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The Commonwealth and International Library 
of Science Technology Engineering end Liberal Studies 


Portrait of Galileo by Susterman, Uflizi Gallery, Florence 


Men of Physics 

Galileo Galilei, his life 
and his works 









Pergamon Press Ltd., Headington Hill Hall, Oxford 

4 & 5 Fitzroy Square, London W.l 

Pergamon Press (Scotland) Ltd., 2 & 3 Teviot Place, Edinburgh 1 

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Copyright © 1966 Pergamon Press Ltd. 

First Edition 1966 

Library of Congress Catalog Card No. 66-23858 

Printed in Great Britain by Sydenham and Co., Bournemouth, Hants. 

This book is sold subject to the condition 
that it shall not, by way of trade, be lent, 
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Muriel Jane and John Mark 





Part 1. HIS LIFE 

1. Curious Student 3 

2. Research Professor 7 

3. Clever Courtier 16 

4. Popular Author 26 
Bibliography 38 

Part 2. HIS WORKS 

5. Interpreting Sense Impressions 

(a) Primary Qualities 44 

(b) Mathematical Language 50 

6. Continua — Mathematical and Physical 52 

7. Magnetism 63 

8. The Pump that Failed 74 

9. Apparent Lightness 76 

10. Weighing Air 115 

11. Floating Ebony 120 

12. Analyzing an Alloy 133 

13. The Screw as a Machine 140 

14. Strength of Materials 

(a) Scaling 151 

(b) Galileo's Problem 154 

(c) Similar Beams 157 

(d) A Cracked Column 163 

(e) Tubes 167 

15. Natural Oscillations 

(a) Simple Pendulum 170 

(b) Vibrating Freely 175 

(c) Resonance 176 

(d) Musical Intervals 176 


16. Falling Bodies 

(a) How Fast? 184 

(b) Inertia 187 

(c) A Thought Experiment 192 

(d) The Medium's Role 194 

(e) Changing Speed 206 

(f) Projectiles 221 

(g) GaUleian Relativity 235 

17. Spots on the Sun 239 

18. New Moons 247 

19. Parallax of a Star 260 

20. Nature — God's Handiwork 269 
Outline of Life and Works 277 
Index 281 


The author and the publisher are grateful to all who have kindly 
allowed quotation from copyright material in this book. Their 
thanks are due to : 

American Institute of Physics for permission to quote from 
Galileo Galilei — Outline of His Life and Works. 

Basic Books, Inc. for permission to quote from Galileo and the 
Scientific Revolution by Laura Fermi and Gilberto Bernadini, 
© 1961 by Basic Books, Inc., Publishers, New York. 

Doubleday & Co., Inc. for permission to quote from Discoveries 
and Opinions of Galileo by Stillman Drake, © 1957 by Stillman 

Northwestern University Press for permission to quote from 
Dialogues concerning Two New Sciences by Galileo Galilei, 
translated by H. Crew and A. de Salvio. 

University of California Press for permission to quote from 
Dialogue concerning Two Chief World Systems — Ptolemaic and 
Copernican by Galileo Galilei, translated by Stillman Drake. 

University of Illinois Press for permission to quote from 
Discourse on Bodies in Water by Galileo Galilei, translated by 
T. Salusbury. 

University of Pennsylvania Press for permission to quote from 
The Controversy on the Comets of 1618 by Galileo Galilei, 
translated by Stillman Drake and C. D. O'Malley. 

University of Wisconsin Press for permission to quote from On 
Motion and on Mechanics by Galileo Galilei, translated by I. E. 
Drabkin and Stillman Drake, © 1960 by the Regents of the 
University of Wisconsin. 


The distinctive feature of modern Western culture has been the 
growing social importance of science and technology. No one, 
however, can hope to comprehend the historical development of 
science without taking into account the role of Galileo Galilei. 
He lived during the critical transition period between the medieval 
and modern eras. He pioneered the path to our own understand- 
ing of physical phenomena. He grappled with complex problems 
that still face us; his methodology and ideas are embedded in the 
very foundations of physics today. As we examine directly the 
thinking inherent in his works, from the vantage point of the 
present, we encounter new light on old experiences. The tacit 
assumptions of crystallized science today stand out vividly in the 
fumbling procedures of its amorphous beginnings. They need to 
be re-evaluated to give us a sense of direction for their future use. 

Galileo began his career in academic institutions and concluded 
it in the service of a powerful city-state. His prominent position 
afforded many contacts with intellectuals in the universities and 
in the Church. His radical approach and dominating personality 
led inevitably to friction with both the philosophers and the 
theologians of his day. The Church trial of Galileo is one of the 
celebrated cases of history. Despite his social problems, Galileo 
was sincerely a religious man of science. 

Galileo, with his broad interests in art and music and with his 
skill as a popular writer, was a typical renaissance man — with an 
interest also in science. His integrated life, therefore, is well 
worth studying in view of our current concern about the 
fragmentation of the culture of our own time. 

Thanksgiving Day 1965. 
Washington, D.C. 

Part 1. HIS LIFE 



Galileo Galilei,* 1 - 2 - 4 ' the founder of modern physics, was born 
near the Arno River in Pisa (a Roman colony of the second 
century b.c, conquered by Florence in 1405) on 15 February 
1564 when Cosimo de' Medici (1519-74) the Great (not Cosimo I, 
the Elder, (1389-1464) who founded the Medici dynasty), was 
Duke of Florence (first Grand Duke of Tuscany as of 1569). 
Galileo's family, which was of the lower nobility, had originally 
been named Bonajuti. Two of his ancestors had held high offices 
in the Republic of Florence. The burial-place of one of these, a 
physician also named Galileo Galilei, is marked in the floor of the 
nave in Florence's Franciscan Church of Santa Croce. Galileo's 
own father, Vincenzio (1520-91), was the great-grandson of the 
brother of this physician. He was traditionally a musician; he 
played the lute. He had broad intellectual interests, including 
mathematics and the classics; he studied music in a spirit of free 
inquiry. In the Introduction of his Dialogue on Ancient and 
Modern Music (1571) he claimed: "They who in proof of any 
assertion rely simply on the weight of authority, without adducing 
any argument in support of it, act very absurdly." Vincenzio's 
own outlook was thus indicative of the independence of mind and 
the spirit of combat that later characterized his son. In 1562 he 
married Giulia Ammannati (1538-1620) of nearby Pescia; they 
had seven children (three sons and four daughters), the eldest of 
whom was Galileo. 

Galileo's early education was at the Pisan school of Jacopo 
Borghini, supplemented by his father's help in the classics. As a 
boy, Galileo showed a certain mechanical inventiveness. Some- 


time after his family moved to the ancient Etruscan town of 
Florence in 1574, he himself attended the monastery school 
(founded in 1060) of Santa Maria di Vallombrosa (25 miles east 
of Florence), but he was withdrawn by his father in 1579 — 
possibly owing to a seemingly undue religious influence. In these 
beautiful wooded heights, he received the typical literary education 
of that period, viz. primarily the classics, including his favorite 
authors like Ovid, Seneca, and Virgil. He enjoyed even then and 
throughout his whole life the popular Italian poetry of his day for 
its own value (in contrast to learning per se); for example, that of 
Dante Alighieri (1265-1321), Francesco Petrarca (1304-74), 
Ludovico Ariosto (1474-1533), and Torquato Tasso (1544-95). 
In his youth he preferred Ariosto and is said to have known by 
heart the "Orlando Furioso" (popular in every century except the 
twentieth); in the "Two New Sciences" he referred to Tasso as a 
"divine poet". In addition, he was interested and skillful in 
music; he played the lute best, but was also adept in other 
instruments such as the organ. He showed such skill in drawing 
and painting that, as he himself confessed later, he would 
probably have chosen painting as a career if circumstances had 
permitted. (Ludovico Cardi da Cigoli (1559-1613) admitted his 
own indebtedness for Galileo's views on painting; in 1612 he 
painted the Virgin, with Galileo's sketch of the moon at her feet, 
for Santa Maria Maggiore in Rome.) Galileo, in short, was a 
typical humanist of the Italian Renaissance, who had also a 
genuine curiosity about physical phenomena. 

On 5 September 1581 Galileo matriculated as a student in the 
Faculty of Arts of the University of Pisa, presumably to study 
medicine. His interest, however, gradually became focused upon 
natural philosophy, toward which he adopted so critical an 
attitude that he was dubbed "the wrangler". Throughout his 
life, indeed, he was opposed to authority as expressed in mere 
dogma, unsupported by experiential evidence. He undoubtedly 
became acquainted in 1583 with mathematics through the extra- 
curricular interest of a Tuscan court tutor, Ostilio Ricci, a pupil 
of Nicolo of Brescia, Tartaglia (1500-57), who, in 1543, had trans- 


lated the Greek writings of Archimedes (287-212 B.C.) into Latin. 
There is a well-known legend about the discovery of the 
isochronism of a pendulum by Galileo, whose curiosity was 
aroused while watching a swinging lamp during a service in the 
Cathedral of Pisa in 1583 (not the 1587 Possenti bronze one now 
hanging there). Although the amplitude of the vibration was 
observed to decrease steadily, the period of a complete swing to 
and fro (timed by his pulse) remained the same. He later 
designed an instrument, known as the pulsilogium, which enabled 
one to adjust the length of a pendulum for a fast or slow oscilla- 
tion, corresponding to the frequency of a patient's pulse — 
typical of his innate desire to translate his discoveries into use. 
His physical insight is exhibited in his replacement of the complex 
chandelier by a simple pendulum (i.e. a thin cord with a bob at 
the end) as an approximate model. He was not satisfied with 
merely describing the phenomenon qualitatively, but subse- 
quently investigated the quantitative factors that determine the 
period (cf. Section 15a). His whole scientific approach was 
motivated by a desire to know "how" and "how much" in order 
to understand "why". Pendulum motion remained a stimulant 
to Galileo's thinking throughout his entire life. 

Owing to a lack of funds and his failure to obtain a needed 
scholarship, in 1585 he had to leave the University and return to 
Florence, an academic dropout; he never did receive a university 
diploma. He intermittently tutored students in mathematics both 
at Florence and the neighbouring Tuscan town of Siena. While 
at home he attempted also to write plays and poems. Having 
studied Euclid (c. 365 to c. 275 B.C.), Galileo turned his attention 
to the works of another Greek who had been adept in geometry, 
namely, the Syracusan Archimedes. In 1586, accordingly, he 
constructed a hydrostatic balance (cf. Section 12) to determine 
accurately the relative amounts of two metals in an alloy mixture, 
which he described, in Italian, in a paper entitled, "The Little 
Balance",f n °t published until 1644. About the same time he 
became interested in another Archimedian concept, the center 
t Ref. (2), Appendix, p. 137. 


of gravity, and wrote, in Latin, a scholarly paper on "Theorems 
about the Center of Gravity of Solids" (not published until 1638, 
and then as an appendix to the Two New Sciences). In 1587 he 
made his first trip to Rome, presumably to make himself known 
to scholars there and to enlist their support for his studies ; he met 
the German astronomer, Father Christopher Clavius (1537-1612), 
professor at the Jesuit Romano Collegio (founded in 1552, 
succeeded by the modern pontifical Gregorian University after 
governmental occupation of the buildings), who had been 
responsible for the Gregorian revision of the calendar in 1582. 
In 1588 he gave two lectures on the site and dimensions of Dante's 
Inferno to the Florentine Accademia della Crusca (1582, word 
means chaff). Upon the recommendation of Guidobaldo, Marquis 
del Monte (1545-1607), a nobleman with scholarly interests, in 
July 1589 he was appointed by Ferdinand I de'Medici (1549-1609) 
to the Chair of Mathematics at the University of Pisa, although 
he was only 25 years old — not at all surprising in view of the 
general lack of Italian interest in mathematics at this time and 
the exceptional skill of Galileo in its use. 



As a professor, Galileo combined his teaching with research. He 
was sincerely and enthusiastically interested in communicating 
his ideas to serious minds. He was never a neutral teacher; 
convinced of the correctness of his own conclusions, he was 
eager to have others share his views. Continuing his geometrical 
studies, in 1590 he discovered the cycloid, i.e. the curve traced by 
a point on the circumference of a wheel rolling on a horizontal 
plane. Although he surmised (possibly from the relative weights 
of cut-out pasteboard figures) that the area under this curve was 
probably three times that of a circle, he himself was never able 
to demonstrate this fact mathematically. Shortly thereafter, the 
Ponti de Mezzo bridge across the Arno at Pisa was constructed 
from such a cycloidal design. 

He turned his attention now to Aristotle's natural philosophy, 
which had been critically examined by many before him, including 
the Pisan Professor of Philosophy, Francesco Buonamico 
(d. 1603), and the Venetian mathematician Giovanni Battista 
Benedetti (1530-90). Galileo, however, impressed critically upon 
all previous speculations a new criterion, namely, the checking of 
all conclusions directly with nature to ascertain if those obtained 
by pure reasoning agree with the actual observations. Nature 
was to be both source and resource of physical theory. One must 
not only look and listen for clues in nature's universal broadcast, 
but also formulate leading questions and ferret out nature's 
answers. Metaphysics might be appropriate, if at all, after 
physics — certainly not before physics. Anthropocentric specu- 
lations had to give way to nature-centered thinking. Inability to 


deduce physical phenomena from a priori first principles of 
general philosophy necessitated a search for experiential first 
principles of a restricted physics that make theoretical deductions 

Tradition relates that Galileo performed his celebrated experi- 
ment of dropping two different weights from the Campanile, the 
179-foot high Leaning Tower of Pisa, in 1590-1. Although this 
event is truly legendary, there being no specific record by Galileo 
or by anyone else as to its actual occurrence, it seems to be quite 
in keeping with Galileo's own interests, opportunity, and 
character. The Greek philosopher Aristotle (384-322 b.c.) had 
reasoned that the speed of a freely falling body should be in 
proportion to its weight, the cause of its natural motion. 
Experience, however, shows that the speed of fall is independent 
of the weight of a body, if the frictional resistance of the medium 
is negligible. 

During this same period (c. 1590), Galileo compiled some 
scholarly notes (in Latin), "On Motion" < 3 > (not published com- 
pletely until 1883). It was a peculiar mixture of old conceptions, 
current misgivings, and new speculations; unquestionably rooted 
in medieval soil, but growing toward modern light. For example, 
he definitely retained the Aristotelian idea of natural places for 
different earthly materials (cf. Section 9): water ever-present, the 
earth below, the air above, and fire above all. Aristotle's idea 
that there must always be a cause for motion was combined with 
the medieval notion that some vague property, impetus, is 
retained by a moving body, even when the external cause has been 
removed. Galileo reported some of his early investigations about 
bodies falling in different media. Here one finds also the first 
record of his pendulum observations. To understand motion, 
indeed, may be said to have been a primary lifetime interest of 
Galileo. It is preferable, accordingly, to judge his dynamics 
ideas on the basis of the Two New Sciences (1638), which expressed 
his cumulatively mature judgments. Galileo's inquiring mind 
was certainly not unique, but rather typical of the critical atmo- 
sphere prevailing in Europe, both in the southern Renaissance 


countries and in the northern Reformation ones. 

The death of Galileo's father in 1591 brought with it the ever- 
growing financial needs of a family of which he was now the 
responsible head — exemplified in the dowry requisite for the 
marriage of his oldest sister Virginia to Benedetto di Luca 
Landucci. Occasional tutoring, indeed, had already been under- 
taken to supplement the meager salary he had been receiving at 
Pisa. The University, therefore, which had not been particularly 
hospitable, became less and less attractive. Matters came to a 
head when he could not honestly recommend the design of a 
dredging machine for the Leghorn harbor by Giovanni de' 
Medici (d. 1621), a natural son of Cosimo. In the summer of 
1592, consequently, he resigned from the University of Pisa and 
returned to Florence. 

Fortunately, the mathematical chair at the University of Padua 
(founded traditionally by the Trojan Antenor, but part of Venice 
as of 1405) had been vacant since 1588 — indicative of the little 
interest in mathematics there. He secured appointment to it in 
September 1592. There he found some kindred spirits in free 
thinking. He soon became friends with the famous Averroes 
(1126-98) philosopher Cesare Cremonini and the distinguished 
anatomicist and surgeon Geronimo Fabrizio d'Acquapendente 
(c. 1533-1619), for whom the famous anatomical theater was built 
in 1594; one of his students was Gustavus Adolphus (1594-1632), 
later king of Sweden. William Harvey (1578-1657), the English 
physician who later discovered the circulation of the blood, was 
also a student there about that time. In many respects the eighteen 
years that Galileo spent at Padua were the most productive in the 
development of his own understanding of physical phenomena. 
Early (1593, 1594, 1600) he wrote, in Italian, some notes, "On 
Mechanics"/ 3 ) which represented the best systematic summary of 
the statics of simple machines then known (not published until 
1634, and then in French by Pere Marin Mersenne (1588-1648); 
published in Italian in 1649). Nowadays it is particularly valuable 
as a source of the knowledge of this interesting subject during 
that period. 


Machines all operate on the same physical principle so that a 
complete understanding of any one of them is adequate for the 
deduction of the mechanical properties of all others. Galileo, 
for example, chose the lever as fundamental, and used it to derive 
the law for an inclined plane (cf. Section 13). In this connection, 
he recognized the experiential characteristic of all machines, 
namely, that whatever is gained in force is lost in speed (some- 
times loosely said to be the principle of virtual velocities, which 
was not enunciated clearly until 1717 by Jean Bernoulli (1667- 
1748)). Even in this instance Galileo made some significant 
contributions, namely, the importance of the direction of the 
motion, and the relation between a machine's input and its 
output — more precisely stated later by Evangelista Torricelli 
(1608-47), viz. that any spontaneous motion of an isolated 
system's center of gravity can only be downward. About the 
same time, Galileo obtained a patent from the Venetian Republic 
for a machine to raise water — apparently successful, but not 
much used. 

In 1593, in company with some friends, one of whom died 
therefrom, he received a severe chill while asleep in a room "air- 
conditioned" from a nearby conduit leading to a cave. He 
suffered an arthritic condition which bothered him the rest of 
his life. 

In 1597 Galileo designed and constructed a popular "Geo- 
metric and Military Compass", which served as a compass, a 
divider, and a quadrant. In addition, various lines were marked 
off numerically for different calculated results : for example, an 
arithmetical line containing the "rule of three" among others, a 
geometrical line for mean proportion, etc., a stereometrical line 
for cube roots, etc. ; on the other side, a polygraphic line as an aid 
for drawing regular polygons, etc., and a pentagonal geometrical 
line for "squaring" figures. Galileo not only designed this 
instrument but, with the aid of a full-time mechanic, actually 
manufactured it as well as magnetic compasses, drawing instru- 
ments, and, later, other devices — all for sale. Galileo's curiosity 
about natural phenomena was undoubtedly closely linked with 


his own skill in handling materials. At Venice, indeed, he was a 
frequent visitor at the celebrated Arsenal with its galleys and 
shipyards; he was a consultant on military engineering. In 1597 
he obtained a Venetian patent for a machine to hoist water. 

About 1602, having obtained a copy of the first English 
scientific book, On Magnetism (De Megnete) by William Gilbert 
of Colchester (1544-1603), physician to Queen Elizabeth I, he at 
once set about investigating magnetic phenomena, in particular, 
the production of powerful armatures (described later in the 
Two Chief World Systems, cf. Section 7). 

An interesting letter was written by Galileo in 1604 to his 
friend Fra Paolo Sarpi (1552-1623) of the Order of Servites, a 
Venetian councillor. In it he gave the correct law for the increase 
in speed of freely falling bodies, but he deduced it quite 
incorrectly. Scholars today still debate whether Galileo arrived 
at this law primarily inductively, or deductively, or by mixture 
of these two — certainly not the second in view of the wrong 
proof. At any rate, later he derived the law satisfactorily, as is 
evident in the Two New Sciences. 

In 1606 he made a thermoscope for measuring relative changes 
of temperature. Despite the fact that not all quantities can or 
should be measured, it is amazing that no one had previously 
produced a more objective means of determining temperature 
than the human body. Specifically, he took a glass bulb with a 
long slender stem, heated it to drive out much of the air, and then 
inserted the inverted stem into a container of water. The amount 
of the rise of the water in the stem is dependent upon the tem- 
perature of the residual air. Unfortunately, barometric pressure 
is also a factor — not corrected until 1653 when the practice of 
hermetically sealing a tube was introduced by Leopold de' 
Medici (later Cardinal, d. 1675), cofounder with his brother, 
Ferdinand II (1610-70), of the Florentine Accademia del Cimento 
(1657-67) with its cautious motto, "Probando e Reprobando" 
(testing and re-testing). 

About the same time he wrote a manual of instructions for his 
Geometric and Military Compass and dedicated it to young 


Cosimo, whom he had been tutoring during the previous summer 
vacation at Florence. This book was plagiarized by a student 
from Milan, Baldassare Capra, so that Galileo felt compelled in 
1607 to write a defense of his own priority; he received a favorable 
judgment from the University. In the same book he mentioned his 
own views about the 1604 Nova (cf. the 1572 Nova is Cassiopeia) 
in the Constellation Ophiuchus, which had also been attacked 
by Capra. (Actually both these so-called novae were supernovae.) 

In passing, we note some facts about Galileo's personal life 
during this period. At first, he lived in a house on Pratto delle 
Valle (now the Piazza Vittorio Emmanuele) near the Benedictine 
Church of Santa Giustina. In 1599, however, he established 
separate quarters for his Venetian mistress, Marina Gamba. His 
first daughter, Virginia, was born in August 1600; his second, 
Livia, in August 1601; and a son Vincenzio in August 1606 
(d. 1649). At the same time he moved to a larger house (No. 9) 
in the Via Vignali (now Via Galileo), where he had as many as 
twenty students rooming with him. Here he was able to garden 
and to enjoy vines, fruits, and flowers. His family problems, 
however, were accentuated by the need to find a dowry for the 
marriage of his sister Livia to Taddeo Galletti — not to mention 
the continual drain upon his resources by his musical brother 
Michelangelo, who was quite selfish and remained financially 
dependent upon Galileo during most of his life. 

Although Galileo achieved outstanding success in physics 
research during his Paduan sojourn, he is usually remembered 
more for his dramatic astronomical discoveries there. To be sure, 
he had always been interested in astronomical phenomena. As 
early as 1597, in acknowledging receipt of a copy of the 
Mysterium Cosmographicum by Johannes Kepler (1571-1630), the 
German astronomer, he confessed, "Many years ago, I became 
a convert to the opinions of Copernicus". Galileo began corre- 
spondence with the Danish astronomer Tycho Brahe (1546-1601) 
in May of 1600 — terminated soon thereafter by the latter's death. 
The 1604 appearance of the (super) Nova for 18 months afforded 
him an opportunity in January 1605 to give three extraordinary 


lectures on this phenomenon in the Aula Magna of the University. 
Such popular lectures would fill this room, which seated more 
than 1000. He himself believed the Nova to be truly a new star. 
From the determination of its parallax, indeed, he concluded that 
it was beyond the atmosphere, farther away even than the moon. 

Galileo became involved also in court horoscopes; for instance, 
at the request of the dowager Grand Duchess Christina, on 16 
January 1609, Galileo predicted a long and active life for 
Ferdinand I, who unfortunately died only 22 days later. How 
much Galileo and other scientists of his time sincerely believed in 
such astrology is difficult for us to ascertain now. Pope Paul III, 
Alessandro Farnese (1468-1549), kept a private astrologer at the 
Vatican. In view of Galileo's general scepticism, however, one 
might suspect that he had his tongue in his cheek whenever he 
made such horoscopes. 

Galileo's lasting astronomical fame, however, developed out of 
his instinctive curiosity. He happened to hear of a spy-glass 
which had been made by a Fleming lens-grinder Johannes 
Lippershey (d. 1619) with two spectacle lenses. Galileo himself 
reasoned that the combination of a concave eyepiece for magni- 
fication and a plane convex object-glass for distinctness might be 
more successful. His second telescope (so-named later) consisted 
of two such lenses at the end of a lead tube; it had a magnification 
of 9 times (9 x ). He used it for his first public appearance before 
the Doge, Leonardo Donati, and the Venetian Senate in August 
1609; from the San Marco Campanile one could see Santa 
Giustina in Padua, 21 miles away. In January 1610, by use of 
such a telescope, he discovered mountains on the moon (contrary 
to the belief in its perfect sphericity). With his fourth telescope 
(20 x) the planets appeared to be moonlike discs rather than 
starlike points. With his fifth telescope (33 x) he unveiled the 
starry structure of the Milky Way and the existence of moons 
(named the Medicean planets by Galileo) revolving about the 
planet Jupiter. On 4 March 1610 he promptly published all 
these discoveries in the Starry Messenger ^ which he dedicated 
to the new Grand Duke, Cosimo II (1590-1621), with great 


expectations of his patronage. 

Here again his personal curiosity led him to careful obser- 
vations with startling results (cf. Section 18). On 7 January he 
happened to notice three bright "stars" lying on a straight line 
in the vicinity of Jupiter, one on the west side and two on the east. 
On the next night, turning to Jupiter again, he noticed that all 
three were now on the west. On 10 January, however, there were 
only two, both on the east. On 11 January these two were still 
on the east but the outer one was twice as large. He continued to 
observe these celestial objects until 2 March, when he had com- 
pleted sixty-six observations. He had then concluded that these 
bodies were moons revolving about Jupiter. The popular excite- 
ment caused by this announcement is comparable only to that of 
Sputnik in our own times. It was, however, more important 
philosophically inasmuch as Aristotle had maintained that the 
heavens are unchangeable. It was even more significant in that it 
lent credence to the notion of the Pole Nicolaus Copernicus 
(1473-1543) that the earth itself might not be the unique center for 
all celestial revolutions; that it, too, might be a planet shining 
on the moon. The prejudices of some of Galileo's contemporaries 
were as deep-rooted as those of many people today. Giulio Libri 
(1550-1610), the leading philosopher of Pisa, for example, refused 
even to look through the telescope — not to mention Cremonini 
of Padua. On 30 July 1610 Galileo noted a peculiar broadening 
of the planet Saturn. It looked to him as if three planets were 
touching (this triple nature was later shown to be merley a 
confused appearance of Saturn's rings ; Galileo himself noted a 
disappearance of the projections during a certain period). In 
order to safeguard his priority claim he announced this discovery 
enigmatically in a group of scrambled letters, which read when 
transposed: "Altissimum planetam tergeminum observavi" 
(I have observed the most distant planet is triple). 

With these demonstrations Galileo became an internationally 
known figure. No longer was he an individual scientist simply 
carrying on his personal research and casually communicating 
the results to friendly scholars and students; he had now become 


a spokesman for a wholly new point of view, which stressed the 
importance of phenomena themselves. Not everyone, indeed, 
would have — nor could have — transmuted a hearsay story 
about a curious toy into a powerful research tool. Not uncon- 
nected with his success was the fact that throughout most of his 
life Galileo ground his own lenses — a task not too easy, particu- 
larly in the case of the object — glass. By March 1610 he had made 
100 telescopes, only ten of which showed Jupiter's moons. It was 
more than 20 years before anyone else in Europe could make even 
one telescope as satisfactory as those of Galileo. Kepler, for 
example, who had a deeper understanding of the optical function- 
ing of a telescope and who designed a terrestrial one of his own, 
never actually made one himself. 

Meanwhile, Galileo's nostalgia for his native Tuscany had 
come to the fore; he found himself still an alien amid the Venetian 
culture — so different from the Florentine. Once more he had 
found the teaching "load" burdensome, particularly that done on 
the side to supplement his income — necessary despite continual 
increases in salary. Taking advantage of his world-wide renown, 
he approached Belisario Vinta, secretary of state for the Grand 
Duke of Tuscany, about the possibility of a position there. In 
July 1610 Cosimo II appointed Galileo chief mathematician and 
philosopher to himself as Grand Duke, as well as head mathe- 
matics professor at the University of Pisa, without any teaching 
duties — quite modern. In September Galileo left his mistress 
but took his children and returned to Florence. A new period of 
Galileo's life had begun. 



Traditionally Guelfic Florence was much more closely identified 
with the established Roman Catholic Church than Venice, which 
had long been respected for its encouragement of freedom of 
expression. For example, in 1606, when Paul V, Camillo Borghese 
(1552-1621) of Siena, had put Venice under interdict, the Venetian 
Republic successfully defied the interdict and retaliated by expel- 
ling all the Jesuits, Capuchins, and Theatines from its entire 
domain. This change of atmosphere turned out unfortunately to 
be critical for Galileo in view of his unorthodox (but Christian) 
opinions. In Florence he continued his astronomical investi- 
gations; in particular, in December 1610, he announced in a letter 
to Kepler the crescent phase of the planet Venus in the form of an 
anagram: "Cynthiae figuras aemulatur mater amorum" (the 
mother of love emulates the shapes of Cynthia). 

In the spring of the following year, Galileo decided to make a 
good-will trip to Rome to inform people there about his celestial 
discoveries. He did so at the expense of the Grand Duke, who 
permitted him also to lodge at the Florence embassy (No. 27 
Piazza Firenze). He had a letter of introduction from Michel- 
angelo Buonarrati, Junior, to the Florentine, Jesuit-educated 
Cardinal Maffeo Barberini (1568-1644). He was received with 
honor by all, particularly after confirmation of his astronomical 
findings (but not their interpretations) by a Commission, including 
Clavius and his successor Father Christopher Grienberger (1561- 
1 636) of the Romano Collegio, which had been set up at the request 
of the Jesuit Cardinal Robert Bellarmine (1547-1621, later canon- 
ized) of Montepulciano. He had a long audience with Pope Paul V. 



