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GALILEO'S MANUSCRIPT 72 : GENESIS OF THE NEW SCIENCE OF MOTION
(Padua, ca. 1600 -1609)
Preprint · August 1996
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GALILEO'S MANUSCRIPT 72 :
GENESIS OF THE NEW SCIENCE OF MOTION
(Padua, ca. 1600 -1609)
With a General Presentation of the Manuscript
Mohamed ABATTOUY *
University of Fez, Morocco
Max Planck Institute for the History of Science, Berlin
* Address of the author:
- Max Planck Instiute for History of Science, Wilhelmstrasse, 44. D-10117 Berlin. E-mail:
abattouy@mpiwg-berlin.mpg.de.
- Department of Philosophy, Faculty of Letters and Human Sciences, Dhar el-Mehraz - Fez
University. P. O. Box 50. 30001. Fez, Morocco.
Acknowledgment
Originally, this text has been presented in a lecture given at the Max Planck
Institut für Wissenschaftsgeschichte in Berlin in March 27, 1996. I expanded and
revised it during a stay of research of four months (July through October 1996) in
the MPIWG .
I thank Professor Dr. Jürgen RENN, Director of the Institute, who invited
me for this stay of research in the MPIWG in Berlin during which, among other
things, I revised this text and participated to the rich work of research which is
being done in the Institute. This stay was the beginning of a collaboration that
lasted for eight years (1996-2003) and which was very fruitful on the scientific
level.
I also want to thank all the staff of the Institute for the help they afforded to
me, and specially Jochen SCHNEIDER, the Research Coordinator, and Urs
Schoepflin, the head Librarian and their collaborator. Should this work be the
expression of my gratitude to all of you.
Finally, I would like to thank my colleagues and friends Sadik RADDAD
(University of Fez) and Paul WEINIG (Max Planck Institute for History of
Science, Berlin) who revised my English. Of course, I am responsible of all the
remaining errors
M. A.
_____________________________________________
Abbreviations
* degV (A) = degree of velocity in A.
* m (AB, AC) = mean proportional between AB and AC.
* mom grav (CE) = momentum gravitatis along CE.
* mom vel (A) or mom vel (AB) = momentum velocitatis in A or along AB.
* mom (C) = momentum or momento at C.
* t (AB) = time of motion along AB from rest.
* t (BC, A) = time of motion on BC, after a previous descent from A.
* tard (FB) = tarditas or tardity on FB.
* tot mom = totale momentum or momento totale.
* v (C) = instantaneous velocity in C.
* V(AB) = velocity on AB or speed acquired in descent from A to B.
TABLE OF CONTENTS
Acknowledgment ....................................................................................................... 2
Abbreviations .............................................................................. ............................... 2
Introduction .............................................................................................................. 4
I. Prolegomena ......................................................................................................... 7
1. Description of the Manuscript ........................................................................... 7
2. On the problem of datation ................................................................................ 11
3. Preliminary work in natural philosophy and mechanics .................................. 13
3. 1. De motu antiquiora.............................................................................................. 13
3. 2. Le Mecaniche......................................................................................................... 17
II. First Discoveries, First Demonstrations ………........................................ 21
1. Isochronism and paths of least-time
.......................................................... 22
1. 2. Attempts to demonstrate the law of chords .............................................. 25
1. 3. Preliminary demonstrations of Theorem XXII ......................................... 28
2. Comparison of motions on different slopes ....................................................... 29
2. 1. Folio 173r : geometrical exercises ...................................................................... 30
2. 2. Folios 177 r-v and 180r : the Right-Angle Theorem 30
III. INVESTIGATIONS ABOUT VELOCITY AND FREE FALL 35
1. Speculations and paradoxes about velocity and acceleration
35
1.1.Dynamical approach to the problem of velocity 35
1.2.Emergence of the Rule of Double Distance
37
1.3. A paradox on velocity
40
2. Derivation of the Law of fall: The crisis of 1604
41
2.1. Official Announcement of the Times-squared theorem
42
2.2. Kinematical demonstration of the Mean Proportional 43
2.3. Physical proof and indications on the resolution of the crisis 45
2.4. Formal deduction of the Times-squared theorem
46
IV. AFTER 1604: NEW RESEARCHES ON FREE FALL AND
FURTHER RESULTS
1. Demonstration of the Length-Time Theorem
53
2. Return to a dynamical foundation 58
3. Deduction of the proportionality Time-Velocity 60
4. Final stage
70
BIBLIOGRAPHY
75
INTRODUCTION
In this text, I will expose a general view of Galileo's work on motion in Padua
between ca. 1600 and 1609. During this period, the pisan scientist developed a new
theory of motion without equivalent in the past of the scientific thought. Before he
was distracted by the news of the telescope and before he began his astronomical
observations and the subsequent Copernican campaign he conducted after his
move to Florence in 1610, Galileo –as a "professor of mathematics" in Padua
university and as a private researcher in physics and natural philosophy– carried
on/executed an intense program of work in which he studied different physical
questions, centered essentially on the problems of motion. In this laborious
research, he discovered many of the theorems and propositions that constituted
the nucleon of his last great publication, the Discorsi e dimostrazioni matematiche (1638)
where he delivered to the scientific community the foundations of two new
sciences, the theoretical mechanics (or "scientia de motu locali") and the resistance of
materials.
In the following reconstruction of Galileo's scientific work on motion in the
decisive period of the first decade of the XVIIth century, I will have recourse
systematically to some manuscript documents preserved in the Manoscritti Galileiani
in Florence –especially to a group of notes preserved in the famous Volume 72. In
order to fill the gaps left by these manuscripts, I will also use the data furnished by
a group of letters written or received by Galileo between 1602 and 1610 which
provide valuable information on the evolution of his physical thought. After an
introduction, the first section will describe the Manuscript Volume which contains
the notes on motion, with the presentation of some historiographical elements
related to the study of these documents. This section will also present the
conceptual background in which the first discoveries accomplished by Galileo in
the field of motion were deeply rooted, that is the framework constituted by the
pisan De motu and the paduan Mecaniche. Section II deals with the early scientific
discoveries realized by Galileo and with his attempts in order to found them on
mathematical proofs. Section III is devoted to the presentation of the work on free
fall in the period gravitating about the last quarter of 1604, as it is extant in some
documents associated with the letter to Sarpi. Finally, Section IV is imparted to the
investigations that can be dated approximately in the limits of the period after
1604, until the beginning of the astronomical program in late 1609. After Galileo
left Padua in 1610, he was occupied almost continually by other things than his
work on motion, to which he returned only occasionally. This situation didn't find
an end before his engagement in the redaction of the Dialogo and in the preparation
of a systematic treatise on motion about 1630, which was published in its final
version in the Discorsi e dimostrazioni matematiche (1638).
I don't pretend to provide an extensive study of the subject-matter. In reality,
a complete reconstruction of Galileo's work in the first quarter century of his
scientific career can hardly be said to be in the scope of a unique scholar. In the
actual state of the Galilean studies, a thorough analysis of the notes on motion
bounded in Vol. 72 requires a collective undertaking and a close collaboration
between researchers of various competences (historical, mathematical, philological,
paleographical...) In the following brief and limited panorama, I propose to analyze
some selected documents with the aim of drawing a general picture describing the
evolution of Galileo's theory of motion. I think that we don't actually need to
produce supplementary scenarii for partial orderings of the manuscript material in
Vol. 72 and to describe the possible ways along which Galileo's thought may have
evolved. There are enough of these partial reconstructions. What can give a new
departure to the Galilean studies is the concentration of the endeavours of
specialists in the elaboration of a global and well documented interpretation of all
the material which can inform us on the genesis of the Galilean science of motion.
Evidently, it is an immense task and a fascinating one. Such an endeavour is
undoubdetdly an international undertaking and requires an intense collaboration
on various levels. After the large debate raised by the studies on Galileo's notes on
motion in the last three decades, the future of the Galilean studies depends
probably on the realisation of such a mission.
I. PROLEGOMENA
1. Description of the Manuscript
The Vol. 72 or Codex A of the Manoscritti Galileiani (Divisione 2a-Parte 5a,
Tomo 2) in the Biblioteca Nazionale di Firenze (BNF) is a volume of Galileo's original
papers that comprises specifically all the manuscript material of his work on
motion and some other sheets concerned with other topics. It is composed of 241
miscellaneous folios of 345-235 mm in size. The numbering of the sheets begins
from 1 to 196. Actually, folio 1 bears the title of the volume: Opere / di / Galileo
Galilei / Parte 5 / Tomo 2 / Meccanica1, while folio 2 contains the table of contents of
the volume. The first 32 folios are devoted to diverse topics. Folios 9-26 are an
incomplete copy of Le Mecaniche, a Galilean treatise on simple machines, while
another complete copy of the same treatise is inserted between folios 27bis and 28
1
According to the classification made by V. Antinori in the 19th century, Vol. 72 was classified
as Tomo II of the Parte V (that comprises the ten volumes of the Galilean Collection in the
BNF devoted to the writings on "Meccanica" (science of motion and related topics) of the
Divisione II. The attribution of numbers to the successive volumes of the Manoscritti
Galileiani evokes a chronological ordering. For example, we note the strict succession
between Vol. 71 (Div. 2a - Parte 5a, t. 1) and Vol. 72 (Div. 2a - Parte 5a, t. 2). The former
contains the pisan De motu antiquiora, while the latter gathers together Le Mecaniche and the
notes on motion.
–in the form of a treatise of 40 folios that bears independent numbering from 1 to
402. The other components of these preliminary sheets are: a copy of an early work
on centers of gravity (ff. 3-6), a copy of the treatment of the properties of the lever
(f. 27r-v) (Opere, VIII, pp. 366.1-367.14)3, two drafts of the original dedication of
the Discorsi by Galileo to the "Conte de Noailles" (f. 28r-v), and the preface to the
same Discorsi (ff. 31-32) 4.
The notes on motion begin properly from f. 33r to f. 194r. The great part of
these notes are autograph drafts in the original hand of Galileo, but almost 30
sheets exist which are copies written by two of his disciples, Mario Guiducci and
Nicolò Arrighetti, perhaps under the direction of the Master in Florence betwenn
1616 and 16185. These copies were made from Galilean original material and many
of these copies also contain corrections and additions in the hand of Galileo. All
the notes of Vol. 72 –whether original material or later copies– have a direct
relation to propositions, theorems, scholia, lemma and problems presented and
discussed in the Discorsi, and especially to those of the De motu locali, the
mathematical treatise of the third and fourth days of this book 6. In general, they
are more or less complete versions of the published propositions.
The importance of this manuscript material for the study and the
reconstruction of the development of Galileo's science of motion has been
recognized since the work of Caverni (1891-1900). In the eighth volume of the
Edizione Nazionale of the Opere di Galileo Galilei, Favaro edited the great part of
these documents as footnotes to the corresponding propositions of DML or as an
Appendix to the Discorsi (Galilei, 1898). It was not however until the last quarter of
this century that they became the centre of Galilean studies and focalised upon
them the attention of historians concerned with the question of the genesis of the
new science of motion in the XVIIth century7. Effectively, these private notes were
Procissi (1959, p. 152) remarks in the two cases: "Copia corrispondente a quanto è stato
pubblicato in Gal., Ediz. Naz., II, 155-181 lin. 19" and "copia del tempo. Pubbl. in Gal., Ediz.
Naz., II, 155-190".
3 All the references to the writings of Galileo will be to the Edizione Nazionale (Galileo, 18901909) in the following manner: Opere, n° of volume in Roman figures, n° of page in Arab
figures and n° of lines (if necessary). For all the bibliographical references, see the
Bibliography following this text.
4 For a complete description of the material in Vol. 72, cf. Procissi (1959, pp. 151-154), the
critical apparatus added by Drake to his edition of Galileo's Notes on Motion (Galilei, 1979, pp.
XLIII-LII) and Camerota ,1995.
5 On the question of the copying of some paduan notes in Florence, cf. Favaro's Avvertimento to
Galilei, 1898, p. 34; Drake, 1978, p. 77 and Drake's "Introduction" to Galilei, 1979, p. XIII.
6 The De Motu Locali being composed of three books: "On Uniform Motion", "On Accelerated
Motion" and "On the Motion of Projectiles", each one of these books will be refered to by DML
1, 2 or 3.
7 A great number of the notes of Vol. 72 were translated in Italian and published by Carugo and
Geymonat in their edition of the Discorsi (Galilei, 1958). This "edition" of the fragments,
2
composed at various dates and reflect with a high degree of fidelity the changes
and the transformations of Galileo's physical thought in the large amount of time
that separated the old pisan dynamics of De motu antiquiora (Opere, I, pp. 251-419)
and the paduan analysis of the machines in Le Mecaniche (ibid, II, pp. 155-190) from
the mathematical physics expounded in the Discorsi.
From this point of view the papers of Vol. 72 provide important tools for the
reconstrucion of the intellectual itinerary of their author. "These manifestations of
Galileo's thinking, which accompanied the ongoing work on motion leading from
De motu to the Discorsi and thus document the transition, offer an invaluable
opportunity to study the cognitive background of the origins of classical
mechanics" (J. Renn, 1992, p.132). On the other hand, the analysis of these papers
constitutes an excellent example of the substitution of a study of the context of
discovery to the traditional examination of the context of proof. In fact, this recent
tendency within Galilean studies displays a fundamental methodological
presupposition implemented by recent history of science: that science is not only
what is published by the scientists and presented to pairs in a highly structured
form; it is also, and perhaps principally, what scientists write and think when they
conduce their research, what they record when they perform experiments and
make observations, when they mark notes and ideas... This method happened to
be very fecond. It helped displace the interest of historians of science from the
published and, consequently, public scientific production, towards the private
papers. These incomplete and obscure records of calculations, theoretical
meditations and experiments, although they are later abandoned, became a
privileged instrument for any attempt to reconstruct a theoretical structure.
The recent studies on Galileo’s notes on motion were inaugurated by the
pioneer work of W. Wisan (1972, 1974), who carefully examined the great part of
the folios, with the explicit objective of elucidating the gaps and the blanks in the
evolution of the Galilean theory of motion. Although her work has been
challenged by other historians (Drake, Naylor), her main results are still an
excellent piece of analysis of the Vol. 72. The work of S. Drake shed new light on
several papers of the manuscript, especially those which had been left unpublished
until he had the privilege of being the first scholar to submit them to an
interpretative investigation. The conclusions of Drake had been systematically
criticized by R. Naylor and the debate between the two historians has greatly
enriched the recent Galilean studies. P. Souffrin, for his part, explored a group of
fragments concerned principally with the relation velocitas–momentum and
reinterpreted them in a new way. In the same very recent period, J. Renn
performed original work with the scope of a general reconstruction of the Galilean
theory of motion, and within this framework paid special attention to the analysis
which constitutes an important stage in the story of the papers bounded in Vol. 72, is rarely
mentioned by the historians.
of Vol. 72 papers devoted to projectile motion (Renn, 1990, 1992). These last two
contributions initiated interesting ideas on the theoretical foundation and evolution
of Galilean theoretical mechanics, and led to a new asses-sment of its status within
the general framework of classical mechanics8.
In order to show the exceptional interest of the manuscript material in Vol.
72, an important element in this context is worthy of mention. It is related to the
new elements of information this material may provide on Galileo's experimental
activity, a topic largely controversed in the history of science. In 1961, T. Settle
repeated successfully the famous experiment of the motions on the inclined plane
described in the Discorsi (Opere, VIII, pp. 212-213). Subsequently, S. Drake –within
his systematic campaign of study of Galileo's notes on motion– identified some of
the fragments in Vol. 72 as records of experiments performed by the pisan
scientist. He reproduced many of these experiments (Drake, 1975c; Drake & MacLachlan, 1975). These two historiographical performances made an end to a long
debate on whether Galileo was a real experimentalist or not, and what were the
results he drew from his experimental activity. Since then, besides the general
recognition of the effectiveness of the experimental dimension of Galileo’s work,
another debate has been opened concerning the interpretation of the documents
bearing records of this empirical investigation. In this context, some of the
fragments which were discarded (partially or totally) from the Edizione Nazionale
(as the folios 114r, 116v, 117r, 180r ...), became the subject of intense studies and
their interpretation passionately controversed. This debate led to a new
understanding of the whole Galilean scientific enterprise, and above all of the the
interaction between theoretical and experimental investigations.
But, as J. Renn pointed out (Renn 1990, p. 78; 1991, p. 110), there is another
group of manuscripts contained in Vol. 72 that, to some extent still remain to be
rediscovered. Those are the sheets that neither include a coherent piece of text nor
any numbers or sketches related to experiments. In effect, besides the vast majority
of Galileo's notes on motion which are direct evidence on the intensive and long
struggle of our scientist with theoretical problems of mechanics, this last category
of documents may lead to the rediscovery of "Galileo's theoretical laboratory",
since they are concerned essentially with theoretical problems, as some conceptual
and mathematical questions of the work of Galileo on the motion of projectiles
and its relations to the free fall on the vertical or on inclined planes9.
