a more eﬀective way. (See the proportion (iv) of Section 3A.)

As to 81r, Hill does not think that the middle and outer curves of the diagram

represent actual trajectories. Instead, he argues that the second set of measurements

, 130

, 177

, 250 as well as the third 89, 131, 178, 250 are each the horizontal dis-

tances (from the table) of a downward projected ball. In this view, the three trajec-

tories that Galileo considers diﬀer only slightly, with the ball landing at a distance

of 250 punti from the table in each case. Hill summarizes his study of 81r with the

statement ([35, p. 656]) that Galileo

”can conclude that for projections with a ﬁxed baseline value, trajecto-

ries approach semiparabolic status as (or very nearly as) their projec-

tive angles approach the horizontal, just what one would expect if the

parabolic trajectory hypothesis were correct.”

Refer again to the coordinate system discussed in Section 4. The semiparabola that

Hill refers to is the parabolic trajectory through the points (0, 0) and (−250, −329.5)

obtained by horizontal projection. Check that its equation is y = −0.005272x

.It

follows that this semiparabolic trajectory is 100.3, 141.8, 186.6 and (as we already

know) 250 punti from the table at the respective distances of 53, 106, 183.5, and 329.5

punti below the table top. In terms of the respective horizontal distances, the step

from the ﬁrst trajectory (this is the inner trajectory corresponding to the data, 81, 121,

170, 250) to the second trajectory (corresponding to the data 87

, 130

, 177

, 250) is

indeed - as Hill asserts - an advance towards the semiparabola just described. Observe,

however, that the second trajectory and the third trajectory (corresponding to the

data 89, 131, 178, 250) are virtually identical. The diﬀerences between the respective

horizontal distances - the largest diﬀerence is less that 1.5 millimeters - fall well within

the error range already discussed in Section 4. Therefore, the step from the second

trajectory to the third is not an advance towards the semiparabola. So contrary to

Hill’s interpretation, there is no approach to semiparabolic status. Even if Galileo

had carried out the experiment as described, he would only have had very indirect

evidence for the parabolic hypothesis for an oblique trajectory.

In summary, each of the analyses of the experiments of folios 81r and 114v dis-

cussed in this section encounters diﬃculties. In particular, neither the scenarios of

success that they describe nor the speciﬁc information about the experiments that

they provide ﬁnd support in the evidence supplied by the folios.

7. Conclusions. The role that experiment and measurement have historically played

in the validation of a scientiﬁc theory, especially a developing theory, is a subtle

one in general. See Kuhn [21, Chapter 8]. But in the case of folio 116v, we have

seen compelling evidence in Section 3 that this experiment is - in terms of both the

validity of the concept and the precision of the execution - a compelling conﬁrmation

of Galileo’s account of motion. This folio refutes the contention of Koyr´e to the eﬀect

that Galileo’s experiments are woefully inadequate. It also conﬁrms that Galileo’s

analyses on day three of the Discorsi as well as the proof on day four of the parabolic

trajectory in the case of a horizontal projection are, at least in a substantial way,

the achievements of Galileo’s work in Padua. Full credit goes to Stillman Drake for

uncovering the meaning of folio 116v and for recognizing its relevance.

While there is consensus in the literature that folio 116v describes a successful

experiment, there is disagreement as to its purpose. Drake and MacLachlan [16] be-

lieve that it is Galileo’s aim to test his principle of horizontal inertia. Most historians

think that Galileo is in pursuit of the time-squared law of fall. See Hill [35, p. 662].

The discussion of Section 3A shows that the experiment tests none of Galileo’s in-

sights independently, but that it in fact tests Galileo’s account of motion as a whole.

Therefore, it tests neither his law of fall nor his principle of inertia directly. Whereas

the test of either the law of fall or the principle of inertia necessarily involves the

measurement of time, the experiment of 116v bypasses any need to measure this elu-

sive variable. This point is of historical signiﬁcance. The inclined plane experiment

that Galileo describes in the Discorsi is designed to shore up the law of fall and some

of its consequences. (See Proposition II. Theorem II and its Corollaries in [2, pp.

166-171]). It therefore focusses on the connection between distance and time. Settle

[8, p. 21] points out in the context of his reconstruction of this experiment that ”the

measurement of time is the most controversial and the most diﬃcult.” Indeed, it is

the measurement of time that is an explicit target of the skepticism expressed by

Koyr´e (and also Descartes and Mersenne) about Galileo’s experiments with inclined

planes. See [5, pp. 106-107 and 126].