He was elected to the Accademia dei Lincei (lynx-eyed), which had 
been founded in 1603 by the 18-year old Prince Federigo Cesi, 
Marquis di Monticelli and son of the Duke of Aquasparta. The 
analogous Royal Society of London was chartered in 1662 by 
Charles II, and the Academie des Sciences in 1666 by Louis XIV. 

In Rome he announced his discovery of sunspots (apparently 
first noted by him in November 1610). In January 1612 the 
German Jesuit Father Christopher Scheiner (1575-1650), a pro- 
fessor of mathematics at Ingolstadt, published his own observa- 
tions on sunspots together with an interpretation of them as stars 
revolving about the sun. Galileo's opinion (cf. Section 17) about 
his views was requested by Mark Welser, a banker of Augsburg. 
Galileo wrote three letters in reply in May, August, and December 
of that year, all from the villa delle Selve (at Signa, about 9 miles 
west of Florence) of a Florentine nobleman and Lincean, Filippo 
Salviati (1542-1614), where he was often a guest. The dispute with 
Father Scheiner was unfortunate; it ignited a spreading fire of 
misunderstandings with the politically powerful Jesuits (it turned 
out that credit for the sunspot discovery belongs strictly to still 
another person, namely, Johannes Fabricius of Wittenberg, who 
was the first to publish his findings in June 1611). Although 
Scheiner was quite contentious and the Jesuits increasingly 
irritated, there is little evidence as to the part that they may have 
played in the final development of the celebrated Galileo case. 
In 1613 Galileo's letters were published in a book,* 4 ) entitled 
History and Demonstrations Concerning Sunspots and their 
Phenomena; it was dedicated to Salviati under the auspices of the 
Accademia dei Lincei. 

Being in a public position Galileo was naturally called upon for 
pronouncements on various subjects ; his publications invariably 
encountered opposition. A major controversy was critically 
reviewed at a dinner party of the Grand Duke in the fall of 161 1. 
A discussion had arisen as to the factors physically significant in 
the case of a floating body. From the current Aristotelian point 
of view, shape was all important; it was supposed to account for 
the floating of a piece of ice, which being frozen water, should 


presumably be denser and sink. Noting that ice is not more 
dense, Galileo pointed out that ice of any shape will float in 
water. Cardinal Maffeo Barberini approved his argument. 
A more critical question later concerned the floating of a thin 
piece of ebony (cf. Section 11). Here Galileo noted shrewdly that 
a depression in the surface is always associated with such floating 
materials that are not wet by the liquid. He showed considerable 
experimental ingenuity in investigating this phenomenon. 
Remarking that his own complete ignorance compelled him to 
seek information directly from nature itself, he employed wax 
models (size of an orange) for observing free fall in specific cases; 
he impregnated them with lead or sand, and varied the tempera- 
ture of the water (cf. Ref. (9), pp. 68-70). At the request of the 
Grand Duke he compiled his findings in a book entitled Discourse 
on Bodies in Water, < 5 > dedicated to the Duke himself — his first 
publication (1612) on experimental physics. Modern humanists 
who seek in Galileo primarily a Platonic philosopher find it 
convenient to ignore this experiential work. They fail to recog- 
nize that Galileo's greatest contribution was probably not so 
much his specific findings, but rather his general methodology. 
Galileo, indeed, preferred Aristotle's emphasis upon experience to 
Plato's glorification of pure reasoning; he insisted, however, upon 
supplementing it with logic as an arbiter for sense errors and with 
mathematical reasoning for physical quantities in lieu of quali- 
tative speculations. With a few significant exceptions (cf. Sections 
5b, 6) he had little interest in mathematics per se except in 
conjunction with observations; he regarded it practically as a 
handmaiden of science. Under any circumstances he was indif- 
ferent to popular metaphysical verbiage such as caused the 
downfall of Giordano Bruno (1548-1600), who was burned at the 
stake in 1600, a heretic both to Catholics and to Protestants. 

The peripatetic philosophers, now placed on the defensive, 
decided to take the offensive ; they were led chiefly by Ludovico 
delle Colombe (b. 1 565), who thus became the arch-villain of the 
Galileo case. The uncompromising boldness of Galileo and the 
secret envy of some contemporaries, coupled with deep-rooted 


prejudices of many, gradually led all his enemies to unite 
informally against him: disappointed Jesuits and expedient 
Dominicans, Aristotelian professors, and conservative Church- 
men. Failing to defeat Galileo with logic they found a common 
bond in theology — in this instance, truly more of a political 
issue than an intellectual one. This so-called League was nick- 
named the Pigeon League by Galileo inasmuch as Colombe 
derives from the word for dove. 

Here again, in 1613, it was table talk at a Grand Ducal dinner 
party at Pisa that sparked an inflammatory controversy. This 
dinner was attended by a former pupil of Galileo, the Benedictine 
Father Benedetto Castelli ((1578-1643), then Professor at Pisa, 
later Papal Mathematician), but not by Galileo himself. Upon 
praising Galileo's discoveries Castelli found Cosimo Boscalgia, 
philosophy professor at the University of Pisa, insidiously remark- 
ing that any double motion of the earth would be contrary to the 
Scriptures. The Grand Duchess Christina immediately reacted 
unfavorably — sensitive to possible theological repercussions. 
After dinner she gathered together a small group to discuss the 
matter in more detail. Castelli reported this conversation to 
Galileo, who replied at once in a letter in which he emphasized 
that the object of the Bible is not to teach astronomy, and that the 
understanding of natural phenomena must begin preferably with 
experience itself. On the fourth Sunday in Advent (20 December 
1614) Fra Tommaso Caccini (1574-1648), a friar of the Dominican 
Convent of Santa Maria Novella in Florence, preached against 
the Copernican hypothesis as being unbiblical, and particularly 
against the reprehensible failure of laymen to abide by the 
orthodox interpretations handed down from the Church Fathers. 
The Dominicans, of course, were loyal to Thomas Aquinas 
(c. 1225-74), who had effectually encompassed Christian faith in 
a modified Aristotelian framework. Caccini was immediately 
answered in a sermon at the cathedral of Santa Maria del Fiore by 
a preacher, who, in turn, defended the Copernican hypothesis and 
praised Galileo as a good Catholic. (In general, after 1616 the 
Jesuits preferred the compromise system of Tycho Brahe.) 


A formal apology was immediately made to Galileo by Fra Luigi 
Maraffe, Florentine patrician and a Preacher-General of the 
Dominican Order in Rome. 

Meanwhile, Castelli lent Galileo's apology to Father Niccolo 
Lorini, a professor at the Dominican Convent of San Marco, 
and thus unknowingly played into the hands of his opponents, 
who objected generally to the raising of any question as to the 
universal authority of the Bible (a critical issue for both Protest- 
ants and Catholics during the post-Reformatian period, and 
specifically to Galileo's ad hoc interpretation of the reported 
Joshua incident about the sun apparently standing still). 

Early in 1615 Lorini actually denounced Galileo to the Sacred 
Congregation of the Holy Office (the Inquisition) in Rome. 
Galileo, meanwhile, decided to clarify, somewhat arrogantly, his 
own views in a formal letter < 4 > (16 February 1615) to the Grand 
Duchess Christina — not published until 1636 at Strasbourg, 
after the Galileo trial in 1633. The central tenet (cf. Section 20) 
of his whole philosophy was the ultimate unity of truth; it allowed 
for no permanently bad consequences of any temporary, practical 
separation of the world of everyday phenomena and the eternal 
world beyond phenomena, which, he believed, must be rationally 
similar. At any rate, theological and metaphysical implications of 
physics were never his major concern or main forte; rather, he 
considered only those relevant to his investigations of physical 
phenomena themselves. Nevertheless, Galileo's handling of any 
apparent conflicts between biblical statements and scientific find- 
ings would be generally acceptable now to both Catholics and 
Protestants; his opinions are well worth reviewing from our 
present standpoint. It was not until 1893 that Pope Leo XIII, 
Gioacchino Pecchi (1810-1903), announced in the Encyclical 
Providentissimus Deus the official position of the Roman Church 
on the relation of science and religion ; it is not much different 
from Galileo's own posture (cf. also the 1943 encyclical on 
modern biblical scholarship by Pope Pius XIII, Eugenio Pacelli 
(1876-1958), on "Divino Afflante Spiritu"). 

Galileo became increasingly uneasy about the whole state of 


natural philosophy and, naively optimistic, decided to make 
another good-will visit to Rome in order to win friends for the 
Copernican theory, which he so ardently admired and aggressively 
promoted after his first public endorsement of it in his third sun- 
spot letter. He was genuinely surprised to find deep-rooted 
opposition, both to this teaching and to his own arguments. 
While there, he was shocked to learn that Qualifiers (official 
experts) of the Holy Office had just taken (ill advisedly) action on 
two points: namely, (1) that "the sun is the center of the world, 
and, therefore, immovable from its place", which was "unani- 
mously declared to be false and absurd philosophically, and 
formally heretical"; and (2) that "the earth is not the center of 
the world and is not immovable, but moves and also with a 
diurnal motion", which was "declared unanimously to deserve 
the like censure (as the first) in philosophy, and, as regards its 
theological aspect, to be at least erroneous in faith". This report, 
given to the Holy Office on 24 February, resulted in Cardinal 
Bellarmine being instructed "to summon before him the said 
Galileo, and admonish him to abandon the said opinion; and in 
case of refusal the Commissary is to intimate to him, before a 
notary and witnesses, a command to abstain altogether from 
teaching or defending the said opinion and even from discussing 
it". The recorded, but strangely unsigned, minute of 26 February 
reported more broadly that Galileo was "to relinquish altogether 
the said opinion, that the sun is the center of the world, and 
immovable, and that the earth moves; nor henceforth to hold, 
teach, or defend it in any way whatsoever, verbally or in writing". 
This document, cited and apparently accepted, at the trial in 1633, 
became the critical evidence in that phase of the Galileo case. 
Opinions have differed not only as to its actual content, but even 
as to its genuineness: was the minute maliciously fabricated in 
1616 or deliberately forged later in 1632, or was it actually bona 
fide? (Ultraviolet testing has shown it to be similar to the 
accredited 1616 materials.) Some would interpret it as merely a 
clerical overemphasis inasmuch as Copernicus's book itself was 
not prohibited at that time, provided only that certain minor 


details were corrected. As to what actually did transpire in 
Galileo's interview with Bellarmine on 25 February, who knows? 
In May of that year, he himself obtained a paper from the 
Cardinal which indicated that there had been no official censure. 
The stage, however, was undoubtedly being set for the next act in 
this developing tension between old traditional ideas and new 
experiential findings, between the concensus of a group and the 
innovation of an individual. Upon the recommendation of the 
anxious Tuscan ambassador, Piero Guicciardini, Galileo's 
tenacious and contentious, but fruitless, efforts were finally 
temporarily curbed by the Grand Duke's order for his return to 

Galileo, meanwhile, was discussing with the Spanish govern- 
ment a proposed use of the Jupiter satellites for the determination 
of longitude at sea, a perennially troublesome problem. Specifi- 
cally, he suggested a comparison of the precise local time of the 
frequent (more than 1000 a year) eclipses of these moons with 
their predicted occurrence, say, at Florence. The time difference 
(in hours) multiplied by 15° would then give the angular distance 
(from Florence). In order to insure good observations from the 
deck of a moving ship, he proposed to "float" the telescope. 
A major handicap, however, was the lack of a good chronometer 
(not available until 1736). Galileo's negotiations were without 

After his return to Florence in June 1616, Galileo suffered his 
chronic malady, which was aggravated by a long seizure of hypo- 
chondria, owing undoubtedly to his failure to persuade the 
Roman dignitaries and scholars to adopt the Copernican view. 
During much of the years 1617-18 he was ill so that he missed the 
exciting appearance of three comets in August 1618. 

In the following year one of his Florentine disciples, Mario 
Guiducci, wrote a Discourse on Comets, in which he presented 
Galileo's ideas about their nature. (It so happened that Galileo 
was wrong in his own Aristotelian view of a comet as a terrestrial 
exhalation — an atmospheric phenomenon like a rainbow.) 
Immediately (1619) a Jesuit, Father Orazio Grassi of Savona, 


Professor of Mathematics at the Collegio Romano and builder of 
Sant' Ignazio, made an anonymous reply under the pseudonym 
Lothario Sarsi. In The Astronomical and Philosophical Balance he 
attacked directly and vigorously many of Galileo's ideas. The 
Jesuits had already adopted Tycho Brahe's view of comets being 
in the highest heaven based upon the parallax determination of 
the comet of 1577. Galileo was urged by friends to prepare his 
own answer. 

About this time Galileo lost a good friend and loyal supporter 
in 1621 upon the death of Cosimo II (his successor, Ferdinand II, 
became of age only in 1627). He felt greatly encouraged, however, 
when Maffeo Barberini, who had written a poem in his honor in 
1620, was made Pope Urban VIII in 1623. 

In that year, under sponsorship of the Lincei and with a 
dedication to Pope Urban VIII, himself, Galileo published The 
A sayer,W-(>) who deliberately used not just any balance, but a 
precise one like that for gold. This supposed letter to a Lincean 
Academician, Virginio Cesarini, Lord Chamberlain to the Pope, 
turned out to be a sharp, clever polemic and, what is more, an 
outstanding philosophical defense (virtually a manifesto) of 
science, emphasizing chiefly the need for first-hand experience 
rather than second-hand authority, for intellectual freedom and 
an experimental approach, for identification of physical quantities 
and the use of mathematics. Here (cf. Section 5b) was formulated 
his practical distinction between primary (objective) qualities (like 
size) and secondary (subjective) ones (like color), which became a 
source of controversy among philosophers (cf. John Locke (1632- 
1704) and George Berkeley (1685-1753)) during the following 
centuries. Here Galileo exhibited his unusual ability to present an 
opponent's arguments in a most plausible manner — and then to 
annihilate them with devastating logic. The opposition he thus 
often aroused resulted more from the manner of his presentation 
than from the doctrines themselves. Never has such excellent 
scientific methodology been advocated — in defense of such poor 
scientific conclusions. 

In the spring of 1624, Galileo made a fourth visit to Rome to 


pay his personal respects to the new Pope. Despite six long 
interviews, in which he was able to present his own Copernican 
arguments, he failed to change the Pope's fixed opinions. Never- 
theless, the latter wrote a commendatory letter to the Grand Duke 
in which he praised Galileo's "virtue and piety". It was during 
this visit that Galileo designed and constructed a compound 

A few words about Galileo's personal living about this time. 
From 1617 to 1631 he rented the villa (later called Villa Albrizzi 
and then Villa L'Ombrellino) of Lorenzo Giovanni Battista Segni 
at Bellosguardo, a western hill across the Arno. This dwelling was 
not too far from the poor Carmelite Convent (161 1) of San Matteo 
in Arcetri where, owing in large measure to his selfishness and own 
unwillingness to assume the parental burden of responsibility, 
Galileo had conveniently committed his two daughters at an 
early age — and they later had become nuns : Virginia as Sister 
Maria Celeste in 1616, and in 1617 Livia as Sister Arcangela, 
who became an invalid under the conditions of poverty there. 
Throughout her entire life the former showed continually a 
genuine love for her father and a great anxiety for his problems. 
She exhibited simple piety; her filial correspondence/ 1 ) extant 
from May 1623, is quite revealing (Galileo's replies are not 
available; they were probably destroyed because of a fear for his 
safety). His son Vincenzio, legitimized in 1619 by the Grand 
Duke, is often confused with Vincenzio, the son of his brother, 
Michelangelo, who was idle and careless with money, particularly 
that of other people (in 1628, for example, he moved with his wife 
and seven children into Galileo's house). In 1628 Galileo's son 
received his law degree from the University of Pisa and in 1629 
married Sestilia Bocchineri. They had three sons: Galileo (b. 
1629), Carlo (b. 1632), and Cosimo (b. 1638) — the end of 
Galileo's direct family line. Some time earlier (1621) his brother- 
in-law Landucci had left the country, and for a long period 
abandoned his family to Galileo's responsibility. Galileo's 
personal problems were compounded by his own continual illness 
(serious in 1628, aggravated by hernia in 1633), as well as by his 


family's perpetual and burdensome financial requests. (He was 
much more solicitous about his sisters and brother than about his 
own children.) At the same time, however, he had to maintain 
his social position as a courtier and to respond to the demands 
made daily upon a prominent personage such as himself, particu- 
larly requests to explain his scientific views. 



Galileo had ended the Starry Messenger with a promise to 
continue his report on the new astronomy at a later date. In his 
1610 application to the Grand Duke he had declared, "I wish to 
gain my bread by my writings". His difficulties in 1616, however, 
had made him cautious. Hope was born anew in the 1624 dis- 
cussion with the new Pope, who himself suggested a title more 
strictly in line with the actual contents of the book, which 
obviously compared the old Ptolemaic and the new Copernican 
theories, rather than the proposed "Dialogues on the Flux and 
Reflux of the Tides". The Pope urged also that the subject be 
treated hypothetically, with special consideration of his own 
supposedly unanswerable argument, namely, that inasmuch as an 
all-powerful God can do anything, to assert only the possibility of 
a double motion for the earth would appear to limit His omni- 
potence. Galileo did not actually begin writing until 1626, when 
his attention became focused again on magnetism and resulted in 
further investigations along this line. The Two Chief World 
Systems, < 7 - 8 > indeed, was not completed until early in 1630. 

Galileo then made a fifth visit to Rome, primarily to solicit 
personally a license, for printing the book with the aid of the 
Tuscan Ambassador, Marquis Francesco Niccolini (his wife, 
Caterina, was Riccardi's cousin), and Monsignor Giovanni 
Ciampoli, private secretary to the Pope. Unfortunately, he no 
longer had the support of his deceased friend and sponsor Prince 
Cesi, and many delays were encountered. The Imprimatur was fin- 
ally authorized by the Florentine Dominican Master of the Sacred 
Palace, Father Niccolo Riccardi, a friend of Galileo — not defini- 



tive, however, owing to the need for certain minor revisions such 
as a Preface to emphasize the hypothetical character of the Coper- 
nican thesis and a conclusion to embody the Pope's "unanswer- 
able" argument. The plague now contributed to poor communica- 
tions, aggravated by the aggressive delaying tactics of his enemies, 
so that little progress was made in printing the book at Rome. 
Galileo, accordingly, sought and finally obtained permission in 
late 1631 to have the work done by the press of Landini in 
Florence. The book was published at last on 21 February 1632; 
it was dedicated to Ferdinand II. 

Meanwhile, in the summer of 1631, he sought to lease a country 
house closer to the convent of his daughters, and in November of 
that year he moved to II Gioiello (the Jewel, later called the Villa 
Galileo, No. 29), which belonged to Esau Martellini and bordered 
on the convent (not far from the home of Giusto Susterman 
(1597-1681), who painted in 1635 the celebrated portrait (cf. 
Frontispiece) of Galileo that still hangs in the Galleria degli Uffizi). 

The Two Chief World Systems was a literary masterpiece (in 
Italian); it presented a dialogue taking place in Venice among 
three people: Sagredo (cf. Galileo's friend diplomat and mathe- 
matician, Giovanni Francesco Sagredo (1571-1620)), an urbane 
Venetian patrician, supposedly neutral, listening to the arguments 
of a genial Aristotelian philosopher Simplicio (presumably named 
after a distinguished commentator of the sixth century) and of an 
astute Copernican enthusiast, the nobleman Salviati of Florence, 
representing Galileo himself. (Note that the adjective "two" 
deliberately ruled out the Tychonic system.) It was a clever 
creation, written in popular style for the cultured layman. It had 
a light tone (Galileo had a good sense of humor, cf. his Pisan 
burlesque poem, "In Abuse of Gowns") with interesting digres- 
sions — typical of the Italian Renaissance. In view of its avowed 
purpose to destroy old notions and introduce new ideas it was 
decidedly more philosophical and pedagogical than scientific (cf . 
continual repetitions and multiple illustrations). It did not 
succeed in winning the Copernican war; but it was a critical 
battle in the history of science. 


The First Day started with a comparison of Aristotle's notion 
of celestial perfection and the actual imperfections like sunspots 
revealed through a telescope. Consideration was then given to 
many resemblances among the earth, the moon, and the planets; 
for example, earthshine and moonshine, the phases both of the 
moon and Venus, the universal spherical form, etc. Salviati 
suggested that similar movements of all planets about the sun 
might imply a simple common explanation. He cited the moons 
of Jupiter as celestial evidence of a non-geocentric system. 
(Galileo was so fascinated by the Copernican theory that he failed 
to consider seriously the equally satisfactory explanation of 
Jupiter's moons and of Venus' phases offered by Tycho Brahe's 
compromise model of a sun-centered planetary system revolving 
about the earth.) 

The Second Day was devoted to a consideration of traditional 
reasons, pro and con, for the motion of the earth, particularly its 
daily rotation. The major objection was not astronomical, but 
physical, viz. the expected sternward lag of a stone falling from 
the mast of such a moving ship. Salviati emphatically made the 
radical declaration that such a body would fall just as if the ship 
were not moving at all — an example of what has been later 
called Galileian relativity (cf. Section 16g). He went further and 
predicted that a ball would actually fall slightly eastward, owing 
to its greater speed at a greater distance from the center of 
the earth. 

On the Third Day the group considered the annual revolution 
and stellar parallax, the apparent displacement of stars owing to 
the annual motion of the earth (cf. Section 19). They ascribed the 
failure to detect any such shift as due to the relatively great dis- 
tance of the stars. In 1838 the German scientist Friedrich Wilhelm 
Bessel (1784-1846) actually measured the parallax of 61 Cygni by 
this very method. 

The Fourth Day was devoted to Galileo's primary scientific 
reason for his belief in the Copernican theory, namely, the 
explanation of tidal phenomena — a false hypothesis, based upon 
his incomplete understanding of gravitational attraction and of 


dynamics itself (he supposed the effect due merely to the com- 
bining of the earth's steady annual revolution with its daily 
periodic rotation). Nevertheless, the local terrain is an important 
factor in determining the actual behaviour of disturbed water in 
terrestrial basins, as Galileo indicated. 

All in all, the book reads well. Despite our own greater 
knowledge today, there is still a novelty of expression for the 
modern reader. One is impressed particularly with Galileo's 
skillful ferreting out of errors, as well as his instinctive ability to 
uncover the truth. It is ironic that the best evidence for the 
Copernican theory was ignored by Galileo although he had it in 
his own hands, namely, Kepler's laws (cf. his New Astronomy, 
1609) for planetary orbits, which were quantitative — not just 
qualitative — announced in the very same year as Galileo's own 
telescopic discoveries. Strangely enough, too, Galileo would have 
fared better, both scientifically and socially, if he been less 
dogmatic, more tolerant, and had followed the prudent advice of 
churchmen like the Catholic Bellarmine, i.e. in the cautious spirit 
of the Lutheran Andreas Osiander (1498-1552) in his unsigned 
Preface to Copernicus' epoch-making book, On the Revolutions of 
Celestial Orbs (1543), to accept the Copernican scheme as a true 
mathematical description of planetary motions, but as a yet 
unverified physical hypothesis. 

Sales were officially stopped six months after the book's 
publication, owing to the appointment of a Papal Commission to 
examine it (no scientifically knowledgeable person was on the 
Commission). It reported that the presentation of the two 
theories was hardly hypothetical; on the contrary, the tides were 
seriously being proposed as physical evidence of the earth's 
motion; and, what was seemingly worse, in the very seeking of the 
Imprimatur Galileo had evidently disobeyed the injunction given 
him in 1616 — both charges were cited in the final judgment. 
As we have seen, there is some reasonable doubt as to whether the 
latter conclusion was legally justified ; Galileo, it would seem, was 
to receive an injunction only if he persisted in refusing not to hold 
the Copernican view. There is little doubt, however, that Galileo 


had shrewdly used all his influential contacts to gain the license. 
And yet, the Church, too, was not blameless inasmuch as it had 
given official approval when it issued the Imprimatur. 

Despite numerous presentations of the Galileo case, Catholic 
and Protestant, national and scientific, the true story remains 
hidden amid the debris of history. Certainly the Roman Church 
had a right to discipline the breach of a formal promise by one of 
its members or of, at best, a roguish compliance. Much of the 
procedure of the Church must be ascribed to the personal involve- 
ment and apparent animosity of the Pope himself, who had 
presumably been persuaded that his own "unanswerable argu- 
ment" had been put in the mouth of a simpleton (translation of 
the name Simplicio). In a letter to the Grand Duke the Tuscan 
ambassador Niccolini reported that the Pope had referred to 
Galileo as one "who did not fear to make game of me". 
Urban VIII, who was certainly not one of the better popes, was 
undoubtedly the guiding mind behind the entire course of action 
adopted (otherwise the judgment would probably have been more 
moderate). An extenuating circumstance was the 1632 political 
pressure being brought by Spain against his seeming assistance to 
heretics, e.g. Gustavus Adolphus, and his unquestionable 
nepotism (cf. Cardinal Antonio Barberini, the Pope's brother, 
and the Lincean Cardinal Francesco Barberini, his nephew 
(friendly to Galileo)). Galileo, moreover, was too prominent a 
person to be allowed to defy openly the established church; his 
pupils held many influential academic positions. The case was 
formally handed over by the Pope to the Sacred Congregation of 
the Holy Office, and Galileo was so notified on 1 October. 

Galileo's immediate reaction was one of naive surprise. It was 
long before he himself realized the seriousness of the charges. 
Meanwhile, by various delaying tactics, he tried to evade the 
order to appear before the Inquisition. Only under threat of 
transportation under chains did the feeble 69-year-old man 
obediently set out on his last visit to Rome on 20 January 1633. 
Although he could have found refuge at Venice, he remained 
loyal to the Church — probably convinced of the truth of his own 


reasoning and the skill of his own persuasiveness. Galileo arrived 
in Rome on 13 February. 

In all fairness, it should be emphasized that throughout his 
confinement and the whole hearing the Inquisition treated Galileo 
quite well, and gave him all the consideration that might be due to 
a distinguished scholar who was a sincere churchman. 

In the first examination, on 12 April, under the Dominican 
Commissary General of the Holy Office, Fra Vincenzio Maculano 
da Firenzuola, Galileo claimed that he had not been given any 
specific injunction with respect to the Copernican doctrine being 
held or defended, and he presented Cardinal Bellarmine's note as 
evidence that he had not been prohibited from discussing it. 
Furthermore, had he not spoken about the book with the Pope 
himself who had not objected, although he was certainly familiar 
with the action of 1616? Galileo, I believe, was sincere in his 
denial of any remembrance of such a specific command having 
been given him earlier by the Dominican Father Commissary. 
Nevertheless, Galileo's absurd pretense of not favoring Coperni- 
canism in the book weakened his whole position. The Inquisition, 
moreover, was particularly concerned over Galileo's failure even 
to mention the 1616 inquiry at the time of his request for the 
Imprimatur. A preliminary secret report, submitted on 17 April, 
concluded that Galileo was guilty. 

By the time of the second interrogation on 30 April, friends had 
apparently persuaded Galileo to acquiesce to the requests of the 
Inquisition. He must have undergone an agonizing spiritual 
struggle between his old religious ties and his new scientific 
aspirations before yielding as a loyal and sincere Catholic. At any 
rate, Galileo now admitted that he had truly defended the 
Copernican doctrine — contrary to his previous denial: "My error 
then had been — and I confess it — one of vainglorious ambition 
and of pure ignorance and of inadvertence." After dismissal he 
returned voluntarily and offered to re-do some of the book in 
order to make certain that there could be no possible misunder- 
standing of his present point of view — and hopefully also to add 
one or two more desirable Days to the Dialogue. He was 


returned this time to the care of Niccolini. 

At his third appearance, 10 May, he was formally asked if he 
had any defense at that time ; he repeated his admitted failure to 
obey the command "not to hold, not to defend and not to 
teach". He again insisted, however, that he did not recall any 
precept other than that in Bellarmine's note. The old man 
appeared pitiful as he now pleaded for leniency in view of his old 
age and poor health. The Inquisition, with the Pope in the Chair, 
made its final decision on 16 June, and informed Galileo (confined 
now at the Inquisition) on 21 June, when they asked him three 
specific questions in order to confirm his present intent. On 
Wednesday morning, 22 June, in the hall of the Dominican 
Convent of Santa Maria sopra Minerva, garbed as a penitential 
criminal and kneeling, Galileo was sentenced by the Inquisition 
and compelled to renounce his beliefs under oath. Galileo 
confessed publicly, "I abjure, curse, and detest the said errors and 
heresies". As for the controversial Copernican theory, Blaise 
Pascal (1623-62) foresaw the final verdict of history, when he 
wrote in the Provincial Letters: "It is in vain that you have 
procured against Galileo a decree from Rome condemning his 
opinion of the earth's motion." The real intellectual crisis in 
Galileo's own life had probably occurred earlier when he realized 
that his clever arguments had failed to win converts and when he 
had accordingly pleaded guilty. The legend that he arose mutter- 
ing "Eppur si muove" (it moves, nevertheless) sounds like a 
natural afterthought which he may well have expressed in 
reminiscing later about the incident. It truly expressed his lifelong 
feeling. He certainly had the courage of his convictions, but it 
was tempered by shrewd expediency, including possibly lying and 
even perjury. Galileo was in no sense a martyr — either for 
science or for religion. His human limitations, though under- 
standable, are evident; his personal and social conscience, 
I believe, was sadly wanting. Every student of Galileo, however, 
has to form his own personal judgment on the basis of the 
incomplete evidence submitted by history. 