These indications are not intended to describe exhaustively the rich and live and still ongoing
debate on the Vol. 72. To be exhaustive, it must include mention of the original writings
published during the last quarter of century by P. Galluzzi (1979), E. Giusti (1981), D. Hill
(1986, 1988, 1994), K. Takahashi (1993) ...
9 In his analysis of some of these sheets (ff. 83v, 86v, 90ar ...), J. Renn reconstructed
comprehensively Galileo's original work on projectile motion and its relationship to the
problems of free fall. For more information, see Renn (1984), (1988), (1990), (1992, pp. 213225).
8
As metioned above, almost all Galileo's notes on motion were edited in the
eighth volume of the Opere, but this publication was more a transcription of the
documents than a real edition, since two major questions were left unanswered: the
chronology in which they were composed and their interpretation on the basis of a
careful analysis of their mathematical, conceptual and experimental contents.
The research conducted by the scholars in the two last decades tended to
resolve a great deal of these unfinished missions. S. Drake realized a facsimile
edition (Galilei 1979) of the notes relatively well supplied by critical tools, but the
value of this work was diminished by some errors and inaccuracies in the
references system (especially the refrences to the sources and to the list of
watermarks) and by the criticable procedures which oriented all the editorial
enterprise. Indeed, the main feature of this edition was that it reproduced the
documents in their probable order of composition. In this perspective, the
photographs of the original papers were cut in many pieces, assumed to be non
contemporaneous. In most cases this procedure was not justified by any legitimate
argument, save the ones issued principally from Drake’s own studies, who
neglected to take into account almost systematically the available suggestions of
other researchers.
Until now, a real edition of the Vol. 72 is still lacking. In this context, the
project announced by J. Renn in 1988 and which is actually in course of execution
will provide the Galilean scholars with a magnificent tool of work and will surely
initiate a new departure in the studies on the genesis of Galileo’s mechanics.
This project consists of the production of an electronic edition of Vol. 72
material. Conducted by scholars at the Max Planck Institut für Wissenschaftsgeschichte in
Berlin, in collaboration with the Istituto e Museo di Storia della Scienza and the
Biblioteca Nazionale at Florence, the first part of this ambitious project has by
now been completed in Berlin at the Max Planck Institute.
Using electronic tools developed by the editors, a carefully-controlled
transcription of the texts related to motion (folios 33-194), encompassing a markup of the text indicating emendations, deletions, marginal notes, and insertions has
been prepared. The relation of the transcription to digital images of the manuscript
pages has been documented by means of frames and hypertext links, the diagrams
of the manuscript redrawn, and a preliminary transcription of the calculations
prepared.
The result of this work will be made accessible on different computer
platforms, by means of specific programs developed for this aim, in order to
convert data in the standardized format of the HTML of the World Wide Web. In
the next future, this electronic edition of Mss. 72 will be put on the Web for free
access and could be viewed via Internet.
Further work is being executed now in the second stage of the project. This
part will be devoted to the English translation of the documents, the analysis of
their deductive structure and the accomplishment of the critical apparatus for texts
and calculations, besides completing the physical description of the manuscript 10.
2. On the problem of datation
No page of the Vol. 72 is explicitly dated. Now the study of the documents
contained in it requires that they should be put in a logical order corresponding as
much as possible to the order of their composition. For this purpose, different
methods and criteria have been proposed. Ca-verni (1891-1900, IV, p. 341)
founded his examination of the notes in question on a double criterion of datation:
their conceptual content and "material indications" –in particular the calligraphical
variations of writing of the different fragments. Without recognizing it explicitly,
Favaro was greatly influenced by the work of Caverni, as it is now clearly stated by
the historians, and this influence was evident in the relative chronology in which
he edited the documents of Vol. 7211.
As far as the question of the criteria of datation is concerned, recent scholars
are divided in two groups: these who prefer theoretical clues, as W. L. Wisan and
R. H. Naylor, and the others who adher to material clues, especially S. Drake and
W. Hooper.
In her pioneering work, Wisan declared her preference for theoretical
methods for the establishment of a relative chronology of Galileo's notes :
"neither handwriting nor watermarks suffice to determine the order
of undated fragments within a given period. To reconstruct the
relative chronology of Galileo's drafts within the different periods,
other evidence is needed. For my own study the main evidence
used is taken from variations in terminology and mathematical
techniques" (Wisan, 1974, p. 127).
For Drake, on the contrary, material tools provide the best way for the
ordering of the fragments:
The informations on this electronic edition are extracted from the Annual Report for the year
1995 of the Max Planck Institute for History of Science in Berlin. I thank Jürgen Renn and
Peter Damerow, MPIWG reponsibles for the project for the Vol. 72project, for providing to
me this information and allowing me to publish it.
11 Drake remarked (in his "Introduction" to Galilei, 1979, p. VIII) that the arrangement of
Galileo's notes of motion in the eighth volume of the Opere "was based partly on the order of
appearance of topics in Galileo's last book, and partly on conjectures published in 1895 by
Raffaelo Caverni, who had selected a few documents to support his theory of Galileo's
sources". More recently, Camerota (1995, p. 9) asserts that "Favaro seguì in modo piuttosto
stretto le indicazioni caverniane, mutuandone in più punti le opzioni nella pubblicazione di
importanti sequenze di stralci del Ms. 72".
10
"To make sure that any ordering is free of serious
implausibilities requires attention to handwriting, vocabulary,
watermarks in paper, color and condition of ink, and in addition to
the implications in that ordering of what was known to Galileo at
every stage, and what remained to be discovered" (Drake, 1978, p.
77).
In reality, it seems that the most efficacious method has to make use of a
synthesis of these two choices and try to combine theoretical and material
indications. Since the aim behind any ordering is logically not the establishing of an
absolute chronology, but only a relative one, the combination of such two methods
would be very valuable.
The question of the criteria of datation is a fundamental one. Without the
realization of significant porgress in this field, the study of Galileo’s notes on
motion will always suffer of incompleteness. Actually, the currently executed
electronical edition of Vol. 72 will undoubtedly shed a new light on the ordering
and datation of these documents. The examination of the physical properties of
the Manuscript is, indeed, an important part of this unprecedented project. For
this investigation, scholars of the collaborative institutions make use of new and
sophisticated technology in the analysis of the ink and paper, in the identification
and classification of watermarks, in the photographying of the original
documents... Undoubtedly, this work will bring to light much hidden information
that remained unsuspected by previous research.
3. Preliminary work in natural philosophy and mechanics
Before we start the analysis of some Vol. 72 documents, let us begin with a
rapid presentation of the texts elaborated by Galileo during the XVIth century part
of his scientific life and where is manifest his familiarity with the traditional
problems of the medieval natural philosophy and the classical theory of machines.
The texts to which I refer are precisely the De motu antiquiora and Le Mecaniche. In
these two sets of writings, our hero obtained some assumptions and positive
results of which he will make use in his later research. Among these results : the De
motu theorem as a general proposition demonstrated twice in the context of the
inclined-plane analysis; the idea of conservation of motion on the horizontal plane,
the conception of acceleration as an accidental feature of the descents of bodies.
3. 1. De motu antiquiora
De motu antiquiora is the generic title of six latin documents written by Galileo
in Pisa ca. 1590 on the problems of natural philosophy12. In these texts, we can
distinguish a main version composed of 23 chapters which is the most developed
quantitatively. It contains a section on the movement on inclined planes where
Galileo, after a laborious mathematical demonstration, obtained a proposition on
descents. Following Wisan (1974, p. 152), this proposition is known as "De motu
theorem".
The analysis of the inclined plane in De motu is founded on an explicit
principle: a heavy body is moved downward with as much force as it is necessary
to draw it up, or, it is moved downward with the same force with which it resists
being raised (Opere, I, p. 297.4-6). In this perspective, the problem of the inclined
plane takes the form of a search for the force with which a body descends a given
plane; this force will be the same as that required to hold the body at rest on the
plane. Making use of this assumption and applying the principle of the lever to the
case of the bent lever, Galileo assumes that the motion of a body at any point
around the lower quadrant of a circle may be identified with the motion along an
inclined plane tangent to that point.
E
C
A
Z
P
S
B
T
G
D
Q
[Fig. 1]
N
De motu antiquiora {1}
R
H
F
In the demonstration, he tried to find out the ratio of the effective weight of
the body on the respective planes. For this purpose, he elaborated the famous
demonstration with the bent lever applied to a vertical circle. After the preliminary
steps, a general proportionality is deduced in terms of the diagram [fig. 1]:
celeritas on EF is to the celeritas on GH as QS to SP.
In other terms, the descents on a vertical and on an oblique path are as the length
of oblique descent to the vertical height of that length. This is the first version of
De motu theorem, enunciated in terms of inverse proportionality of celeritates to
the oblique and vertical fall of the same height. Then the theorem is enunciated
generally :
The texts De motu were edited by Favaro in Opere, I, pp. 251-419. An important
historiographical problem related to these writings is the conceptual and chronological
order of their composition. For a synthesis on this topic, see Abattouy & Mazet (1996,
forthcoming).
12
"And it is clear that the same weight can be drawn up an inclined
plane with less force than vertically, in proportion as the vertical
ascent is smaller than the oblique. Consequently, the same heavy
body will descend vertically with greater force than on an inclined
plane in proportion as the length of the descent on the incline is
grea-ter than the vertical fall" (ibid, I, p. 298.26-30; Galilei, 1960, p.
65)13 .
Continuing his argument, Galileo deduced the theorem in a more general
form: ratios of celeritates along planes of any inclination, but with the same vertical
height, are inversely proportional as the lengths of the planes. As in diagram 2,
celeritas on CA / celeritas on AD = AD / AC. Since DA / AB = tarditas on AD /
tarditas on AB, and since tard (AB) / tard (AC) = AB / AC, it follows, ex aequali,
that DA / AC = tard (AD) / tard (AC).
A
[Fig. 2]
De motu antiquiora{2}
B
C
D
"And therefore line DA will be to line AC also as the celeritas on AC
is to celeritas on AD. It is clear, then, that the celeritas of the same
body moving on different inclinations are to each other inversely as
the lengths of the oblique paths, if these entail equal vertical
descents" (Opere, I, p. 301.24-26; Galilei, 1960, p. 68).
The De motu theorem constitutes the first proposition of Galileo's work on
motion. It left indelibile and evident traces in the subsequent Galilean
investigations. Precisely, the law of chords (which will be discussed below) is
nothing else than one of its geometrical consequen-ces. Recently, the
understanding of this theorem has been renewed by P. Souffrin who rejected the
prevailing view about its falsity. He affirms (1991, pp. 93-96; 1993, p. 122 sq.) that
the central proposition elaborated by Galileo in chapter 1414 is deprived of any
amibiguity concerning the signification of celeritas; for this reason, the theorem is
correct and the classical mechanics gives the same result. In this section of De motu,
nothing is said nor explicitly assumed concerning a law of descent on a given
inclined plane; the problem posed is typically standard in pre-classical treatment of
The excerpts of Galileo's writings reproduced in the present text are in most cases
adaptations of the available English translations of his works (by Wisan, Drabkin, Drake,
Renn...) In some cases, I have revised the translations of some passages
14 That is the celeritas of the same movable on planes of different inclinations are between
them as the inverse of the inclined descents corresponding to the same vertical height.
13
motion. In fact, the problem consists of comparing the spaces traversed "in the
same time or in equal times" by two movables15.
In another part of the treatise De motu, Galileo elaborated a rule of motion
intented to take into account the material in which the movable is made and the
media in which it descends. According to such a rule, the velocity of motion of
falling bodies is quantitatively characterized as a ratio between the specific gravity
of the body and that of the medium, but the acceleration is not mentioned. On the
contrary, it is considered as an accident, the manifestation of an external quality
which impels the agreement between the observed motions of the bodies and the
theoretical predictions. In general, the motion of fall is considered as accelerated,
but it is explained by the difference between the falling body's weight and a
gradually decreasing impressed force. In this perspective, the phenomenon of
acceleration is said to intervene only towards the end of motion, when the body
moves swiftly, after it has regained more weight. This change of speed is accounted
for by the action of an external force that is impressed on the falling body and
which has nothing to do with any influence on the part of the medium (Opere, I,
pp. 315-318).
The impressed force remains in the body when it begins to fall naturally, but
its progressive disappearence explains why the movable is accelerated in a later
stage of its fall. With the same reasoning is also explained how acceleration is
stopped when the impressed force is so weakened that the inherent weight of the
body recovers a great amount of its vigor (ibid, pp. 319-323).
In a subsequent chapter, Galileo adds supplementary elements to explain why
bodies that are less heavy move more swiftly in the beginning of their motion than
heavier ones (ibid, pp. 333-337). According to this pisan theory of acceleration,
heavy bodies must destroy more of the impressed force than must lighter bodies,
and this is the cause of their slowness; but since the contrary quality is weakened
by itself, the heavy bodies get rid of it more rapidly than lighter ones (thanks to the
great opposition manifested by their matter) and regain their inherent great weight.
An important aspect of the interpretation elaborated by P. Souffrin is that he doesn't agree
with the consideration of De motu theorem as applying only to uniform motions. "Les
lectures critiques modernes ont ceci en commun qu'elles entendent celeritas dans l'un des
sens que nous donnons actuellement à "vitesse" en cinématique : soit "vitesse moyenne", soit
"vitesse instantanée". Or ...il s'agit là d'acceptions spécifiques de la tradition classique, et de
ce fait les interprétations se trouvent marquées par un anachronisme fondamental relatif au
concept central de la discussion". In order to justity this point of view, P. Souffrin restated
the theorem in the preclassical conceptual terms by the attribution of the standard
preclassical signification to celeritas and enunciated the theorem as follows : "Il est donc clair
que le rapport des espaces parcourus dans un même temps par un même mobile sur des
plans d'inclinaisons différentes est égal à l'inverse du rapport des descentes obliques sur ces
mêmes plans correspondant à une même descente verticale... ce qui est évidemment...
théoriquement parfaitement exact" (Souffrin, 1991, pp. 95-96). For more details, see Souffrin
(1992a), (1993).
15
In this later stage, they increase the speed of their motion and catch up the lighter
bo-dies, with which they reach the ground at the same time, at least for short
distances of fall16.
The treatment of projectile motion in chapter 17 shares the general lines of
the above reasoning. The motion of projection is considered as a violent motion
and it is explained by the same accidental quality used to account for the temporary
acceleration, that is an impressed force. In this context, the trajectory of oblique
projection is composed of two parts: a straight one followed by a curved part,
generated when the projectile begins to descend downward [fig. 3]17.
B
C
D
[Fig. 3]
E
De motu antiquiora{3}
A
In the conception of acceleration as in the analysis of the motion of
projectiles, the theoretical structure of De motu proves to be tightly linked with the
medieval tradition of natural philosophy. The treatise of mechanics composed at
Padua many years later, although manife-sted some conceptual progress, reveals
the continuation of a strict adhesion to the same fundamental theses stated in De
motu.
3. 2. Le Mecaniche
Galileo’s texts on mechanics of simple machines provide important and
useful elements of information for the reconstruction of his thought in the period
preceding the researches recor-ded in Vol. 72. Indeed, his paduan production in
In Chap. 14, just after the general enunciation of the De motu theorem, Galileo warns that the
theoretical ratios stated in this proposition are not observed because of the imperfections of
matter and of the "accident" of acceleration (a lighter body descends more quickly than a
heavier one at the beginning of its motion). Because of the difficulty of these two elements,
and because their interferences intervene in so many ways, "rules cannot be given for these
accidental factors since they can occur in countless ways" (ibid, p. 302.12-13).
17 The last section of De motu (chap. 23) devoted to the exploration of a specific question:
"Why objects projected by the same force move farther on a straight line the less acute are
the angles they make with the plane of the horizon" (Opere, I, pp. 337-340). As is it remarked
by J. Renn (1992, p. 144) , the way the question is posed presupposes that an obliquely
projected body follows a straight line before it statrts to turn downward, as it is illustrated by
the diagram drawn by Galileo in his text. For an analysis of Galileo's theory of projectile
motion in De motu, see J. Renn, 1992, pp. 144-147.
16
the theory of machine is extant in three texts of which many copies were preserved
up till now. But none of these copies is an autograph of Galileo. The three texts
may be classified in fact into two main categories: two of them are abridged
memoranda which served eventually as booknotes for Galileo's private and public
lessons on mechanics, while the third is a developed and more extended revision
of the same material contained of the early concise outlines, with the addition of
some new sections. This revision was executed very probably around 160018.