From our modern perspective, 116v is a successful illustration of experiment test-

ing theory. Was Galileo’s perspective the same? What did his experiments mean to

him? Why is there no reference to the folio 116v experiment in the Discorsi? There

have been eﬀorts to answer these questions. In both the Dialogo and the Discorsi

experiments quite clearly serve a didactic even rhetorical function (at least in part)

and Galileo stretches his descriptions of them beyond the limits of his observations.

Naylor [19] discusses the contrast between these descriptions and Galileo’s actual

experimental procedures, and suggests plausibly that ”the problems to be met in

making the folio 116v experiment intelligible for his readers were ... [an] obstacle to

its inclusion in the Discorsi.” Machamer [44] develops the point that for Galileo to

have an intelligible explanation, he needs to have an accompanying experience - often

provided by a mechanical device - that illustrates the phenomenon. In particular,

Machamer regards the idea that Galileo’s insistence on making his thinking about

motion intelligible by mechanical models ”ﬁts well” with the practice in geometry

(the mathematical environment in which Galileo worked) of proof by geometrical

construction.

The evident failure of the experiments of folios 81r and 114v does not detract from

the success of 116v. But it does call for a reassesment of Galileo the experimental

scientist. Recall Koyr´e’s assertion ([5, p. 106]) ”Galileo’s experiment is beautifully

conceived; the idea of substituting a body rolling down an inclined plane for a body

in free fall is truly a mark of genius. But, we are obliged to note, its execution is not

of the same order as its conception.” It is indeed the case that Galileo’s experiments

with inclined planes are ingenious in their concept, but rather than their execution,

it is Galileo’s understanding of the underlying physical principles and mathematical

relationships that is not always up to the task.

REFERENCES (in chronological order)

For much more extensive bibliographies consult McMullin [9] and Damerow-Freudenthal-

McLaughlin-Renn [38].

1. Galileo Galilei, Dialogo doue ne i congressi di quattro giornate si discorre sopra

i due Massimi Sistemi del Mondo,: Tolemaico, e Copernicano, Florence, 1632.

Stillman Drake translation, Dialogue Concerning the Two Chief World Systems

- Ptolemaic and Copernican, University of California Press, 1967.

2. Galileo Galilei, Discorsi e dimostrazioni matematiche, intorno a due nuove

scienze atteninti alla mecanica & i movimenti locali, Leyden, at the Elzeviers,

1638. H. Crew and A. de Salvio, translation, Dialogues Concerning Two New

Sciences, Northwestern University, Evanston and Chicago, 1950; reprinted by

McGraw Hill, New York, 1963. Stillman Drake translation, Discourses & Math-

ematical Demonstrations Concerning Two New Sciences Pertaining to Mechan-

ics & Local Motion, University of Wisconsin Press, 1974, second edition, Wall

and Thompson, Toronto, 1989.

3. I. Kant, Critik der Reinen Vernunft, B edition 1787. N. K. Smith translation,

Critique of Pure Reason, Second Edition, London, 1956.

4. E. Mach, The Science of Mechanics, Chicago, 1893.

5. A. Koyr´e, Etudes Galil´eennes, Librarie Scientiﬁques Hermann et Cie, 1939.

John Mepham translation, Galileo Studies, Humanities Press, New Jersey, 1978.

6. A. Koyr´e, An Experiment in Measurement, Proc. Am. Phil. Soc. 97 (1953),

222-237.

7. R. Dugas, Mechanics of the seventeenth century, from the scholastic antecedents

to classical thought, Neuchatel,

Editions du Griﬀon; New York, Central Book

Co., 1958.

8. T. Settle, An Experiment in the History of Science, Science, Vol. 133 (1961),

19-23.

9. E. McMullin, editor, Galileo: Man of Science, New York, NY, Basic Books,

1967.

10. E. McMullin, Galileo: Man of Science, in Galileo: Man of Science, E. McMullin,

editor, 1967, 3-51.

11. T. Settle, Galileo’s use of experiment as a tool of investigation, in Galileo: Man

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12. S. Drake, Galileo’s Experimental Conﬁrmation of Horizontal Inertia: Unpub-

lished Manuscripts, Isis 64 (1973), 291-305.

13. R. H. Naylor, Galileo and the Problem of Free Fall, The British Journal for the

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University Press, Cambridge, MA, 2001.

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA.

e-mail: [email protected]