Galileo's books, of course, were placed on the Index Expurga- 


torius; they remained there until its 1835 edition when they were 
removed pursuant to an 1822 action based upon the Holy Office's 
1820 decision not to oppose Copernican views any more. 

Galileo, a heresy suspect, was given life imprisonment by the 
Inquisition — an unexpected measure; however, he was allowed 
to return immediately to the Villa Medici (the Academie Nationale 
de France since 1863) on Pincian hill. Although permitted by the 
Pope to leave Rome on 30 June he could go only to Siena, where 
he had to remain under house arrest with the kind Archbishop 
Ascania Piccolomini (one of his former students). Here, with the 
respect and stimulation of local visitors, he began to write his 
scientifically most significant work, the Two New Sciences, which 
he had listed as another project in his letter to the Grand Duke's 
secretary in 1610 and which Sagredo had mentioned at the 
end of the Two Chief World Systems. In December he was finally 
given permission to return to his home in Arcetri under house 
confinement and without visitors (opposed by his enemies after 
the Siena experience). His comfort, however, was short-lived 
in view of the death of his favorite daughter, Sister Maria Celeste, 
on 2 April 1634. 

The Two New Sciences, begun when the ever-industrious Galileo 
was almost 70, was completed in 1636, but had to wait until 1638 
for transmission by Fra Fulgenzio Micanzio of Venice for publi- 
cation at non-Catholic Amsterdam by Louis Elzevier. It was 
dedicated to the French ambassador Comte de Noaillis, who had 
been a pupil of Galileo at Padua. Like the Two Chief World 
Systems it consisted of four dialogues, with the same group of 
persons, viz. Sagredo, Simplicio, and Salviati; it lacked, however, 
the literary value and crusader-spirit of the older book. The two 
new sciences were the strength of materials and dynamics. 

The book began with an everyday question as to the relative 
strength of scaled structures (cf. Section 14), notably the failure 
of geometry per se to solve this problem. Galileo, it will be 
recalled, had always been consulted on practical matters owing to 
his prominent position as a science adviser (for example, he had 
been Superintendent of Tuscan waters). Strictly speaking, this 


analysis was the beginning of engineering science, and was 
certainly treated de now by Galileo. 

The first two days were presented as a real dialogue, involving 
many incidental experiments, whereas the last two were con- 
cerned primarily with formal, mathematical deduced theorems 
about motion, read supposedly from a manuscript of an 
academician friend (Galileo). 

Throughout the book, one is impressed with Galileo's keen 
physical insights into everyday phenomena (cf. Sections 8, 10). 
He compared physical atomism with a mathematical continuum 
(cf. Section 6), the speed of light with that of sound, acoustical 
vibrations with musical intervals (cf. Section 15), falling bodies 
(cf. Section 16) with downhill rollings, natural motion with 
violent ones (projectiles). He was at once experiential in his 
outlook and theoretical in his understanding; he used mathe- 
matics wherever practicable (probably not his mode of discovery), 
but subject to natural observations. 

With respect to the second new science, as Joseph-Louis, 
Comte de Lagrange (1736-1813), noted: Galileo undoubtedly laid 
the foundations of dynamics as a science — with due credit to the 
many particular results achieved by various precursors. Newton 
acknowledged his own indebtedness to Galileo for the laws of 
motion. Galileo certainly understood the principle of inertia, viz. 
the motion of a force-free body with constant velocity (though not 
in the generalized form of Newton's first (axiomatic) law for an 
isolated body). He also had some inkling of the relationship 
between force and acceleration, but here his understanding was 
definitely much less. Although neither a devastating polemic like 
The Assayer, nor a great literary creation like the Two Chief 
World Systems, at least it communicated more accurately the 
fundamental scientific ideas of the author. 

In 1636 Galileo continued his discussions about longitude, only 
this time with the Dutch, who presented him with a golden chain, 
which he felt obliged to decline in view of the watchful eye of the 
Inquisition. In June 1637 he lost the sight of his diseased right 
eye (glaucoma) and in December that of his left. Meanwhile, in 


November, in a letter to Fra Fulgenzio Micanzio he announced 
his last observation, namely, his discovery of lunar librations. 
In March 1638 he finally received papal permission to get medical 
attention while staying at his son's house (No. 1 1 Via della Costa 
near the Porta San Giorgio in Florence), where he was visited by 
the Grand Duke himself in September (cf. the plaque there 
commemorative of this visit). In December he was at last per- 
mitted to have his long-time friend Castelli as a visitor. The next 
month he returned to Arcetri, probably not voluntarily in view of 
his initial eagerness to get to the city. At this time, however, 
there was a definite relaxation of supervision on the part of the 
Inquisition. In March of that year he was visited by the English 
poet John Milton (1608-74), who returned to Florence after 
having stopped there in September 1638, when he probably called 
upon Galileo at his son's house. It is significant that the French 
philosopher and mathematician Rene Descartes did not go to see 
Galileo when in Florence, owing presumably to Descarte's 
indifference to his somewhat empirical approach to nature; the 
Frenchman Pierre Gassendi (1592-1655), however, did visit him. 

In the summer of 1639, 18-year old Vincenzio Viviani (1622- 
1703, F.R.S. 1696) came to live with him as his "last disciple". 
In 1641 blind Galileo was still thinking purposefully. He was 
developing a scheme to use a pendulum to regulate the movement 
of an ordinary clock. In September of that year, he was again 
visited by Castelli; and in October Torricelli, a former student of 
Castelli, came to stay with him at his invitation (and later 
succeeded Galileo at the Court of the Grand Duke of Tuscany). 
Bonaventura Cavalieri (1598-1647), a former pupil of Galileo, 
who had become professor of mathematics at Bologna, and 
Micanzio sent regrets that they could not join Galileo during this 
period. Galileo was always idolized by his friends. 

On 5 November Galileo went to bed with an illness from which 
he never recovered. On 20 December, he wrote his last letter — 
to the thrice-married Alessandra Bocchineri, sister of Sestilia, 
with whom he had corresponded continually after her return to 
Italy in 1630. Almost 78, Galileo died on the evening of 8 January 


1642, blessed with the Last Rites of the Church. Rome, however, 
refused to allow the public funeral and honor proposed by 
Florence. Accordingly, he was buried in the small Cappella del 
Campanile del Noviziato in Santa Croce, the Florentine West- 
minster Abbey. On 12 March 1737 his remains (together with 
those of Viviani, who had been buried next to him) were finally 
transferred to the main part of Santa Croce, across from Michel- 
angelo's tomb; the memorial monument has a bust of Galileo 
flanked by a figure of astronomy and by a figure of geometry 
(physical). At the exhumation various souvenirs of Galileo were 
removed. The left index finger, taken by Professor Antonio 
Francesco Gorri of Florence, may still be seen in the Instituto 
e Museo di Storia delle Scienze di Firenze, which replaced the 
Museo di Fisica e Storia Naturale, established in 1841 by 
Leopold II (1797-1870), the last Grand Duke of Tuscany, when 
he opened the Tribuna di Galileo in Florence (this museum still 
contains a statue of Galileo by Costoli and some remarkable 
frescoes painted from his life — at present (1965) not open to the 
public, pending reorganization). The thumb and right forefinger, 
taken by Canon Giovanni Vincenzio Capponi, President of the 
Florentine Academy, are said to be in Rome. At the University 
of Padua one will find Galileo's fifth lumbar vertebra, taken by 
Professor Antonio Cocchi. 

Some Galileo enthusiasts at the Vatican Council II in 1964 
proposed that the Church review the Galileo case. In 1965 Pope 
Paul VI, Giovanni Battista Montini, visited Santa Croce and 
mentioned Galileo graciously — at last. 

Despite the fact that the Galileo story has become historically 
celebrated in the relations between scientists and churchmen and 
that most of Galileo's work is extant, much of his personal 
thinking remains an unsolved puzzle. Recently there has 
developed a controversy even as to the public significance of his 
private scientific investigations. It has become fashionable for 
some professional historians of science to relegate Galileo to a 
minor position in a social movement of science beginning in the 
thirteenth century and culminating with Newton. Professional 


physicists, on the other hand, still recognize in Galileo the 
founder of physics as an experimental and theoretical discipline; 
he stressed both the necessity of experiential data and the role of 
quantitative theory. For us all his works still live in the modern 
era of science — symbolic of his inquiring life. 


(1) Brodrick, James, s.j., Galileo: the Man, his Work, his 

Misfortunes, Harper & Row, New York, 1964. 
Celeste, Sister Maria, The Private Life of Galileo, edited by 

M. C. Olney, Nichols & Noyes, Boston, 1870. 
De Santillana, Giorgio, The Crime of Galileo, University 

of Chicago Press, 1955. 
Geymonat, Ludovico, Galileo Galilei: A Biography and 

Inquiry into his Philosophy of Science, McGraw-Hill, New 

York, 1965. 
Harsanyi, Zsolt de The Star-Gazer, translated by P. Tabor, 

S. P. Putnam Sons, New York, 1939. 
Langford, Jerome J., O.P., Galileo, Science and the Church, 

Declee, New York, 1966. 
Taylor, Frank Sherwood, Galileo and the Freedom of 

Thought, Watts, London, 1938. 

(2) Laura Fermi and Gilberto Bernadini, Galileo and the 

Scientific Revolution, Basic Books, New York, 1961. 

(3) Galileo Galilei, On Motion and on Mechanics, translated 

by I. E. Drabkin and S. Drake, University of Wisconsin 
Press, Madison, 1960. 

(4) Drake, Stillman, Discoveries and Opinions of Galileo, 

Doubleday, Garden City, 1957. 

(5) Galileo Galilei, Discourse on Bodies in Water, translated by 

T. Salusbury with notes by S. Drake, University of 
Illinois Press, Urbana, 1960. 

* Selected books (in English) of Galileo's life and works in Florence, 
1890-1909, edited by Antonio Favaro (Opere di Galileo Galilei — Nationale 



(6) Galileo Galilei, The Controversy on the Comets of 1618, 

translated by Stillman Drake and C. D. O'Malley, 
University of Pennsylvania Press, Philadelphia, 1960. 

(7) Galileo Galilei, Dialogue Concerning Two Chief World 

Systems ~ Ptolemaic and Copernican, translated by Still- 
man Drake, University of California Press, Berkeley, 1953. 

(8) Galileo Galilei, Dialogue on the Great World Systems, trans- 

lated by T. Salusbury and revised by G. de Santillana, 
University of Chicago Press, 1953. 

(9) Galileo Galilei, Dialogues concerning Two New Sciences, 

translated by H. Crew and A. de Salvio, Northwestern 
University Press, Evanston, 1914. 

Part 2. HIS WORKS 


The following sixteen sections contain excerpts from Galileo's 
important writings. They have been selected for their significant 
contributions to the methodology and concepts of cumulative 
science — not for their illustrative value of intellectual and social 
history. In each case introductory comments are provided to 
indicate the modern viewpoint and to make up for any short- 
comings of translation or brevity. 




(a) Primary Qualities 

Our experiences of physical phenomena are transmitted through our 
senses. Some of our sense data we can share with others, namely, those 
geometrical and external properties which seem to belong to a particular 
object, e.g. its size, its shape, and its position. Others vary more with the 
viewer's own physiological and psychological perception, e.g. hotness, 
sound, color, etc. The former are more objective — or primary, as Locke 
later called them — the latter are seemingly secondary, i.e. more subjective. 
Galileo discussed this distinction in connection with his idea of heat. 

Primary qualities have inherently a common denominator; therefore, they 
can be used as the basis of a common description. Hence physical scientists 
have used them to seek acceptable standards for communication, and by 
allowing certain tolerances compatible with inherent errors of measurement 
have been remarkably successful in establishing intersubjective universal 

Galileo initially included sensory experiences like hotness and sound 
among secondary qualities. He himself, however, discovered some measur- 
able characteristics of hot bodies and of musical harmony so that the vague 
line of demarcation gradually shifted. Philosophers, who delight in logical 
distinctions, debated the essential meaning of this nebulous dichotomy for 
centuries. Scientists, on the other hand, have recognized more and more 
that each experience has both objective and subjective aspects that cannot 
be wholly dissociated. 

The Assayerf 

It now remains, in accordance with the promise made above to 
your Excellency, for me to tell you some of my thoughts about 
the proposition, "Motion is the cause of heat," and to show in 
what sense this may be true in my opinion. But first I must give 
some consideration to what we call "heat," for I much suspect 

t Ref. (6), pp. 308-13. 



that in general people have a conception of this which is very 
remote from the truth, believing heat to be a real attribute, 
property, and quality which actually resides in the material by 
which we feel ourselves warmed. 

Therefore I say that upon conceiving of a material or corporeal 
substance, I immediately feel the need to conceive simultaneously 
that it is bounded and has this or that shape; that it is in this 
place or that at any given time; that it moves or stays still; that 
it does or does not touch another body; and that it is one, few, or 
many. I cannot separate it from these conditions by any stretch 
of my imagination. But that it must be white or red, bitter or 
sweet, noisy or silent, of sweet or foul odor, my mind feels no 
compulsion to understand as necessary accompaniments. Indeed, 
without the senses to guide us, reason or imagination alone would 
perhaps never arrive at such qualities. For that reason I think 
that tastes, odors, colors, and so forth are no more than mere 
names so far as pertains to the subject wherein they reside, and 
that they have their habitation only in the sensorium. Thus, if 
the living creature (Vanimale) were removed, all these qualities 
would be removed and annihilated. Yet since we have imposed 
upon them particular names which differ from the names of those 
other previous real attributes, we wish to believe that they should 
also be truly and really different from the latter. 

I believe I can explain my idea better by means of some 
examples. I move my hand first over a marble statue and then 
over a living man. Now as to the action derived from my hand, 
this is the same with respect to both subjects so far as the hand is 
concerned; it consists of the primary phenomena of motion and 
touch which we have not designated by any other names. But the 
animate body which receives these operations feels diverse 
sensations according to the various parts which are touched. 
Being touched on the soles of the feet, for example, or upon the 
knee or under the armpit, it feels in addition to the general sense 
of touch another sensation upon which we have conferred a 
special name, calling it tickling; this sensation belongs entirely to 
us and not to the hand in any way. It seems to me that anyone 


would seriously err who might wish to say that the hand had 
within itself, in addition to the properties of moving and touching, 
another faculty different from these; that of tickling — as if the 
tickling were an attribute which resided in the hand. A piece of 
paper or a feather drawn lightly over any part of our bodies 
performs what are inherently quite the same operations of moving 
and touching; by touching the eye, the nose, or the upper lip 
it excites in us an almost intolerable titillation while in other 
regions it is scarcely felt. Now this titillation belongs entirely to us 
and not to the feather; if the animate and sensitive body were 
removed, it would remain no more than a mere name. And 
I believe that many qualities which we come to attribute to 
natural bodies, such as tastes, odors, colors, and other things, 
may be of similar and no more solid existence. 

A body which is solid and, so to speak, very material, when 
moved and applied to any part of my person produces in me that 
sensation which we call touch; this, though it pervades the entire 
body, seems yet to reside principally in the palms of the hands and 
especially in the fingertips, by means of which we sense the most 
minute differences of roughness, smoothness, and hardness, these 
being not so clearly distinguished by other parts of the body. 
Of these sensations, some are more pleasant to us and some less 
so, and they are smooth or rough, acute or obtuse, hard or 
yielding, according to the differences of shape which exist among 
tangible bodies. This sense being more material than the others 
and having its rise in the solidity of matter, it seems to be related 
to the earthy element. And since some bodies continually dissolve 
into minute particles of which some are heavier than air and 
descend, while others are lighter and rise on high, perhaps herein 
lies the origin of two other senses, accordingly as these particles 
strike us upon two parts of our bodies which are very much more 
sensitive than the skin and which feel the invasion of such subtle, 
tenuous, and yielding matter. The tiny descending particles are 
received upon the upper surface of the tongue, penetrating and 
mixing with its moisture; and their substance gives rise to tastes, 
sweet or unsavory accordingly as the shapes of these particles 


differ, as they are few or many, and as they are fast or slow. The 
other particles, ascending, enter by our nostrils and strike upon 
some small protuberances which are the instrument of smell; and 
here likewise their touch and passage are received to our liking or 
our dislike accordingly as they have this or that shape, move 
quickly or slowly, and are few or many. The tongue and the nasal 
passages are indeed seen to be providently arranged for the above 
as to location ; the former extends from below to receive the incur- 
sions of descending particles, while the latter are accommodated 
for those which ascend. Perhaps the excitation of tastes may be 
likened with a certain analogy to fluids, which descend through 
the air, and odors to fires, which ascend. 

Next the element of air remains available for sounds; these 
come to us indifferently from below, from above, and from all 
sides, since we are situated in the air and its movements within 
its own domain displace it equally in all directions. And the 
situation of the ear is most fittingly accommodated to all positions 
in space. Sounds are created and are heard by us when — without 
any special "sonorous" or "transonorous" property — a rapid 
tremor of the air, ruffled into very minute waves, moves certain 
cartilages of a tympanum within our ear. External means capable 
of producing this ruffling of the air are very numerous, but for 
the most part they reduce to the trembling of some body which 
strikes upon the air and disturbs it; waves are thereby very rapidly 
propagated, and from their frequency originates a high pitch, or 
from their rarity a deep sound. 

I do not believe that for exciting in us tastes, odors, and sounds 
there are required in external bodies anything but sizes, shapes, 
numbers, and slow or fast movements; and I think that if ears, 
tongues, and noses were taken away, shapes and numbers and 
motions would remain but not odors or tastes or sounds. These, 
I believe, are nothing but names, apart from the living animal — 
just as tickling and titillation are nothing but names when armpits 
and the skin around the nose are absent. And as the four senses 
considered here are related to the four elements, I believe that 
vision, the sense which is eminent above all others, is related to 


light, but in that ratio of excellence which exists between the finite 

and the infinite, the temporal and 

and the indivisible; between darkness and light. Of this sense 

the instantaneous, the quantity 

I pretend to understand but a 
still not suffice for me to explain 
explanation in writing — I pass 

and the matters pertaining to it, 
trifle, and since a long time would i 
that trifle — or even to hint at its i 
this over in silence. 

Returning now to my original purpose, and having already seen 
that many sensations which are deemed to be qualities residing in 
external subjects have no real existence except in ourselves, and 
outside of us are nothing but names, I say that I am inclined to 
believe that heat is of this character. Those materials which 
produce heat in us and make us feel warmth, which we call by 
the general name fire, would be a multitude of minute particles 
having certain shapes and moving with certain velocities. Meeting 
with our bodies, they penetrate by means of their consummate 
subtlety, and their touch which We feel, made in their passage 
through our substance, is the sensation which we call heat. This 
is pleasant or obnoxious according to the number and the greater 
or lesser velocity of these particles which thus go pricking and 
penetrating; that penetration is pleasant which assists our 
necessary insensible transpiration, and that is obnoxious which 
makes too great a division and dissolution of our substance. To 
sum up, the operation of fire by means of its particles is merely 
that in moving it penetrates all bodies by reason of its great 
subtlety, dissolving them more qiu'ckly or more slowly in propor- 
tion to the number and velocity of the fire-corpuscles (ignicoli) 
and the density or rarity of the material of these bodies, of which 
many are such that in their decomposition the major part of them 
passes over into further tiny corpuscles (ignei), and the dissolution 
goes on so long as it meets wi:h matter capable of being so 
resolved. But I do not believe at all that in addition to shape, 
number, motion, penetration, and touch there is any other quality 
in fire which is "heat" ; I believe that this belongs to us, and so 
intimately that when the animate and sensitive body is removed, 
"heat" remains nothing but a simple vocable. And since this 


sensation is produced in us by the passage and touch of the tiny 
corpuscles through our substance, it is obvious that if they were 
to remain at rest their operation would remain null. Thus we see 
that a quantity of fire retained in the pores and narrow channels 
of a piece of quicklime does not warm us even when we hold it in 
our hands, because it rests motionless. But place the quicklime 
in water, where the fire has a greater propensity to motion than 
it has in air — because of the greater gravity of this medium, and 
because the fire opens the pores of water as it does not those of 
air — and the little corpuscles will escape; and, touching our hand, 
they will penetrate it and we shall feel heat. 

Since, then, the presence of the fire-corpuscles does not suffice to 
excite heat, but we need also their movement, it seems to me that 
one may very reasonably say that motion is the cause of heat. 
This is that motion by which arrows and other sticks are burned 
and by which lead and other metals are liquefied when the little 
particles of fire penetrate the bodies, being either moved by them- 
selves or, their own strength not sufficing, being driven by the 
impetuous draught of a bellows. Of these bodies, some resolve 
into other flying particles of fire and some into a most minute 
powder; some liquefy and become as fluid as water. But I hold 
it to be foolish to take that same proposition from the common 
point of view — that a stone or a piece of iron or a stick must 
heat up if moved. 

The rubbing together and friction of two hard bodies, either by 
resolving their parts into very subtle flying particles or by opening 
an exit for the tiny fire-corpuscles within, finally set these in 
motion and, upon their encountering our bodies and penetrating 
and coursing through them, our conscious mind {anima sensitiva) 
feels that pleasant or obnoxious sensation which we have named 
heat, burning, or scalding. And perhaps when the thinning and 
attrition stop at or are confined within the tiniest particles 
(i minimi quanti), their motion is temporal and their action is 
calorific only, but, when their ultimate and highest resolution into 
truly indivisible atoms is reached, light is created which has 
instantaneous motion — or let us say instantaneous expansion 


and diffusion — and is capable off occupying immense spaces by 
its — I do not know whether to say by its subtlety, its rarity, 
its immateriality, or yet some other property different from all 
these, and nameless. 

(b) Mathematical Language 

Qualities that have quantitative (numerical) aspects can be compared 
relatively easily. What is more, the highly developed techniques of mathe- 
matics can then be applied in their analysis. Galileo, therefore, sought to 
replace vague (and speculative) qualities, wherever possible, with measurable 
(and repeatable) physical quantities. He was deeply impressed with the 
applicability of abstract (and symbolic) mathematics to concrete (and phenom- 
enological nature). 

Is the universe essentially mathematical, i.e. amenable to particular mathe- 
matical reasoning, or is it wholly mathematics, i.e. a physical realization of 
some universal mathematical ideas? Or, is its mathematical appearance 
solely man's impression? It is obviously quite different, on one hand, to 
admit that the universe has mathematical properties (natural or human), 
and, on the other, to claim that is it nothing but mathematics. The connota- 
tions about mathematics should not be read out of context. In this instance 
it is not even the chief topic of discussicjn. The real foe is authoritarianism. 
Mathematics is used merely as a contrasting illustration, i.e. rigorous 
reason (in the form of mathematical logic) in comparison with poetic 
(speculative or metaphysical) imagination, the book of nature interpretable 
in quantitative terms with the books of men expressing qualitative feelings. 
Mathematics is not being compared with phenomena, Platonic idealism with 
Aristotelian experience. 

Most physicists today consider that mathematical proofs are validated 
only in terms of their experiential assumptions and conclusions — physical 
meaning per se cannot be found in the meaningless symbols of pure 

The Assay erf 

But getting back to the point, you 
have it that I deemed it a great defect 
to have adhered to Tycho's doctrine, 
"Whom then should he have followed 
has been revealed to be false by the 

t Ref. (6), pp. 183-7. 

see how once more he will 

on the part of Father Grassi 

and he asks resentfully: 

? Ptolemy, whose doctrine 

recent observations of Mars ? 


Or perhaps Copernicus ? But he must rather be rejected by every- 
one, in view of the ultimate condemnation of his hypothesis." 
Here I note several things, and I reply first that it is quite false 
that I have ever criticized anyone for following Tycho, even 
though I might very reasonably have done so, as will indeed at 
last become clear to his adherents from the Anti-Tycho of the 
distinguished Chiaramonti. Hence, so far as this remark is 
concerned, Sarsi is very wide of the mark. Even more irrelevant 
is his introduction of Ptolemy and Copernicus, who are never to 
be found writing a word relative to the distances, magnitudes, 
movements, and theory of the comets, which (and which alone) 
are here under consideration. He might with as much reason 
have brought in Sophocles, Bartoli, or Livy. 

It seems to me that I discern in Sarsi a firm belief that in philo- 
sophizing it is essential to support oneself upon the opinion of 
some celebrated author, as if when our minds are not wedded to 
the reasoning of some other person they ought to remain com- 
pletely barren and sterile. Possibly he thinks that philosophy is 
a book of fiction created by some man, like the Iliad or Orlando 
Furioso — books in which the least important thing is whether 
what is written in them is true. Well, Sig. Sarsi, that is not the 
way matters stand. Philosophy is written in this grand book — 
I mean the universe — which stands continually open to our gaze, 
but it cannot be understood unless one first learns to comprehend 
the language and interpret the characters in which it is written. 
It is written in the language of mathematics, and its characters 
are triangles, circles, and other geometrical figures, without which 
it is humanly impossible to understand a single word of it; without 
these, one is wandering about in a dark labyrinth. Sarsi seems to 
think that our intellects should be enslaved to that of some other 
man (I shall disregard the fact that in thus making everyone, 
including himself, an imitator, he will praise in himself what he 
has blamed in Sig. Mario), and that in the contemplation of the 
celestial motions one should adhere to somebody else. 



In considering the cohesion of materials, which renders them resistant to 
fracture, Galileo discussed also a mathematical continuum as contrasted 
with physical atomism. By drawing radii from a common center through 
concentric circles he noted that the different circumferences always have 
corresponding points so that, in a certain sense, they are equivalent. He 
concluded that it is meaningless to speak of relative magnitudes of infinite 
classes of objects. An infinite class, ir deed, is quite different from a finite 
class — a distinction that became significant only in connection with infinite 
sets in the nineteenth century. All numbers, too, as he noted, can be put 
in one-to-one correspondence with thefr squares. 

All in all, however, Galileo himself admitted the incomprehensibility of 
the infinite and its association with divisibility. Although Galileo showed 
little interest in perusing such questions of abstract mathematics (preferring 
applications of mathematics to natural phenomena), he exhibited shrewd 
understanding of some of the difficulties of the continuum that have worried 
man from ancient times to today. 

The inquisitive Florentine Salviati represents Galileo in the following 
discussion. He is instructing the urbafle Venetian Sagredo and the conserv- 
ative Aristotelian Simplicio. 

Dialogues Concerning Two New tfciences^ 

Salv. Let us return to the 
tioned polygons whose behavior 
the case of polygons with 10000Q 
perimeter of the greater, i.e., 
sides one after another, is equal 
100000 sides of the smaller, provided 

t Ref. (9), pp. 24-34. 

consideration of the above men- 

we already understand. Now in 

sides, the line traversed by the 

line laid down by its 100000 

to the line traced out by the 

we include the 100000 



vacant spaces interspersed. So in the case of the circles, polygons 
having an infinitude of sides, the line traversed by the continuously 
distributed [continnamente disposti] infinitude of sides is in the 
greater circle equal to the line laid down by the infinitude of 
sides in the smaller circle but with the exception that these latter 
alternate with empty spaces; and since the sides are not finite in 
number, but infinite, so also are the intervening empty spaces not 
finite but infinite. The line traversed by the larger circle consists 
then of an infinite number of points which completely fill it; 
while that which is traced by the smaller circle consists of an 
infinite number of points which leave empty spaces and only partly 
fill the line. And here I wish you to observe that after dividing 
and resolving a line into a finite number of parts, that is, into a 
number which can be counted, it is not possible to arrange them 
again into a greater length than that which they occupied when 
they formed a continuum [continuate] and were connected without 
the interposition of as many empty spaces. But if we consider the 
line resolved into an infinite number of infinitely small and 
indivisible parts, we shall be able to conceive the line extended 
indefinitely by the interposition, not of a finite, but of an infinite 
number of infinitely small indivisible empty spaces. 

Now this which has been said concerning simple lines must be 
understood to hold also in the case of surfaces and solid bodies, 
it being assumed that they are made up of an infinite, not a finite, 
number of atoms. Such a body once divided into a finite number 
of parts it is impossible to reassemble them so as to occupy more 
space than before unless we interpose a finite number of empty 
spaces, that is to say, spaces free from the substance of which the 
solid is made. But if we imagine the body, by some extreme and 
final analysis, resolved into its primary elements, infinite in 
number, then we shall be able to think of them as indefinitely 
extended in space, not by the interposition of a finite, but of an 
infinite number of empty spaces. Thus one can easily imagine 
a small ball of gold expanded into a very large space without the 
introduction of a finite number of empty spaces, always provided 
the gold is made up of an infinite number of indivisible parts. 


who denied Divine Provi- 
a similar occasion by a certain 

Simp. It seems to me that you ape travelling along toward those 
vacua advocated by a certain ancient philosopher. 

Salv. But you have failed to a^dd, 
dence," an inapt remark made on 
antagonist of our Academician. 

Simp. I noticed, and not without indignation, the rancor of 
this ill-natured opponent; further references to these affairs 
I omit, not only as a matter of good form, but also because 
I know how unpleasant they are to the good tempered and well 
ordered mind of one so religions and pious, so orthodox and 
God-fearing as you. 