The definition of the term momento in Le Mecaniche constituted an important
conceptual achievement by Galileo. In Section II of the treatise, this term is
defined as:
"the tendency to move downward caused not so much by the
heaviness [gravità] of the movable body as by the arrangement
which different heavy bodies have among themselves... Momento is
that impeto to go downward composed of heaviness, position, and
of anything else by which this tendency may be caused" (Opere, II,
p. 159.17-28; Galilei, 1960, p. 151).
Two aspects of momento emerge from this definition. First the signification of
static moment, meaning the product or combination of weight and distance on a
balance. The second aspect is represented by the dynamical implication of the
concept, which assimilate it to the product of weight and velocity. In fact, it is
possible that Galileo considered the velocity as a component of the second aspect
This last text alone has been edited in the Opere (vol. II, pp. 155-190) under the title Le
Mecaniche, while one of the two early versions has been published elsewhere by Favaro (cf.
Favaro, 1899). The third version was discovered and published by S. Drake in the middle of
this century (cf. Drake, 1958). The datation of the three versions of the Mechanics is a topic of
debate between historians, but the standard opinion holds that the two first versions were
written at Padua about 1593-1594, while the revision was accomplished ca. 1600 (Drake
(1958); (1978, p. 55)).
I am preparing a French annotated edition of Le Mecaniche that I began in September 1995 at
the Observatoire de la Côte d'Azur (Nice) where I was appointed as a "chercheur associé du
CNRS" for three months (sept.-nov. 1995). This edition will be published in Paris, at Les
Belles Lettres. In this work, I pay a careful attention to the structure of the texts Delle
Meccaniche. I have already localised new elements for the datation and the organisation of
the different sections of the main text and its relations with the earlier versions. In the same
context, I will try to elucidate the "mystery" of the presence of the two copies of Le Mecaniche
in Vol. 72 among the notes on motion : what significance can we give to this presence? Who
insert them there and when? This problem is similar to the one raised by Fredette (1969, pp.
85-111) about the presence of the draft De motu accelerato in Vol. 71 among the various
documents of De motu antiquiora. On the basis of various elements, it is very probable that
no one else than Galileo himself who proceeds to the organisation of the contents of Vol. 72
as it is now extant. So we can suppose logically that it was Galileo too who inserted the copies
of his work on simple machines in the file of his notes and fragments on motion. I think that
the signification of this act is of chrnonological nature, as an indication of a sequential
ordering. It can mean, in effect, that all the work on motion now extant in Vol. 72 was
brought into play after the revision of Le Mecaniche.
18
of momento. But this dynamic meaning of the concept is not sufficiently valorised in
Le Mecaniche. It was made a fundamental element of Galileo’s theory of motion
only in subsequent work19.
That the definition of momento in Le Mecaniche may be seen as having a
dynamical implication is strongly suggested by the analysis of the inclined plane in
Section 7 "Della vite" of the treatise.
The analysis of the motion on inclined planes in Le Mecaniche begins with the
enunciation of a much more developed version of the proto-inertial idea if
compared with the one edicted in De motu. Effectively, it is observed here that a
body would move downward on a surface only slightly tilted; on the contrary, if
posed on a horizontal plane, it would be indifferent to motion or to rest, but it
would have resistance toward the ascentional motion upward (Opere, II, pp. 179.9180.6). Then two weights of equal momenti are considered which are balanced at
A and C; the arm of the bent lever BC is supposed to be rotated to the position
indicated by the line BF, with the weight at F [fig. 4]. The momento of the weight in
this new position is as if it were suspended from K, and its ratio to the weight at A
is as KB to BA. If the arm BC continues to rotate until it coincides with BL, the
momento of the weight will go on diminishing proportionally.
Until this point, the demonstration has been conducted along the same lines
of the precedent one elaborated in De motu. But at this point a new approach is
inaugurated in the proof developed in Le Mecaniche.
D
[Fig. 4]
B
A
M
K
CH
Le Mecaniche {1}
O
F
I
N
L
G
E
Many passages of Le Mecaniche testify that Galileo made a sort of parallel between the static
and the dynamic meanings of momento. But it was only in the Discorso on floating bodies
(1612) that the dynamic aspect of the concept was made fully explicit as follows: "Moment
among mechanicians signifies that force, that power, that efficacy, with which the mover
moves and the moved body resists, which force depends not only upon simple heaviness, but
upon the speed of motion and upon the varying inclinations of the space over which the
motion is made, as a descending body has more impeto in a very steep descent that in one
less steep" (Opere, IV, p. 68).
19
"You see, then, how the weight placed at the end of line BC,
inclining downward along the circumference CFLJ, comes
gradually to diminish its momento and its impeto to go downward,
being sustained more and more by the lines BF and BL. But to
consider this heavy body as descending and sustained now less and
now more by the radii BF and BL, and as constrained to travel
along the circumference CFL, is not different from imagining the
same circumference CFLJ to be a surface of the same curvature
placed under the same movable body, so that this body, being
supported upon it, would be constrained to descend along it. For in
either case the movable body traces out the same path, and it does
not matter whether it is suspended from the center B and sustained
by the radius of the circle, or whether this support is removed and
it is supported by and travels upon the circumference CFLJ... if the
movable body is located at the point F, then its heaviness is partly
sustained by the circular path placed under it, and... at the first
point of its motion it is as if it were on an inclined plane according
to the tangent line GFH, since the tilt of the circumference at the
point F does not differ from the tilt of the tangent FG, apart from
the insensible angle of contact" (Opere, II, pp. 181-183; Galilei,
1960, pp. 173-174).
Then, similarly, the mom (L) / mom (C) = BM / BC; and mom (HG) / total
impeto on DCE = KB / BC = KB / BF. From the similarities of the triangles
KBF and KFH, it is concluded that "the whole and absolute moment" acquired by
the movable body in the perpendicular is in the same proportion to that which it
has on the inclined plane HF as the length of the inclined plane HF is to the
vertical line FK(ibid, 183.5-9).
In other terms, mom (FH) / tot mom (FK) = KF / FH [fig. 5]
F
[Fig. 5]
Le Mecaniche {2}
K
H
Namely,
"upon the inclined plane the force has the same proportion to the
weight as the perpendicular dropped to the horizontal from the
end of the plane has to the length of the plane" (ibid, p. 183.22-24).
The specific aspects of the demonstration of De motu theorem in Le Mecaniche
can be summarized as follows. An important point of this geometrical deduction is
indeed the diminution of the momento as the body descends on the internal face of
the circumference. In the second part of the proof, Galileo indicates clearly that he
considers the motion of the body continuously about the lower quadrant of a
circle. He assumes also that at each point of its trajectory, the motion of the body
is the same as if it were on an inclined plane tangent to that point20.
Nevertheless, the main aspect of the above deduction is the fact that the
momento of the movable is diminished as the body descends down and approaches
the horizontal. The diminution of the moment downward of the body operates in
two ways: if the body is sustained by the arm of the bent lever along the lines BF
and BL (in this case the momento is measured on the projected points M and K) or
if it travels freely along the circumference CFL. Besides, the momento being total
and integral on the perpendicular, on a horizontal plane the body has no tendency
to move and "would remain indifferent and questioning between motion and rest"
(Opere, II, p. 180.3; Galilei, 1960, p. 171) and would be made in motion "da
qualunque minima forza" (ibid, p. 180.9-10).
Several notes recorded on the papers bounded in of Vol. 72 make use of the
diminution of momento during the descent in the lower quadrant of a vertical circle.
Similarly, some typical expressions –like totale momento– or the geometrical
technique of representation by the bent lever representing the rotating arm of a
balance are manifest evidences of the influence of the theory of machines on the
early Galilean researches on motion in the period following the revision of Le
Mecaniche.
II. FIRST DISCOVERIES, FIRST DEMONSTRATIONS
In the second half of 1602, Galileo obtained his first three mathematical
propositions while he was investigating the properties of isochronism. Just after
this important achievement, he attempted laboriously to build mathematical
demonstrations of these theorems. He tried geometrical and experimental
procedures, working under the direct and tenacious influence of the concepts,
methods and terminology of the theory of machines.
1. Isochronism and paths of least-time
According to an hypothesis formulated by S. Drake (Galileo, 1960, p. 174n.; Drake, 1978, p.
61), a possible consequence of this demonstration may have been the establishment in the
mind of Galileo of points of similarity between the motion of a body on the internal face of
the circumference while it is sustained by the arm of the balance and the oscillation of a
pendeulum
20
On 29 November 1602 Galileo sent to Guidobaldo dal Monte a letter (Opere,
X, pp. 97-100) which constitutes a document of great importance. It is precisely
dated and affords, consequently, an excellent means for the datation of other
documents associated with it of which the date of composition is not precisely
known. This letter contains the mention of the first three results discovered by the
Pisan scientist in his new process of research on motion: the isochronism of the
pendulum, the law of chords (or Theorem VI, DML2) and a proposition on the
descents in the least time along the chords of a quadrant (Theorem XXII, DML2).
In terms of the diagrams in fig. 6-7, as they are drawn in the letter, the three
propositions can be expressed as follows:
 Pendulum isochronism: t(FIG) = t(BCD) [fig. 6];
 law of chords21: t(FA) = t(EA)... = t(BA) = t(IA) [fig. 7];
 Theorem XXII t(SIA) < t(SA) [fig. 7] 22.
C
A
E
D
S
B
E
F
D
C
G
I
Letter to Guidobaldo {1}
[Fig. 6]
I
F
A
Letter to Guidobaldo {2}
[Fig. 7]
Galileo didn't communicate to Guidobaldo the proofs of the three
propositions he said he had discovered, but he specified that he demonstrated
them "without transgressing the bounds of mechanics". His demonstrations are
extant in some sheets preserved in Vol. 72. These autograph documents reveal, in
effect, that his attempts to demonstrate the pendulum isochronism were essentially
The law of chords or Theorem VI-DML2 (Opere, VIII, pp. 221 sq.) is known as "Galileo's
Theorem": the time of descent along any chord of a vertical circle to its lowest point remains
the same, regardless of the length and slope of the plane; in other words, descents along arcs
of the lower quadrant should be completed in the same time. This is true of chords (paths of
pendulum oscillations) as of arcs (isochronism of the chords). On the work of Galileo on
isochronism , cf. Abattouy , 1996c (forthcoming)
22 Theorem XXII (ibid, p. 262 sq.) states that a descent along conjugate chords in the lower
quadrant of a vertical circle to the lowest point takes less time than along the single chord
connecting their endpoints. On Galileo's work on Theorem XXII, see below, Section II. 2. 2.
21
calculatory and experimental (ff. 115v, 154r, 166r, 183r, 189v...)23 On the contrary,
the main feature of his work on Theorems VI and XXII took the form of an
extensive theoretical study, in which the empirical and numerical approach were
not missing (ff. 121v, 131r, f. 150v, 151r, 160r, 163r, 186v, 189r).
1. 2. Attempts to demonstrate the law of chords
The recto of f. 154 contains traces of Galileo's research on the isochronism,
superficially analysed by means of analytical tools made up in the preceding
investigations on the theory of machines. Beside some scattered calculations
disseminated on the page, the principal diagram [fig. 8] shows a suspended body,
supported from the center and from a point located in the arm of a balance,
touching the circumference at a point where a line is tangent to the circle.
Folio 121v
[Fig. 9]
Folio 154r
[Fig. 8]
Folio 121v (Galilei, 1979, p. 13) bears probably the traces of geometric
exercises on the law of equality of times for descents along chords of a vertical
circle [fig. 10]. This diagram is similar to another diagram drawn in W. Gilbert's De
Magnete and may then be dated after 1600 (Drake, 1978, p. 67). Another fragment
on the recto of the folio (Galilei, 1979, p. 20) shows vestiges of investigations on
pendulum oscillations. The diagram drawn on the verso of f. 150 24 [fig. 10] reveals
the same concerns: the determination of the geometrical means for the demonstration of the law of isochronism. But the essential part of the work conducted by
Analysing folios 166r, 183r and 189v, Hill (1994) affirmed that these papers are records of
careful and impressive experiments in which Galileo determines empirically the times of
descent for arcs and chords (pendulums and planes), and thus develops a method for
determining time of descent for any height, given the quarter period of any pendulum. In this
analysis, he found that Galileo was well aware of the non-isochronism of the pendulum,
despite his published claims of the opposite. For a different assessment of Galileo's
manuscript material related to pendulum (ff. 151v, 154r, 198v), see S. Drake, 1990, pp. 12-31
24 Galilei, 1979, p. 12. Folio 150v is dated 1603 by Drake.
23
Galileo on the law of chords in this early period is reflected in folios 151r and
160r. These documents contain two demonstrations of Galileo's theorem which
will be reproduced almost verbatim in the Discorsi.
G
Folio 150v
[Fig.10]
D
E
B
A
C
Here is the translation of the fragment written on f. 151r:
"Let GD be erected [perpendicular] to the horizon, but DF
inclined [to it]: I say that motion takes place from G to D in the
same time as from F to D. Thus the moment along FD is the same
as that along the tangent in E, which is parallel to that same FD;
therefore, the moment along FD is to the total moment as CA to
AB, that is, to AE; but as CA to AE so is ID to DA and also the
double [of ID] FD to the double [of DA] DG; therefore the
moment along FD is to the total moment, namely, along GD, as
FD to GD; therefore, the motion along FD and GD takes place in
the same time" (Opere, VIII, p. 378.1-11; Galilei, 1979, p. 25).
The relation to be deduced is t (GD) = t (FD). Constructing the line tangent to the
circle in E, mom (FD) = momento on the tangent, while the totale momentum is the
same on all GD, so mom (FD) / mom (GD) = CA / AB. The momento on FD
being equal to that on the tangent, it is measured on the balance AB as if it were
suspended from C , so mom (FD) / mom (GD) = CA / AE . But CA / AE = ID
/ DA = FD/ DG. Therefore mom (FD) / mom (GD) = FD / GD. Finally, t (FD)
= t (GD).
The deduction is left unfinished, but it can be completed by the same elements
used in the demonstration of the inclined plane in Le Mecaniche. As noted by Wisan
(1974, p. 164), the conclusion follows from the assumption that in equal time
intervals, speeds are proportional to distances traversed. Speed is not distinguished
from momentum and in the last step of the proof, it is not distinguished from
distance either.
The other part of the early work of Galileo on the law of chords is
documented on folio 160r, where is elaborated a complete proof of the theorem.
This time, Galileo intended to prove the proposition mechanically. The use of
expressions like "constat ex elementis mecanicis", "momentum ponderis" and
"momentum suum totale" denotes of the early date of the document, and they are in
the same time evident reminiscences of the conceptual framework of the theory of
machines25.
1. BA being equal to DA, construct BE and DF perpendicularly to the
horizontal; "from the elements of mechanics", momento on ABC /
momentum totale on the vertical = BE/BA (De motu theorem
enunciated in terms of moments).
2. Since mom(DA)/mom(DF) = DF/DA or DF/BA,
3. therefore ("constat ex elementis mechanicis") that momentum ponderis
(DA)/mom(CA) = DF/BE.
4. Now, CA/DA = BE/DF.
5. Therefore, t(CA) = t(DA).
In the second part of the demonstration, the core of the proof is
concentrated in the relation AC/AD = BE/DF, which is proved by geometrical
tools introducing geometry of angles and triangles, as in the proof on f. 151r26.
1. 3. Preliminary demonstrations of Theorem XXII
Many papers of Vol. 72 document the long search of Galileo in order to
provide a rigorous proof for the other theorem discovered in 1602 and numbered
later Theorem XXII in DML2. This theorem affirms that less time is required in
The text on f. 160r has not been published by Favaro among the Frammenti attinenti ai
Discorsi, but in the footnotes to Theorem VI in DML2: Opere, VI, pp. 221-222n. Later, it was
reconstructed by W. Wisan (1974, p. 164) and reproduced by Drake (in Galilei, 1979, p. 5).
The date of composition of the note recorded on this folio is evidently an early date. This
conclusion is supported by many elements, material and theoretical, as the mathematical
technique, the handwriting with which the fragment is written and by terminological data.
For example, it contains one of the rare occurences of the expression "momentum ponderis",
to which Galileo substituted later the concept of "momentum gravitatis".
26 The material on f. 160r is published in the DML2 as the second demonstration of Theorem
VI (the famous "mechanical proof"). In DML, the result -at step 4- is said to follow from
Theorem II of uniform motion (DML1). This reference had been considered by the
commentators as a false assumption, since it relies on an illegitimate use of a proposition
valid exclusively for uniform motions (cf. M. Clavelin in Galilée, 1970, n. 90 to p. 155). But a
recent study did prove that there is no problem in the use of Theorem II of De motu æequabili
in this demonstration of the law of chords (Souffrin & Gauthero, 1992).