But to return to our subject, your previous discourse leaves 
with me many difficulties which] I am unable to solve. First 
among these is that, if the circuBfiferences of the two circles are 
equal to the two straight lines, C£ and BF, the latter considered 
as a continuum, the former as interrupted with an infinity of 
empty points, I do not see how it| is possible to say that the line 
AD described by the center, and nWe up of an infinity of points, 
is equal to this center which is a single point. Besides, this build- 
ing up of lines out of points, divisibles out of indivisibles, and 
finites out of infinites, offers me an obstacle difficult to avoid; 
and the necessity of introducing a vacuum, so conclusively 
refuted by Aristotle, presents the same difficulty. 

Salv. These difficulties are rdal; and they are not the only 
ones. But let us remember that wfe are dealing with infinities and 
indivisibles, both of which transcend our finite understanding, 
the former on account of their magnitude, the latter because of 
their smallness. In spite of this, njien cannot refrain from discus- 
sing them, even though it must be^ done in a roundabout way. 

Therefore I also should like to jake the liberty to present some 
of my ideas which, though not necessarily convincing, would, on 
account of their novelty, at least, prove somewhat startling. But 
such a diversion might perhaps c^rry us too far away from the 
subject under discussion and rdight therefore appear to you 
inopportune and not very pleasing. 

Sagr. Pray let us enjoy the advantages and privileges which 


come from conversation between friends, especially upon subjects 
freely chosen and not forced upon us, a matter vastly different 
from dealing with dead books which give rise to many doubts but 
remove none. Share with us, therefore, the thoughts which our 
discussion has suggested to you; for since we are free from urgent 
business there will be abundant time to pursue the topics already 
mentioned; and in particular the objections raised by Simplicio 
ought not in any wise to be neglected. 

Salv. Granted, since you so desire. The first question was, 
How can a single point be equal to a line ? Since I cannot do more 
at present I shall attempt to remove, or at least diminish, one 
improbability by introducing a similar or a greater one, just as 
sometimes a wonder is diminished by a miracle. 

And this I shall do by showing you two equal surfaces, together 
with two equal solids located upon these same surfaces as bases, 
all four of which diminish continuously and uniformly in such a 
way that their remainders always preserve equality among them- 
selves, and finally both the surfaces and the solids terminate their 
previous constant equality by degenerating, the one solid and the 
one surface into a very long line, the other solid and the other 
surface into a single point; that is, the latter to one point, the 
former to an infinite number of points. 

Sagr. This proposition appears to me wonderful, indeed; but 
let us hear the explanation and demonstration. 

Salv. Since the proof is purely geometrical we shall need a 
figure. Let AFB be a semicircle with center at C; about it describe 
the rectangle ADEB and from the center draw the straight lines 
CD and CE to the points D and E. Imagine the radius CF to be 
drawn perpendicular to either of the lines AB or DE, and the 
entire figure to rotate about this radius as an axis. It is clear that 
the rectangle ADEB will thus describe a cylinder, the semicircle 
AFB a hemisphere, and the triangle CDE, a cone. Next let us 
remove the hemisphere but leave the cone and the rest of the 
cylinder, which, on account of its shape, we will call a "bowl." 
First we shall prove that the bowl and the cone are equal; then 
we shall show that a plane drawn parallel to the circle which 


forms the base of the bowl and wh|ch has the line DE for diameter 
and F for a center — a plane whc-se trace is GN — cuts the bowl 
in the points G, I, O, N, and the \x>ne in the points H, L, so that 
the part of the cone indicated by ^HL is always equal to the part 
of the bowl whose profile is represented by the triangles GAI and 
BON. Besides this we shall prove that the base of the cone, i.e., 
the circle whose diameter is HL, is equal to the circular surface 
which forms the base of this portion of the bowl, or as one might 
say, equal to a ribbon whose wid^th is GI. (Note by the way the 

nature of mathematical definitions which consist merely in the 
imposition of names or, if you prefer, abbreviations of speech 
established and introduced in ordfcr to avoid the tedious drudgery 
which you and I now experienc^ simply because we have not 
agreed to call this surface a "circular band" and that sharp solid 
portion of the bowl a "round rafcor.") Now call them by what 
name you please, it suffices to understand that the plane, drawn 
at any height whatever, so long ^s it is parallel to the base, i.e., 
to the circle whose diameter is Dp, always cuts the two solids so 
that the portion CHL of the corje is equal to the upper portion 
of the bowl; likewise the two arenas which are the bases of these 
solids, namely the band and the Circle HL, are also equal. Here 
we have the miracle mentioned! above; as the cutting plane 
approaches the line AB the portions of the solids cut off are 
always equal, so also the areas of) their bases. And as the cutting 
plane comes near the top, the twc| solids (always equal) as well as 


their bases (areas which are also equal) finally vanish, one pair 
of them degenerating into the circumference of a circle, the other 
into a single point, namely, the upper edge of the bowl and the 
apex of the cone. Now, since as these solids diminish equality is 
maintained between them up to the very last, we are justified in 
saying that, at the extreme and final end of this diminution, they 
are still equal and that one is not infinitely greater than the other. 
It appears therefore that we may equate the circumference of a 
large circle to a single point. And this which is true of the solids 
is true also of the surfaces which form their bases; for these also 
preserve equality between themselves throughout their diminution 
and in the end vanish, the one into the circumference of a circle, 
the other into a single point. Shall we not then call them equal 
seeing that they are the last traces and remnants of equal magni- 
tudes ? Note also that, even if these vessels were large enough to 
contain immense celestial hemispheres, both their upper edges and 
the apexes of the cones therein contained would always remain 
equal and would vanish, the former into circles having the 
dimensions of the largest celestial orbits, the latter into single 
points. Hence in conformity with the preceding we may say that 
all circumferences of circles, however different, are equal to each 
other, and are each equal to a single point. 

Sagr. This presentation strikes me as so clever and novel that, 
even if I were able, I would not be willing to oppose it; for to 
deface so beautiful a structure by a blunt pedantic attack would 
be nothing short of sinful. But for our complete satisfaction pray 
give us this geometrical proof that there is always equality 
between these solids and between their bases; for it cannot, 
I think, fail to be very ingenious, seeing how subtle is the philo- 
sophical argument based upon this result. 

Salv. The demonstration is both short and easy. Referring to 
the preceding figure, since IPC is a right angle the square of the 
radius IC is equal to the sum of the squares on the two sides IP, 
PC; but the radius IC is equal to AC and also to GP, while CP is 
equal to PH. Hence the square of the line GP is equal to the sum 
of the squares of IP and PH, or multiplying through by 4, we have 


the square of the diameter GN equal to the sum of the squares 
on IO and HL. And, since the afeas of circles are to each other 
as the squares of their diameters, it follows that the area of the 
circle whose diameter is GN is equal to the sum of the areas of 
circles having diameters IO and HL, so that if we remove the 
common area of the circle having IO for diameter the remaining 
area of the circle GN will be equal to the area of the circle whose 
diameter is HL. So much for the first part. As for the other part, 
we leave its demonstration for the present, partly because those 
who wish to follow it will find it in the twelfth proposition of the 
second book of De centro gravitates solidorum by the Archimedes 
of our age, Luca Valerio, who made use of it for a different object, 
and partly because, for our purpose, it suffices to have seen that 
the above-mentioned surfaces are always equal and that, as they 
keep on diminishing uniformly, they degenerate, the one into a 
single point, the other into the circumference of a circle larger 
than any assignable; in this fact lies our miracle. 

Sagr. The demonstration is ingenious and the inferences 
drawn from it are remarkable. And now let us hear something 
concerning the other difficulty raised by Simplicio, if you have 
anything special to say, which, however, seems to me hardly 
possible, since the matter has already been so thoroughly 

Salv. But I do have something special to say, and will first 
of all repeat what I said a little vyhile ago, namely, that infinity 
and indivisibility are in their very fiature incomprehensible to us ; 
imagine then what they are when combined. Yet if we wish to 
build up a line out of indivisible points, we must take an infinite 
number of them, and are, therefore, bound to understand both 
the infinite and the indivisible at the same time. Many ideas 
have passed through my mind concerning this subject, some of 
which, possibly the more important, I may not be able to recall 
on the spur of the moment; but in the course of our discussion it 
may happen that I shall awaken in you, and especially in 
Simplicio, objections and difficulties which in turn will bring to 
memory that which, without su#i stimulus, would have lain 


dormant in my mind. Allow me therefore the customary liberty 
of introducing some of our human fancies, for indeed we may so 
call them in comparison with supernatural truth which furnishes 
the one true and safe recourse for decision in our discussions and 
which is an infallible guide in the dark and dubious paths of 

One of the main objections urged against this building up of 
continuous quantities out of indivisible quantities [continuo 
d' indivisibili] is that the addition of one indivisible to another 
cannot produce a divisible, for if this were so it would render the 
indivisible divisible. Thus if two indivisibles, say two points, can 
be united to form a quantity, say a divisible line, then an even 
more divisible line might be formed by the union of three, five, 
seven, or any other odd number of points. Since however these 
lines can be cut into two equal parts, it becomes possible to cut 
the indivisible which lies exactly in the middle of the line. In 
answer to this and other objections of the same type we reply that 
a divisible magnitude cannot be constructed out of two or ten or a 
hundred or a thousand indivisibles, but requires an infinite 
number of them. 

Simp. Here a difficulty presents itself which appears to me 
insoluble. Since it is clear that we may have one line greater than 
another, each containing an infinite number of points, we are 
forced to admit that, within one and the same class, we may have 
something greater than infinity, because the infinity of points in 
the long line is greater than the infinity of points in the short line. 
This assigning to an infinite quantity a value greater than infinity 
is quite beyond my comprehension. 

Salv. This is one of the difficulties which arise when we 
attempt, with our finite minds, to discuss the infinite, assigning to 
it those properties which we give to the finite and limited; but 
this I think is wrong, for we cannot speak of infinite quantities as 
being the one greater or less than or equal to another. To prove 
this I have in mind an argument which, for the sake of clearness, 
I shall put in the form of questions to Simplicio who raised this 


I take it for granted that you know which of the numbers are 
squares and which are not. 

Simp. I am quite aware that a> squared number is one which 
results from the multiplication of another number by itself; thus 
4, 9, etc., are squared numbers which come from multiplying 2, 3, 
etc., by themselves. 

Salv. Very well; and you also know that just as the products 
are called squares so the factors £re called sides or roots; while 
on the other hand those number^ which do not consist of two 
equal factors are not squares. [Therefore if I assert that all 
numbers, including both squares and non-squares, are more than 
the squares alone, I shall speak the truth, shall I not? 

Simp. Most certainly. 

Salv. If I should ask further h(j>w many squares there are one 
might reply truly that there are ^s many as the corresponding 
numbers of roots, since every square has its own root and every 
root its own square, while no square has more than one root and 
no root more than one square. 

Simp. Precisely so. 

Salv. But if I inquire how many roots there are, it cannot be 
denied that there are as many as tfyere are numbers because every 
number is a root of some square. This being granted we must 
say that there are as many squares as there are numbers because 
they are just as numerous as their roots, and all the numbers are 
roots. Yet at the outset we said there are many more numbers 
than squares, since the larger portion of them are not squares. 
Not only so, but the proportionate number of squares diminishes 
as we pass to larger numbers. Thug up to 100 we have 10 squares, 
that is, the squares constitute 1/10 (part of all the numbers; up to 
10000, we find only 1/100 part to be squares; and up to a million 
only 1/1000 part; on the other hand in an infinite number, if one 
could conceive of such a thing, he would be forced to admit that 
there are as many squares as there are numbers all taken together. 

Sagr. What then must one conclude under these circum- 
stances ? 

Salv. So far as I see we can only infer that the totality of all 


numbers is infinite, that the number of squares is infinite, and 
that the number of their roots is infinite; neither is the number of 
squares less than the totality of all numbers, nor the latter greater 
than the former; and finally the attributes "equal," "greater," 
and "less," are not applicable to infinite, but only to finite, 
quantities. When therefore Simplicio introduces several lines of 
different lengths and asks me how it is possible that the longer 
ones do not contain more points than the shorter, I answer him 
that one line does not contain more or less or just as many points 
as another, but that each line contains an infinite number. Or if 
I had replied to him that the points in one line were equal in 
number to the squares; in another, greater than the totality of 
numbers; and in the little one, as many as the number of cubes, 
might I not, indeed, have satisfied him by thus placing more 
points in one line than in another and yet maintaining an infinite 
number in each? So much for the first difficulty. 

Sagr. Pray stop a moment and let me add to what has already 
been said an idea which just occurs to me. If the preceding be 
true, it seems to me impossible to say either that one infinite 
number is greater than another or even that it is greater than a 
finite number, because if the infinite number were greater than, 
say, a million it would follow that on passing from the million to 
higher and higher numbers we would be approaching the infinite; 
but this is not so; on the contrary, the larger the number to which 
we pass, the more we recede from [this property of] infinity, 
because the greater the numbers the fewer [relatively] are the 
squares contained in them; but the squares in infinity cannot be 
less than the totality of all the numbers, as we have just agreed; 
hence the approach to greater and greater numbers means a 
departure from infinity, f 

Salv. And thus from your ingenious argument we are led to 
conclude that the attributes "larger," "smaller," and "equal" 

t A certain confusion of thought appears to be introduced here through 
a failure to distinguish between the number n and the class of the first n 
numbers; and likewise from a failure to distinguish infinity as a number 
from infinity as the class of all numbers. [Trans.] 


have no place either in comparing infinite quantities with each 
other or in comparing infinite with finite quantities. 

I pass now to another consideration. Since lines and all 
continuous quantities are divisible into parts which are themselves 
divisible without end, I do not see how it is possible to avoid the 
conclusion that these lines are built up of an infinite number of 
indivisible quantities because a division and subdivision which 
can be carried on indefinitely presupposes that the parts are 
infinite in number, otherwise the subdivision would reach an end; 
and if the parts are infinite in number, we must conclude that they 
are not finite in size, because an infinite number of finite quantities 
would give an infinite magnitude. And thus we have a continuous 
quantity built up of an infinite number of indivisibles. 



The attraction of magnetite ore (an oxide of iron) was first mentioned by 
Thales of Miletus (b. 624 B.C.), one of the seven sages of ancient Greece 
sometimes called also the father of physics. Comparatively few additional 
facts were noted until 1600 when William Gilbert published his systematic 
study, marking the beginning of magnetism as a science; Gilbert, accordingly 
has been called "the father of magnetism and electricity". 

Galileo apparently read this book shortly afterwards. He proceeded at 
once to make some investigations of his own, e.g. the behavior of magnetic 
poles. In particular, he experimented with armatures, i.e. a thin hemisphere 
of iron fitting a lodestone like a jacket. By this means the attractive force 
could be made considerably greater. Galileo boasted that he could thus 
make an armature sustain 26 times its own weight — in comparison with 
Gilbert's maximum of only 4 times. 

Galileo was convinced by Gilbert's experiments with a lodestone model 
of the earth, a so-called terella, that the earth itself is a large magnet, but 
he did not exhibit much interest in Gilbert's speculations as to this force 
being applicable to planetary motions. He expressed regrets, moreover 
that Gilbert failed to quantify his work. 

Dialogue Covering Two Chief World Systems — 
Ptolemaic and Copernican\ 

Simp. Then you are one of those people who adhere to the 
magnetic philosophy of William Gilbert? 

Salv. Certainly I am, and I believe that I have for company 
every man who has attentively read his book and carried out his 
experiments. Nor am I without hope that what has happened to 
me in this regard may happen to you also, whenever a curiosity 
similar to mine, and a realization that numberless things in nature 
remain unknown to the human intellect, frees you from slavery 

t Ref. (7), pp. 400-8. 



to one particular writer or another on the subject of natural 
phenomena, thereby slackening the reins on your reasoning and 
softening your stubborn defiance of your senses, so that some day 
you will not deny them by giving ear to voices which are heard 
no more. 

Now, the cowardice (if we may be permitted to use this term) 
of ordinary minds has gone to such lengths that not only do they 
blindly make a gift — nay, a tribute — of their own assent to 
everything they find written by those authors who were lauded 
by their teachers in the first infancy of their studies, but they 
refuse even to listen to, let alone examine, any new proposition 
or problem, even when it not only has not been refuted by their 
authorities, but not so much as examined or considered. One of 
these problems is the investigation of what is the true, proper, 
basic, internal, and general matter and substance of this ter- 
restrial globe of ours. Even though neither Aristotle nor anybody 
else before Gilbert ever took it into his head to consider whether 
this substance might be lodestone (let alone Aristotle or anybody 
else having disproved such an opinion), I have met many who 
have started back at the first hint of this like a horse at his 
shadow, and avoided discussing such an idea, making it out to be 
a vain hallucination, or rather a mighty madness. And perhaps 
Gilbert's book would never have come into my hands if a famous 
Peripatetic philosopher had not made me a present of it, I think 
in order to protect his library from its contagion. 

Simp. I frankly confess myself to have been one of these 
ordinary minds, and it is only since I have been allowed during 
the past few days to take part in these conferences of yours that 
I am aware of having wandered somewhat from the trite and 
popular path. But I do not yet feel so much awakened that the 
roughness of this new and curious opinion does not make it seem 
to me very laborious and difficult to master. 

Salv. If what Gilbert writes is true, it is not an opinion ; it is a 
scientific subject; it is not a new thing, but as ancient as the earth 
itself; and if true, it cannot be rough or difficult, but must be 
smooth and very easy. If you like, I can make it evident to you 


that you are creating the darkness for yourself, and feeling a 
horror of things which are not in themselves dreadful — like 
a little boy who is afraid of bugaboos without knowing anything 
about them except their name, since nothing else exists beyond 
the name. 

Simp. I should enjoy being enlightened and removed from error. 

Salv. Then answer the questions I am about to ask you. First, 
tell me whether you believe that this globe of ours, which we 
inhabit and call "earth," consists of a single and simple material, 
or an aggregate of different materials. 

Simp. I can see that it is composed of very diverse substances 
and bodies. In the first place, I see water and earth as its major 
components, which are quite different from each other. 

Salv. For the present let us leave out the oceans and other 
waters, and consider just the solid parts. Tell me whether these 
seem to you to be all one thing, or various things. 

Simp. As to appearances, I see them various, finding great fields 
of sterile sand, and others of fertile and fruitful soil; innumerable 
barren and rugged mountains are to be seen, full of hard rocks 
and stones of the most various kinds, such as porphyry, alabaster, 
jasper, and countless sorts of marble; there are vast mines of 
many species of metal, and, in a word, such a diversity of 
materials that a whole day would not suffice to enumerate 
these alone. 

Salv. Now of all these different materials, do you believe that 
in the composition of this great mass they occur in equal propor- 
tions ? Or rather that among them all there is one part which far 
exceeds the others and is in effect the principal matter and 
substance of this huge bulk? 

Simp. I believe that the stones, the marbles, the metals, the 
gems, and other materials so diverse are exactly like jewels and 
ornaments, external and superficial to the original globe, which 
I think immeasurably exceeds in bulk all these other things. 

Salv. Now this vast principal bulk, of which the things you 
have named resemble excrescences and ornaments : Of what do 
you believe this to be made? 


Simp. I think it is the simple, less impure, element of earth. 

Salv. But what is it that you understand by "earth"? Is it 
perhaps that which is spread over fields, which is broken with 
spades and plows, in which grain and fruit are sown and great 
forests spring up spontaneously? Which, in a word, is the 
habitat of all animals and the womb of all vegetation ? 

Simp. This, I should say, is the primary substance of our globe. 

Salv. Well, that does not seem to me to be a very good thing 
to say. For this earth that is broken, sown, planted, and that 
bears fruit is one part of the surface of the globe, and quite a 
shallow part. It does not go very deep in relation to the distance 
to the center, and experience shows that by digging not far down 
materials are to be found very different from the external crust; 
harder, and not any good for producing vegetation. Besides, the 
more central parts may be supposed, from being compressed by 
the very heavy weights which rest upon them, to be compacted 
together and to be as hard as the most solid rock. Add to this 
that it would be vain to endow with fertility material never 
destined to produce crops, but merely to remain buried forever 
in the deep dark abysses of the earth. 

Simp. Who is to say that the interior parts, close to the center, 
are sterile? Perhaps they also have their produce of things 
unknown to us. 

Salv. Why, you, of all people, since you understand so well 
that all the integral parts of the universe are produced for man's 
benefit alone — you ought to be most certain that this above all 
should be destined for the sole convenience of us inhabitants of 
it. And what good could we get out of materials so hidden from 
us and so remote that we can never make them available ? The 
interior substance of this globe of ours, then, cannot be material 
which can be broken or dissipated, or is loose like this topsoil 
which we call "earth," but must be a very dense and solid body; 
in a word, very hard rock. And if it must be such, what reason 
have you for being more reluctant to believe that it is lodestone 
than that it is porphyry, jasper, or some other hard stone? If 
Gilbert had written that the inside of this globe is made of 


sandstone, or chalcedony, perhaps the paradox would seem less 
strange to you ? 

Simp. I grant that the most central parts of this globe are much 
compressed, and therefore compacted together and solid, more 
and more so as they go deeper; Aristotle also concedes this. But 
I am not aware of any reasons which oblige me to believe that 
they degenerate and become other than earth of the same sort 
as this on the surface. 

Salv. I did not interject this argument for the purpose of 
proving conclusively to you that the primary and real substance 
of this globe of ours is lodestone, but merely to show you that 
there is no reason for people to be more reluctant to grant that it 
is lodestone than any other material. And if you think it over, 
you will find that it is not improbable that merely a single and 
arbitrary name motivated men to believe that this substance is 
earth, from the name "earth" being commonly used to signify 
that material which we plow and sow, as well as to name this 
globe of ours. But if the name for the latter had been taken from 
stone (as it might just as well have been as from earth) then 
saying that its primary substance was stone would surely not have 
met resistance or contradiction from anybody. Indeed, this is 
much more probable; I think it certain that if one could husk 
this great globe, taking off only a bulk of one or two thousand 
yards, and then separate the stones from the earth, the pile of 
rocks would be much, much larger than that of fertile earth. 

Now I have not adduced for you any of the reasons which 
conclusively prove de facto that our globe is made of lodestone, 
nor is this the time to go into those, the more so as you may look 
them up in Gilbert at your leisure. I am merely going to explain, 
with a certain likeness to my own, his method of procedure in 
philosophizing, in order that I may stimulate you to read it. 
I know that you understand quite well how much a knowledge of 
events contributes to an investigation of the substance and 
essence of things; therefore I wish you to take care to inform 
yourself thoroughly about many events and properties that are 
found uniquely in lodestone. Examples of this are its attraction 


of iron, and its conferring this same power upon iron merely by 
its presence; likewise its communicating to iron the property of 
pointing towards the poles, just as it retains this power in itself. 
Moreover, I want you to make a visual test of how there resides 
in it a power of conferring upon the compass needle not only the 
property of pointing toward the poles with a horizontal motion 
under the meridian — a property long since known — but also 
a newly observed faculty of vertical dip when it is balanced upon 
a small sphere of lodestone on which this meridian has been 
previously marked. I mean that the needle declines from a given 
mark, a greater or less amount according as the needle is taken 
closer to or farther from the pole, until at the pole itself it stands 
erect and perpendicular, while in the equatorial regions it remains 
parallel to the axis. 

Next, make a test of the power of attraction being more active 
in every piece of lodestone, nearer the poles than at the middle, 
and noticeably stronger at one pole than at the other, the stronger 
pole being the one which points toward the south. Note that in a 
small lodestone this stronger south pole becomes weaker when- 
ever it is required to support some iron in the presence of the 
north pole of a much larger lodestone. To make a long story 
short, you may ascertain by experiment these and many other 
properties described by Gilbert, all of which belong to lodestone 
and none to any other material. 

Now, Simplicio, suppose that a thousand pieces of different 
materials were set before you, each one covered and enclosed in 
cloth under which it was hidden, and that you were asked to find 
out from external indications the material of each one without 
uncovering it. If, in attempting to do this, you should hit upon 
one which plainly showed itself to have all the properties which 
you had already recognized as residing only in lodestone and not 
in any other material, what would you judge to be the essence of 
that material? Would you say that it might be a piece of ebony, 
or alabaster, or tin ? 

Simp. There is no question at all that I should say it was a piece 
of lodestone. 


Salv. In that case, declare boldly that under this covering or 
wrapper of earth, stone, metal, water, etc. there is concealed a 
huge lodestone. For in regard to this there are recognized, by 
anyone who observes carefully, all the same events which are 
perceived to belong to a true and unconcealed sphere of lodestone. 
If nothing more were to be observed than the dipping of the 
needle, which, carried around the earth, tilts more upon its 
approach to the pole and less as it goes towards the equator, where 
it finally becomes balanced, this alone ought to persuade the 
most stubborn judgment. I say nothing of another remarkable 
effect which is plainly seen in all pieces of lodestone and causes 
the south pole of a lodestone to be stronger than the otherf for us 
inhabitants of the Northern Hemisphere. This difference is found 
to be the greater, the more one departs from the equator; at the 
equator, both sides are of equal strength, though noticeably 
weaker. But in the southern regions, far from the equator, it 
changes its nature and the side which is the weaker for us acquires 
power over the other. All this conforms with what we see done 
by a little piece of lodestone in the presence of a big one whose 
force prevails over the smaller and makes it subservient, so that 
according as it is held near to or far from the equator of the large 
one, it makes just such variations as I have told you are made by 
every lodestone carried near to or far from the earth's equator. 

Sagr. I was convinced at my first perusal of Gilbert's book, 
and, having found an excellent piece of lodestone, I made many 
observations over a long period, all of which merited the greatest 
wonder. But what seemed most astonishing of all to me was the 
great increase in its power of sustaining iron when provided with 
an armaturef in the manner taught by this same author. By thus 
equipping my piece I multiplied its strength by eight, and where 
previously it would scarcely hold up nine ounces of iron, with 
the armature it would sustain more than six pounds. Perhaps 
you have seen this very piece, sustaining two little iron anchors, 
in the gallery of your Most Serene Grand Duke, on whose behalf 
I parted with it.f 

Salv. I used to look at it frequently with great amazement, 


until a still greater admiration seized me because of a little 
specimen in the possession of our Academician. This, being not 
over six ounces in weight and sustaining no more than two ounces 
unarmatured, supports one hundred sixty ounces when so 
equipped. Thus it bears eighty times as much with an armature 
as without, and holds up twenty-six times its own weight. This is 
a greater marvel than Gilbert was able to behold, since he writes 
that he was never able to get a lodestone which succeeded in 
sustaining four times its own weight. 

Sagr. It seems to me that this stone opens to the human mind 
a large field for philosophizing, and I have often speculated to 
myself on how it imparts to the iron which arms it a force so 
greatly superior to its own. But I was unable ever to find any 
satisfactory solution, nor did I find anything to much advantage 
in what Gilbert has to say on this particular. I wonder whether 
the same is true of you. 

Salv. I have the highest praise, admiration, and envy for this 
author, who framed such a stupendous concept regarding an 
object which innumerable men of splendid intellect had handled 
without paying any attention to it. He seems to me worthy of 
great acclaim also for the many new and sound observations 
which he made, to the shame of the many foolish and mendacious 
authors who write not just what they know, but also all the 
vulgar foolishness they hear, without trying to verify it by experi- 
ment; perhaps they do this in order not to diminish the size of 
their books. What I might have wished for in Gilbert would be a 
little more of the mathematician, and especially a thorough 
grounding in geometry, a discipline which would have rendered 
him less rash about accepting as rigorous proofs those reasons 
which he puts forward as verae causae for the correct conclusions 
he himself had observed. His reasons, candidly speaking, are not 
rigorous, and lack that force which must unquestionably be 
present in those adduced as necessary and eternal scientific 

I do not doubt that in the course of time this new science will 
be improved with still further observations, and even more by 


true and conclusive demonstrations. But this need not diminish 
the glory of the first observer. I do not have a lesser regard for 
the original inventor of the harp because of the certainty that his 
instrument was very crudely constructed and more crudely played ; 
rather, I admire him much more than a hundred artists who in 
ensuing centuries have brought this profession to the highest 
perfection. And it seems to me most reasonable for the ancients 
to have counted among the gods those first inventors of the fine 
arts, since we see that the ordinary human mind has so little 
curiosity and cares so little for rare and gentle things that no 
desire to learn is stirred within it by seeing and hearing these 
practiced exquisitely by experts. Now consider for yourself 
whether minds of that sort would ever have been applied to the 
construction of a lyre or to the invention of music, charmed by 
the mere whistling of dry tortoise tendons, or the striking of 
four hammers If To apply oneself to great inventions, starting 
from the smallest beginnings, and to judge that wonderful arts 
he hidden behind trivial and childish things is not for ordinary 
minds; these are concepts and ideas for superhuman souls. 

Now, in answer to your question, I say that I also thought for 
a long time to find the cause for this tenacious and powerful 
connection that we see between the iron armature of a lodestone 
and the other iron which joins itself to it. In the first place, I am 
certain that the power and force of the stone is not increased 
at all by its having an armature, for it does not attract through 
a longer distance. Nor does it attract a piece of iron as strongly 
if a thin slip of paper is introduced between this and the armature; 
even if a piece of gold leaf is interposed, the bare lodestone will 
sustain more iron than the armature. Hence there is no change 
here in the force, but merely something new in its effect. 