25
the descent along conjugate chords to the lowest point than along the single chord
connecting their endpoints.
In some early entries recorded on folios 131r and 189r, we find Galileo’s first
attempts to demonstrate the theorem by geometrical means, arithmetical
calculations and probably experimental measurements also.
A
B
C
D
[Fig. 13]
Folio 189v
E
F
On the verso of folio 189 (Galilei, 1979, p. 28), there is a diagramn containing
traces of preliminary work on the problems of isochronism with relation to
Theorem XXII [fig. 13]. The diagram is similar to the one used for the final and
successful demonstration of this proposition recorded on the folios 186v and 163r.
Folio 131r is an early draft composed of four fragments, the first of which is
deleted by cross lines [fig. 14]27. That these fragments may be attenpts to prove
Theorem XXII is testified by the diagram, almost identical to the one on f. 189v.
Here is a paraphrase of the four fragments:
T
R
D
F
O
V
M
B
S
A
Folio 131r
[Fig. 14]
L
G
1. Suppose DO C= m (RC,
BT);
CO will be = m (CD, DF) which is the
same DO;
2. As in Le Mechanice, Sect. 7, the momentum diminishes as the body
moves downward along the quadrant, as if it is weighed in a balance
suspended from DR; and the weight of the body diminishes as T (the
fulcrum of the balance) approaches to the center R.
3. BC / CD = CD ...
Folio 131r (Opere, VIII, p. 417.8-17; Galilei, 1979, pp. 7, 9, 50) was cut by S. Drake in four
non contemporaneous fragments. It is evident that this document is of early composition, as
it is attested by the incompleteness of the argument, the terminology (totale momentum) and
the similarities existing between its graphical representation and the analysis of the inclined
plane in Le Mecaniche. Besides, as in folio 154r, the line of momenti is a dotted one. It is
possible that the first fragment was added later while the third was probably abandoned for
the fourth. For an analysis of this document, cf. Wisan, 1974, pp. 176-177.
27
4. Momentum (DC) / total moment = TR / RD, with LB parallel to CD.
The third fragment can be completed in this way: BC / CD = CD/CA, and it
becomes an expression of De motu Theorem. It is possible that this proportionality
was abandoned for the last and fourth fragment, which is aonther version of the
reasoning outlined on f. 160r, where the law of chords was demonstrated by the
use of the tangent to the circle and the balance.
What Galileo sought to prove was precisely a relation of the type : t (DBC) <
t (DC). As it can be reconstructed from the diagram and from the fragments
disseminated on the folio, the mathematical argument is identical to the one used
for the final proof of the theorem on f. 163r (the letters O and V, the relation of
the first fragment : m (RC, BT) = DO = m (CD, DF). To prove the theorem, it
was sufficient to introduce the mean proportional in order to show that if VA = m
(AC, BA), then the relation between CO and CV is the solution of the problem.
The work on Theorem XXII required the mobilisation of Galileo's efforts for
many years as it is revealed by several padouan documents. In this laborious
endeavour, worth of independent and minutious study, he obtained and
demonstrated three other propositions concerned with descents accomplished in
least time: theorems XIX, XX, XXI of DML 2. The final conclusion of this family
of propositions was reached in the scholium of the brachistochrone28.
In the folios 140r, 127v, 168r, where we find the demonstrations of these
theorems, the proofs have the form of geometrical exercises on the paths of the
quickest descents. Developing mathematical consequences of the law of chords,
they deal with an identical problem: the determination of the trajectory where is
realized the swiftest descents between two points, between a line and a point and
between a point and a line. The method employed is based on elementary
geometrical procedures, issued principally from the geometry of the circle.
Reference is made to abstract mathematical relations of times that are not
represented on the diagrams. Finally, in the three cases, the result is deduced by
simple application of the law of chords.
On the other hand, the work on Theorem XXII was accomplished in two
other directions: geometrical exercises and experimental investigations. Indeed, on
folios 166r, 183r, 184r, 189r and 192r abundant numerical calculations founded on
empirical measures related to this proposition are preserved, while the geometrical
The proposition of the brachistochrone is enunciated in the DML2 as a scholium to Theorem
XXII : Opere, VIII, pp. 263-264. It asserts that the path of quickest descent in the lower
quadrant of a vertical circle is the arc and not along the chord subtending the arc. There is no
manuscript version of this proposition in Vol. 72, which may mean that this proposition is a
late discovery made up by Galileo when he prepared the Discorsi for publication. W. Wisan
has made of the scholium of the brachistochrone a fundamental piece of her interpretation of
the development of Galileo's work on motion and related it even to his cosmological
concerns : see Wisan, 1974, pp. 184-187, 1977, p. 155sq., 1983, 1984, pp. 270-271 and the
critics of Drake (1978, pp. 503-504n. 19) and Naylor (1976, pp. 95- 96) .
28
research was recorded on the folios 129r, 140r, 149v, 150r-v , 157v, 185v and 188.
Finally, on the folios 163r, 172r and 186v complete mathematical demonstrations
were constructed. The proof composed on the recto of f. 163, where Galileo
succeeded in the deduction of this difficult theorem, is specifically elaborated along
an euclidean model and on the basis of physical principles. Later, he reproduced it
in full in the third book of De motu locali29.
2. Comparison of motions on different slopes
In this sub-section our attention will be focused on a group of documents
(folios 173r, 180r, 177r, 172v, 163v...) in which the work of Galileo on a specific
problem is reflected: the ratio existing between motions of bodies moving under
natural acceleration along different planes from the same point. An aspect of this
question is the relation of the accelerated and the uniform motions. These
investigations led Galileo to the discovery of many important results: the RightAngle Theorem, the Length-Time proportionality and the Rule of Double
Distance. In the same context, emerged in the mind of our scientist some
speculations and paradoxes on the concept of velocity and on the way of finding a
rational foundation for the velocity in accelerated and uniform motions.
Although they are presented in a sequential succession, the order in which
these papers will be discussed doesn’t pretend to be a chronological one. The
succession they evoke is rather of conceptual nature. Effectively, it is possible that
Galileo's treatment of the ratios of velocity, time and distance in the different cases
of motion has followed an itinerary similar to the one described here. In this
perspective, it doesn't matter so much if these papers were written in this order or
not.
2. 1. Folio 173r : geometrical exercises
Discarded from the publication amongst the Frammenti in the Edizione
Nazionale of Galileo's works, folio 173r remained unpublished until the recent
period. It reflects the survival of the mechanical conceptual framework. An
important feature of the material recorded in this document is represented by
some cogitations on motions along the vertical and the incline in the case of two
bodies beginning their descents from the same point along different slopes30.
For more information on the papers documenting the work on least-time theorems, see
Wisan (1974, pp. 177-187); (1983); Drake (1974b); (1987), (1989, chap. 1); and Abattouy
(1989, pp. 345-349, 470-489).
30 This document has not been edited by Favaro; it was the merit of S. Drake to signal it to the
attention of historians (Drake, 1978, p. 66; Galileo, 1979, pp. XIV-XV, 4). I reproduce the
main parts of the folio below in fig. 14. For more convenience, I added numbers desides the
29
The first diagram [fig. 14] represents two balls of the same volume
descending along two different slopes; below, a note is concerned by the ratio of
momenti, in the vein of the basic relation of De motu theorem. At the right side, a
diagram shows a vertical height which is to the inclined path in the ratio of 1 to 2.
If two balls were moving from rest in the same instant on the incline and on the
vertical, in the same time in which the first will attain the extremity of the plane,
the second will traverse all the length of the perpendicular. The extreme point
attained on the inclined plane will be the beginning of a perpendicular to the
vertical path which represents geometrically the ratio existing between the two
paths of descent. The diagrams marked 3 and 6 in my reproduction of f. 173 [fig.
14] are concerned respectively with planes of various inclinations related to vertical
and horizontal lines, and with an inclined plane tangent to a vertical circle.
The examination of the contents of the folio reveals the multiform presence
of relations describing the behaviour of motions on different slopes. The inverse
proportionality relation emerges sometimes and the De motu theorem is applied to
momenta : totale momentum on BC/ momentum on BD = BD/ BC [3]. Similarly, the
presence of the inclined plane tangent to a vertical circle is an evident reminiscence
of the analysis of the inclined plane in De motu and Le Mecaniche, but this time the
bent lever seems to be replaced by a body oscillating about the lower part of the
dotted line circle [6].
2. 2. Folios 177 r-v and 180r : the Right-Angle Theorem
Folio 180r begins with the enunciation of the theorem, followed by a
relatively detailed demonstration. Such a careful enunciation and demonstration is
not usual in the early documents of Vol. 72. Here Galileo stated and demonstrated
a proposition called by the historians "the Right-Angle theorem" and which is
similar to the sequence of theorem IX31.
"If a vertical and an inclined plane are drawn from the same
point on the horizontal, and on the inclined plane is taken any
point from which a line perpendicular to it is extended toward the
vertical, [then] the motions [both] along the part of the vertical
intercepted between the horizontal and the [point of] intersection
with the perpendicular and along the part of the inclined plane
intercepted between the same perpendicular and the horizontal are
completed in the same time" (Opere, VIII, p. 385.21-26).
After that the proposition was enunciated in terms of the diagram, the
demonstration was laid down as follows:
different fragment parts. The diagram corresponding to number 6-fig. 14 was reproduced by
Drake (in Galilei, 1979, p. 9) in a false way. The circle is not on the inclined plane but under it.
31 The sequence of theorem IX is composed of theorems such as Propositions IX-XVI (DML 2).
"From the ... point F draw FH perpendicular to the horizontal,
which is be parallel to the vertical CD. Then angle HFC will be
equal to the alternate angle FCG, and right angle CHF to right
angle CFG. Therefore triangles CHF and CFG will be equiangular,
so that as HF is to FC, so is FC to CG. But just as HF is to FC, so
the moment of heaviness [momentum gravitatis] of a movable on
plane CE to its total moment [totale suum momentum] on the
vertical CD. Therefore distance CF has to distance CG the same
ratio that the moment of heaviness on the plane CE has to the total
moment along the vertical CG. Wherefore the motions along CF
and CG will be accomplished in the same time" (Opere, VIII, pp.
385.21-386.11)32.
What was to be proved is that t(CG) = t(CF). This was demonstrated by
geometrical means and by the use of the moment of heaviness. The main steps of
the deduction can be summarized as follows:
1. ° HFC = ° FCG and ° CHF = ° CFG = Right angle.
2.
3. Consequence: HF / FC = FC / CG.
4. But HF / FC = mom grav (CE) / tot mom (CD).
5. Therefore from 3 and 4, FC/CG= mom grav (CE) / mom grav (CG).
6. Wherefore t (CF) = t (CG).
The result may have been demonstrated by the application of the law of
chords, after the drawing of a circle around the diameter CG. But Galileo
discarded this solution and preferred to deduce his result as a relation of velocities
where he invested De motu theorem. From the proportionality between momenta
gravitatis and velocities, assumed elsewhere (as in f. 172v), the equation [5] may be
transcribed as
FC / CG = V (CF) / V (CG),
assuming that the moment of heaviness is constant along all the parts of the
inclined plane and along all the parts of the vertical [mom grav (CE) = mom grav
(CF) and mom grav (CD) = mom grav (CG)].
On the basis of these implicit assumptions, Galileo didn't pose the question if
the moments of heaviness make the instantaneous velocities or the total velocity
increase. This question threatened to ruin all his reasoning, but for him it wasn't
worthy of attention; he contented himself with establishing a proportionality
between the momenta gravitatis and the velocity, wether instantaneous or total.
Cut by S. Drake in two pieces (Galilei, 1979, pp. 4, 8), the text of folio 180r was neatly and
elegantly copied by Arrighetti on the verso of f. 177 (Galilei, 1979, p. 148).
32
Later, in Florence around 1618, according to the available results of the recent
critical li-terature, Nicolò Arrighetti made a copy of the contents of folio 180r on
the verso of f. 177v. On the recto of this last document, he copied another
Galilean fragment, composed of two parts. The first one was crossed out by cross
lines and interrupted in the middle of a word. Its transcription in the Ediz. Naz.
didn't reproduce the letter "b" present on the diagram in the manuscript; although
this letter plays an important role in recent interpretations of this document, it has
been neglected by previous studies.
"The velocities of moving bodies which begin motion with
such unequal moment[s] that, for example,the body begins its
motion along AC with a moment [momentum] which is to the
moment along AD as AD is to AC, are always to one another in
the same ratio as if they were to move with uniform motion [ac si
aequabili motu progrederetur]. If it were to move with uniform
motion [si aequabili motu progrederetur], the time along AC would
be to the time along AD as AC is to AD, which I doubt in the case
of accelerated [motion]; and therefore demontrate [quod in
accelerato dubito quidem, et ideo demonstra...]"
Thus, otherwise [Aliter sic]33:
"From the preceding [proposition], the time along AC is to
the time along AB as line AC to line AB. But it has the same ratio
also with respect to the time along AD, since AB is the mean
proportional between AC and AD. Therefore, the times along AD
and along AB will be equal" (Opere, VIII, pp. 386.12-24, Galilei,
(1979), pp. 10, 26).34
In this translation of "aliter sic" I follow the "ainsi autrement" proposed by P. Souffrin (1991,
p. 101).
34 Folio 177r is a copy in the hand of Arrighetti. The first paragraph is canceled by cross lines,
probably by Galileo, whom the traces of the revisions and corrections on the copies made by
his disciples are visible in many documents of Vol. 72. Favaro estimated that Arrighetti did
add at the beginning of the fragment : "addere hanc propositionem ad praecendetem
existimo", then he cancelated the first word "addere" to which he substituted "necessariam"
(Opere, VIII, p. 386, n.2). Drake (1978, p. 81, n.16), considering that f. 177r shares with f. 180r
the same theoretical concern, considered that this phrase is a part of the original galilean
note copied by Arrighetti. So, in his translation of the fragment, it begins with : "I deem this
proposition necessary to the preceding". If we pay attention to the substitution of "addere" to
"necessarium", the sentence should be: "I consider [or I deem] that I should add this
proposition to the preceding". Logically and very probably, the "preceding" to which the
fragment refer is nothing else than the fragment of f. 180r copied by Arrighetti on the verso
of f. 177. As it is remarked by P. Souffrin (1991, p. 101), the two fragments are considered in
the historiographical literature as independent and without relation with each other. In this
perspective, the "aliter sic" is seen to have no precise signification. But the results of the
recent investigations show that such an attitude doesn't help to understand this document.
33
A
D
A
B
C
D
[Fig. 16]
C
B
[Fig. 17]
Folio 177r
Stated in the first fragment is a relation of momenti: Mom (AC) / Mom (AD) =
DA / AC. Then, and under the explicit condition of the uniformity of motion, the
Theorem III is said to follow: t (AD) / t (AC) = AD / AC. The apparent
evokation of uniform motion does not mean that Galileo considered that the
theorem holds only in this type of motion.
Indeed, according to a point of view widely accepted, this first part of f. 177r
reveals the doubts Galileo raised concerning the validity of Theorem III in the case
of accelerated motion. According to this view, it was for this reason that he
returned to its demonstration in the second fragment. But the results of
contemporary research show that no question was raised in the mind of Galileo on
the applicability of the theorem in accelerated motion. From the time in which he
revised Le Mecaniche and discovered the three propositions communicated to
Guidobaldo dal Monte, he couldn't ignore the acceleration any more. Therefore,
the sort of motion he was occupied with after 1602 –as in f. 177r– must be nothing
else than the accelerated motion. In this document, he stated, in a standard
medieval way of speaking, that the Theorem III holds in the case of uniform
motion (of which it is a sort of definition) and proposed to demonstrate it also in
the case of accelerated motion.
The disappointment he manifested is relevant to his incredulity of the fact
that the theorem holds in the case of the two kinds of motions. In order to find his
way out of the puzzle, he paraphrased the theorem in terms of momenti (Mom
(AC) / Mom (AD) = DA / AC). Assuming that the momenti generate equal
velocities, this relation may be expressed in terms of velocity: V(AC) / V (AD) =
AD / AC. But his supposition is founded on the assumption of the double equality
AD = AC and V (AD) = V (AC).
Seeking to demonstrate the Length-Time proportionality in the second
fragment, Galileo began by stating the theorem:
1. By the preceding, it is assumed that t (AC) / t (AB) = AC / AB.
2. But t (AC) / t (AD) = AC / AB, AB being m (AC, AD).
3. Therefore, t (AD) = t (AB).
Referring to the diagram of the second fragment [fig. 17], the last equality is
attesting the Right-Angle theorem demonstrated in f. 180r and copied on the verso
of f. 177. If this interpretation is exact, the two fragments cited above are attempts
to resolve puzzles and doubts gene-rated by the Right-Angle theorem and the
Length-Time proportionality. In these puzzles lays the question of the status of
velocity in the relations of times and distances35.