And since for a new effect there must be a new cause, we seek 
what is newly introduced by the act of supporting the' iron via 
the armature, and no other change is to be found than a difference 
in contact. For where iron originally touched lodestone, now iron 
touches iron, and it is necessary to conclude that the difference 
in these contacts causes the difference in the results. Next, the 


difference between the contacts must come, so far as I can see, 
from the substance of the iron being finer, purer, and denser in 
its particles than is that of the lodestone, whose parts are coarser, 
less pure, and less dense. From this it follows that the surfaces 
of the two pieces of iron which are to touch, when perfectly 
smoothed, polished, and burnished, fit together so exactly that 
all the infinity of points on one touch the infinity of points on 
the other. Thus the threads which unite the pieces of iron are, 
so to speak, more numerous than those which join lodestone to 
iron, on account of the substance of lodestone being more porous 
and less integrated, so that not all the points and threads on the 
surface of the iron find counterparts to unite with on the surface 
of the lodestone. 

Now we may see that the substance of iron (especially when 
much refined, as is the finest steel) is much more dense, fine, 
and pure in its particles than is the material of lodestone, from 
the possibility of bringing the former to an extremely thin edge, 
such as a razor edge, which can never be done to a piece of lode- 
stone with any success. The impurity of the lodestone and its 
adulteration with other kinds of stone can next be sensibly 
observed; in the first place by the color of some little spots, gray 
for the most part, and secondly by bringing it near a needle 
suspended on a thread. The needle cannot come to rest at these 
little stony places; it is attracted by the surrounding portions, and 
appears to leap toward these and flee from the former spots. 
And since some of these heterogeneous spots are large enough to 
be easily visible, we may believe that others are scattered in great 
quantity throughout the mass but are not noticeable because of 
their small size. 

What I am telling you (that is, that the great abundance of 
contacts made between iron and iron is the cause of so solid an 
attachment) is confirmed by an experiment. If we present the 
sharp point of a needle to the armature of a lodestone, it attaches 
itself no more strongly than it would to the bare lodestone; this 
can result only from the two contacts being equal, both being 
made at a single point. But now see what follows. A needle is 


placed upon the lodestone so that one of its ends sticks out some- 
what beyond, and a nail is brought up to this. Instantly the 
needle will attach itself to it so firmly that upon the nail being 
drawn back, the needle can be suspended with one end attached 
to the lodestone and the other to the nail. Withdrawing the nail 
still farther, the needle will come loose from the lodestone if the 
needle's eye is attached to the nail and its point to the lodestone; 
but if the eye is toward the lodestone, the needle will remain 
attached to the lodestone upon withdrawing the nail. In my 
judgment, this is for no other reason than that the needle, being 
larger at the eye, makes contact in more places than it does at 
its very sharp point. 

Sagr. The entire argument looks convincing to me, and I rank 
these experiments with the needle very little lower than mathe- 
matical proof. I frankly admit that in the entire magnetic science 
I have not heard or read anything which gives so cogently the 
reasons for any of its other remarkable phenomena. If their 
causes were to be explained to us this clearly, I can think of 
nothing pleasanter that our intellects could wish for. 

Salv. In investigating the unknown causes of our conclusions, 
one must be lucky enough right from the start to direct one's 
reasoning along the road of truth. 



The lift pump, often used to raise water in wells, has a piston that moves up 
and down; a flap valve opens on the down-stroke but closes on the up-stroke. 
The first few strokes force air out of the cylinder, thus creating a partial 
vacuum into which water flows up from the well. The lift pump had been 
used for wells of various depths. It was quite unexpected, therefore, to find 
a practical limit (about 27 ft) to the depth, regardless of the size of the lift 
pump. In order to account for this phenomenon Galileo pictured a column 
of water breaking under its own weight. The true explanation, however, 
had to await his successor Torricelli's explanation in terms of the external 
atmospheric pressure (due to the weight of air) on the water in the well with 
the invention of the barometer as a consequence. 

Dialogues Concerning Two New Sciences^ 

Sagr. Thanks to this discussion, I have learned the cause of a 
certain effect which I have long wondered at and despaired of 
understanding. I once saw a cistern which had been provided 
with a pump under the mistaken impression that the water might 
thus be drawn with less effort or in greater quantity than by means 
of the ordinary bucket. The stock of the pump carried its sucker 
and valve in the upper part so that the water was lifted by attrac- 
tion and not by a push as is the case with pumps in which the 
sucker is placed lower down. This pump worked perfectly so 
long as the water in the cistern stood above a certain level; but 
below this level the pump failed to work. When I first noticed 
this phenomenon I thought the machine was out of order; but 
the workman whom I called in to repair it told me the defect was 
not in the pump but in the water which had fallen too low to be 

t Ref. (9), pp. 16-17. 



raised through such a height; and he added that it was not 
possible, either by a pump or by any other machine working on 
the principle of attraction, to lift water a hair's breadth above 
eighteen cubits; whether the pump be large or small this is the 
extreme limit of the lift. Up to this time I had been so thoughtless 
that, although I knew a rope, or rod of wood, or of iron, if 
sufficiently long, would break by its own weight when held by the 
upper end, it never occurred to me that the same thing would 
happen, only much more easily, to a column of water. And really 
is not that thing which is attracted in the pump a column of 
water attached at the upper end and stretched more and more 
until finally a point is reached where it breaks, like a rope, on 
account of its excessive weight? 

Salv. That is precisely the way it works; this fixed elevation 
of eighteen cubits is true for any quantity of water whatever, be 
the pump large or small or even as fine as a straw. We may 
therefore say that, on weighing the water contained in a tube 
eighteen cubits long, no matter what the diameter, we shall 
obtain the value of the resistance of the vacuum in a cylinder of 
any solid material having a bore of this same diameter. 



In his early lectures, although Galileo accepted many traditional views about 
motion, he was not wholly Aristotelian. Later he became increasingly 
critical of Aristotle's static universe, particularly as interpreted by fanatic 
followers; he himself showed an inquiring spirit and sought to understand 
phenomena themselves. 

Thus while accepting the Aristotelian doctrine of a place for everything 
and a natural tendency for everything to go to its proper place, he explained 
the actual motion of a body in a medium, i.e. its apparent lightness or 
heaviness (he early regarded these as relative terms inasmuch as all materials 
have weight and hence would naturally move down if free), as dependent 
basically upon the buoyant force of the medium itself. By Archimedes' 
principle the unbalanced force, i.e. the apparent weight Wa of a body A of 
volume Va and density Da in a medium of density dnt is 

W A = (D A -d M ) V A g , 
where g is the acceleration due to gravity. The actual cause of motion in this 
instance was regarded by Galileo as analogous to the law of a balance (e.g. 
a lever), which moves downward on the apparently heavier side. Intuitively, 
he assumed that the resulting speed va is proportional to the unbalance 
force, i.e. 

W A = (D A Va) va 
(true according to Newton's second law of motion for a uniformly accelerated 
body starting from rest). Galileo, of course, recognized that "the points set 
forth . . . cannot very well be further elucidated mathematically; they 
require rather a physical explanation". Likewise for a second body B in 
a medium N, 

W B = (D B V B )v B . 
Hence, dividing and substituting for the apparent weights, we obtain 

va_ = Da D A —d M 
v B D A ' D B -d M ' 
Let us apply this analysis to several different problems. 

(1) A body having the same density as the medium, i.e. Da = dju . In 
this case Wa = ; hence /a = 0. The body will move neither upward nor 

(2) A body less dense than the medium. Here DA<d M so that Wa is 
negative, signifying upward motion. 



(3) A body more dense that the medium. Wa is now positive, inasmuch 
as DA>dM, so that the body moves downward. 

(4) Two different bodies of the same material A in the same medium M, 

vj_ _ DA—dM = 

vb DA—dM 
In other words, regardless of their size, if dropped simultaneously from rest, 
these bodies of different weights will nevertheless fall at the same rate — 
contrary to Aristotle's opinion that a larger body moves more swiftly. 

(5) For any body in a vacuum, i.e. rfjvf = 0., there is a finite speed, namely, 

v =g 
owing to dependence upon the difference of densities — not any absolute 
value. Aristotle, on the contrary, believed that the speed of a body in a 
vacuum would be infinite inasmuch as it varied inversely only as the "resis- 
tance" of a medium (zero for a vacuum). Considering such instantaneous 
motion to be impossible, Aristotle concluded that a vacuum cannot exist. 

On Motion^ 


That natural motion is caused by heaviness or lightness. 
In the previous chapter we asserted, and assumed it as well 
known, that nature has so arranged it that heavier bodies remain 
at rest under lighter. We must, therefore, now note that bodies 
which move downward move because of their heaviness, while 
those that move upward move because of their lightness. For, 
since heavy bodies have, by reason of their heaviness, the property 
of remaining at rest under lighter bodies — inasmuch as they are 
heavy, they have been placed by nature under the lighter — they 
will also have the property, imposed by nature, that, when they 
are situated above lighter bodies, they will move down below 
these lighter bodies, lest the lighter remain at rest under the 
heavier, contrary to the arrangement of nature. And, in the same 
way, light bodies will move upward by their lightness, whenever 
they are under heavier bodies. For if they have, by reason of their 
lightness, the property of remaining at rest above heavier bodies, 
they will also, by that same lightness, have the property of 

t Ref. (3), pp. 16-17. 


not remaining at rest below heavier bodies, unless they are 

Now it is clear from this that, in the case of motion, we must 
consider not merely the lightness or heaviness of the moving 
body, but also the heaviness or lightness of the medium through 
which the motion takes place. For if water were not lighter than 
stone, a stone would not sink in water. But a difficulty might 
arise at this point, as to why a stone cast into the sea moves 
downwards naturally, despite the fact that the [sum total of] 
water of the sea is far heavier than the stone that was thrown. 
We must therefore recall what we pointed out in chapter [1], viz., 
that the stone is, in fact, heavier than the water of the sea, if we 
take a quantity of water equal in volume to that of the stone; and 
so the stone, being heavier than the water, will move downward 
in the water. 

But again a difficulty will arise as to just why we must consider 
a quantity of water equal in volume to the volume of the stone, 
rather than of the whole sea. In order to remove this difficulty, 
I have decided to adduce some proofs, on which not only the 
solution of this difficulty, but the whole discussion will depend. 
Though, to be sure, there are many media through which motions 
take place, e.g., fire, air, water, etc., and in all of them the same 
principle applies, we shall assume that water is the medium in 
which the motion is to take place. And first we shall prove that 
bodies equally heavy with the water itself, if let down into the 
water, are completely submerged, but then move neither down- 
ward nor upward. Secondly, we shall show that bodies lighter 
than water not only do not sink in the water, but are not even 
completely submerged. Thirdly, we shall prove that bodies 
heavier than water necessarily move downward. 


First demonstration, in which it is proved that bodies of 
the same heaviness as the medium move neither upward nor 
downward. Coming now to the proofs, let us first consider a 


body of the same heaviness as water, i.e., one whose weight is 
equal to the weight of a quantity of water equal in volume to the 
volume of the given body. Let the body be ef. It must then be 
proved that body ef, if let down into the water, is completely 
submerged, and then moves neither upward nor downward. Let 
abed be the position of the water, before the body is let down 
into it. And suppose that body ef, after being let down into the 
water, is not completely submerged, if that is possible, but that 
some part of it, say e, protrudes, only part / being submerged. 
While part /of the body is being submerged, the level of the water 
is necessarily raised. Thus, suppose surface ao of the water is 
raised to surface st. Clearly, the volume of water so is exactly 
equal to the volume of/, the submerged part of the body. For it is 
necessary that the space into which the body enters be vacated by 
the water, and that a volume of water be removed equal to the 
volume of that portion of the body that is submerged. Therefore 

s t 



b c 

Fig. 2 

the volume of water so is equal to the volume of the submerged 
part/of the body. Hence also the weight of/ will equal the weight 
of water so. 

Now the water so strives by its weight to return downwards to 
its original position, but cannot achieve this unless solid ef is 
first lifted out of the water, i.e., raised by the action of the water. 
And the solid resists such raising with all the weight that it has ; 
moreover, both the solid body and the water are assumed to be 


at rest in this position. Therefore the weight of water so, by which 
the water strives to raise the solid upward, must necessarily be 
equal to the weight with which the solid resists and presses down- 
ward. (For if the weight of water so were greater than the weight 
of solid ef, ef would be raised and forced out of the water; if, on 
the other hand, the weight of solid <?/were greater, the water level 
would be raised. But everything is assumed to be at rest in this 
position.) Hence the weight of water so is equal to the weight of 
the whole magnitude ef. But that is impossible, for the weight of 
so is equal to that of part/. It is clear, therefore, that no part of 
the solid magnitude ef will protrude [above the water level], but 
that the whole will be submerged. 

This is the complete demonstration which I have set forth in 
fuller detail so that those who come upon it for the first time may 
be able to understand it more easily. But it might be better 
explained more briefly, so that the entire heart of the proof would 
be as follows. We must prove that magnitude ef, which is 
assumed to be of the same weight as water, is completely sub- 
merged. Suppose, then, if it is not completely submerged, that 
some part of it protrudes. Let e be the part protruding, and let 
the water level be raised to the surface st; and, if possible, let the 
water and the body both be at rest in this position. Then, since 
magnitude ef presses by its weight and tends to raise water so, 
and water so by its weight resists being further raised, it must 
follow that the weight of ef pressing down is exactly equal to the 
weight of water so which resists. For since they are assumed to 
be at rest in this position, neither will the pressure be greater than 
the resistance, nor the resistance greater than the pressure. Hence 
the weight of water so is equal to the weight of magnitude ef. 
But this is impossible. For, since the volume of the whole body ef 
is greater than the volume of the water so, the weight of body ef 
will also be greater than that of the water so. It is therefore clear 
that bodies of the same heaviness as water will be completely 
submerged in water. 

And I say further that they will move neither upward nor 
downward, but will remain at rest wherever they are placed. For 


there is no reason why they should move downward or upward. 
Since they are assumed to be of the same heaviness as water, to 
say that they sink in water would be the same as saying that 
water, when placed in water, sinks underneath this water; and 
then that the water which rises above the first-mentioned water 
again moves downward, and that the water thus continues to 
move alternately downward and upward forever. This is 


Second demonstration, in which it is proved that bodies 
lighter than water cannot be completely submerged. now 
since the demonstration in the previous chapter had to do with a 
state of rest, we must now consider a case which involves motion 
upward. I say, then, that bodies lighter than water, when let 
down into water, are not completely submerged, but that some 
portion protrudes. Let the first level of the water, before the body 
is let down, be along surface ef; and suppose that body a, lighter 
than water, is let down into the water, and, if possible, is com- 
pletely submerged, and the water level raised to surface cd. And 
suppose, if it is possible, that both the water and the body remain 
at rest in this position. Now the weight with which the body 

c d 


e f 

Fig. 3 

exerts pressure and tends to raise water cf will be equal to the 
weight with which water cf exerts pressure to raise body a. But 
the volume of water cf is equal to that of body a. There are 


therefore two magnitudes, one the body a, the other the water cf; 
and the weight of a is equal to that of cf, and also the volume of 
a is equal to the volume of the water cf Therefore body a is of 
the same heaviness as the water. But this is impossible : for the 
body was assumed to be lighter than water. Therefore body a 
will not remain completely submerged under the water. It will 
therefore of necessity move upward. 

It is clear, then, why and how motion upward results from 
lightness. And, from what has been said in this and in the 
previous chapter, it can easily be understood that bodies heavier 
than water are completely submerged and must keep moving 
downward. That they are completely submerged is a necessary 
conclusion. For if they were not completely submerged, they 
would be lighter than water, and this would be contrary to our 
assumption. For it follows from the converse of the proposition 
just proved that bodies which are not completely submerged are 
lighter than water. Moreover, these bodies [i.e., those heavier 
than water] must continue to move downward. For if they did 
not move downward, they would either be at rest or move 
upward. But they could not be at rest : for it was proved in the 
preceding chapter that bodies having the same heaviness as water 
remain at rest and move neither upward nor downward. And it 
has just been shown that bodies lighter than water move upward. 
Therefore, from all these considerations, since bodies which move 
downwards must be heavier than the medium through which they 
move, it is clear that heavy bodies move downward by reason of 
their weight. And it is clear that, in the case of the stone thrown 
into the sea, we must reckon not with all the water of the sea, but 
only with that small part which is removed from the place into 
which the stone enters. 

But the points set forth in these last two chapters cannot very 
well be further elucidated mathematically; they require rather a 
physical explanation. For this reason I propose, in the next 
chapter, to reduce the matter to a consideration of the balance, 
and to explain the analogy that holds between bodies that move 
naturally and the weights of the balance. My aim is a richer 


comprehension of the matters under discussion, and a more 
precise understanding on the part of my readers. 



consider what happens in the case of the balance, so that we may 
then show that all these things also happen in the case of bodies 
moving naturally. Let line ab, then, represent a balance, whose 
center, over which motion may take place, is the point c bisecting 
line ab. And let two weights, e and o, be suspended from points 
a and b. Now in the case of weight e there are three possibilities: 



Fig. 4. 

it may either be at rest, or move upward, or move downward. 
Thus if weight e is heavier than weight o, then e will move down- 
ward. But if e is less heavy, it will, of course, move upward, and 
not because it does not have weight, but because the weight of o 
is greater. From this it is clear that, in the case of the balance, 
motion upward as well as motion downward takes place because 
of weight, but in a different way. For motion upward will occur 
for e on account of the weight of o, but motion downward on 
account of its own weight. But if the weight of e is equal to that 
of o, then e will move neither upward nor downward. For e will 
not move downward unless weight o, which tends to raise it, is 
less heavy; nor will e move upward unless weight o, by which it 
must be raised, is heavier. 

Having examined the case of the balance, we return to naturally 
moving bodies. We can assert this general proposition: that the 
heavier cannot be raised by the less heavy. On this assumption 
it is easy to understand why solids lighter than water are not 
completely submerged. 


If, for example, we let a piece of wood down into water, then, 
if the wood is to be submerged, water must necessarily leave the 
place into which the wood enters, and this water must be raised, 
that is, must be moved in a direction away from the center of the 
universe. If, then, the water which has to be raised is heavier than 
the wood, it surely will not be able to be raised by the wood. But 
if the whole piece of wood is submerged, then from the place 
into which the wood enters a volume of water must be removed 
equal to the volume of the whole piece of wood. But a volume of 
water equal to the volume of the wood is heavier than the wood 
(for it is assumed that the wood is lighter than water). It will 
therefore not be possible for the wood to be completely sub- 
merged. And this is in agreement with what was said in the case 
of the balance, namely, that a smaller weight cannot raise a larger. 

But if the wood were of the same heaviness as the water, that 
is, if the water which is raised by the complete submerging of the 
wood is not heavier than but only just as heavy as the wood, the 
wood will of course then be completely submerged, since it does 
not meet [sufficient] resistance from the lifting action of the 
water. But once it is entirely submerged it does not continue to 
move either upward or downward. And this corresponds 
analogically to what was said, in the case of the balance, about 
equal weights neither of which moves upward or downward. 

But if, on the other hand, the wood is heavier than that part 
of the water which tends to be lifted by the wood, that is to say, 
if the wood is heavier than a volume of water equal to the volume 
of the wood (for, as has often been said, the volume of water 
that is lifted by the submerged wood is equal to the volume of 
the wood), then surely the wood will continue to move down- 
ward. And this corresponds analogically to what was said in the 
case of the balance — namely, that one weight moves down and 
sends the other one up when it is heavier than that other. 

Besides, in the case of bodies moving naturally, as in the 
weights in a balance, the cause of all motions, up as well as down, 
can be referred to weight alone. For when something moves up 
it is at that time being raised by the weight of the medium. Thus, 


if a piece of wood lighter than water is forcibly held under water, 
then, since the submerged wood displaces a volume of water 
equal to its own volume, and since a volume of water equal to 
the volume of the wood is heavier than the piece of wood, doubt- 
less the wood will be lifted by the weight of that water and will 
be impelled upward. Thus upward motion will occur because of 
the heaviness of the medium and the [relative] lightness of the 
moving body; and downward motion because of the heaviness of 
the moving body and the [relative] lightness of the medium. 

And from this one can easily understand (contrary to Aristotle 
De Caelo 1.89) that what moves moves, as it were, by force and 
by the extruding action of the medium. For when the wood is 
forcibly submerged, the water violently thrusts it out when, with 
downward motion, it moves toward its own proper place and is 
unwilling to permit that which is lighter than itself to remain at 
rest under it. In the same way, the stone is thrust from its position 
and impelled downward because it is heavier than the medium. 
It is therefore clear that this kind of motion may be called 
"forced," although commonly the upward motion of wood in 
water and the downward motion of stone in water are called 
"natural." And Aristotle's argument is invalid when he says: 
"If the motion were forced, it would lose speed at the end and not 
gain it, as it does." For forced motion loses speed only when the 
moving body leaves the hand of the mover, not while it is still in 
contact with the mover. 

It is therefore clear that the motion of bodies moving naturally 
can be suitably reduced to the motion of weights in a balance. 
That is, the body moving naturally plays the role of one weight 
in the balance, and a volume of the medium equal to the volume 
of the moving body represents the other weight in the balance. 
So that, if a volume of the medium equal to the volume of the 
moving body is heavier than the moving body, and the moving 
body lighter, then the latter, being the lighter weight, will move 
up. But if the moving body is heavier than the same volume of 
the medium, then, being the heavier weight, it will move down. 
And if, finally, the said volume of the medium has a weight equal 


to that of the moving body, the latter will move neither up nor 
down, just as the weights in the balance, when they are equal to 
each other, neither fall nor rise. 

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


The cause of speed and slowness of natural motion. Since 
it has been quite fully explained above that natural motions are 
caused by heaviness and lightness, we must now consider how the 
greater or lesser speed of such motion comes about. In order to 
be able to accomplish this more easily, we must make the follow- 
ing distinction, viz., that inequalities in the slowness and speed of 
motion occur in two ways: either the same body moves in 
different media, or else there are different bodies moving in the 
same medium. We shall show presently that in both these cases 
of motion the slowness and speed depend on the same cause, 
namely, the greater or lesser weight of the media and of the 
moving bodies. But first we shall show that the cause given by 
Aristotle to account for this effect is insufficient. 

Aristotle wrote (Physics 4.71) that the same body moves more 
swiftly in a rarer than in a denser medium, and that therefore the 
cause of slowness of motion is the density of the medium, and 
the cause of speed its rareness. And he asserted this on the basis 
of no other reason than experience, viz., that we see a moving 
body move more swiftly in air than in water. 

But it will be easy to prove that this reason is not sufficient. 
For if the speed of motion depends on the rareness of the medium, 
the same body will always move more swiftly through rarer media. 
But this is erroneous. For there are many moving bodies that 
move more swiftly with natural motion in denser media than they 
do in rarer ones, e.g., more swiftly in water than in air. If, for 


example, we take a very thin inflated bladder, it will descend 
slowly with natural motion in air. But if we release it in deep 
water, it will fly up very fast, again with natural motion. At this 
point I know that someone may reply that the bladder moves in 
air and is swiftly carried down, but in water not only does it not 
fall faster, it does not fall at all. I would say in answer that the 
bladder moves up very swiftly in the water, but then does not 
continue moving in the air. But, not to prolong the argument, 
I say that in the rarer media not every motion, but only downward 
motion, is swifter; and upward motion is swifter in denser media. 
And this is reasonable. For in a place where a downward motion 
takes place with difficulty, an upward motion necessarily takes 
place with ease. Clearly, then, the statement of Aristotle that 
slowness of natural motion is due to the density of the medium 
is inadequate. 

Therefore, dismissing his opinion, so that we may adduce the 
true cause of slowness and speed of motion, we must point out 
that speed cannot be separated from motion. For whoever asserts 
motion necessarily asserts speed; and slowness is nothing but 
lesser speed. Speed therefore proceeds from the same [cause] 
from which motion proceeds. And since motion proceeds from 
heaviness and lightness, speed or slowness must necessarily 
proceed from the same source. That is, from the greater heaviness 
of the moving body there results a greater speed of the motion, 
namely, downward motion, which comes about from the heavi- 
ness of that body; and from a lesser heaviness [of the body], a 
slowness of that same motion. On the other hand, from a greater 
lightness of the moving body will result a greater speed in that 
motion which comes about from the lightness of the body, 
namely, upward motion. 

Thus it is clear that a difference in the speed and slowness of 
motion occurs in the case of different bodies moving in the same 
medium. For if the motion is downward, the heavier substance 
will move more swiftly than the lighter; and if the motion is 
upward, that which is lighter will move more swiftly. But whether 
two bodies moving in the same medium maintain the same ratio 


between the speed of their motions as there is between their 
weights, as Aristotle believed, will be considered below. 

And in the case of the speed and slowness of the same body 
moving in different media, the situation is similar. The body 
moves downward more swiftly in that medium in which it is 
heavier, than in another in which it is less heavy; and it moves 
upward more swiftly in that medium in which it is lighter, than in 
another in which it is less light. Hence it is clear that if we find 
in what media a given body is heavier, we shall have found media 
in which it will fall more swiftly. And if, furthermore, we can 
show how much heavier that same body is in this medium than 
in that, we shall have shown how much more swiftly it will move 
downward in this medium than in that. Conversely, in consider- 
ing lightness, when we find a medium in which a given body will 
be lighter, we shall have found a medium in which it will rise 
more swiftly; and if we find how much lighter the given body is in 
this medium than in that, we shall also have found how much 
more swiftly the body will rise in this medium than in that. 

But in order that all this may be more precisely grasped in any 
particular case of motion, we shall first speak of the motions of 
different bodies in the same medium, and show what ratio there 
is between these motions, with respect to slowness and speed. 
We shall then consider motions of the same body moving in 
different media, and show, likewise, what ratio there is between 
the motions. 

In which it is shown that different bodies moving in the 


more easily with the matters under investigation, we must under- 
stand, in the first place, that a difference between two such bodies 
can arise in two ways. They may be of the same material, e.g., 
both lead, or both iron, and differ in size [i.e., volume]; or else 
they may be of different materials, e.g., one iron, the other wood, 


and differ either in size and weight, or in weight but not size, or 
in size but not weight. 

Of those [naturally] moving bodies which are of the same 
material, Aristotle said that the larger moves more swiftly. This 
is found in De Caelo 4.26, where he wrote that any body of fire 
moves upward, and that body which is larger moves faster; also 
that any body of earth moves downward, and, similarly, that that 
body which is larger moves faster. Aristotle also wrote (De Caelo . 
3.26) : "Suppose a heavy body b moves on line ce, which is divided 
at point d. If, then, body b is divided in the same ratio as line 
ce is divided by d, clearly a part of b will move over line cd in the 
same time as the whole of b moves over the whole line ce." From 

Fig. 5. 

this it is obvious that Aristotle holds that, in the case of bodies 
of the same material, the ratio of the speeds of their [natural] 
motion is equal to the ratio of the sizes of the bodies. And he 
puts this most clearly when he says (De Caelo 4.16) that a large 
piece of gold moves more swiftly than a small piece. 

But how ridiculous this view is, is clearer than daylight. For 
who will ever believe that if, for example, two lead balls, one a 
hundred times as large as the other, are let fall from the sphere of 
the moon, and if the larger comes down to the earth in one hour, 
the smaller will require one hundred hours for its motion? Or 


that, if two stones, one twice the size of the other, are thrown 
from the top of a high tower at the same moment, the larger 
reaches the ground when the smaller is only halfway down from 
the top of the tower? Or, again, if a very large piece of wood and 
a small piece of the same wood, the large piece being a hundred 
times the size of the small one, begin to rise from the bottom of 
the sea at the same time, who would ever say that the large piece 
would rise to the surface of the water a hundred times more 

But, to employ reasoning at all times rather than examples (for 
what we seek are the causes of effects, and these causes are not 
given to us by experience), we shall set forth our own view, and 
its confirmation will mean the collapse of Aristotle's view. We 
say, then, that bodies of the same kind (and let "bodies of the 
same kind" be defined as those that are made of the same material, 
e.g., lead, wood, etc.), though they may differ in size, still move 
with the same speed, and a larger stone does not fall more swiftly 
than a smaller. Those who are surprised by this conclusion will 
also be surprised by the fact that a very large piece of wood can 
float on water, no less than a small piece. For the reasoning is 
the same. 

Thus, if we imagine that the water on which a large piece of 
wood and a small piece of the same wood are afloat, is gradually 
made successively lighter, so that finally the water becomes lighter 
than the wood, and both pieces slowly begin to sink, who could 
ever say that the large piece would sink first or more swiftly than 
the small piece ? For, though the large piece of wood is heavier 
than the small one, we must nevertheless consider the large piece 
in connection with the large amount of water that tends to be 
raised by it, and the small piece of wood in connection with the 
correspondingly small amount of water. And since the volume of 
water to be raised by the large piece of wood is equal to that of 
the wood itself, and similarly with the small piece, those two 
quantities of water, which are raised by the respective pieces of 
wood, have the same ratio to each other in their weights as do 
their volumes (for portions of the same substance are to each 


other in weight as they are in volume; which would have to be 
proved) — i.e., the same ratio as that of the volumes of the large 
and the small piece of wood. Therefore the ratio of the weight of 
the large piece of wood to the weight of the water that it tends 
to raise is equal to the ratio of the weight of the small piece of 
wood to the weight of the water that it tends to raise. And the 
resistance of the large amount of water will be overcome by the 
large piece of wood with the same ease as the resistance of the 
small amount of water will be overcome by the small piece of 

Again, if we imagine, for example, a large piece of wax floating 
on water, and we mix this wax either with sand or with some 
other heavier substance, so that it ultimately becomes heavier than 
the water and just barely begins to sink very slowly, who could ever 
believe, if we take a small piece of that wax, say one-hundredth 
part, that it would either not sink at all or would sink a hundred 
times more slowly than the whole piece of wax? Surely no one. 
And one may make the same experiment with the balance. For 
if the weights on both sides are equal and very large, and then 
some weight, but a small one, is added to one side, the heavier 
side will fall, but not any faster than if the weights were small. 
Similarly in the case of the water: the large piece of wood repre- 
sents one weight in the balance, and the other weight is represented 
by a volume of water equal to the volume of the wood. Now if 
this volume of water is of equal weight with the large piece of 
wood, the wood will not sink. But if the wood is made a little 
heavier so that it sinks, it will not sink any faster than will a small 
piece of the same wood, which at first weighed the same as an 
[equally] small volume of water, and then was made a little bit 

But we may reach this same conclusion by another argument. 
Let us first make this assumption: if there are two bodies of which 
one moves [in natural motion] more swiftly than the other, a 
combination of the two bodies will move more slowly than that 
part which by itself moved more swiftly, but the combination will 
move more swiftly than that part which by itself moved more 


slowly. Thus, if we consider two bodies, e.g., a piece of wax and 
an inflated bladder, both moving upward from deep water, but 
the wax more slowly than the bladder, our assumption is that if 
both are combined, the combination will rise more slowly than 
the bladder alone, and more swiftly than the wax alone. Indeed 
it is quite obvious. For who can doubt that the slowness of the 
wax will be diminished by the speed of the bladder, and, on the 
other hand, that the speed of the bladder will be retarded by the 
slowness of the wax, and that some motion will result intermediate 
between the slowness of the wax and the speed of the bladder ? 