The investigations conducted by Galileo on the problem of the Right-Angle
Theorem led him to the discovery of the Rule of Double Distance and the
Postulate36. Several theorems of DML2 (numbered later as theorems IV-V, VIIXI) emerged within these investigations. It is seen in the passages concerned with
the question of velocity in the First Day of the Dialogo (Opere, VII, pp. 47-50) the
expression of Galileo's remembering of his ancient speculations on the RightAngle Theorem and the question of the motion of two bodies from rest on two
different slopes.
III. INVESTIGATIONS ABOUT VELOCITY AND FREE FALL
In this section a decisive period of the Galilean investigations on motion will
be approached, when Galileo began to deal directly with the problems of the law
of fall. The heart of this work is evidently constituted by the two folios 128 and
85v, which are the documents associated with the letter to Sarpi of October 1604.
But other contemporaneous papers of Vol. 72 are also concerned. For example,
folio 147 documents the effort of Galileo to prove the law of fall according to a
geometrical pattern more in conformity with the Euclidean canons.
For a more detailed analysis of this first fragment, see Souffrin (1991, pp. 100-103) and
Takahashi (1993, pp. 12-14). It is remarkable that the independent researchs of P. Souffrin
and K. Takahashi led them to almost identical interpretations of f. 177r. The main feature of
Souffrin's interpretation is the strict relationship he established between the two fragments
of the document, in order to clear out the physical situation. In this perspective, the
proposition demonstrated in the second fragment is considered to be strictly linked with the
statement proposed in the beginning of the first fragment, i. e. the distances traversed in
equal times are in a constant ratio, while the conclusion of the second fragment is precisely a
geometrical construction of such a ratio. He concluded, finally, that the result established by
Galileo was the Right-angle theorem: in the motion on the vertical and on an inclined plane
from rest, distances that are as the spaces (which are as the moments) are traversed in equal
times or, in other words, the velocities are between them in a constant ratio, which is the
ratio of the moments.
36 The Postulate is the second principle on which was grounded the theory of accelerated
motion in DML, the first being the definition of naturally accelerated motion. According to
this principle, "the degrees of speed acquired by the same moveable over different
inclinations of planes are equal whenever the heights of those planes are equal" (Opere, VIII,
p. 205; Galilei, 1974, p. 162). The Rule of Double Distance as well as the Postulate pose the
problem of the ratios of velocities or of the times along different paths of motions. On the
emergence of the Rule of Double Distance see in the next section the discussion of a fragment
recorded on f. 163v; for the postulate, see below section IV. 1.
35
But before we discuss this episode of the genesis of the Galilean thought, let
us begin with the analysis of some notes that inform us on the paradoxes and the
confusions faced by Galileo in his meditations on velocity and acceleration.
1. Speculations and paradoxes about velocity and acceleration
The folios 172v, 163v and 164v document the early efforts of Galileo to
precise his ideas on the problem of velocity in motion. The general background of
his analysis is still marked by the De motu theorem and by the mechanical
conceptual system. In folio 172v the problem of velocity is posed in relation to the
law of chords and in this document we may see the emergence of a dynamic
principle according to which the velocities are generated by momenta gra-vitatis.
On the other hand, folio 163v reveals the discovery of a positive result that will be
useful in later research: the Double Distance Rule. If this folio has been composed
after the docu-ments associated with the letter to Sarpi, it may mean that Galileo
sought for a while to construct a theory of motion on the basis of the
proportionality velocity-space. Finally, a fragment entered on the verso of folio 164
accounts for a paradox on velocity.
1.1. Dynamic approach to the problem of velocity
Folio 172v reflects Galileo's concerns with the concept of velocity. Here is
the text of this important document:
"Let there be a horizontal plane along line ABC and two
planes inclined to it along lines DB and DA; I say that the same
movable will move more slowly along DA than along DB in the
ratio of length DA to length DB.
Indeed, erect BE from B vertical to the horizontal, and from
D draw DE perpendicular to BD, meeting BE at E. Around
triangle BDE describe a circle tangent to AC at point B; from this,
draw BF parallel to AD, and connect FD. It is clear that the
slowness [tarditatem] along FB will be similar to the slowness along
DA. But since the movable moves in the same time along DB and
FB, clearly the velocities [velocitates] along DB to the velocities
along FB are as DB to FB, so that two moveables coming from
points D and F by lines DB and FB always travel along
proportional parts of the entire lines BD and FB in the same times
[from rest]. But since angle BFD in the segment is equal to angle
DBA with the tangent, while angle DBF is alternate to BDA,
triangles BFD and ABD are similar, and as BD is to BF, so AD is
to BD. Therefore as AD is to BD, so the speed along DB is to the
speed along DA, and conversely, the tardity along DA is [in the
same ratio] to the tardity along DB.
E
Folio 172v
[Fig. 18]
D
F
A
B
C
If this is assumed, the rest can be demonstrated. Therefore it is
to be postulated that velocity of motion is increased or diminished
according to the ratio in which moments of heaviness [gravitatis
momenta] are incresaed or diminished; and since it is clear that the
moments of heaviness of the same moveable on the plane DB are to
the moments [of heaviness] on plane DA as length DA is to length
DB, for that reason the velocity along DB is to the velocity along DA
as DA is to DB" (Opere, VIII, pp. 378.12-379.6; Galilei, 1979, p. 11) 37.
The proposition is first ennunciated generally in terms of the diagram: the
ratio of slowness on two inclined planes is proportional to the lengths of these
planes. The final conclusion affirms instead a relation of velocities. The
demonstration proceeds as follows:
1. Tard (DA) / tard (DB) = DA / DB, the tardity being assumed
inversely proportional to velocity.
2. After the construction of the diagram, tard (FB) = tard (DA), the two
planes having the same slope.
3. t (DB) = t (FB), by the law of chords.
This demonstration has been copied by Guiducci on f. 34r (Galilei, 1979, p. 127). Folio 172v
is a document very controversed in recent galilean studies. W. Wisan (1974, p. 222) linked it
to the letters exchanged by Galileo and Luca Valerio in 1609, but in a subsequent work
(Wisan, 1984, pp. 272-273) she considered it implicitly as belonging to the period before the
letter to Sarpi of 1604. For S. Drake (1978, p. 80) f. 172v is an early document composed
around 1603 in which Galileo sought to express the conclusion of the law of chords in terms
of constant but differing speed along each slope. P. Souffrin (1991, pp. 96-100), on his part,
interpreted it as a document that seeks to prove a proposition which is theoretically exact
and constitutes the De motu theorem: "the spaces traversed in equal times on inclined planes
of the same height are to each other inversely proportional as the lengths of the planes". For
a detailed analysis of f. 172v, see the critical literature refered to in this note and galluzzi
(1979, pp. 288-291); Giust (1981, p. 12) and takahashi (1993, pp. 4-6).
37
4. So it is evident that the velocities along DB / velocities along FB = DB
/ FB, that is V (DB) / V (FB) = DB / FB.
5. Since the ratio of the velocities on two planes of diverse inclinations is
constant in the course of motion, then that parts of DB and FB which
are between them as DB / FB are always traversed in the same times.
6. From elementary geometry, it is infered that BD / BF = AD / DB
7. Therefore, AD / DB = V (DB) / V (DA) and conversely AD / DB =
tard (DA) / tard (DB).
8. In the lat paragraph a postulate is assumed: the increase or decrease of
velocity depends on the increase or decrease of moments of heaviness.
9. Since mom grav (DB) / mom grav (DA) = DA / DB,
10. So V (DB) / V (DA) = DA / DB.
What was to be shown is that the velocities on CA and on DB are in the ratio
of DB to DA. In other terms, that the spaces traversed in equal times on two
inclined planes of the same height are between them inversely as the lengths of
these planes. This is, of course, the De motu theorem. The same result is infered in
the last part of the document, but this time it is expressed in the form of a general
dynamical principle, as a relation of moments of heaviness generating constant
velocities, applied here to free fall.
The analysis is founded on the law of chords and on the assumption that on
two inclined planes, the spaces corresponding to the same time of motion from
rest are in a constant ratio. This last assumption was assumed by Galileo at least
from Le Mecaniche period.
1. 2. Emergence of the Rule of Double Distance
During his Paduan researches on motion, Galileo recorded many entries on
folio 163. Among the numerous notes marked on this document we find the
complete derivation of Theo-rem XXII on the recto. The verso was devoted to a
couple of fragments, written probably at different dates. The first of these is a sort
of derivation of the Rule of Double Distance by means of comparison between
velocities in accelerated and uniform motion. In the second note was attempted a
demonstration of the Length-Time Proportionality, considered as a consequence
of the law of fall and of the theorem of chords38.
The text of the first note is short and illustrated by a double diagram:
"Let there be naturally accelerated [naturaliter acceleratus]
motion from A to B: I say that if velocity [velocitas] at all points
AB were the same as that found at point B, the space AB would be
38
See below for an analysis of this note, Sect. IV. 1.
traversed twice as quickly, because all the velocities [velocitates
omnes] in the single points of line AB have the same ratio to all the
velocities each of which is equal to the velocity BC, as the triangle
ABC has to the rectangle ABCD. From this, it follows that if there
were a plane BA inclined to the horizontal line CD, and BC being
double BA, then the moving body would come from A to B and
successively from B to C in equal times. For after it was in B, it will
be moved along the remaining BC with the same uniform velocity
that it had at terminus B after fall AB. It is further clear that the
whole time through ABE is three-halves [sesquialterum] that
through AB" (Opere, VIII, pp. 383.24-384.13; Galilei, 1979, p. 19)39.
A
D
A
B
[Fig. 19]
C
C
Fo lio 163v {1}
E
B
D
[ Fig. 20 ]
In De motu, Galileo has affirmed a notion of the possibility of continuation of
motion on the horizontal plane; in Le Mecaniche, this idea was well developed: on
such a plane, a frictionless body should be indifferent and questioning between
rest and motion (Opere, II, p. 180.1-5). On the verso of f. 163, a similar idea of
conservation is formulated in the shape of a mathematical relationship, the Rule of
Double Distance, This rule will be so useful in the subsequent work performed by
our scientist later on projectile motion40.
Another assumption guided the reasoning on f. 163v. Since velocity at any
point on AB is to the final velocity on BC as triangle ABC to rectangle ABCD (that
is in the ratio 1 to 2) [fig.19], the space AB would be traversed "twice as fast" in
that uniform motion whose velocity is equal to the final velocity acquired at point
B [fig. 2O]. This result may be called the double velocity rule. It is obvious that
Galileo is assuming here the relation v  d. In a subsequent note on the same f.
163v, the Double Distance Rule is set forth, on the basis of the result stated in the
first note:
This note had been copied by Arrighetti (f. 181r, Galilei, 1979, p. 150). In the bottom of f.
181r, the Galilean note is followed by the following remark: "Huic demonstrationi
necessarium mihi videtur ostendisse antea, motum orizontalem uniformiter progredi in
infinitum".
40 DDR affirms that if a body is moved from rest on an inclined plane and then its motion
continued on an horizontal plane with a uniform motion equal to the maximum velocity
acquired in the accelerated descent, it would complete the space AB "twice as quickly" as in
the first accelerated motion. This rule has been helpful in the process of derivation of the
fundamental relation v  t . On this point, see below the analysis of f. 91v, Sect. IV. 3.
39
if BC= 2AB, then t (AB) = t (BC, A) [fig. 21].
"If a motion is continued through the horizontal plane after a
fall through a certain inclined plane, the time of fall through the
inclined plane will be to the time of fall through any horizontal line
as double the length of the inclined plane is to the taken horizontal
line.
Let a horizontal line be CB, and an inclined plane be AB. And
after the fall through AB let a motion be continued through the
horizontal, in which let any line, BD, taken. I say that the time of
fall through AB is to the time of fall through BD as double AB is
to BD. For, BC be taken double AB, it is clear from what were
demonstrated that the time of fall through AB is equal to the time
of motion through BC. But the time of motion through BC is to
the time of motion through BD as line CB is to line BD. Therefore
the time of motion through AB is to the time of motion through
BD as double line AB is to line BD" (Opere, VIII, p. 384. 14-24).
A
Folio 163v {2}
[ Fig. 21 ]
C
D
B
In the inference of DDR, Galileo sought to find BC such that t(BC, A) =
t(AB). On the basis of the idea of preservation of motion and the result of the first
step, he derived DDR: if BC = 2AB and t (AB) = t (BC, A) = T. Then, two
corollaries are derived on the lower part of the page. The first one stated that if BE
= 1/2 BC, then t (BE, A) = 1/2 t (AB) = 1/2 T; and t(AB) + t (BE, A) = T + 1/2
T = 3/2 T; while we learn from the second that if D is a point between B and C,
then t (AB) / t (BD) = 2 AB / BD. This corollary will be later formalized as
Theorem XVI of DML2.
The Double Distance Rule provided an important element on the the
relationship between accelerated and uniform motions. In the Discorsi, this
relationship will be expressed formally as the Theorem I which opens the Liber II
of DML41. In f. 163v, this rule was founded on the assumption that in free fall the
velocity is directly proportional to distance of fall. In effect, altough the Double
Distance Rule is a correct statement about the motion of fall from the point of
Theorem I de motu naturaliter accelerato enunciated that a body traversing the same
distance under acceleration from rest has half as much velocity as if it moved uniformly over
the whole distance at the maximum speed acquired under acceleration (Opere, VIII, p. 208).
41
view of classical mechanics, Galileo's derivation of it depends on the principle
mentioned in the letter to Sarpi (the proportionality between velocities and
distances in accelerated motion), which is a false principle from the point of view
of classical mechanics42.
If the first fragment of f. 163v was composed after the letter to Sarpi, it may
mean that Galileo considered the possibility of founding a theory of motion on the
basis of the proportio-nality v  s. If this hypothesis is correct, a document of the
same period, the folio 172r (Opere, VIII, pp. 392.16-393.17; Galilei, 1979, p. 45),
shows that he also attempted to derive Theorem III directly from the relation v 
s so that he could demonstrate the law of chords on the basis of the law of fall and
of Theorem III.
1. 3. A paradox on velocity
Folio 164v reveals a paradox related to the status of the concept of velocity in
accelerated motion. Comparing two accelerated motions, one along the vertical
and the other along an inclined plane, Galileo posed the question about the
velocity in the two cases.
"It has to be seen [Mirandum] wether the motion along the
perpendicular AD is not perhaps faster than that along the inclined
plane AB ? It seems so; in fact, equal spaces are traversed more
quickly along AD than along AB. Still it seems not so; in fact,
drawing the horizontal BC, the time along AB is to the time along
AC as AB is to AC, then the moments of velocity [momenta
velocitatis] along AB and along AC are the same. And effectively,
that velocity is one and the same which, in unequal times traverses
unequal spaces which are in the same proportion as the times"
(Opere, VIII, p. 375.14-22; Galilei, 1979, p. 78).
A
B
C
Folio 164v
[ Fig. 22 ]
D
This note can be paraphrased as follows :
1. The motion along the vertical must be faster than the motion along
the inclined plane, since "equal spaces are traversed more quickly" (or
in less time) along the vertical than along the oblique path.
42
Wisan (1974), pp. 206-07; Renn (1992), p. 174.
2. But, according to Theorem III: t (AB) / t (AC) = AB / AC, which
contradicts the first statement.
3. Now, the mom vel (AB) = mom vel (AC). This equality is then
expressed in the form of a general definition of equal velocities, which
are those that "in unequal times traverse unequal spaces which are in
the same proportion as the times".
Such a reflection on velocity with relationship to Theorem III is similar to the
one recor-ded on f. 177r, but without reference here to uniform motion. In the
Mirandum fragment, Galileo is preoccupied with another aspect of his velocity
concept, trying to solve the incompatibility between the two first statements of his
argument. Noticing this incompatibility, he did let it open as one more paradox of
the treatment of velocity: how could it be that in accelerated motion too, equal
velocities are characterized by equal proportions between distances and times43.
2. Derivation of the Law of fall: The crisis of 1604
The law of free fall states that distances from rest are proportional to the
squares of elapsed times from rest. In 1604, Galileo sought to prove this important
theorem on the basis of an erroneous principle: the proportionality between the
growth of velocity and the space traversed from rest. In the Discorsi, he opened the
De motu naturaliter accelerato by the definition attesting the proportionality
between the degrees of velocity and the times of fall, then made an allusion to his
ancient error and tried to refute it by a reasoning considered generally by historians
as erroneous44. Then, the law of free fall was stated and demonstrated as Theorem
II of accelerated motion, followed by two corollaries which constitute different
formulations of the same law: the rule of odd numbers45 and the mean
proportional corollary46.
We don't find a manuscript version of the demonstration of the Timessquared theorem as it was presented in the Discorsi. Instead, there exists in Vol. 72
a direct demonstration of the theorem on f. 128 and two mathematical arguments
related to the mean-proportional corollary on ff. 85v and 147r. It is possible that
I rely in the interpretation of f. 164v on Wisan (1974), pp. 201-204 and Renn (1992), pp.
197-199.
44 On this question, see Cohen (1956), Hall (1958), Drake (1970 a), pp. 28-42, (1970b), pp. 229237, Finnochiaro (1972), (1973), Sylla (1986), pp. 81-83.
45 The successive spaces traversed in successive equal intervalls of time in the motion of fall
are as the odd numbers from the unity (Opere, VIII, p. 210).
46 If at the beginning of motion there are taken any two spaces whatever, run through in any
two times, the times will be to each other as either of these two spaces is to the mean
proportional between these two given spaces (Ibid, p. 214).
43
Galileo performed his final deduction of this important proposition in the stage of
the final composition of his treatise on motion.
The historical and epistemological debate about the 1604 documents has
been thoroughly discussed by historians. In his classical study of this episode of
the Galilean thought, Koyré (1966) concluded that Galileo has accorded to space
what must be attributed to time and his error may be characterized as a
"géométrisation à outrance". Hence, his error was mathematical and conceptual,
and, in sum, a trivial one, because it was favorised by the visual and empirical
advantage of space. But the errors of a Galileo are instructive. They are helpful in
the appre-ciation of the inherent difficulties raised by a scientific problem in a
defined historical context. It does import to characterize them conveniently, and to
scrutinize the functioning or dysfunctioning of the systems of thoughts and
concepts which provoked them47.
The scenario I propose for the way in which Galileo may have discovered
and demonstra-ted the squared-time theorem in Padua around 1604 is the
following. First, he formulated it as a relation of the mean proportional, more
compatible with Euclidean geometry and specially with the theory of proportions
(Elements, Book V), which was the main mathematical theory he worked with.
Then he tried to demonstrate his theorem directly. This line of reasoning is
reflected by his work on ff. 147, 85 and128.
2. 1. Official announcement of the Times-squared theorem
The famous letter addressed by Galileo on October 16, 1604, to his Venitian
friend Paolo Sarpi constitutes the act one of the "1604 affair." In it he announced
his discovery of a law of free fall in the form of the times-squared theorem, and he
stated his theorem on the basis of a "natural and evident principle", namely the
proportionality of degrees of velocity in natural motion to the space traversed (v 
S). The general line along which he presented his thought is as follows: if we
accept as true this principle, then the theorem (the space traversed is proportional
to the square of the time elapsed in fall: S  T2) can be proved.
The 1604-episode in the genesis of Galileo's thought is well known. The documents
associated with the letter to Sarpi were analysed widely. In a previous work (Abattouy, 1989,
chap. V) I examined extensively all the aspects related to this point. In the following, my aim
will be the simple presentation ofa general overview on the debate concerning these
documents.
47
A
B
C
Letter to Sarpi
[ Fig. 23 ]
D
"Thinking again about the matters of motion, in which, to
demonstrate the phenomena [accidenti] observed by me, I lacked a
completely indubitable principle to put as an axiom, I am reduced
to a proposition which has much of the natural and the evident:
and with this assumed, I then demonstrate the rest; that is, that the
spaces passed by natural motion are in double proportion to the
times, and consequently the spaces passed in equal times are as the
odd numbers from one, and the other things. And the principle is
this: that the natural movable goes increasing in velocity with that
proportion with which it departs from the beginning of its motion;
as, for example, the heavy body falling from the terminus A along
the line ABCD, I assume that the degree of velocity that it has at C,
to the degree it had at B, is as the distance CA to the distance BA,
and thus consequently, at D it has a degree of velocity greater than
at C according as the distance DA is greater than CA. I should like
your reverence to consider this a bit, and tell me your opinion. And
if we accept this principle, we not only demonstrate (as I said) the
other conclusions, but I believe we also have it very much in hand
to show that the naturally falling body and the violent porojectile
pass through the same proportions of velocity" (Opere, X, p. 115)48.
In the letter Galileo didn't expose any demonstration of the Times-squared
theorem; the proof is contained in a document of Vol. 72, the famous folio 128r-v
which is chronologically very close to the letter to Sarpi. In effect, many elements,
theoretical and material, confirm that the two documents belong to the same
intellectual stage in the evolution of Galileo's thought, and were written very
probably in the last quarter of 160449.
Namely, that "it is manifest that the impetus at the points D, C, B goes decreasing in the
proportions of the lines DA, CA, BA; whence, if it goes acquiring degrees of velocity in the
same (proportions) of the lines DA, CA, BA; whence, if it goes acquiring of velocity in the
same (proportions) in natural fall, what I have said and believed up to now is true" (ibid, p.
115).
49 Folio 128 is one of the rare documents of Vol. 72 to be dated very undoubtedly. Besides the
similarity of the mahematical argument in the folio and in the letter to Sarpi, the same
watermark is found in a cover sheet of the letter to Sarpi, in folio 128 and in f. 4 of Mss. Gal.
47, on which Galileo recorded some notes related to the nova of October 1604. On this point,
see Drake, 1972a, pp. 60-61 and Drake, 1978, p. 76, n. 6. Moreover, J. Renn (1992, pp. 16948
2. 2. Kinematical demonstration of the Mean Proportional
As conjectured above, it is possible that Galileo dealt with the Times-squared
theorem at first in the form of the mean proportional formula. The traces of this
preliminary investigation is recorded on ff. 147r and 85v. Let's begin by outlining
the reasoning entered on the first of these documents50.
"After it has been demonstrated that the times through AB
and AC are equals, it must be shown that the time through AD is
to time through AE as DA is to the mean [proportional] between
DA and AE. For the time through DA is to the time through AC
as line DA is to AC; but the time through AC (which is that
through AB) is to the time AE as line BA is to AE, which is as SA
is to AD. Therefore, by equidistance of ratios in perturbed
proportionality [ex aequali in analogia perturbata], the time through
AD is to the time through AE as line SA is to line AC. And since
AC, as has been demonstrated, is the mean [proportional] between
SA and AB, while as SA is to AB, so DA is to AE, therefore the
time through AD is to the time through AE as DA is to the mean
[proportional] between DA and AE, which was to be proved"
(Opere, VIII, 380.1-12; Galilei, 1979, 24).
In this document, we have probably indications on how Galileo's work on the
law of fall gave birth to a precise formula for the characterization of acceleration in
natural motion, which was expressed in terms of the Mean Proportional Rule, in
such a form that the times-squared relation can be deduced automatically from it.
The demonstration is founded on two presuppositions: the law of chords and the
Theorem III [Fig. 24]. Beginning by assuming the law of chords, the conclusion to
be deduced is said to follow from it.
For t (AD) /t (AC) = DA / AC,
but t (AC) [= t (AB)] / t (AE) = BA / AE = SA / AD.
Therefore, by the application of a technique of the Euclidean theory of
proportions, the equidistance of ratios in perturbed proportionality, t (AD) / t
(AE) = SA / AC.
170) attaracted attention to the close relationship devoted by the two Galilean documents
(the letter to sarpi and f. 128) to the symmetry between projectile motion and the motion of
fall. This argument can be considered as a supplementary argument of datation.
50 This reconstrcution of Galileo's discovery and demonstration of the law of fall along these
lines (that is, at first in terms of mean-propoprtional, then as the Times-squared relation, and
finally with an experimental corroboration, as the one recorded on f. 107v) was presented in
Abattouy, 1989, p. 367 ff and pp. 431-450.
Assuming that AC = m (SA, AB) and SA / AB = DA / AE, therefore, t (AD) / t
(AE) = DA / m (DA, AE).
As noted by J. Renn (1992, p. 158), in this demonstration Galileo derived "a
quantitative law of acceleration from two premises which by themselves do not
presuppose an analysis of the internal structure of the motion of fall". Although it
didn't constitute a proof in the strict sense, since it said nothing on the law of
motion and on the crucial question of the mode in which the speed grows in fall,
the demonstration developed in f. 147r constituted undoubtedly an important step
in Galileo's search for a complete proof of the law of fall.
2. 3. Physical proof and indications on the resolution of the crisis
The third document which may be associated with the letter to Sarpi is f. 85v, a
Latin copy by Guiducci of a paduan original of Galileo, now lost. The contents of
the note recorded on this paper can be an intermediary stage between the
mathematical arguments constructed on the ff. 147 and 128. On f. 85v, the timessquared theorem was formulated in terms of the Mean Proportional Rule, and its
deductive structure was founded on the proportionality of velocities to spaces
traversed from rest.
"I assume that the acceleration [accelerationem]of the falling body along the line
AL is such that the velocity [velocitas] increases in the ratio of the space traversed
so that the velocity in C is to the velocity in B as the space CA is to the space BA,
etc. Matters standing thus, let the line AX be drawn at some angle to AL, and,
taking the parts AB, BC, CD, DE, etc. To be equal, draw BM, CN, DO, EP, etc. If
therefore the velocities of the body falling along AL in the places B, C, D, E are as
the distances AB, AC, AD, AE, etc., then they will also be as the line BM, CN,
DO, EP.
But because the velocity is successively increased in all points of the line AE, and
not only in B, C and D, which are drawn, therefore all these velocities [velocitates
illae omnes]are to one another as the lines from all the said points of the line AE,
which are gene-rated equidistantly from the same BM, CN, DO.
But those are infinite and constitute the triangle AEP: therefore the velocities in all
points of the line AB are to the velocities in all points of the line AC as the triangle
ABM to the triangle ACN, and so for the remaining, i.e., in double proportion of
the lines AB, AC.
But because in the ratio of the increase [of velocity due] to acceleration the times
in which the motions themselves occur must decrease, therefore the time in which
the moving body traverses AB will be to the time in which it traverses AC as the
line AB is to that line which is the mean proportional between AB and AC" (Opere,
VIII, pp. 383.1-23; Galilei, 1979, 43).
As it is extant now in the Manuscript 72, this note is crossed out by cross lines,
eventually by Galileo himself. It is possible that this Latin memorandum was
wrongly copied in Florence. If this conjecture is correct, the error may have been
done by Galileo himself (who included it among the fragments to be reproduced)
or by Guiducci. It is in effect very difficult to think that Galileo continued to hold
very probably the copies were executed.
On the contrary, that the note on f. 85v was crossed out can be interpreted as a
decisive indication o
Moreover, there is another indication as much significative as this one. Effectively,
the obvious and logical function of the line marked "S" near the diagram [fig. 25]
should be that the vertical line in the triangle of speeds must represent time, while
the graphical representation of the spaces traversed is to be drawn on an
independant line. By this, Galileo corrected implicitly, at least for himself, his old
error, and perthaps conceived at this occasion the argument by which he rejec-ted
in the Discorsi the proportionality of the changing velocities to space.
2. 4. Formal deduction of the Times-squared theorem
The document represeting the attempts of Galileo in 1604 in order to produce a
deductive treatment of acceleration, and precisely to demonstrate mathematically
the Times-squared theorem is recorded on f. 128. The text of this paper is two
pages long and well structured.
"I suppose (and perhaps I shall be able to demonstrate this) that the naturally
falling heavy body goes continually increasing its velocity according as the distance
increases from the terminus from which it parted, as, for example, the heavy body
departing from the point A and falling through the line AB. I suppose that the
degree of velocity at point D is as much greater than the degree than AC; and so
the degree of velocity at E is to the degree of velocity at D as EA to DA, and thus
at every point of the line AB it [the body] is to be found with degrees of velocity
proportional to the distances of these points from the terminus A. This principle
appears to me very natural, and one that corresponds to all the experiences that we
see in the instruments and machines that work by striking, in which the percussent
works so much the greater effect, the greater the height from which it falls; and
this principle assumed, I shall demonstrate the rest.
Draw line AK at any angle with AF, and through points C, D, E and F draw the
parallels CG, DH, EI, FH: and since lines FK, EI, DH, and CG are to one another
as FA, EA, DA, CA, therefore the velocities at points F, E, D and C are as lines
FK, EI, DH and CG. So the degrees of velocity go continually increasing at all
points of line AF according to the increase of parallels drawn from all those same
points. Moreover, since the velocity with which the moving body has come from A
to D is compounded from all the degrees of velocity it had at all the points of line
AD, and the velocity with which it has passed through line AC is compounded
from all points of line AC, therefore the velocity with which it has passed the line
AD has that proportion to the velocity with which it has passed the line AC which
all the parallel lines drawn from all the points of line AD over to AH have to all
the parallels drawn from all the points of line AC over to AG; and this proportion
is that which the triangle ADH has to the triangle ACG, that is the square of AD
to the square of AC. Then the velocity with which the line AD is traversed to the
velocity with which the line AC is traversed has the double proportion that DA has
to CA.
And since velocity to velocity has contrary proportion [contraria proportione] of
that which time has to time (for it is the same thing to increase the velocity as to
decrease the time), therefore the time of the motion along AD to the time of the
motion on AC has half the proportion that the distance AD has to the distance
AC. The distances, then, from the beginning of the motion are as the squares of
the times, and, dividing, the spaces passed in equal times are as the odd numbers
from unity. Which corresponds to what I have always said and to experiences
observed; and thus all the truths are in accord" (Opere, VIII, pp. 373.1-374.22;
Galilei, 1979, pp. 41-42) .
The line ACDEFB represents the trajectory of the body falling from rest at A; the
parallels CG, DH, EI, FK are between them as the degrees of velocity in C, D, E,
K respectively. After the graphical representation is set out, the demonstration can
be resumed in the following way.
1. The degrees of velocity are proportional to the spaces traversed: deg vel (C)/
deg vel (D) = CG / DH.
2. Representing the velocity at any point of AB by a parallel line, V(F), V(E), V(D)
and V(C) are proportional to FK, EI, DH and CG.
3. Moreover, since V(AD) = all the instantaneous velocities on AD, then V(AD) /
V(AC) = all the vi (AD) / vi (AC).
4. That is, D ADH / D ACG = AD2 / AC2.
5. Therefore V(AD) / V(AC) = AD2 / AC2.
6. And since velocity has contrary proportionality to time, then t (AD) / t (AC) =
÷ AD / AC.
7. Therefore AD / AC = t (AD)2 / t (AC)2, and by the division of ratios we
obtain the rule of odd numbers.
In this deduction, Galileo made use of two concepts of velocity: instantaneous
velocity and a velocity acquired when the moveable passes through a given space
in a time. Galileo linked these two concepts in step 3, when he considered that V is
composed by numerous v. This allowed him to represent graphically V by the area
of the triangle constituted of all parallel lines representing v at all points of AB.
Historians considered that the error of Galileo is situated at steps 5 and 6, when he
considered that velocity is proportional to the square of the distance and that
velocity had contrary proportionality to time, from which he deduced the theorem
finally. In other terms, Galileo derived a relation between space and time from a
proportionality between velocity and space. In fact, he was helped in this
procedure by the second of his two concepts of velocity: the velocity acquired on a
distance, which was defined in Galileo's days along the lines of what was allowed
by Euclidean theory of proportionality: in the case of equality of velocities, the
spaces are as the times; in the case of equality of the times, the velocities are
proportional to spaces. From these relations, he could generalize, as he does in
Theorem IV of De motu aequabili .
It remains to understand Galileo's use of the contrary proportionality. It seems that
there Galileo made an error in manipulating ratios, or at least that he made use of a
technique of the theory of proportions that have not yet been clearly identified by
historians. Perhaps he founded his assertion on a trivial conception that the
relation of velocity with time in motion is a relation between an increasing
magnitude with a decreasing one. In effect, conceiving that "it is the same to
increase velocity as to diminish time", he transcribed this proportionality in a
relation of square root. On this ground, step (6) of the paraphrase can be
translated in: Since V1 / V2 = ÷ Tl / T2, then Tl / T2 = ÷ S1 / S2;
therefore S1 / S2 = T12 / T22.
After he has finished his work on free fall as it is documented by the papers of
1604, Galileo must have continued his interrogations and speculations on the
growth of velocity in the motion downward. It is in reality hard to admit that he
was satisfied with the demonstration recorded on the f. 128 or on the paduan
original of f. 85v. But failing to find an alternative way to demonstrate the Timessquared theorem, he gave up his search in this direction, and began working on
other problems of motion, in the direction of researches recorded on the
documents that will be analyzed in the next section .
But before this, let's look briefly at another aspect of Galileo's investigations on the
fall of bodies around 1604. I mean the experimental investigation he conducted in
order to confirm the times-squared theorem. An important element of information
in this perspective is provided by a set of calculations and diagrams on folio 107v
where Galileo recorded probably the data obtained in the course of an experiment
performed in order to confirm the law of fall. In this experiment, he measured the
distances traversed in equal times along an inclined plane, then confronted the
successive distances with the sequence of square numbers starting from the unity.