Similarly, if two bodies move downward [in natural motion], 
one more slowly than the other, for example, if one is wood and 
the other an [inflated] bladder, both falling in air, the wood more 
swiftly than the bladder, our assumption is as follows : if they are 
combined, the combination will fall more slowly than the wood 
alone, but more swiftly than the bladder alone. For it is clear that 
the speed of the wood will be retarded by the slowness of the 
bladder, and the slowness of the bladder will be accelerated by 
the speed of the wood; and, as before, some motion will result 
intermediate between the slowness of the bladder and the speed 
of the wood. 

On the basis of this assumption, I argue as follows in proving 
that bodies of the same material but of unequal volume move [in 
natural motion] with the same speed. Suppose there are two 
bodies of the same material, the larger a, and the smaller b, and 
suppose, if it is possible, as asserted by our opponent, that a moves 


Fig. 6. 

[in natural motion] more swiftly than b. We have, then, two 
bodies of which one moves more swiftly. Therefore, according to 
our assumption, the combination of the two bodies will move 
more slowly than that part which by itself moved more swiftly 
than the other. If, then, a and b are combined, the combination 


will move more slowly than a alone. But the combination of a 
and b is larger than a is alone. Therefore, contrary to the assertion 
of our opponents, the larger body will move more slowly than 
the smaller. But this would be self-contradictory. 

What clearer proof do we need of the error of Aristotle's 
opinion ? And who, I ask, will not recognize the truth at once, if 
he looks at the matter simply and naturally ? For if we suppose 
that bodies a and b are equal and are very close to each other, all 
will agree that they will move with equal speed. And if we imagine 
that they are joined together while moving, why, I ask, will they 
double the speed of their motion, as Aristotle held, or increase 
their speed at all ? Let us then consider it sufficiently corroborated 
that there is no reason per se why bodies of the same material 
should move [in natural motion] with unequal velocities, but every 
reason why they should move with equal velocity. Of course, if 
there were some accidental reason, e.g., the shape of the bodies, 
this will not be considered among causes per se. Moreover, as we 
shall show in the proper place, the shape of the body helps or 
hinders its motion only to a small extent. 

Still we must not immediately go to extremes, as many do, and 
compare, say, a large piece of lead with a very thin plate or even 
leaf of the same substance, which would sometimes even float on 
water. For since there is a certain cohesiveness of the parts both 
of air and of water, and, so to speak, a tenacity and viscosity, this 
cannot be overcome by a very small weight. Our conclusion must 
therefore be understood to apply to [two] bodies when the weight 
and volume of the smaller of them are large enough not to be 
impeded by the small viscosity of the medium, e.g., a leaden 
sphere of one pound. Moreover, as for those scoffers who, 
perhaps, believe that they can defend Aristotle, what happens to 
them if they have recourse to extremes is that they get into deeper 
difficulties, the greater the difference between the bodies which 
they take for comparison. For if one of the bodies is a thousand 
times as large as the other, surely these people must do some 
toiling and sweating before they can show that the velocity of one 
is a thousand times that of the other. 


But, to come to the next point, we must now consider the ratio 
[of the speed] of motion of bodies of different material moving 
[with natural motion] in the same medium. Though such bodies 
may differ from each other in three ways — either in size but not 
in weight, or in weight but not in size, or in both weight and 
size — we must examine only the case of those that differ in 
weight but not in size. For the ratios of those that differ in the 
two other ways can be reduced to this one. Thus in the case of 
bodies differing in size but not in weight, if from the larger we 
take a part equal to the smaller, the bodies will then differ in 
weight, but not in size. And the whole of the larger body will, 
with the smaller body, maintain the same ratio [in the speed of 
their motions] as will the part taken from the larger body. For 
it has been proved that bodies of the same material, though they 
differ in size, move with the same velocity. 

Similarly, in the case of bodies differing both in size and weight, 
if we take from the larger a part equal [in size] to the smaller, we 
shall again have two bodies differing in weight, but not in size. 
And the part [of the larger body] will, with the smaller, keep the 
same ratio [in the speed] of their motions, as will the whole of the 
larger body. For, once again, in the case of bodies of the same 
material, the part and the whole move with the same speed. It is 
therefore clear that, if we know the ratio of the speeds of those 
bodies that differ only in weight, but not in size, we also know the 
ratios of those that differ in every other way. And so, in order to 
find this ratio and to show, in opposition to Aristotle's view, that 
also for bodies of different material this ratio is not equal to the 
ratio of their weights, we shall prove certain propositions. On 
these depends the outcome not only of this investigation but also 
of the investigation of the ratio of the speeds of the same body 
moving in different media. And we shall consider both matters 

Let us therefore proceed to investigate the ratio [of the speeds] 
of the same body moving in different media. And first let us 
examine whether or not Aristotle's view on this is sounder than 
the other view explained above. Now Aristotle held that in the 


case of the same body moving in different media the ratio of the 
speeds was equal to the ratio of the rarenesses of the media. 
Indeed, that is what he wrote quite clearly, saying {Physics 4.71): 
"That medium which is denser interferes with motion more. Thus 
a will move over path b in time c, and over path d, a rarer medium, 
in time e, [the times being] in the ratio of the hindering action of 

Fig. 7. 

the media, provided the lengths of the paths are equal. Thus, if b 
is water and d air, a will move through d more swiftly than 
through b in proportion as air is less dense than water. Thus 
speed has to speed the same ratio as that between air and water. 
That is, if air is twice as rare as water, a will take twice as long to 
traverse path b as to traverse path d; and time c will be twice 
time e." 

These are Aristotle's words, but surely they embrace a false 
viewpoint. And to make this perfectly clear I shall construct the 
following proof. If the ratio of the speeds is equal to the ratio of 
the rarenesses of the media, let there be a moving body o and a 
medium a, whose rareness is 4; let this medium be water, for 
example. Let the rareness of medium b be 16, greater, that is, than 
the rareness of a; and let us say, for example, that b is air. Now 
suppose that body o is such that it does not sink in water, but 
suppose that its velocity in medium Z> is 8. Hence, since the speed 
of o in medium b is 8, but in medium a is zero, some medium can 
surely be found in which the speed of o is 1 . Let such a medium 
be c. Now since o moves more swiftly in medium b than in c, the 
rareness of c must be less than the rareness of b, and it must, 
according to our adversary, be less in proportion as the speed in 


medium c is less than the speed in medium b. But the speed in 
medium b was assumed to be eight times the speed in medium c. 
Therefore the rareness of medium b will be eight times the rare- 
ness of medium c. Hence the rareness of c will be 2. Therefore, 
o moves with speed 1 in the rareness of medium c, which is 2. 
But it was assumed that it cannot move in the rareness of medium 

2 4 16 
Fig. 8. 

a, which is 4. Hence o will fail to move in the medium of greater 
rareness, though it moves in the medium of lesser rareness. This 
is completely absurd. Clearly, then, the speeds of the motions 
are not in the same ratio as the rarenesses of the media. 

But even apart from other proof, can anyone fail to see the 
error in Aristotle's opinion? For if the speeds have the same ratio 
as the [rarenesses of the] media, then, conversely, the [rarenesses 
of the] media will have the same ratio as the speeds. Hence, since 
wood falls in air but not in water, and, consequently, the speed 
in air has no ratio to the speed in water, it follows that the 
rareness of air will have no ratio to the rareness of water. What 
can be more absurd than this ? But someone might think that he 
would be giving a sufficient answer to my argument, if he said: 
"Though wood does not move downward in water, it does move 
upward; and the rareness of water has to the rareness of air the 
same ratio as the speed of the motion upward in water has to the 
speed of the motion downward in air." And he might believe that 
he had skillfully saved Aristotle by such an answer. But we shall 
destroy this subterfuge, too, by considering a body which moves 


neither up nor down in water. Such a body, for example, would 
be water itself. But water moves in air with considerable speed. 
And so, having properly rejected Aristotle's view, let us now 
investigate the ratio [of the speeds] of the motion of the same 
body in different media. And first, in connection with upward 
motion, let us show that, when solids lighter than water are com- 
pletely immersed in water, they are carried upward with a force 
measured by the difference between the weight of a volume of 
water equal to the volume of the submerged body and the weight 
of the body itself. Thus, let the first position of the water, before 
the body is immersed in it, have as its surface ab; and let the 
solid cd be forcibly immersed in it, the surface of the water being 
raised to ef. Since the raised water eb has a volume equal to that 
of the whole submerged body, and the body is assumed to be 
lighter than water, the weight of the water eb will be greater than 
the weight of cd. Then let tb represent that part of the water 




Fig. 9. 

whose weight is equal to the weight of body cd. We must there- 
fore prove that body cd is carried upward with a force equal to 
the weight of tf (for this is the amount by which water eb is 
heavier than water tb, that is to say, than body cd). Now since 
the weight of water tb is equal to the weight of cd, water tb will 
press upward so as to raise cd with the same force with which that 
body will resist being raised. Thus the weight of a part of the 
water that exerts pressure, namely tb, is equal to the resistance of 
the solid body. But the weight of all the water that exerts pressure, 
namely eb, exceeds the weight of water tb by the weight of water //. 


Thus the weight of all the water eb will exceed the resistance of 
solid cd by the weight of water tf. Therefore the weight of all the 
water that exerts pressure will impel the solid upward with a force 
equal to the weight of part tf of the water. Which was to be 

From this proof, first, it is clear that upward motion results 
from the weight not merely of the body, but of the medium, as 
we have shown; and, secondly, the whole purpose of our investi- 
gation can be achieved. For since we are investigating how much 
faster the same body rises in one medium than in another, when- 
ever we know how fast it moves through each medium, we shall 
also know the interval between the two speeds. And this is what 
we seek. If, for example, a piece of wood whose weight is 4 moves 
upward in water, and the weight of a volume of water equal to 
that of the wood is 6, the wood will move with a speed that we 
may represent as 2. But if, now, the same piece of wood is carried 
upward in a medium heavier than water, a medium such that a 
volume of it equal to the volume of the wood has a weight of 10, 
the wood will rise in this medium with a speed that we may 
represent as 6. But it moved in the other medium with a speed 2. 
Therefore the two speeds will be to each other as 6 and 2, and 
not (as Aristotle held) as the weights or densities of the media, 
which are to each other as 10 and 6. It is clear, then, that in all 
cases the speeds of upward motion are to each other as the excess 
of weight of one medium over the weight of the moving body is 
to the excess of weight of the other medium over the weight 
of the body. 

Therefore, if we wish to know at once the [relative] speeds of 
a given body in two different media, we take an amount of each 
medium equal to the volume of the body, and subtract from the 
weights [of such amounts] of each medium the weight of the body. 
The numbers found as remainders will be to each other as the 
speeds of the motions. 

And we also obtain the answer to our second problem, namely, 
the ratio of the speeds of different bodies equal in volume but 
unequal in weight. For if each of them moves upward with a force 


measured by the difference between the weight of a volume of the 
medium equal to the volume of the body and the weight of the 
body itself, the numerical remainders, when the weights of the 
various bodies are subtracted from the weight of the aforesaid 
volume of the medium, will have the same ratio as the speeds. 
For example, if the weight of one body is 4, of a second body 6, 
and of the medium 8, the speed of the body whose weight is 4 will 
be 4, and the speed of the other body will be 2. These speeds, 4 
and 2, do not have the same ratio, as the lightnesses of the bodies, 
6 and 4. For the excesses of one number over two others will 
never have the same ratio to each other as the two numbers 
themselves; nor will the excesses of two numbers over another 
number have the same ratio to each other as the two numbers 
themselves. It is therefore clear that in motion upward the speeds 
of the different bodies are not in the same ratio as the lightnesses 
of the bodies. 

It remains for us to show that also in the [natural] downward 
motion of bodies the ratio of the speeds is not equal to the ratio 
of the weights of the bodies ; and at the same time to determine 
the ratio of the speeds of the same body moving in different 
media. All these results will easily be drawn from the following 
demonstration. I say, then, that a solid body heavier than water 
moves downward [in water] with a force measured by the dif- 
ference in weight between an amount of water equal to the 
volume of the solid body and the body itself. Thus, let the water 
in its first position have the surface de; and let solid bl, heavier 
than water, be let down into it, the surface of the water rising to 
ab, so that water ae has a volume equal to the volume of the solid 
itself. Since the solid is assumed to be heavier than water, the 
weight of the water [ae] will be less than the weight of the solid. 
Thus, let ao be an amount of water that has a weight equal to the 
weight of bl. Now, since water ae is lighter than ao by the weight 
of do, we must prove that body bl moves downward with a force 
measured by the weight of water do. 

Imagine a second solid body, lighter than water and joined to 
the first body ; let its volume be equal to that of water ao and its 


weight equal to the weight of water ae. Let Im represent this 
body. Since the volume of bl is equal to that of ae, and the 
volume of Im is equal to that of ao, the volume of the combined 
bodies, bl and Im, is equal to the sum of the volumes of ea and ao. 


d e 

i I 



Fig. 10. 

But the weight of water ae is equal to the weight of body Im; and 
the weight of water ao is equal to the weight of body bl. There- 
fore the whole weight of both bodies, bl and Im, is equal to the 
weight of water oa and ae. But the volume of the [combined] 
bodies [bl and Im] has been shown to be equal to the [combined] 
volume of water oa and ae. Hence by our first proposition, the 
bodies so joined will neither rise nor sink. Therefore the force 
of the downward pressure of body bl will be equal to the force of 
the upward pressure of Im. But, by the foregoing demonstration, 
magnitude Im tends to move upward with a force equal to the 
weight of water do. Therefore body bl will move downward with 
a force equal to the weight of water do. Which was to be proved. 
Now if this demonstration is grasped, the answer to our 
problems can easily be discerned. For, clearly, in the case of the 
same body falling in different media, the ratio of the speeds of the 


motions is the same as the ratio of the amounts by which the 
weight of the body exceeds the weights [of an equal volume] of 
the respective media. Thus, if the weight of the body is 8, and 
the weight of a volume of one medium equal to the volume of 
the body is 6, the speed of the body can be represented by 2. And 
if the weight of a volume of the second medium equal to the 
volume of the body is 4, the speed of the body in this second 
medium can be represented by 4. Clearly, then, these speeds will 
be to each other as 2 and 4, and not as the densities or weights of 
the media, as Aristotle believed, i.e., as 6 and 4. 

And similarly we have a clear answer to our second problem — 
to find the ratio of the speeds of bodies equal in size, but unequal 
in weight, moving [with natural motion] in the same medium. 
For the speeds of these bodies have the same ratio as do the 
amounts by which the weights of the bodies exceed the weight 
of the medium. For example, if there are two bodies equal in 
volume but unequal in weight, the weight of one of them being 8, 
and of the other 6, and if the weight of a volume of the medium 
equal to the volume of either body is 4, the speed of the first body 
will be 4 and of the second 2. These speeds will have a ratio of 
4 to 2, not the same as the ratio between their weights, which 
is 8 to 6. 

And from all that has been said here, it will not be difficult also 
to find the ratio in the case of bodies of different material moving 
in different media. For we examine first the ratio of the speeds 
of both bodies in the same medium. How this is to be done is 
clear from what has already been said. Then we consider what 
speed [the appropriate] one of the bodies has in the second 
medium (again with the help of what has been stated above): 
and we shall then have what is sought. For example, suppose 
there are two bodies, equal in size but unequal in weight, the 
weight of one being 12, and of the other 8, and we seek the 
ratio between the speed of the one whose weight is 12 sinking in 
water, and the speed of the one whose weight is 8 falling in air. 
Consider first how much faster the body weighing 12 sinks in 
water than the body weighing 8; then how much faster the body 


weighing 8 moves in air than in water, and we shall have what 
we are looking for. Or, alternatively, consider how much more 
swiftly the body weighing 12 falls in air than the body weighing 8 ; 
and then how much more slowly the body weighing 12 moves in 
water than in air. 

These, then, are the general rules governing the ratio of the 
speeds of [natural] motion of bodies made of the same or of 
different material, in the same medium or in different media, and 
moving upward or downward. But note that a great difficulty 
arises at this point, because those ratios will not be observable by 
one who makes the experiment. For if one takes two different 
bodies, which have such properties that the first should fall twice 
as fast as the second, and if one then lets them fall from a tower, 
the first will not reach the ground appreciably faster or twice as 
fast. Indeed, if an observation is made, the lighter body will, at 
the beginning of the motion, move ahead of the heavier and will 
be swifter. This is not the place to consider how these contra- 
dictory and, so to speak, unnatural accidents come about (for they 
are accidental). In fact, certain things must be considered first 
which have not yet been examined. For we must first consider 
why natural motion is slower at its beginning. 


In which all that was demonstrated above is considered 
in physical terms, and bodies moving naturally are reduced 
to the weights of a balance. When a person has discovered 
the truth about something and has established it with great effort, 
then, on viewing his discoveries more carefully, he often realizes 
that what he has taken such pains to find might have been 
perceived with the greatest ease. For truth has the property that 
it is not so deeply concealed as many have thought; indeed, its 
traces shine brightly in various places and there are many paths 
by which it is approached. Yet it often happens that we do not 
see what is quite near at hand and clear. And we have a clear 
example of this right before us. For everything that was demon- 


strated and explained above so laboriously is shown us by nature 
so openly and clearly that nothing could be plainer or more 

That this may be clear to everyone, let us consider how and 
why bodies moving upward [in natural motion] move with a 
force measured by the amount by which the weight of a volume 
of the medium (through which motion takes place) equal to the 
volume of the moving body exceeds the weight of the body itself. 
Consider a piece of wood that rises in water and floats on the 
surface. Now it is clear that the wood moves upward with just as 
much force as is necessary to submerge it forcibly in the water. 
If, therefore, we can find how much force is necessary to hold the 
wood under the water, we shall have what we are looking for. 
But if the wood were not lighter than water, that is, if its weight 
were the same as the weight of a volume of water equal to the 
volume of the wood, it would, of course, remain submerged, and 
it would not rise above the surface of the water. Therefore a force 
equal to the amount by which the weight of the aforesaid volume 
of water exceeds the weight of the piece of wood is sufficient to 
submerge the piece of wood. That is, we have found how much 
weight is required to submerge the piece of wood. But it was just 
determined that the wood moves upward with a force equal to 
that required to submerge it. And the weight just now found is 
what is required to submerge it. Therefore the wood moves 
upward with a force measured by the amount by which the weight 
of a volume of water equal to the volume of the wood exceeds 
the weight of the wood. And this is what was sought. 

We must deal with downward motion by like reasoning. Thus 
we ask with what force a lead sphere moves downward in water. 
Now it is clear, to begin with, that the lead sphere moves down- 
ward with as much force as would be required to draw it upward. 
But if the sphere were made of water instead of lead, no force 
would be necessary to draw it upward, or, more precisely, the 
very smallest of forces. Therefore a weight equal to the amount 
by which the lead sphere exceeds an aqueous sphere of the same 
size measures the resistance of the lead sphere to being drawn 


upward. But the lead sphere also moves downward with the same 
force with which its resists being drawn upward. Therefore the 
lead sphere moves downward with a force equal to the weight 
by which it exceeds the weight of an aqueous sphere [of the 
same size]. 

One can see this same thing in the weights of a scale. For if the 
weights are in balance, and an additional weight is added to one 
side, then that side moves down, not in consequence of its whole 
weight, but only by reason of the weight by which it exceeds the 
weight on the other side. That is the same as if we were to say 
that the weight on this side moves down with a force measured 
by the amount by which the weight on the other side is less than 
it. And, for the same reason, the weight on the other side will 
move up with a force measured by the amount by which the 
weight on the first side is greater than it. 

From what was said in this and in the previous chapter, we 
have the general conclusion that in the case of bodies of different 
material, provided that they are equal in size, the ratio of the 
speeds of their [natural downward] motions is the same as the 
ratio of their weights — and not their weights as such, but the 
weights found by weighing them in the medium in which the 
motion is to take place. Consider, for example, two bodies, a and 
b, equal in size but not in weight. Let the weight in air of a be 
8 and of b 6. The speeds [of the natural motion] of these bodies 

in water will not, as has been said before, have the ratio 8 to 6. 
For if we take a volume of water c equal to the volume of the 
bodies, and its weight is 4, the speed of body a will be represented 
by 4, and the speed of b by 2. And these speeds are in the ratio 


of 2 to 1 , not in the ratio of 4 to 3, the ratio of the weights of the 
bodies in air. Yet the weights of these same bodies in water will 
also be in the ratio of 2 to 1 ; for the weight of a in water would 
only be 4. 

This can be made clear as follows. If the weight of a in air were 
4, it would be zero in water. For a would then be of the same 
weight as water, since 4 was assumed to be the weight in air of a 
volume of water c equal to the volume of a. But the weight of c 
in water would be zero, for it would move neither upward nor 
downward. Therefore the weight of a in water would be zero, if 
it were 4 in air. But because it is 8 in air, it will be 4 in water ; and, 
by the same reasoning, the weight of b in water would be 2. 
Therefore their weights would be in the ratio of 2 to 1, as are also 
the speeds of their motions. And one must deal with lightness by 
a similar argument. 

Now we conclude that, given the weights of two bodies in air, 
their weights in water can be found immediately. For having 
subtracted from each the weight of a volume of water equal to 
the volume of the solid bodies, we shall have as remainders the 
weights of these bodies in water. And similarly with other media. 

Now from what has been said it should be clear to everyone 
that we do not have for any object its own proper weight. For if 
two objects are weighed, let us say, in water, who can say that 
the weights which we then obtain are the true weights of these 
objects, when, if these same objects are weighed in air, the 
weights will prove to be different from those [found in water] 
and will have a different ratio to each other? And if these objects 
could again be weighed in still another medium, e.g., fire, the 
weights would once more be different, and would have a different 
ratio to each other. And in this way the weights will always vary, 
along with the differences of the media. But if the objects could 
be weighed in a void, then we surely would find their exact 
weights, when no weight of the medium would diminish the weight 
of the objects. However, since the Peripatetics, following their 
leader, have said that in a void no motions could take place, and 
that therefore all things would be equally heavy, perhaps it will 


not be inappropriate to examine this opinion and to consider 
its foundations and its proofs. For this problem is one of the 
things that have to do with motion. 


In which, in opposition to Aristotle, it is proved that, if 
there were a void, motion in it would not take place 
instantaneously, but in time. Aristotle, in Book 4 of the 
Physics, in his attempt to deny the existence of a void adduces 
many arguments. Those that are found beginning with section 64 
are drawn from a consideration of motion. For since he assumes 
that motion cannot take place instantaneously, he tries to show 
that if a void existed, motion in it would take place instan- 
taneously; and, since that is impossible, he concludes necessarily 
that a void is also impossible. But, since we are dealing with 
motion, we have decided to inquire whether it is true that, if a 
void existed, motion in it would take place instantaneously. And 
since our conclusion will be that motion in a void takes place in 
time, we shall first examine the contrary view and Aristotle's 

In the first place, of the arguments adduced by Aristotle there 
is none that involves a necessary conclusion, but there is one 
which, at first sight, seems to lead to such a conclusion. This is 
the argument set forth in sections 71 and 72, in which Aristotle 
deduces the following inconsistency — that, on the assumption 
that motion can take place in time in a void, then the same body 
will move in the same time in a plenum and in a void. In order 
to be better able to refute this argument, we have decided to state 
it at this point. 

Thus, Aristotle's first assumption, when he saw that the same 
body moved more swiftly through the rarer than through the 
denser medium, was this: that the ratio of the speed of motion in 
one medium to the speed in the second medium is equal to the 
ratio of the rareness of the first medium to the rareness of the 
second. He then reasoned as follows. Suppose body a traverses 


medium b in time c, and that it traverses a medium rarer than b, 
namely d, in time e. Clearly, the ratio of time c to time e is equal 
to the ratio of the density of b to the density of d. Suppose, then, 
that there is a void /and that body a traverses/, if it is possible, 
not in an instant, but in time g. And suppose that the ratio of the 
density of medium d to the density of some new medium is equal 
to the ratio of time e to time g. Then, from what has been estab- 
lished, body a will move through the new medium in time g, 
since [the density of] medium d has to that of the new medium 

Fig. 12. 

the same ratio as time e to time g. But in the same time g body a 
also moves through the void/ Therefore a will in the same time 
move over two equal paths, one a plenum, the other a void. But 
this is impossible. Therefore the body will not move through the 
void in time; and therefore the motion will be instantaneous. 

Such is Aristotle's proof. And, indeed, his conclusions would 
have been sound and necessary, if he had proved his assumptions, 
or at least if these assumptions, even though unproved, had been 
true. But he was deceived in this, that he assumed as well- 
recognized axioms propositions which not only are not obvious 
to the senses, but have never been proved, and cannot be proved 
because they are completely false. For he assumed that the ratio 


of the speeds of the same body moving in different media is equal 
to the ratio of the rarenesses of the media. But that this is false 
has been fully proved above. In support of that proof, I shall add 
only this. Suppose it is true that the ratio of the rareness of air to 
the rareness of water is equal to the ratio of the speed of a body 
moving in air to the speed of the same body in water. Then, when 
a drop or some other quantity of water falls swiftly in air, but does 
not fall at all in water, since the speed in air has no ratio to the 
speed in water, it follows, according to Aristotle himself, that 
there will be no ratio between the rareness of air and the rareness 
of water. But that is ridiculous. 

Therefore, it is clear that, when Aristotle argues in this way, 
we must answer him as follows. In the first place, as has been 
shown above, it is not true that differences in the slowness and 
speed of a given body arise from the greater or lesser density and 
rareness of the medium. But even if that were conceded, it is still 
not true that the ratio of the speeds of the motion of the body is 
equal to the ratio of the rarenesses of the media. 

And as for Aristotle's statement in the same section that it is 
impossible for one number to have the same relation to another 
number as a number has to zero, this is, of course, true of geo- 
metric ratios [viz., alb], and not merely in numbers but in every 
kind of quantity. Since, in the case of geometric ratios, it is 
necessarily true that the smaller magnitude can be added to itself 
a sufficient number of times so that it will ultimately exceed any 
magnitude whatever, it follows that this smaller magnitude is 
something, and not zero. For zero, no matter how often it is 
added to itself, will exceed no quantity. But Aristotle's conclusion 
does not apply to arithmetic relations [viz., the difference, a—b]. 
That is, in these cases, one number can have the same relation to 
another number as still another number has to zero. For, since 
[two pairs of] numbers are in the same arithmetic relation when 
the difference of the [two] larger is equal to the difference of the 
[two] smaller, it will, of course, be possible for one number to 
have the same [arithmetic] relation to another number, as still 
another number has to zero. Thus, we say that the [arithmetic] 


relation of 20 to 12 is the same as that of 8 to 0: for the excess of 
20 over 12, i.e., 8, is equal to the excess of 8 over 0. 

Therefore, if, as Aristotle held, the ratio of the speeds were 
equal to the ratio, in the geometric sense, of the rarenesses of the 
media, Aristotle's conclusion would have been valid, that motion 
in a void could not take place in time. For the ratio of the time 
in the plenum to the time in the void cannot be equal to the ratio 
of the rareness of the plenum to the rareness of the void, since 
the rareness of the void does not exist. But if the ratio of the 
speeds were made to depend on the aforesaid ratio, not in the 
geometric, but in the arithmetic sense [i.e., as a ratio of differ- 
ences], no absurd conclusion would follow. And, in fact, the 
ratio of the speeds does depend, in an arithmetic sense, on the 
relation of the lightness of the first medium to that of the second. 
For the ratio of the speeds is equal, not to the ratio of the light- 
ness of the first medium to that of the second, but, as has been 
proved, to the ratio of the excess of the weight of the body over 
the weight of the first medium to the excess of the weight of the 
body over the weight of the second medium. 