The fact that he also entered several other simple sequences of integers in this
manuscript page suggests that he had a precise idea about the acceleration along
the plane.
Besides the different sequences of numbers, folio 107v contains indeed two
diagrams which may suggest that when drawing them Galileo was thinking of
acceleration, precisely of the change of velocity in motion; at least one of the
diagrams evokes the graphical representation proper to the medieval technique of
configurations applied to the motion of fall . In fact, in f. 107v as in the other
documents pertaining to 1604, the acceleration is explicitly and entirely admitted.
In sum, these papers reveal an idea of acceleration which is incompatible with the
one elaborated in De motu, in the sense that acceleration is no longer considered as
a transitory accident, but as an essential property of the natural fall, to which it is
worthwhile to look for a precise mathematical formula.
IV. AFTER 1604 : NEW RESEARCHES ON FREE FALL AND FURTHER
RESULTS
The failure of the demonstration of the Times-squared law in folio 128 generated a
sort of crisis in Galileo's work on motion. The obvious decisive consequence of
such a crisis was that our scientist seems to start a new process of researches, in
order to proove some other propositions emerged in the previous work on the fall
of bodies. As far as the documents of Vol. 72 can inform us, these new
investigations led him to the deduction of mathematical proofs of two important
propositions, Theorem III and Theorem XXII . The analysis of the papers on
which he recorded his exercises on these propositions reveals that he couldn't
erase all the traces of the principle v  s. Moreover, it is possible that he sought for
a while to found his theory of motion on this principle, since several of his
propositions were based on the proportionality of the velocities and the spaces
traversed. If this interpretation is correct, we can suppose that the note concerned
with the Rule of Double Distance on f. 163 emerged in this post-1604 crisis
context. Indeed, on the verso of this document was entered a fragment devoted to
the deduction of Theorem III as a consequence of the law of fall and of the law of
chords.
1. Demonstration of the Length-Time Theorem
The Theorem III of accelerated motion in the Discorsi stated a Length-Time
proportiona-lity: in the motion from rest along the vertical and along the inclined
plane of the same height, the times are as the lengths. The full manuscript
demonstration of this proposition is built on the recto of f. 179, where the
theorem is deduced in a way similar to the one in which it was demonstrated finally
in the Discorsi. But we find in Vol. 72 other proofs of this theorem, as the one
recorded on the verso of f. 163 and which seems to be of early date.
The fragment on Theorem III in f. 163v follows the note related to the Rule of
Double Distance discussed above and it is obviously written in a different
handwriting:
"The time of fall through an inclined plane is to the time of fall through the line of
its height as the length of the same plane is to the length of its height.
Let there be an inclined BA to the horizontal line AC such that the vertical line of
heights is BC; I say that the time of fall in which a movable runs through BA is to
the time in which it falls through BC as BA is to BC. Let a perpendicular AD to
the horizontal be erected from A, and let [another] perpendicular to AB drawn
from B, that is BD, meet it at D. Let a circle be circumcribed around triangle ABD.
Then, since DA and BC are both perpendicular to the horizontal, it is clear that the
time of fall through DA is to the time of fall through BC as the mean proportional
between DA and BC is to the same BC. But the time of fall through DA is equal to
the time of fall through BA, while the mean proportional between DA and BC is
BA itself. Therefore the proposition is evident (Opere, VIII, pp. 384.25-386.20).
In order to demonstrate Theorem III, the law of fall and the Mean Proportional
Rule are introduced.
t (DA) = t (BA),
t (DA) / t (BC) = m (DA, BC) / BC.
Since m (DA, BC) = BA, then t (BA) / t (BC) = BA / BC.
On f. 179r, the demonstration of the same theorem followed another way. It was
founded on the consequences of the Postulate and of a proposition on uniform
motion. First, the theorem is stated:
"If the same body is carried in the vertical and in an inclined plane, of which the
height is the same, the times of motions are to one another as the lengths of the
inclined planes and of the vertical".
Then it is enunciated in terms of the diagram:
"I say that the time of motion along AB is to the time of motion along AC as
lenght AB is to length AC."
"For, since it is assumed that in natural descent the {same} moments of velocity
{are always found in points equally distant from the horizontal according to the
vertical distances}, [and] grow continually according to the ratio of the vertical
removal from the horizontal in which the beginning of motion was, it is clear tha
producing the horizontal line AM which will be parallel to BC, and taking in the
vertical any number of points E, G, I, and L, through which are drawn the parallels
to the horizontal ED, GF, IH, and LK, then the moment or degree of velocity at
point E of the body moving along AB will be the same with the degree of velocity
at point D of the body carried along AC, since points E and D have the same
vertical distance from the horizontal AM; and it is likewise concluded that the
moment of velocity is the same at points F and G, and then again at points H and
I, K and L, C and B. (And since velocity is always intensified in the ratio of the
removal from the terminus A, it is clear that in motion AB there are many different
degrees or moments of velocity [tot esse velocitatis gradus seu momenta diversa]
as there are in the same line AB points distant from the terminus A, to which there
correspond as many [points] in line AC, and these are determined by the parallel
lines in which the same degrees of velocity exist).
"Therefore there are in line AB, so to speak, innumerable tiny spaces [innumera
spaciola], and the others, being equal in a certain sense to them in multitude and,
{taken in pair}, corresponding [to them] in the same ratio, are designated in AC by
innumerable parallel lines extended from points on line AB to line AC. {For the
intercepted spaces AD, DF, FH, etc. correspond one by one to the spaces AE,
EG, GI etc., according to the ratio of AC to AB}. And there exist the same
degrees of velocity in particular corresponding pairs.
"Therefore from the preceding, all the times taken together of all the motions
along AB will have to all the times likewise taken of all the motions along AC the
same ratio that all the spaces of line AB [have] to all the spaces of line AC. But this
is indeed identical with the case that the time of fall along AB [tempus casus per
AB] is to the time of fall along AC as line AB is to AC, which was to be
demonstrated" (Opere, VIII, p. 387. 16-388. 28).
This document is composed of many parts, as it is obvious, and one important
feature of its structure is the existence of some signs revealing the revision to
which it was submitted. This revision is manifest by the underlining of two
passages and the addition of marginal notes, destinated to be inserted at precise
places in the text.
The first marginal note was intended evidently to be inserted in the third paragraph
between the words velocitatis momenta and the following underlined passage. In
this note was presented the famous Postulate, according to which the same
moments of velocity are acquired at points equally distant from the horizontal after
descents from rest. On the other hand, the two underlined passages refer to the
principle v  s, attesting that the velocities grow continually according to the
distance from the initial point of motion. The underlining may indicate that Galileo
intended to suppress these two passages, in order to annulate the effects of the
false principle of the law of fall. But it was sufficient to substitute the Postulate to
the first passage, and the effect of this controversed principle should be
counterbalanced. In this case, the second underlined passage should be understood
as developing the consequences of the Postulate, as it is clear from the analysis of
the successive steps of the demonstration.
What was to be proved is the proportionality t (AB) / t (AC) = AB / AC. The
demonstration began properly by the introduction of the Postulate, after which a
first result was obtained: the moment or degree of velocity acquired by the
moveable on AE is the same as the one acquired on AD, since the points D and E
are equally distant from A. Then, mom vel (F) = mom vel (G), mom vel (H) =
mom vel (I), mom vel (K) = mom vel (L) and mom vel (C) = mom vel (B). In the
next step, accounted for by the second underlined passage, AB abd AC are
decomposed in many points in which are acquired different velocities. The points
on AC correspond one-to-one to the points on AB. In the extremities of the
parallel lines joining each pair of them two identical degrees of velocity are
generated.
In the last two paragraphs, a new and radical shift is introduced. Galileo proceeded
to compare infinitesimal times in which are traversed, on the lines AB and AC,
infinitesimal corres-ponding spaces. In this way, the two lines were decomposed in
"innumerable tiny spaces" which correspond between them by pairs, each two
spaces of the same pair holding between them the same ratio of AB to AC. Finally,
he refered to the proposition elaborated on f. 138v: since the moments or degrees
of velocity in each pair of the tiny spaces are equal, we can consider the velocity on
each of these infinitesimal segments as if it was uniform. Therefore, the times on
all the spaciola on AC will have the same ratio to all the spaciola on AB as the
spaciola AC to the spaciola AB.
It was not necessary to introduce in this demonstration any law of motion. As
supposed by Wisan (1974, p. 219),this document reveals probably that at a certain
time Galileo identified his Postulate with the erroneous law v  s.
An important aspect of the conceptual analysis conducted on f. 179r is represented
by the infinitesimal method to which Galileo had recourse. Indeed, he began by
associating the infinitude of instantaneous velocities acquired during the motion on
the points of the vertical line to the infinitude of instantaneous velocities acquired
on the inclined plane. But the points of motion on the two paths are not
considered as mathematical points, but as little tiny spaces, an infinitude of which
exists on each path of motion. Between these two infinite sets of spaciola a one-toone correspondence was established and represented by the infinity of parallel
lines, on the extremities of each one of them two identical degrees of velocity are
supposed to exist. Du-ring the motion along each corresponding two tiny spaces,
the times elapsed are in the same ratio as these spaces and as the inclined plane is
to the vertical. Therefore, on each one of these innumerable little segments, the
motion can be considered as uniform. So the sum of the times along the two paths
of motion would be as the sum of the segments; finally, by summation it could be
concluded that t (AB ) / t (AC) = AB / AC.
The shift from the "infinite points" to the "quasi innumera spaciola" allowed the
application of the theorem recorded on f. 138v. As it was mentioned before, this
proposition is a gene-ralization of Theorem I of uniform motion, elaborated in
order to apply it to any two spaces of which the parts or segments correspond to
each other if they are taken in successive pairs. Indeed, since Galileo specified that
his geometrical reasoning on f. 179r took into account finite tiny spaces traversed
in finite times, it didn’t matter how short was their duration. Even if they were very
short intervalls of time, it was still possible to apply the Theorem stated in folio
138v. In this context, on each segment the velocity could be considered as
uniform, and, hence, the times are as the distances (by Theorem I of DML1). It
remained only to sum up the times and the tiny spaces and to infer the
proportionality between the total times and spaces.
2. Return to a dynamical foundation
On the verso of f. 179 are recorded two notes. The first note establishes a
corollary to Theorem III (direct proportionality between times and lengths along
any two inclined planes of the same height) (Opere, VIII, p. 389.1-14 ; Galilei, 1979,
p. 65 ), while the second one was abondoned before its completion and two slips
of paper were pasted over its text. Here is the text of this second note:
"If in the line of natural descent two unequal distances from the starting point of
the motion are taken, the moments of velocity [momenta velocitatis] with which
the moving body traverses these distances are to one another in double proportion
of those distances.
Let AB be the line of natural descent, in which from the starting point A of the
motion two distances AC and AD are taken: I say, that the moments of velocity
with which the moving body traverses AD are to the moments of velocity with
which it traverses AC in double proportion of the distances AD and AC. Draw line
AE in an arbitrary angle with respect to AB..." (Opere, VIII, p. 380.15-24; Galilei,
1979, p. 65).
Evidently, this unfinished note is related to a new speculation about free fall in
which Galileo tried to demonstrate the relationship between moments of velocity
and the square of the distances through which they are acquired. This
proportionality was one of the crucial steps in the process of deduction of the
Times-squared theorem in f. 128. It is possible that Galileo sought to prove it
independently in order to prepare the tools by which he could return to a new
analysis of the law of fall.
In this fragment on f. 179v, the principle v  s is implicitly assumed, at least in the
geometrical representation, identical to the one which governed the mathematical
reasoning in the 1604 papers. The fact that Galileo abondoned his argument
before completion is a controversed question among historians. He did so
probably because of his sudden awareness that he was repeating the same
reasoning which he developed on f. 128.
As it is preserved in Vol. 72, the incomplete fragment on f. 179v is covered by two
sheets of paper on which are written two propositions. Here are the statements of
these propositions:
"The moments of heaviness [momenta gravitatis] of the same body on the inclined
plane and on the vertical correspond to the length and elevation of the plane.
(...) This is already proved in mechanics.
"The moments of heaviness of the same body on the planes of different
inclinations have to one another the same ratio as the lengths of the planes,
provided that they correspond to the same elevation"(Opere, VIII, p. 376. 1-377. 2;
Galilei, 1979, p. 8).
The first proposition (mom grav (CA) / mom grav (CB) = CB / CA) was familiar
to Galileo, and he declared that he proved it mechanically. It is, in fact, the De motu
theorem, expressed –as in f. 172v– in terms of momenta gravitatis. The second is a
corollary of this same proposition, generalizing it to any two inclined planes of the
same height.
Galileo’s act of pasting the two slips of paper bearing the De motu theorem on the
verso of f. 179 can be interpreted as a search for a dynamical foundation, assuming
probably that in this way he should not have recourse to any law of fall. If this
explanation is correct, such a decision would mean that he decided to discard the
kinematical methods with which he worked, and which led him to such paradoxes
and confusions he experienced in the successive stages of his physical
investigations.
3. Deduction of the proportionality Time-Velocity
The destiny of the fragment written on the recto of f. 179 is illuminating about the
struggle in which Galileo was engaged in order to solve the problems linked to his
work on a law of fall. An important episode of this struggle is recorded on the
verso of f. 164, where we find the paradox on velocity revealed by the Mirandum
fragment (discussed above in Section III.1.3). Just after this fragment, a little note
of two lines was added at the end of the document. It shows the emergence in
Galileo's mind of a new formula for the characterization of velocity in free fall.
This time, velocities in accelerated motion are considered to be proportional to
square root of the distances. This formula was certainly considered by Galileo as a
mean to resolve the 1604 crisis and the subsequent consequences of the analysis of
motion bearing on the principle v  s. His work of this new relation took the
double form of theoretical analysis and experimental test, as it is revealed by some
manuscript documents like the famous two folios 152r and 116v.
The little note marked on the verso of folio 164 is composed of two lines. It stated
that:
"The moments of velocities [momenta velocitatis] of a body falling from a height
are to one another as the square roots [radices] of the distances traversed, namely,
[they are] in a subduplicated ratio of those" (Opere, VIII, pp. 380.13-14; Galilei,
1979, p. 78) .
Written as a result obtained probably after intense investigations, this note
corrected the error of 1604, and provided a mathematical relation that allows to
calculate the rate of variation of the velocity in the motion of fall and the ratio
existing between the velocity and the distance. Mathematically equivalent to the
fundamental relation of the proportionality of velocity to the time (v  t), it
eliminated the possibility of referring directly the growth of velocity to the spaces
traversed from rest and established the kind of relation that must be established
between velo-cities and distances: a relation of square root.
Evidently, this result is very important and should have made an end in principle
to the crisis of 1604. The tortuous and difficult itinerary that led Galileo to the
substitution of time to the space as the independent parameter for the growth of
speed in free fall found here probably its first great achievement. With this new
formula in hand, it was possible for Galileo to obtain the fundamental relation v 
t by a simple combination of his various theorems related to free fall:
The folios 91v and 152r document Galileo’s discovery of the proportionality of
degrees of velocity to the times elapsed in accelerated motion, and the way in
which he demonstrated it. In f. 91v, the deduction of such an important result is
conducted in full and all the steps of the demonstration are made explicit, which
may suggest that the note recorded on this document represents the final stage of a
long investigation, of which the traces are apparent on folio 152r.
Folio 152 is composed of scattered notes, of two diagrams and of the text of a
little geometrical demonstration [fig. 32]. Its fragmentary structure gave rise to
many controversies. But recently, the debate on this document has been renewed
by the discerning analysis of J. Renn (1992, pp. 181-194) who produced the most
comprehensive and coherent analysis of all the material recorded on this paper.
Considering it as one of the documents revealing Galileo’s discovery of the
proportionality of degrees of velocity to times, he recognized the relationship
linking it to f. 91. Consequently, he was able to produce a global interpretation of
all the scattered notes disseminated on the folio and to describe the way in which
Galileo found his way out of the puzzles he met in his long search for a law of fall.
The experimental aspect of Galileo’s work on the formula v   s can be found on
the folio page 116v (Galilei, 1979, p. 79). This manuscript seems to be the most
important unpublished document of the Vol. 72. Since its first publication in 1973,
it was very discussed among historians in the recent Galilean studies. It is very
likely that we have on this page a brief and sketchy protocol of a sophisticated
experiment (according to the standards of Galileo's time and even to his own
work) in which our scientist intended to test the formula which he discovered in
his previous work, namely v   s .
Folio 116v, as f. 152r, seems to have been composed at the end of the Paduan
period. It consists in a simple sheet of paper written only on the verso, comprising
a set of arithmetical operations, two diagrams and some scattered numbers and
words. For the interpretation of the document, it is necessary to account for these
calculations disseminated, to link them to the main diagram and to give a coherent
interpretation of all this material.
The main diagram is particularly important and constitutes the key for the
interpretation of the document. It consists of a rough drawing made of a vertical to
which two parallel line representing the ground and the top of a table are
perpendicular. Between these two lines five curved trajectories are displayed, which
have the same origin and the same horizontal tangent. The dotted part of the
vertical line is crossed by an important assertion: 828 punti, altezza della tavola.
Since the punto is equal to 0.95 cm, this assertion means that the expriment was
very probably performed on a real table of a height of 78cm. This information is
very important because of its signification for the understanding of the experiment
accounted for on f. 116v.
Beside the vertical line, Galileo marked the distances of 300, 600, 800 and1000
punti. Another height of 828 punti was taken into account in the calculations that
we find on the document, but it was not marked explicitly like the others.
Therefore, we have five heights for vertical descent; a curved trajectory
corresponds to each one of them and to each trajectory two numbers are
associated. These numbers were inscribed in two superposed sets along the
horizontal lower line. The first group of values comprises the numbers 800, 1172,
1328, 1340 and 1500, which are the experimental measures of horizontal
projections from the extremity of the table, after initial vertical falls on an inclined
plane from heights of 300, 600, 800, 828 and 1000 punti. The second group of
numbers is composed by the theoretical predictions that Galileo obtained after he
performed the calculations which are disseminated on the page and which are
founded on a precise mathematical formula. These data are represented by the
numbers 1131, 1306, 1330 and 1460. Each one of these is accompanied by the
word "doveria" (should be) and by the difference which separates it from the value
of the experimental measure that corresponds to it in the first group of data. These
differences are, respectively, of 41, 22, 10 and 40 punti. The fact that there are two
sets of numbers between which differences exist and that these differences are
stressed by Galileo himself proves, undoubtedly, that the two sets of data were
obtained by distinct methods.
The experiment may have been performed in the following way. A spherical and
round ball, composed of solid material, is let to descend along a grooved inclined
plane AB put on a table [fig. 35]. In B, the inclined plane is connected to a little
curved deflector intended to direct the trajectory of the ball towards the horizontal.
In C, at the end of the deflector, the ball begins to describe a horizontal motion
which is soon transformed in a semiparabolic trajectory, before reaching the
ground at F. The components of this trajectory are the vertical height h and the
ho-rizontal projection D.
In order to obtain different horizontal projections, Galileo could change the initial
point from which the ball starts to fall on the same inclined plane, or he could use
many inclined planes of different heights. In the two cases, this implies different
values for the height H from which the ball begins to roll down. In the first part of
the experiment, Galileo was concerned exclusively by measuring the data of the
different rollings and projections of the ball. First, he let the ball roll from a height
of 300 punti and obtained a horizontal projection of 800 punti, then he changed the
vertical heights, giving to for H the values of 600, 800 and 1000 punti successively.
The projection D measured after each of these descents was respectively of 1172,
1328 and 1500 punti. Seeking to test the Rule of Double Distance, he performed a
fifth test in which H = 828 punti, which is exactly the height of the table.
According to the Double Distance Rule the ball should be projected along a
distance D equal to 1656 punti. But Galileo obtained 1340 only, and when he
calculated this value with his mathematical formula, he got no more than 1329.
This discrepancy from the theoretical predictions must have disappointed him and
remained one of the mysteries of the experiment recorded on f. 116v. This could
be one of the reasons why this experiment was never described in Galileo’s later
public works.
The second part of the investigation was represented by the control to which the
experimental data were submitted. In this control, Galileo made the calculations
which are recorded on f. 116v, using a specific formula. The results of these
calculations were reported beside the group of experimental data. He considered H
= 300 and D = 800 as the standard against which to compare the other values of
horizontal projections. That this was precisely his choice is clearly attested by the
examination of the arithmetical operations disseminated on the folio page in which
the numbers 300 and 800 play a decisive rôle. As it is obvious on the document
that the number 800 is the only one to which was associated a dotted curved line
and to which is lacking any theoretical prediction [see fig. 34]. Moreover, a remark
connected to the second experimental result (D = 1172 punti) affirms: it "should
be, to correspond with the first, 1131" (doveria per rispondere al primo esser
1131). This primo is no other than D = 800, which is the value of the horizontal
projection D for H = 300.
The calculations performed on f. 116 were founded on two fundamental
assumptions: a) the principle of the independence of vertical and horizontal
motions, which also includes the possibility of their composition, and b) the
principle of conservation of velocity in horizontal projection. On the ground of
these two basic assumptions, known to Galileo at the time of this experiment –as it
is attested by some of his documents belonging to the last years of his Paduan
period–, the time of the projection from C is constant, whatever is the initial height
from which the ball rolls along the inclined plane. This time is the one which
should be consumed if the moveable fell along the height of the table in a free
descent to the ground. In other terms, since the vertical component of the
projection (h) is always the same (i.e. the height of the table), the ball takes the
same time to pass over the different distances of the projection D, according to the
different initial heights H. It follows that the horizontal projections are as the final
velocities of the descent along AC. This presupposes that during the projection on
CF, the velocities are not diminished and remain constant.
The velocity with which the ball leaves the plane ABC is the instantaneous velocity
which should be produced if the fall was through the height H. Since the time of
descent is not affected by the horizontal component of the projection, therefore
the horizontal distance of projection (D) should be the exact measure of the
instantaneous velocity with which the body leaves the inclined plane at C. Thus, D
must be proportional to the instantaneous velocity acquired during the descent on
the plane AC of height H. Now, this velocity is proportional to the square root of
H.
From this we can disengage the mathematical formula which Galileo intended to
test in his experiment: it is the law of fall formulated in the form of direct
proportionality of velocities to the square root of the distances. On f. 116v, this
law was expressed as:
v1 / v2 = ÷ H1 / ÷ H2 .
As D  v (C) and v (C) H,
v1 / v2 = D1 / D2 and D1 / D2 = H1 / H2.
This is the fundamental equation behind the work recorded on f. 116v. It states
that the instantaneous velocities in C are proportional to the square root of the
distance H and are measured by the horizontal projection D. Indeed, the
comparison of this equation is evidently a direct test of the law v  s and of the
Times-squared theorem.
Precisely, the formula with which Galileo performed his calculations is the
following:
v1 / v2 = H1 / H2  v12 / v22 = H1 / H2  v2 = (H2 . v12 )/H1.
H1 and v1 being respectively 300 and 800, and H2 representing the various values
of D = 600, 800, 828 and 1000, the formula became:
D =  ( H . 800 ) 2 / 300,
by which Galileo obtained the values for D = 1131, 1306, 1329 and 1460 .
Whatever was the conclusion Galileo drew from his experiment and the
subsequent calculations by which he tried to control the results he obtained, there
remain evidently notable differences between his experimental and theoretical data.
Especially, the failure of the test of the Double Distance Rule must have
disappointed him. Nevertheless, the main positive result he got was that he became
more and more confident in the validity of v   s and in its mathematical
equivalence to v  t, this last proportionality exactly which he demonstrated
directly on folio 91v. In this document, he deduced successfully the fundamental
relation of the law of fall, namely that in the accelerated free fall the moments or
degrees of velocity grow proportionally to the times elapsed.
Folio 91 v begins with the statement of a proposition on projectile motion which
was numbered in the Discorsi Theorem I DML3; then comes the demonstration of
the relation v  t :
"In motion from rest the moment of velocity [velocitatis momentum]and the time of
this motion are intensified in the same ratio. For let there be a motion through AB
from rest in A, and let an arbitrary point C be assumed; and let it be posited that
AC is the time of fall through AC, and the moment of the acquired speed in C is
also as AC, and assume again any point B: I say that the time of fall through AB to
the time through AC will be as the moment of velocity in B to the moment in C.
Let AS be the mean [proportional] between BA and AC; and since the time of fall
through AC was set to be AC, AS will be the time through AB: it thus has to be
shown that the moment of speed in C to the moment of speed in B is as AC to
AS. Assume the horizontal [line] CD to be double CA, but BE to be double BA; it
follows from what has been shown, that the [body] falling through AC, deflected
into the horizontal CD, will traverse CD in uniform motion in an equal time as it
also traversed AC in naturally accelerated motion. And, similarly, it follows that BE
is traversed in the same time as AB; but the time of AB itself is AS: therefore, the
horizontal [line] BE is traversed in the time AS. But let EB be to BL as the time SA
is to the time AC; and since the motion through BE is uniform, the space BL will
be traversed in the time AC according to the moment of speed in B. But according
to the moment of speed in C, in the same time AC the space CD will be traversed;
but the moments of speed are to one another as the spa-ces, which according to
these moments are traversed in the same time. Therefore the moment of speed in
C is to the moment of speed in B as DC to BL. But as DC to BE, so are their
halves, i. e., CA to AB; but as EB to BL, so BA to AS. Therefore, ex aequali, as DC
to BL, so CA to AS: that is, as the moment of speed in C to the moment of speed
in B, so CA to AS, that is, the time through CA to the time through AB. Which
was to be proved" (Opere, VIII, pp. 281-282n.; Galilei, 1979, p. 83; Wisan, 1974, p.
227) .
What was to be proved is that the velocities acquired in the motion of fall from
rest are in the proportion of the times. Precisely, that t (AB) / t (AC) = mom vel
(B) / mom vel (C). This fundamental proposition is underlined in the course of the
demonstration and it is deduced in terms of direct proportionality between
momenta celeritates and times.
The demonstration is founded on three propositions obtained by Galileo in his
previous work: the Mean Proportional Rule, the Double Distance Rule and
Theorem I of uniform motion. The first rule was introduced in the beginning of
the demonstration in order to represent the times of fall. Since CA = t (AC), then
AS = m (AB, AC) = t (AB). The final conclusion was reformulated in this sense: it
has to be shown that mom vel (C) / mom vel (B) = AC / AS. The Double
Distance Rule was necessary to account for the deflections of the free fall on the
horizontal lines CD and BE traversed in uniform motion. Since CD and BE are
respectively double CA and AB, the times to traverse them in uniform motion are
equal to the times of fall along CA and AB. Of course, the uniform speed on CD
and BE is equal to the degree of velocity acquired in accelerated motion at the
points C and B. Finally, Theorem I on uniform motion was needed to find the
distance along BE that is passed through in t (AC) with the degree of velocity
acquired in C. This distance is BL, since the moments of velocity are to one
another as the spaces traversed in the same times. Therefore, mom vel (C) / mom
vel (B) = DC / BL.
In the last part of the demonstration, where he deduced his conclusion, Galileo
returned once again tot the law of fall, making use this time too of the Mean
Proportional Rule:
DC / BE = CA / AB,
but EB / BL = BA / AS,
therefore ex aequali DC / BL = CA / AS,
that is mom vel (C) / mom vel (B) = CA / AS = t (AC ) / t (AB).
With this result in hand, Galileo corrected the ancient principle v  s and was able
to make rapid progress in the study of different properties of projectile motion.
His work in this perspective resulted in a complete theory presented later in the
Fourth Day of the Discorsi. Indeed, there exists a tight link between the
proportionality v  t and the parabolic shape of the trajectory of projectile motion.
In the Discorsi, the demonstration of this property was grounded on the basis of the
composition of a vertical accelerated motion and a horizontal uniform motion
(Opere, VIII, DML3, Theorem I, p. 269).
4. Final stage
In the last years of his stay in Padua, Galileo seems to have performed a lot of
work on projectile motion, as it is revealed by many documents of Vol. 72. His
interest in "violent motion" of projectiles is quite apparent in the Pisan De motu
(see Section I.3.1). Moreoever, it is almost sure that he participated to an
experiment performed by Guidobaldo about 1600. Consisting of the throwing of a
ball colored with ink on a plane where it traces the shape of a semiparabolic
trajectory, it is very similar to the one described by Galileo in the Second Day of
the Discorsi (Opere, VIII, p. 185), by which –he said– is demonstrated the parabolic
form of the trajectory of projectile motion.
So the first hint of Galileo on the parabolic trajectory can be traced to the first
years of the XVIIth century. J. Renn (1992, pp. 151-156) suggested convincingly
that this experiment must have drawn Galileo's attention to the falsity of his
treatment of projectile motion in the old De motu as on the falsity of his ancient
treatment of acceleration in the natural fall of bodies. Moreover, it must have
played an important rôle in the concomitant discovery of the parabolic trajectory
and the quadratic relation between times and spaces in the fall: in effect, the
recognition of the parabolic shape of the trajectory for projection may have
implied the recognition of the Times–squared relationship for the motion of falling
bodies and vice versa.
One important element of Galileo’s work on projectile motion is the idea of
conservation of motion. Now, we know that there existed in his papers, since the
pisan De motu and paduan Mecaniche a proto-inertial notion according to which
on a horizontal plane the movable bodies can be put in motion by a very small
force, and that on this plane the body may be indifferent to rest as well as to
motion. In a letter of Benedetto Castelli to Galileo of April 1607 we learn that the
Master has taught to his pupils a similar concept of proto-inertia:
"From your Excellency's doctrine that although to start motion the mover is
necessary, yet to continue it the absence of opposition is sufficient..." (Opere, X, p.
170).
At about 1609, Galileo, having the feeling that his work on motion is very near to
completion, he began to entertain a campaign of public relations about his
discoveries. He behaved in this sense in two directions: towards the scientific
authorities (Luca Valerio, in instance) and political power (some members of the
Medici family and some officials of the Florentine court).
In this context, he sent to Valerio, the Roman mathematician, whom he knew
from many years ago, some of his results and asked him for his opinion.
Unfortunately, Galileo’s letters are lost, but the subject of his correspondence with
Valerio can be roughly reconstructed from the extant answers he received from
him. In his letter dated 23 May 1609, Galileo’s correspondent said:
"Your Lordship complies to send me his excellent work on bodies naturally moved
and on projectiles, a subject that Your Lordship rightly thinks never dealt with so
far. Therefore, I beg Your Lordship to go on with it, and to bring it to an end
sooner as possible, because he is going to give to the world a very useful and
admirable thing" (Opere, X, pp. 244-245).
Receiving another letter from Galileo, Valerio answered on July 18, 1609. In this
letter, he is concerned with the discussion of two "principles" submitted to him by
Galileo.
On the other hand, Galileo’s correspondence with the representatives of
Florentine autho-rities can be illustrated by the letters he exchanged with Antonio
de’ Medici and with Belisario Vinta, secretary of the Grand-Duke of Tuscany.
These letters beared on scientific matters and on projects and research programs
for the future as well.
For illustration, let’s quote two excerpts of Galileo’s intense and rich
correspondence in 1609-1610. The first one is extracted from a letter to Antonio
de' Medici of February 1609:
"Since my return from Florence I have been occupied in some contemplations of
various experiments pertaining to my treatise on mechanics, in which I hope that
the greater part will consist of new things not touched on by others before. And
just recently I have completed the discovery of all the conclusions, with proofs,
pertaining to the strengths and resistances of wooden beams of various lengths,
sizes and shapes; how much weaker those are in the middle than at the ends, and
how much greater weight they can sustain if distribu-ted evenly rather than put in
one plane, and what shape they should have in order to be equally strong
throughout –a science most necessary in the construction of machines and of all
sorts of buildings, nor has anyone treated of this" (Opere, X, p. 229).
In one of his many letters to Belisario Vinta, he proposed his services at the
Medicean Court and described the program of the publications that he wished to
complete if he was appointed as "mathematician and philosopher" in Florence:
"Three books on local motion, an entirely new science in which no one else,
ancient or modern,has discovered any of the most remarkable laws which I
demonstrate to exist in both natural and violent movement, whence I may call this
a new science and one discovered by me from its first principles. Three books on
mechanics, two relating to its principles and one concerning its problems; and
though other men have written on this subject, what has been done is not onequarter of what I write, either in quantity or otherwise" (Opere, X, pp. 351-352).
As it is clearly testified by these letters and by many other documents, before he
left to Florence, Galileo has accomplished the greatest part of the program of his
work on motion in Padua, namely, the creation of a new science of motion.
Founded on results linked to the medie-val discussions on natural philosophy, it
was mathematical in method and in spirit. The program announced in the letter to
Belisario Vinta was in no sense mere exaggeration, but effectively Galileo had in
hand already a new treatise on problems of de motu locali. The manuscripts of
Vol. 72 show without any doubt that what was still missing concerned only the
general conceptual framework and some particular propositions, obtained by
Galileo in Florence during the following years, and especially during the creative
years 1630-1634 where he composed the major part of the Discorsi.
In the middle of 1609, the news of the telescope reached Galileo’s ears at the
crucial time of completion of his work on motion. If he had had some pause, he
would have envisaged to publish his De motu locali in Padua, at about 1610. But the
events went differently, and this publication had to wait until the mitigated end of
the Copernican campaign.
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