So that this may be clearer, here is an example. Suppose there 
is a body a whose weight is 20, and two media unequal in weight, 






i c 

« , 


14 ' 

Fio. 13. 

be and de. Let the volume of b be equal to that of a, and the 
volume of d also equal to that of a. Since we are now discussing 


downward motion that takes place in a void, let the media be 
lighter than the body a, and let the weight of b be 12, and of d 6. 
It is clear, then, from what was proved above, that the ratio of the 
speed of body a in medium be to the speed of the same body in 
medium de will be equal to the ratio of the excess of the weight 
of a over the weight of b to the excess of the weight of a over the 
weight of d, that is, as 8 is to 14. Thus if the speed of a in medium 
be is 8, its speed in medium de would be 14. Now it is clear that 
the ratio of the speeds, 14 to 8, is not the same as the ratio (in the 
geometric sense) of the lightness of one medium to the lightness 
of the other. For the lightness of medium de is double that of 
medium be (for since the weight of b is 12, and of d 6, i.e., since 
the weight of b is double the weight of d, the lightness of d will be 
double the lightness of b); but a speed of 14 is less than twice a 
speed of 8. Yet the speed 14 has to the speed 8 the same relation, 
in the arithmetic sense, as the lightness of d to the lightness of b, 
since the difference between 14 and 8 is 6, and 6 is also the 
difference between the lightness of d (12) and the lightness of b (6). 
Furthermore, if medium de should be lighter, so that the 
weight of dis 5, the speed/ will be 15 (for 15 will be the difference 
between the weight of body a and the weight of the medium d). 
And again the relation [i.e., arithmetic difference] of speed 15 and 
speed 8 will be the same as between the weight of medium b (12) 
and the weight of medium d (5), that is, the same as the relation 
of the lightness of d and the lightness of b. For the difference in 
each case will be 7. Furthermore, if the weight of d is only 4, the 
speed / will be 16 : and the relation of speed 16 and speed 8 (with 
a difference of 8) is the same arithmetic relation as between the 
weight of b (12) and the weight of d (4), i.e., between the lightness 
of d and the lightness of b, the difference being also 8. If, again, 
medium de becomes lighter, and the weight of d is only 3, the 
speed / will now be 17. And between the speed / (17) and the 
speed 8 (a difference of 9), the difference is the same as between 
the weight of b (12) and the weight of d (3), i.e., as between the 
lightness of d and the lightness of b. If, again, medium de becomes 
lighter, and the weight of d is only 2, the speed/ will now be 18. 


And the arithmetic difference between that speed and the speed 8 
will be the same as the difference between the weight of b (12) and 
the weight of d (2), i.e., between the lightness of d and the light- 
ness of b. In each case the difference will be 10. If, again, medium 
de becomes lighter, and the weight of d is only 1, the speed /will 
now be 19. And there will be the same arithmetic difference 
between this speed and the speed 8 as between the weight of b (12) 
and the weight of d (1), i.e., between the lightness of d and the 
lightness of b. In each case the difference will be 11. Now if, 
finally, the weight of d is 0, so that the difference between the 
weight of body a and of the medium d is 20, the speed/ will be 20 ; 
and the arithmetic difference between the speed / (20) and the 
speed 8 will be the same as that between the weight of b (12) and 
the weight of d (0), the difference in each case being 12. 

It is clear, therefore, that the relation of speed to speed is the 
same as the relation of the lightness of one medium to the light- 
ness of the other, not geometrically [i.e., as a quotient] but 
arithmetically [i.e., as a difference]. And since it is not absurd for 
this arithmetic relation [i.e., difference] to be the same between 
one quantity and a second quantity as between a third quantity 
and zero, it will similarly not be absurd for the relation of speed 
to speed to be the same, in this arithmetic sense, as the relation of 
a given lightness [of medium] to zero. 

Therefore, the body will move in a void in the same way as in a 
plenum. For in a plenum the speed of motion of a body depends 
on the difference between its weight and the weight of the 
medium through which it moves. And likewise in a void [the 
speed of] its motion will depend on the difference between its 
own weight and that of the medium. But since the latter is zero, 
the difference between the weight of the body and the weight of 
the void will be the whole weight of the body. And therefore the 
speed of its motion [in the void] will depend on its own total 
weight. But in no plenum will it be able to move so quickly, since 
the excess of the weight of the body over the weight of the 
medium is less than the whole weight of the body. Therefore its 
speed will be less than if it moved according to its own total weight. 


From this it can clearly be understood that in a plenum, such as 
that which surrounds us, things do not weigh their proper and 
natural weight, but they will always be lighter to the extent that 
they are in a heavier medium. Indeed, a body will be lighter by 
an amount equal to the weight, in a void, of a volume of the 
medium equal to the volume of the body. Thus, a lead sphere 
will be lighter in water than in a void by an amount equal to the 
weight, in a void, of an aqueous sphere of the same size as the 
lead sphere. And the lead sphere is lighter in air than in a void by 
an amount equal to the weight, in a void, of a sphere of air 
having the same size as the lead sphere. And so also in fire, and 
in other media. And since the speed of a body's motion depends 
on the weight the body has in the medium in which it moves, 
its motion will be swifter, the heavier the body is in relation to 
the various media. 

But the following argument is invalid: "A void is a medium 
infinitely lighter than every plenum; therefore motion in it will 
be infinitely swifter than in a plenum; therefore such motion will 
be instantaneous." For it is true that a void is infinitely lighter 
than any [nonvacuous] medium; but we must not say that such 
a [nonvacuous] medium is of infinite weight. We must instead 
understand [the applicability of the term "infinite"] in this way, 
that between the lightness of air, for example, and a void there 
may exist an unlimited number of media lighter than air and 
heavier than a void. And if we understand the matter in this way, 
there may also exist, between the speed in air and the speed in a 
void, an unlimited number of speeds, greater than the speed in 
air and less than the speed in a void. And so also between the 
weight of a body in air and its weight in a void, an unlimited 
number of intermediate weights may exist, greater than the weight 
of the body in air, but less than its weight in a void. 

And the same is true of every continuum. Thus between lines 
a and b, of which a is greater, an unlimited number of inter- 
mediate lines, smaller than a, but greater than b may exist (for 
since the amount by which a exceeds b is also a line, it will be 
infinitely divisible). But we must not say that line a is infinitely 


greater than line b, in the sense that even if b were to be added to 
itself without limit, it would not produce a line greater than a. 
And by similar reasoning, if we suppose a to be the speed in a 
void, and b the speed in air, an unlimited number of speeds, 
greater than b and smaller than a, will be able to exist between 

Fig. 14. 

a and b. Yet we must not conclude that a is infinitely greater 
than b, in the sense that the time in which [the motion with] 
speed a is accomplished, when added to itself any number of times 
without limit, can still never exceed the time corresponding to 
speed b, and that, therefore, the speed corresponding to time a 
is instantaneous. 

It is therefore clear how the argument is to be understood. 
"The lightness of a void infinitely exceeds the lightness of a [non- 
vacuous] medium; therefore the speed in the void will infinitely 
exceed the speed in a plenum." All that is conceded. What is 
denied is the conclusion: "Therefore the speed [i.e., the motion] 
in the void will be instantaneous." For such motion can take 
place in time, but in a shorter time than the time corresponding 
to the speed in a plenum; so that between the time in the plenum 
and the time in the void an unlimited number of times, greater 
than the latter and smaller than the former, may exist. Hence it 
follows, not that motion in a void is instantaneous, but that it 
takes place in less time than the time of motion in any plenum. 

Therefore, to put it briefly, my whole point is this. Suppose 
there is a heavy body a, whose proper and natural weight is 1000. 
Its weight in any plenum whatever will be less than 1000, and 
therefore the speed of its motion in any plenum will be less than 


1000. Thus if we assume a medium such that the weight of a 
volume of it equal to the volume of a is only 1, then the weight of 
a in this medium will be 999. Therefore its speed too will be 999. 


Fig. 15. 

And the speed of a will be 1000 only in a medium in which its 
weight is 1000, and that will be nowhere except in a void. 

This is the refutation of Aristotle's argument. And from this 
refutation it can readily be seen that motion in a void does not 
have to be instantaneous. 



A liter of air weighs about 1 .3 g, which is small in comparison with that 
of an ordinary container so that any accurate determination of its density 
even today requires considerable care. What is more, the buoyant force in 
the atmosphere is the same amount. Therefore, one has to produce an 
increment of weight of air either by compressing more into a container or 
by partially evacuating it. Galileo did the former; he used sand to adjust 
the balance finely and then determined the initial volume of the compressed 
air. In this way he found water to be about 400 times (actually about 767) 
heavier than air, in comparison with Aristotle's estimate of 10. 

In this connection, Galileo discussed the general problem of determining 
the weight of a body in a vacuum by simply weighing it in air. 

Dialogues Concerning Two New Sciences f 

As to the other question, namely, how to determine the specific 
gravity of air, I have employed the following method. I took a 
rather large glass bottle with a narrow neck and attached to it 
a leather cover, binding it tightly about the neck of the bottle: in 
the top of this cover I inserted and firmly fastened the valve of a 
leather bottle, through which I forced into the glass bottle, by 
means of a syringe, a large quantity of air. And since air is easily 
condensed one can pump into the bottle two or three times its 
own volume of air. After this I took an accurate balance and 
weighed this bottle of compressed air with the utmost precision, 
adjusting the weight with fine sand. I next opened the valve and 
allowed the compressed air to escape; then replaced the flask 
upon the balance and found it perceptibly lighter : from the sand 
which had been used as a counterweight I now removed and laid 
aside as much as was necessary to again secure balance. Under 

t Ref. (9), pp. 79-82. 



these conditions there can be no doubt but that the weight of the 
sand thus laid aside represents the weight of the air which had 
been forced into the flask and had afterwards escaped. But after 
all this experiment tells me merely that the weight of the com- 
pressed air is the same as that of the sand removed from the 
balance; when however it comes to knowing certainly and 
definitely the weight of air as compared with that of water or any 
other heavy substance this I cannot hope to do without first 
measuring the volume [quantitd] of compressed air; for this 
measurement I have devised the two following methods. 

According to the first method one takes a bottle with a narrow 
neck similar to the previous one; over the mouth of this bottle is 
slipped a leather tube which is bound tightly about the neck of 
the flask; the other end of this tube embraces the valve attached 
to the first flask and is tightly bound about it. This second flask 
is provided with a hole in the bottom through which an iron rod 
can be placed so as to open, at will, the valve above mentioned 
and thus permit the surplus air of the first to escape after it has 
once been weighed: but this second bottle must be filled with 
water. Having prepared everything in the manner above 
described, open the valve with the rod; the air will rush into the 
flask containing the water and will drive it through the hole at the 
bottom, it being clear that the volume [quanttia] of water thus 
displaced is equal to the volume [mole e quanttia] of air escaped 
from the other vessel. Having set aside this displaced water, 
weigh the vessel from which the air has escaped (which is 
supposed to have been weighed previously while containing the 
compressed air), and remove the surplus of sand as described 
above; it is then manifest that the weight of this sand is precisely 
the weight of a volume [mole] of air equal to the volume of water 
displaced and set aside; this water we can weigh and find how 
many times its weight contains the weight of the removed sand, 
thus determining definitely how many times heavier water is than 
air; and we shall find, contrary to the opinion of Aristotle, that 
this is not 10 times, but, as our experiment shows, more nearly 
400 times. 


The second method is more expeditious and can be carried 
out with a single vessel fitted up as the first was. Here no air is 
added to that which the vessel naturally contains but water is 
forced into it without allowing any air to escape; the water thus 
introduced necessarily compresses the air. Having forced into 
the vessel as much water as possible, filling it, say, three-fourths 
full, which does not require any extraordinary effort, place it 
upon the balance and weight it accurately; next hold the vessel 
mouth up, open the valve, and allow the air to escape; the volume 
of the air thus escaping is precisely equal to the volume of water 
contained in the flask. Again weigh the vessel which will have 
diminished in weight on account of the escaped air; this loss in 
weight represents the weight of a volume of air equal to the volume 
of water contained in the vessel. 

Simp. No one can deny the cleverness and ingenuity of your 
devices; but while they appear to give complete intellectual 
satisfaction they confuse me in another direction. For since it is 
undoubtedly true that the elements when in their proper places 
have neither weight nor levity, I cannot understand how it is 
possible for that portion of air, which appeared to weigh, say, 
4 drachms of sand, should really have such a weight in air as the 
sand which counterbalances it. It seems to me, therefore, that 
the experiment should be carried out, not in air, but in a medium 
in which the air could exhibit its property of weight if such it 
really has. 

Salv. The objection of Simplicio is certainly to the point and 
must therefore either be unanswerable or demand an equally 
clear solution. It is perfectly evident that that air which, under 
compression, weighed as much as the sand, loses this weight 
when once allowed to escape into its own element, while, indeed, 
the sand retains its weight. Hence for this experiment it becomes 
necessary to select a place where air as well as sand can gravitate; 
because, as has been often remarked, the medium diminishes the 
weight of any substance immersed in it by an amount equal to 
the weight of the displaced medium; so that air in air loses all its 
weight. If therefore this experiment is to be made with accuracy 


it should be performed in a vacuum where every heavy body 
exhibits its momentum without the slightest diminution. If then, 
Simplicio, we were to weigh a portion of air in a vacuum would 
you then be satisfied and assured of the fact? 

Simp. Yes truly: but this is to wish or ask the impossible. 

Salv. Your obligation will then be very great if, for your sake, 
I accomplish the impossible. But I do not want to sell you 
something which I have already given you; for in the previous 
experiment we weighed the air in vacuum and not in air or other 
medium. The fact that any fluid medium diminishes the weight 
of a mass immersed in it, is due, Simplicio, to the resistance 
which this medium offers to its being opened up, driven aside, 
and finally lifted up. The evidence for this is seen in the readiness 
with which the fluid rushes to fill up any space formerly occupied 
by the mass; if the medium were not affected by such an 
immersion then it would not react against the immersed body. 
Tell me now, when you have a flask, in air, filled with its natural 
amount of air and then proceed to pump into the vessel more 
air, does this extra charge in any way separate or divide or change 
the circumambient air? Does the vessel perhaps expand so that 
the surrounding medium is displaced in order to give more room? 
Certainly not. Therefore one is able to say that this extra charge 
of air is not immersed in the surrounding medium for it occupies 
no space in it, but is, as it were, in a vacuum. Indeed, it is really 
in a vacuum; for it diffuses into the vacuities which are not 
completely filled by the original and uncondensed air. In fact 
I do not see any difference between the enclosed and the surround- 
ing media : for the surrounding medium does not press upon the 
enclosed medium and, vice versa, the enclosed medium exerts no 
pressure against the surrounding one; this same relationship 
exists in the case of any matter in a vacuum, as well as in the case 
of the extra charge of air compressed into the flask. The weight 
of this condensed air is therefore the same as that which it would 
have if set free in a vacuum. It is true of course that the weight 
of the sand used as a counterpoise would be a little greater in 
vacuo than in free air. We must, then, say that the air is slightly 


lighter than the sand required to counterbalance it, that is to say, 
by an amount equal to the weight in vacuo of a volume of air 
equal to the volume of the sand. 



According to Archimedes' principle the buoyant force Fon a body immersed 
in a fluid, i.e. its apparent loss of weight, is equal to the weight of the displaced 
fluid. Thus 

Buoyant force = (d F V F )g, 

where d F and V F are, respectively, the density and volume of the displaced 
fluid, and g is the acceleration due to gravity. Now the weight of a body is 

Weight = (DV)g, 
where D and V are, respectively, its own density and volume. Consider the 
unbalanced force, i.e. the difference between the weight of the body down- 
ward and the buoyant force on it upward, i.e. 

Weight— Buoyant force = (DV—d F V F )g. 

If the body is completely submerged, then V= V F . Hence 
Weight— Buoyant force = (D—dp)Vg- 
The weight, accordingly, will be greater or less than the buoyant force as D 
is greater or less than d F . In other words, a more dense body will sink in 
such a fluid. If D = d F , e.g. the same fluid, then the submerged body will 
remain at rest; as Galileo remarked, "Water hath no gravity in water". If 
D is less than D F , then the body will rise until the displaced volume V F is 
such that DV-d F V F = 0, the body will float. In this case, the resulting 
volumes are inversely proportional to the densities. Galileo expressed 
continually his amazement that a body can float in a volume of liquid even 
less than its own volume. 

What is the effect of a body's shape? An empty brass kettle, for example, 
will float in water — despite the fact that brass is denser than water. As 
Galileo pointed out, in this instance, the floating body is not just brass, but 
virtually a combination of brass and the enclosed air. 

A more puzzling effect is the floating of a chip of solid ebony, which is 
slightly denser than water, but which contains no air. Galileo did not 
correctly solve this problem. He did observe, however, that in such cases 
the liquid does not wet the body. If the same ebony is wet all over, it sinks. 
(His descriptive drawings were quite accurate.) It was not until the sixteenth 
century that surface tension was finally recognized as the missing physical 



Discourse on Bodies in Waterf 

Moreover, it seemed to me convenient to inform your Highness 
of all the sequel, concerning the controversy of which I treat, as 
it hath been advertised often already by others : and because the 
doctrine which I follow, in the discussion of the point in hand, is 
different from that of Aristotle; and interferes with his principles, 
I have considered that against the authority of that most famous 
man, which amongst many makes all suspected that comes not 
from the schools of the Peripatetics, it is far better to give one's 
reasons by the pen than by word of mouth, and therefore 
I resolved to write the present discourse : in which yet I hope to 
demonstrate that it was not out of capriciousness, or for that 
I had not read or understood Aristotle, that I sometimes swerve 
from his opinion, but because several reasons persuade me to it, 
and the same Aristotle hath taught me to fix my judgement on 
that which is grounded upon reason, and not on the bare authority 
of the master; and it is most certain according to the sentence of 
Alcinoos, that philosophating should be free. Nor is the reso- 
lution of our question in my judgement without some benefit to 
the universal, forasmuch as treating whether the figure of solids 
operates, or not, in their going, or not going to the bottom in 
water, in occurrences of building bridges or other fabrics on the 
water, which happen commonly in affairs of grand import, it may 
be of great avail to know the truth. 

I say therefore, that being the last summer in company with 
certain learned men, it was said in the argumentation; that 
condensation was the propriety of cold, and there was alledged 
for instance, the example of ice : now I at that time said, that, in 
my judgement, the ice should be rather water rarified than 
condensed, and my reason was, because condensation begets 
diminution of mass, and augmentation of gravity, and rarification 
causeth greater lightness, and augmentation of mass : and water 
in freezing, increaseth in mass, and the ice made thereby is lighter 
than the water on which it swimmeth. 

t Ref. (5), pp. 3-5, 22, 26-30. 


What I say, is manifest, because, the medium subtracting from 
the whole gravity of solids the weight of such another mass of the 
said medium; as Archimedes proves in his first book De Insidentibus 
Humido; whenever the mass of the said solid increaseth by distrac- 
tion, the more shall the medium detract from its entire gravity; and 
less, when by compression it shall be condensed and reduced to a 
less mass. 

It was answered me, that that proceeded not from the greater 
levity, but from the figure, large and flat, which not being able to 
penetrate the resistance of the water, is the cause that it sub- 
mergeth not. I replied, that any piece of ice, of whatsoever figure, 
swims upon the water, a manifest sign, that its being never so 
flat and broad, hath not any part in its floating: and added, that 
it was a manifest proof hereof to see a piece of ice of very broad 
figure being thrust to the bottom of the water, suddenly return to 
float on top, which had it been more grave, and had its swimming 
proceeded from its form, unable to penetrate the resistance of the 
medium, that would be altogether impossible; I concluded there- 
fore, that the figure was in sort a cause of the natation or 
submersion of bodies, but the greater or less gravity in respect of 
the water: and therefore all bodies heavier than it of what figure 
soever they be, indifferently go to the bottom, and the lighter, 
though of any figure, float indifferently on the top: and I suppose 
that those which hold otherwise, were induced to that belief, by 
seeing how that diversity of forms or figures, greatly altereth the 
velocity, and tardity of motion; so that bodies of figure broad and 
thin, descend far more leisurely into the water, than those of a 
more compacted figure, though both made of the same matter : 
by which some might be induced to believe that the dilation of the 
figure might reduce it to such ampleness that it should not only 
retard by wholly impede and take away the motion, which I hold 
to be false. Upon this conclusion, in many days' discourse, was 
spoken much, and many things, and divers experiments produced, 
of which your Highness heard, and saw some, and in this discourse 
shall have all that which hath been produced against my assertion, 


and what hath been suggested to my thoughts on this matter, and 
for confirmation of my conclusion: which it shall suffice to 
remove that (as I esteem hitherto false) opinion, I shall think 
I have not unprofitably spent my pains and time, and although 
that come not to pass, yet ought I to promise another benefit to 
myself, namely, of attaining the knowledge of the truth, by 
hearing my fallacies confuted, and true demonstrations produced 
by those of the contrary opinion. 

And to proceed with the greatest plainness and perspicuity 
that I can possible, it is, I conceive, necessary, first of all to 
declare what is the true, intrinsic, and total cause, of the ascending 
of some solid bodies in the water, and therein floating; or on the 
contrary, of their sinking and so much the rather in asmuch as 
I cannot satisfy myself in that which Aristotle hath left written on 
this subject. 

I say then the cause why some solid bodies descend to the 
bottom of water, is the excess of their gravity, above the gravity 
of the water; and on the contrary, the excess of the water's 
gravity above the gravity of those, is the cause that others do not 
descend, rather that they rise from the bottom, and ascend to the 
surface. This was subtlely demonstrated by Archimedes in his 
book of the Natation of Bodies: conferred afterwards by a very 
grave author, but, if I err not invisibly, as below for defence of 
him, I shall endeavour to prove. 

I, with a different method, and by other means, will endeavour 
to demonstrate the same, reducing the causes of such effects to 
more intrinsic and immediate principles, in which also are 
discovered the causes of some admirable and almost incredible 
accidents, as that would be, that a very little quantity of water, 
should be able, with its small weight, to raise and sustain a solid 
body, a hundred or a thousand times heavier than it. 

And because demonstrative order so requires, I shall define 
certain terms, and afterwards explain some propositions, of which, 
as of things true and obvious, I may make use of to my present 


Definition I 

I then call equally grave in specie, those matters of which equal 
masses weigh equally. 

As if for example, two balls, one of wax, and the other of some 
wood of equal mass, were also equal in weight, we say, that such 
wood, and the wax are in specie equally grave. 

Definition II 

But equally grave in absolute gravity, we call two solids, weighing 
equally, though of mass they be unequal. 

As for example, a mass of lead, and another of wood, that 
weigh each ten pounds, I call equal in absolute gravity, though 
the mass of the wood be much greater than that of the lead. 

And, consequently, less grave in specie. 

Definition III 

I call a matter more grave in specie than another, of which a 
mass, equal to a mass of the other, shall weigh more. 

... a piece of wood which by its nature sinks not in water, 
shall not sink though it be turned and converted into the form of 
any vessel whatsoever, and then filled with water: and he that 
would readily see the experiment in some other tractable matter, 
and that is easily reduced into several figures, may take pure 
wax, and making it first into a ball or other solid figure, let him 
add to it so much lead as shall just carry it to the bottom, so that 
being a grain less it could not be able to sink it, and making it 
afterwards into the form of a dish, and filling it with water, he 
shall find that without the said lead it shall not sink, and that 
with the lead it shall descend with much slowness: and in short 
he shall satisfy himself, that the water included makes no altera- 
tion. I say not all this while, but that it is possible of wood to 
make barks, which being filled with water, sink; but that proceeds 
not through its gravity, increased by the water, but rather from 
the nails and other iron works, so that it no longer hath a body 
less grave than water, but one mixed of iron and wood, more 
grave than a like mass of water. Therefore let Signor Buonamico 
desist from desiring a reason of an effect, that is not in nature : 
yea if the sinking of the wooden vessel when it is full of water, 


may call in question the doctrine of Archimedes, which he would 
not have you to follow, is on the contrary consonant and agree- 
able to the doctrine of the Peripatetics, since it aptly assigns a 
reason why such a vessel must, when it is full of water, descend 
to the bottom; converting the argument the other way, we may 
with safety say that the doctrine of Archimedes is true, since it 
aptly agreeth with true experiments, and question the other, 
whose deductions are fastened upon erroneous conclusions. As 
for the other point hinted in this same instance, where it seems 
that Buonamico understands the same not only of a piece of 
wood, shaped in the form of a vessel, but also of massy wood, 
which filled, scilicet, as I believe, he would say, soaked and 
steeped in water, goes finally to the bottom that happens in some 
porous woods, which, while their porosity is replenished with air, 
or other matter less grave than water, are masses specifically less 
grave than the said water, like as is that vial of glass whilst it is 
full of air: but when, such light matter departing, there 
succeedeth water into the same porosities and cavities, there 
results a compound of water and glass more grave than a like 
mass of water: but the excess of its gravity consists in the matter 
of the glass, and not in the water, which cannot be graver than 
itself: so that which remains of the wood, the air of its cavities 
departing, if it shall be more grave in specie than water, fill but 
its porosities with water, and you shall have a compost of water 
and wood more grave than water, but not by virtue of the water 
received into and imbibed by the porosities, but of that matter of 
the wood which remains when the air is departed: and being 
such it shall, according to the doctrine of Archimedes, go to the 
bottom, like as before, according to the same doctrine it did swim. 
As to that finally which presents itself in the fourth place, 
namely, that the ancients have been heretofore confuted by 
Aristotle, who denying positive and absolute levity, and truly 
esteeming all bodies to be grave, said, that that which moved 
upward was driven by the circumambient air, and therefore that 
also the doctrine of Archimedes, as an adherent to such an 
opinion was convicted and confuted: I answer first, that Signor 


Buonamico in my judgement hath imposed upon Archimedes, 
and deduced from his words more than ever he intended by them, 
or may from his propositions be collected, in regard that Archi- 
medes neither denies, nor admitteth positive levity, nor doth 
he so much as mention it: so that much less ought Buonamico 
to infer, that he hath denied that it might be the cause and 
principle of the ascension of fire, and other light bodies : having 
but only demonstrated, that solid bodies more grave than water 
descend in it, according to the excess of their gravity above the 
gravity of that, he demonstrates likewise, how the less grave 
ascend in the same water, according to its excess of gravity, above 
the gravity of them. So that the most that can be gathered from 
the demonstration of Archimedes is, that like as the excess of the 
gravity of the moveable above the gravity of the water, is the 
cause that it descends therein, so the excess of the gravity of the 
water above that of the moveable, is a sufficient cause why it 
descends not, but rather betakes itself to swim: not enquiring 
whether of moving upwards there is, or is not any other cause 
contrary to gravity: nor doth Archimedes discourse less properly 
than if one should say : If the South Wind shall assault the bark 
with greater impetus than is the violence with which the stream of 
the river carries it towards the South, the motion of it shall be 
towards the North : but if the impetus of the water shall overcome 
that of the wind, its motion shall be towards the South. The 
discourse is excellent and would be unworthily contradicted by 
such as should oppose it, saying: Thou mis-alledgest as cause of 
the motion of the bark towards the South, the impetus of the 
stream of the water above that of the South Wind; mis-alledgest 
I say, for it is the force of the North Wind opposite to the South, 
that is able to drive the bark towards the South. 

Theorem V 

The diversity of figures given to this or that solid, cannot any way 
be a cause of its absolute sinking or swimming. 
So that if a solid being formed, for example, into a spherical 
figure, doth sink or swim in the water, I say, that being formed 


into any other figure, the same figure in the same water, shall sink 
or swim: nor can such its motion by the expansion or by other 
mutation of figure, be impeded or taken away. 

The expansion of the figure may indeed retard its velocity, as 
well of ascent as descent, and more and more according as the 
said figure is reduced to a greater breadth and thinness: but that 
it may be reduced to such a form as that that same matter be 
wholly hindered from moving in the same water, that I hold to 
be impossible. In this I have met with great contradictors, who 
producing some experiments, and in particular a thin board of 
ebony, and a ball of the same wood, and showing how the ball in 
water descended to the bottom, and the board being put lightly 
upon the water submerged not, but rested; have held, and with 
the authority of Aristotle, confirmed themselves in their opinions, 
that the cause of that rest was the breadth of the figure, unable 
by its small weight to pierce and penetrate the resistance of the 
water's crassitude, which resistance is readily overcome by the 
other spherical figure. 

This is the principal point in the present question, in which 
I persuade myself to be on the right side. 

Therefore, beginning to investigate with the examination of 
exquisite experiments that really the figure doth not a jot alter 
the descent or ascent of the same solids, and having already 
demonstrated that the greater or less gravity of the solid in 
relation to the gravity of the medium is the cause of descent or 
ascent: whenever we would make proof of that, which about this 
effect the diversity of figure worketh, it is necessary to make the 
experiment with matter wherein variety of gravities hath no place. 
For making use of matters which may be different in their specific 
gravities, and meeting with varieties of effects of ascending and 
descending, we shall always be left unsatisfied whether that 
diversity derive itself really from the sole figure, or else from the 
divers gravity also. We may remedy this by taking one only 
matter, that is tractable and easily reduceable into every sort of 
figure. Moreover, it will be an excellent expedient to take a kind 
of matter, exactly alike in gravity unto the water: for that matter, 


as far as pertains to the gravity, is indifferent either to ascend or 
descend; so that we may presently observe any the least difference 
that derives itself from the diversity of figure. 

Now to do this, wax is most apt, which, besides its incapacity 
of receiving any sensible alteration from its imbibing of water, is 
ductile or pliant, and the same piece is easily reduceable into all 
figures : and being in specie a very inconsiderable matter inferior 
in gravity to the water, by mixing therewith a little of the filings 
of lead it is reduced to a gravity exactly equal to that of the water. 

This matter prepared, and, for example, a ball being made 
thereof as big as an orange or bigger, and that made so grave as 
to sink to the bottom, but so lightly, that taking thence one only 
grain of lead, it returns to the top, and being added, it submergeth 
to the bottom, let the same wax afterwards be made into a very 
broad and thin flake or cake; and then, returning to make the 
same experiment, you shall see that it being put to the bottom, it 
shall, with the grain of lead rest below, and that grain deducted, 
it shall ascend to the very surface, and added again it shall dive 
to the bottom. And this same effect shall happen always in all 
sort of figures, as well regular as irregular: nor shall you ever 
find any that will swim without the removal of the grain of lead, 
or sink to the bottom unless it be added: and, in short, about the 
going or not going to the bottom, you shall discover no diversity, 
although, indeed, you shall about the quick and slow descent: for 
the more expatiated and distended figures move more slowly as 
well in the diving to the bottom as in the rising to the top; and 
the other more contracted and compact figures, more speedily. 
Now I know not what may be expected from the diversity of 
figures, if the most contrary to one another operate not so much 
as doth a very small grain of lead, added or removed. 

Methinks I hear some of the adversaries to raise a doubt upon 
my produced experiment. And first, that they offer to my 
consideration, that the figure, as a figure simply, and disjunct 
from the matter works not any effect, but requires to be conjoined 
with the matter; and, furthermore, not with every matter, but 
with those only, wherewith it may be able to execute the desired 


operation. Like as we see it verified by experience, that the acute 
and sharp angle is more apt to cut, than the obtuse; yet always 
provided, that both the one and the other, be joined with a 
matter apt to cut, as for example, with steel. Therefore, a knife 
with a fine and sharp edge, cuts bread or wood with much ease, 
which it will not do, if the edge be blunt and thick: but he that 
will instead of steel, take wax, and mould it into a knife, 
undoubtedly shall never know the effects of sharp and blunt 
edges: because neither of them will cut, the wax being unable by 
reason of its flexibility, to overcome the hardness of the wood and 
bread. And, therefore, applying the like discourse to our purpose, 
they say, that the difference of figure will show different effects, 
touching natation and submersion, but not conjoined with any 
kind of matter, but only with those matters which, by their 
gravity, are apt to resist the velocity of the water, whence he that 
would elect for the matter, cork or other light wood, unable, 
through its levity, to superate the crassitude of the water, and of 
that matter should form solids of divers figures, would in vain 
seek to find out what operation figure hath in natation or sub- 
mersion; because all would swim, and that not through any 
property of this or that figure, but through the debility of the 
matter, wanting so much gravity, as is requisite to superate and 
overcome the density and crassitude of the water. 

It is needful, therefore, if we would see the effect wrought by 
the diversity of figure, first to make choice of a matter of its 
nature apt to penetrate the crassitude of the water. And, for this 
effect, they have made choice of such a matter, as fit, that being 
readily reduced into spherical figure, goes to the bottom; and it 
is ebony, of which they afterwards making a small board or 
splinter, as thin as a lathe, have illustrated how that this, put 
upon the surface of the water, rests there without descending to 
the bottom: and making, on the other side, of the same wood a 
ball, no less than a hazel nut, they show, that this swims not, but 
descends. From which experiment, they think they may frankly 
conclude, that the breadth of the figure in the flat lathe or board, 
is the cause of its not descending to the bottom, forasmuch as a 


ball of the same matter, not different from the board in anything 
but in figure, submergeth in the same water to the bottom. The 
discourse and the experiment hath really so much of probability 
and likelihood of truth in it, that it would be no wonder, if many 
persuaded by a certain cursory observation, should yield credit to 
it; nevertheless, I think I am able to discover, how that it is not 
free from fallacy. 

Beginning, therefore, to examine one by one, all the particulars 
that have been produced, I say, that figures, as simple figures, not 
only operate not in natural things, but neither are they ever 
separated from the corporeal substance : nor have I ever alledged 
them stripped of sensible matter, like as also I freely admit, that 
in our endeavouring to examine the diversity of accidents, 
dependent upon the variety of figures, it is necessary to apply 
them to matters, which obstruct not the various operations of 
those various figures: and I admit and grant, that I should do 
very ill, if I would experiment the influence of acuteness of edge 
with a knife of wax, applying it to cut an oak, because there is no 
acuteness in wax able to cut that very hard wood. But yet such 
an experiment of this knife, would not be besides the purpose, to 
cut curded milk, or other very yielding matter : yea, in such like 
matters, the wax is more commodious than steel; for finding the 
diversity depending upon angles, more or less acute, for that milk 
is indifferently cut with a razor, and with a knife, that hath a 
blunt edge. It needs, therefore, that regard be had, not only to 
the hardness, solidity or gravity of bodies, which under divers 
figures, are to divide and penetrate some matters, but it forceth 
also, that regard be had, on the other side, to the resistance of 
the matters, to be divided and penetrated. But since I have in 
making the experiment concerning our contest, chosen a matter 
which penetrates the resistance of the water; and in all figures 
descends to the bottom, the adversaries can charge me with no 
defect; yea, I have propounded so much a more excellent method 
than they, in as much as I have removed all other causes, of 
descending or not descending to the bottom, and retained the 
only sole and pure variety of figures, demonstrating that the same 


figures all descend with the only alteration of a grain in weight: 
which grain being removed, they return to float and swim; it is 
not true, therefore, (resuming the example by them introduced) 
that I have gone about to experiment the efficacy of acuteness, in 
cutting with matters unable to cut, but with matters proportioned 
to our occasion; since they are subjected to no other variety, than 
that alone which depends on the figure more or less acute. 

But let us proceed a little farther, and observe, how that indeed 
the consideration, which, they say, ought to be had about the 
election of the matter, to the end, that it may be proportionate 
for the making of our experiment, is needlessly introduced, 
declaring by the example of cutting, that like as acuteness is 
insufficient to cut, unless when it is in a matter hard and apt to 
superate the resistance of the wood or other matter, which we 
intend to cut; so the aptitude of descending or not descending in 
water, ought and can only be known in those matters, that are 
able to overcome the renitence, and superate the crassitude of the 
water. Unto which, I say, that to make distinction and election, 
more of this than of that matter, on which to impress the figures 
for cutting or penetrating this or that body, as the solidity or 
obdurateness of the said bodies shall be greater or less, is very 
necessary: but withall I subjoin, that such distinction, election 
and caution would be superfluous and unprofitable, if the body to 
be cut or penetrated, should have no resistance, or should not at 
all withstand the cutting or penetration: and if the knife were to 
be used in cutting a mist or smoke, one of paper would be equally 
serviceable with one of Damascus steel: and so by reason the 
water hath not any resistance against the penetration of any solid 
body, all choice of matter is superfluous and needless, and the 
election which I said above to have been well made of a matter 
reciprocal in gravity to water, was not because it was necessary, 
for the overcoming of the crassitude of the water, but its gravity, 
with which only it resists the sinking of solid bodies: and for 
what concerneth the resistance of the crassitude, if we narrowly 
consider it, we shall find that all solid bodies, as well those that 
sink, as those that swim, are indifferently accomodated and apt 


to bring us to the knowledge of the truth in question. Nor will 
I be frightened out of the belief of these conclusions, by the 
experiments which may be produced against me, of many several 
woods, corks, galls, and, moreover, of subtle slates and plates of 
all sorts of stone and metal, apt by means of their natural gravity, 
to move towards the center of the earth, the which, nevertheless, 
being impotent, either through the figure (as the adversaries think) 
or through levity, to break and penetrate the continuity of the 
parts of the water, and to distract its union, do continue to swim 
without submerging in the least: nor on the other side, shall the 
authority of Aristotle move me, who in more than one place, 
afnrmeth the contrary to this, which experience shows me. 



Hiero, King of Syracuse, asked his science adviser, Archimedes, how to 
ascertain if a crown made by the royal goldsmith contained only the royal 
gold or also some cheaper silver. No reliable account exists as to just how 
Archimedes actually solved this problem. In thinking about it, however, the 
22-year old Galileo devised a balance utilizing the concept of specific gravity, 
the law of the lever, and the principle of buoyancy. (Archimedes had invented 
the first, attempted a proof of the second, and discovered the last.) Galileo's 
paper was written in Italian for popular consumption. 

We shall consider this method from a modern point of view. Let the 
weight of a body A in air be A a and its apparent weight in water be A w . 
If the body is balanced in air with a given counterpoise C (cf. Galileo's own 
figure below), the law of the lever requires that 

A w . ca = C . cb , 

where ca and cb are the lever arms of the body in air and of the counterpoise, 
respectively. Likewise, for the same counterpoise, but body balanced in 

A w . eg = C . cb , 

where eg is the new lever arm of the body in this instance. Dividing these 
two expressions, we obtain 

A a _ eg 

■Aw ca 

The specific gravity SG of a body, by definition, is the ratio of its weight in 
air to that of an equal volume of water, which, by Archimedes' principle, is 
equal to the apparent loss of weight in water, i.e. 


Hence, the specific gravity SG^ for body A is 

SGa A -A = =~~- — 
A a A w ca—cg 



Likewise, for a gold body E of the same volume, with a balance point of the 
counterpoise at e, 

SG £ = =^= ; 
ca — ce 

and for a similar silver body F, with the counterpoise at /: 

— ca 

SG f = . 


Let G be the unknown fraction, by weight, of gold in an alloy mixture of 
gold and silver, then (1— G) will be the fraction of silver. 

The weight W (strictly speaking, its mass) of a material M with volume V 
and density D, by definition, is given by 

W M = V M ■ D M . 
Hence the specific gravity SGm is 

__ W M D M 


W M 1 Wm 


D m SGjvf Dw 

In this case, we obtain for the body A 

ca — cg W A 
Va = _ • —z — ; 
ca D W 

for the gold body E, 

ca—ce Gb Wa 

and for the silver body F, 

ca-Tf 0—F)W A 

Vf = 

« D * 

Assuming that the volumes of the constituents combined in the alloy are 
the same (not always true) as those in the free state, we have for the total 
volume of the body 

Va = V E +V F , 

or upon substitution, 

ca—cg = G(ca—ce)+(l—GXca—cf) . 

Solving for G, the fraction of gold, we find 

cf-ce fe 

Note: the problem has been solved without actually weighing the sample 

t J nhrM C V f l hlS me , th ° d WOuld probabl y not have b ^ ^ good as 
that obtainable by chemical analysis in the 16th century. As in Archimedes' 

c™d„g ouTapfece" thlS ^ * e b ° dy W ° Uld ^ ^ t0 ta defaCed h * 

The Little Balance} 

Just as it is well known to anyone who takes the care to read 
ancient authors that Archimedes discovered the jeweler's theft in 
Hiero's crown, it seems to me the method which this great man 
must have followed in this discovery has up to now remained 
unknown. Some authors have written that he proceeded by 
immersing the crown in water, having previously and separately 
immersed equal amounts [in weight] of very pure gold and of 
silver, and, from the differences in their making the water rise or 
spill over, he came to recognize the mixture of gold and silver of 
which the crown was made. But this seems, so to say, a crude 
thing, far from scientific precision; and it will seem even more so 
to those who have read and understood the very subtle inventions 
of this divine man in his own writings; from which one most 
clearly realizes how inferior all other minds are to Archimedes's 
and what small hope is left to anyone of ever discovering things 
similar to his [discoveries]. I may well believe that, a rumor 
having spread that Archimedes had discovered the said theft by 
means of water, some author of that time may have then left a 
written record of this fact; and that the same [author], in order 
to add something to the little that he had heard, may have said 
that Archimedes used the water in that way which was later 
universally believed. But my knowing that this way was 
altogether false and lacking that precision which is needed in 
mathematical questions made me think several times how, by 
t Ref. (2), pp. 134-40. 


means of water, one could exactly determine the mixture of two 
metals. And at last, after having carefully gone over all that 
Archimedes demonstrates in his books On Floating Bodies and 
Equilibrium, a method came to my mind which very accurately 
solves our problem. I think it probable that this method is the 
same that Archimedes followed, since, besides being very 
accurate, it is based on demonstrations found by Archimedes 

This method consists in using a balance whose construction 
and use we shall presently explain, after having expounded what 
is needed to understand it. One must first know that solid bodies 
that sink in water weigh in water so much less than in air as is 
the weight in air of a volume of water equal to that of the body. 
This [principle] was demonstrated by Archimedes, but because 
his demonstration is very laborious I shall leave it aside, so as not 
to take too much time, and I shall demonstrate it by other means. 
Let us suppose, for instance, that a gold ball is immersed in water. 
If the ball were made of water it would have no weight at all 
because water inside water neither rises nor sinks. It is then clear 
that in water our gold ball weighs the amount by which the 
weight of the gold [in air] is greater than in water. The same can 
be said of other metals. And because metals are of different 
[specific] gravity, their weight in water will decrease in different 
proportions. Let us assume, for instance, that gold weighs twenty 
times as much as water; it is evident from what we said that gold 
will weigh less in water than in air by a twentieth of its total 
weight [in air]. Let us now suppose that silver, which is less heavy 
than gold, weighs twelve times as much as water; if silver is 
weighed in water its weight will decrease by a twelfth. Thus the 
weight of gold in water decreases less than that of silver, since 
the first decreases by a twentieth, the second by a twelfth. 

Let us suspend a [piece of] metal on [one arm of] a scale of 
great precision, and on the other arm a counterpoise weighing as 
much as the piece of metal in air. If we now immerse the metal 
in water and leave the counterpoise in air, we must bring the said 
counterpoise closer to the point of suspension [of the balance 


beam] in order to balance the metal. Let, for instance, ab be the 
balance [beam] and c its point of suspension; let a piece of some 
metal be suspended at b and counterbalanced by the weight d. 
If we immerse the weight b in water the weight d at a in the air 
will weigh more [than b in water], and to make it weigh the same 
we should bring it closer to the point of suspension c, for instance 
to e. As many times as the distance ac will be greater than the 
distance ae, that many times will the metal weigh more than water. 
Let us then assume that weight b is gold and that when this is 
weighed in water, the counterpoise goes back to e; then we do the 
same with very pure silver and when we weigh it in water its 
counterpoise goes in/. This point will be closer to c [than is e], 
as the experiment shows us, because silver is lighter than gold. 
The difference between the distance af and the distance ae will 
be the same as the difference between the [specific] gravity of gold 
and that of silver. But if we shall have a mixture of gold and 
silver it is clear that because this mixture is in part silver it will 
weigh less than pure gold, and because it is in part gold it will 
weigh more than pure silver. If therefore we weigh it in air first, 
and if then we want the same counterpoise to balance it when 
immersed in water, we shall have to shift said counterpoise closer 
to the point of suspension c than the point e, which is the mark 
for gold, and farther than/, which is the mark for pure silver, and 
therefore it will fall between the marks e and/. From the propor- 
tion in which the distance ef will be divided we shall accurately 
obtain the proportion of the two metals composing the mixture. 
So, for instance, let us assume that the mixture of gold and silver 
is at b, balanced in air by d, and that this counterweight goes to g 
when the mixture is immersed in water. I now say that the gold 
and silver that compose the mixture are in the same proportion 
as the distances fg and ge. We must however note that the 
distance gf, ending in the mark for silver, will show the amount 
of gold, and the distance ge ending in the mark for gold will 
indicate the quantity of silver; so that, if/g will be twice ge, the 
said mixture will be of two [parts] of gold and one of silver. And 
thus, proceeding in this same order in the analysis of other 


mixtures, we shall accurately determine the quantities of the 
[component] simple metals. 

Fig. 16. The Little Balance. A drawing based on Galileo's description. 

To construct this balance, take a [wooden] bar at least two 
braccia long — the longer the bar, the more accurate the instru- 
ment. Suspend it in its middle point; then adjust the arms so 
that they are in equilibrium, by thinning out whichever happens 
to be heavier; and on one of the arms mark the points where the 
counterpoises of the pure metals go when these are weighed in 
water, being careful to weigh the purest metals that can be found. 
Having done this, we must still find a way by which easily to 
obtain the proportions in which the distances between the marks 
for the pure metals are divided by the marks for the mixtures. 
This, in my opinion, may be achieved in the following way. 


On the marks for the pure metals wind a single turn of very 
fine steel wire, and around the intervals between marks wind a 
brass wire, also very fine: these distances will be divided in many 
very small parts. Thus, for instance, on the marks e, f I wind 
only two turns of steel wire (and I do this to distinguish them 
from brass); and then I go on filling up the entire space between 
e and /by winding on it a very fine brass wire, which will divide 
the space e/into many small equal parts. When then I shall want 
to know the proportion between fg and ge I shall count the 
number of turns in fg and the number of turns in ge, and if 
I shall find, for instance, that the turns in fg are 40 and the turns 
in ge 21, 1 shall say that in the mixture there are 40 parts of gold 
and 21 of silver. 

Here we must warn that a difficulty in counting arises: Since 
the wires are very fine, as is needed for precision, it is not possible 
to count them visually, because the eye is dazzled by such small 
spaces. To count them easily, therefore, take a most sharp 
stiletto and pass it slowly over said wires. Thus, partly through 
our hearing, partly through our hand feeling an obstacle at each 
turn of wire, we shall easily count said turns. And from their 
number, as I said before, we shall obtain the precise quantity of 
pure metals of which the mixture is composed. Note, however, 
that these metals are in inverse proportion to the distances: Thus, 
for instance, in a mixture of gold and silver the coils toward the 
mark for silver will give the quantity of gold, and the coils 
toward the mark for gold will indicate the quantity of silver; and 
the same is valid for other mixtures. 



Man has experienced the advantages of many simple machines from earliest 
known times. He sought subsequently to ascertain theoretically the machine 
that might be regarded as fundamental, the operating principle of which 
would be sufficient for deriving the specific laws of all other machines. 

Galileo, following his acknowledged "master" Archimedes, chose the 
lever — which both considered (wrongly) could be understood physically 
merely in terms of mathematical symmetry. First principles, however, for 
explaining natural phenomena must be basically experiential. 

A screw can be regarded as a totality of inclined-plane elements, which 
thus become fundamental for its comprehension. Galileo derived the law 
of the inclined plane from that of the lever. Although his manner is less 
ingenious than that of Simon Stevin (1548-1620), it is more natural and 
more profound. (Nemorius Jordanus (c. 1220), who had been the first to 
derive the law, also did so on the basis of machine experience itself.) 

Consider a bent lever (cf. Galileo's Fig. 18) with a load (weight, say) L 
at the point F, having a lever arm BK and with an effort (applied force) E 
at A having a lever arm BA (=FB), the fulcrum being at B. Now the inclined 
plane GH, with length 1 (=FA) and vertical height h (=FK), is tangent to 
the circle with center B and radius FG. In the case of equilibrium law of the 
lever requires that 

Since right triangle KBF is similar to right triangle HBF, we have for corres- 
ponding sides 



substituting, we obtain 



which is the law of the inclined plane. 



Galileo, moreover, realized that the load L travels effectively only through 
the vertical height h, while the effort E is effective along the whole length / 
of the inclined plane. He concluded, "It is very important to consider along 
what lines the motions are made". 

On Mechanics^ 

Of the Screw. Among all the mechanical instruments devised 
by human wit for various conveniences, it seems to me that for 
ingenuity and utility the screw takes first place, as something 
cleverly adapted not only to move but also to fix and to press 
with great force; and it is constructed in such a manner as to 
occupy but a very small space and yet to accomplish effects that 
the other instruments could perform only if made into large 
machines. The screw thus being among the most beautiful and 
useful of contrivances, we may rightly take the trouble to explain 
as clearly as we may both its origin and its nature. To do this we 
shall start from a theory which, though at first it may appear to 
be somewhat remote from the consideration of this instrument, is 
nevertheless its basis and foundation. 

There can be no doubt that the constitution of nature with 
respect to the movements of heavy bodies is such that any body 
which retains heaviness within itself has a propensity, when free, 
to move toward the center; and not only by a straight perpendic- 
ular line, but also, when it cannot do otherwise, along any other 
line which, having some tilt toward the center, goes downward 
little by little. And thus we see, for instance, that water from 
some high place not only drops perpendicularly downward, but 
also runs about the surface of the earth on lines that are inclined, 
though but very little. This is seen in the course of rivers whose 
waters, though the bed is very little slanted, run freely dropping 
downward; which same effect, just as it is perceived in fluid 
bodies, appears also in hard solids, provided that their shapes and 
other external and accidental impediments do not prevent it. So 
that if we have a surface that is very smooth and polished, as 

t Ref. (3), pp. 169-77. 


would be that of a mirror, and a perfectly smooth and round ball 
of marble or glass or some such material capable of being 
polished, then if this ball is placed on that surface it will go 
moving along, provided that the surface has some little tilt, even 
the slightest; and it will remain still only on that surface which is 
most precisely leveled, and equidistant from the plane of the 
horizon. This, for example, might be the surface of a frozen lake 
or pond, upon which such a spherical body would stand still, 
though with a disposition to be moved by any extremely small 
force. For we have understood that if such a plane tilted only by 
a hair, the said ball would move spontaneously toward the lower 
part, and on the other hand it would have resistance toward the 
upper or rising part, nor could it be moved that way without 
some violence. Hence it is perfectly clear that on an exactly 
balanced surface the ball would remain indifferent and question- 
ing between motion and rest, so that any the least force would be 
sufficient to move it, just as on the other hand any little resistance, 
such as that merely of the air that surrounds it, would be capable 
of holding it still. 

From this we may take the following conclusion as an in- 
dubitable axiom : That heavy bodies, all external and adventitious 
inpediments being removed, can be moved in the plane of the 
horizon by any minimum force. But when the same heavy body 
must be driven upon an ascending plane, having a tendency to 
the contrary motion and commencing to oppose such an ascent, 
there will be required greater and greater violence the more 
elevation the said plane shall have. For example, the movable 
body G being placed on the line AB parallel to the horizon, it will 
stand there, as was said, indifferent to motion or to rest, so that it 
may be moved by the least force; but if we have the inclined 
planes AC, AD, and AE, upon these it will be driven only by 
violence, more of which is required to move it along the line AD 
than along AC, and still more along AE than AD. This comes 
from its having greater impetus to go downward along the line 
EA than along DA, and along DA than along CA. So that we 
may likewise conclude heavy bodies to have greater resistance to 


being moved upon variously inclined planes, according as one is 
more or less tilted than another; and finally the resistance to 
being raised will be greatest on the part of the heavy body in the 
perpendicular AF. But what proportion the force must have to 
the weight in order to draw it upon various inclined planes must 
be explained precisely before we proceed further, so that we may 
completely understand all that remains to be said. 

From the points C, D, and E, therefore, let fall the perpendicu- 
lars CH, DJ, and EK upon the horizontal line AB. It will be 
demonstrated that the same weight will be moved upon the 
inclined plane AC by less force than in the perpendicular AF 
(where it will be raised by a force equal to itself), in proportion 
as the perpendicular CH is less than AC; and upon the plane AD 
the force will have to the weight the same proportion as the 
perpendicular line JD has to DA; and finally in the plane AE the 
ratio of the force to the weight will be that of KE to EA. 

This present theory was attempted also by Pappus of Alex- 
andria in the eighth book of his Mathematical Collections, but in 
my opinion he missed the mark, being defeated by the assump- 
tion which he made when he supposed that the weight would 
have to be moved in the horizontal plane by a given force. This 
is false, no sensible force being required (neglecting accidental 
inpediments, which are not considered by the theoretician) to 
move the given weight horizontally, so that it is vain thus to seek 


the force with which it will be moved on the inclined plane. It 
will be better, given the force that would move the object perpen- 
dicularly upward (which would equal the weight of the object), to 
seek the force that will move it on the inclined plane. This we 
shall attempt to achieve, with an attack different from that 
of Pappus. 

Consider, then, the circle AJC and in this the diameter ABC 
with center B, and two weights of equal moments at the extremities 

Fig. 18. 

A and C, so that the line AC being a lever or balance movable 
about the center B, the weight C will be sustained by the weight 
A. But if we imagine the arm of the balance BC to be inclined 
downward along the line BF in such a way that the two lines AB 
and BF are fixed together at the point B, then the moment of the 
weight C will no longer be equal to the moment of the weight A, 
the distance of the point F from the line BJ, which goes from the 
support B to the center of the earth, being diminished. 

Now if we draw from the point F a perpendicular to BC, which 
is FK, the moment of the weight at F will be as if it were hung 
from the line KB; and as the distance KB is made smaller with 
respect to the distance BA, the moment of the weight F is accord- 
ingly diminished from the moment of the weight A. Likewise, as 
the weight inclines more, as along the line BL, its moment will go 


on diminishing, and it will be as if it were hung from the distance 
BM along the line ML, in which point L a weight placed at A will 
sustain one as much less than itself as the distance BA is greater 
than the distance BM. 

You see, then, how the weight placed at the end of the line BC, 
inclining downward along the circumference CFLJ, comes grad- 
ually to diminish its moment and its impetus 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. Whence we may undoubtedly affirm that 
the heavy body descending from the point C along the circum- 
ference CFLJ, its moment of descent at the first point C is total 
and integral, since it is in no way supported by the circumference; 
and at this first point C it has no disposition to move differently 
from what it would freely do in the perpendicular tangent DCE. 
But if the movable body is located at the point F, then its heavi- 
ness is partly sustained by the circular path placed under it, and 
its moment downward is diminished in that proportion by which 
the line BK is exceeded by the line BC. Now when the movable 
body is at F, 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. 

And in the same way we shall find that at the point L the 
moment of the same movable body is diminished as the line BM 
is diminished from BC, so that in the tangent plane to the circle 
at L, represented by the line NLO, the moment of descent is 


lessened in the movable body in the same proportion. Therefore 
if on the plane HG the moment of the movable body is diminished 
from its total impetus (which it has in the perpendicular DCE) 
in the proportion of the line KB to the line BC or BF, the similarity 
of the triangles KBF and KFH making the proportion between 
the lines KF and FH the same as between KB and BF, we conclude 
that the whole and absolute moment that the movable body has 
in the perpendicular to the horizon is in the same proportion to 
that which it has on the inclined plane HF as the line HF is to 
the line FK, which is that of the length of the inclined plane to the 
perpendicular dropped from this on the horizontal. So that, 
passing to the present separate diagram, the moment downward 
of the movable body on the inclined plane FH has the same 
proportion to the total moment with which it presses down in the 

Fig. 19. 

perpendicular FK as this line KF has to FH. This being the case, 
it is clear that the force that sustains the weight on the perpen- 
dicular FK must be equal to the weight, so that to sustain it on 
the inclined plane FH there will suffice one as much less than that 
as this perpendicular FK is less than the line FH. And since, as 
has been mentioned at other times, the force to move the weight 
need only insensibly exceed that which sustains it, we derive this 
general conclusion: That 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. 

Returning now to our original purpose, which was to investi- 
gate the nature of the screw, let us consider the triangle ACB, of 
which the line AB is horizontal, BC is perpendicular to it, and 


AC is the inclined plane upon which the movable body D will be 
drawn by a force as much less than itself as the line BC is shorter 
than CA. Now to raise the same weight on the same plane AC 
with the triangle CAB standing still and the weight D being 

Fig. 20. 

moved toward C, is the same thing as if the weight D were not 
moved from the perpendicular DJ while the triangle was being 
driven forward toward H, for when the triangle has reached the 
place FHG the movable body will have climbed to the altitude AJ. 
Now finally the form and first essence of the screw is no other 
than such a triangle ACS which, driven forward, slips under the 
heavy body to be raised, and boosts it or jacks it up (as they say); 

Fio. 21. 

and such was its first origin. Whoever was its first inventor 
considered that as the triangle ABC coming forward raised the 
weight D, so an instrument could be constructed similar to the 
said triangle of some solid material which, driven forward, would 
elevate the given weight; but then considering better how such a 
machine could be reduced to a much smaller and more convenient 
form, he took the same triangle and wound it round the cylinder 


ABCD in such a manner that the altitude CB of the triangle 
should be the height of the cylinder. Thus the ascending plane 
generates upon the cylinder the helical line denoted by the line 
AEFGH, which is commonly called the thread of the screw; and in 
this way there was created the instrument called by the Greeks 
cochlea, and by us the screw; which, turning round, comes to 
bear with its thread beneath the weight and easily raises it. 

Fig. 22. 

Now having demonstrated that upon the inclined plane the 
force has to the weight the same proportion as the perpendicular 
height of the inclined plane has to its length, we thus understand 
that the force is multiplied by the screw ABCD according to the 
ratio that the length of its entire thread AEFGH has to its height 
CB. In this way we learn that the more dense the threading of 
the screw by its helices, the more powerful it becomes, as being 
generated by a less steeply inclined plane, whose length is in 
greater proportion to its own perpendicular height. But let us not 
neglect to mention that if we wish to find the force of a given 
screw we do not have to measure the length of its entire thread, 
and the height of its whole cylinder, but it will suffice that we 
examine the number of times the distance between any two 
contiguous elements divides a single revolution of the thread. 


This, for example, will be the number of times the distance AF 
is contained in the length of the turn AEF, for this is the same 
proportion that the entire height CB has to the whole thread. 

When one understands all we have said about the nature of 
this instrument, I do not doubt that all its other properties can be 
understood without trouble — as, for instance, that instead of 
raising the weight upon the screw, a female thread is accommo- 
dated to it with a concave helix in which the male thread of the 
screw enters and is then turned round, raising and lifting this nut 
together with the weight attached to it. 

Finally one must not ignore the consideration which from the 
beginning has been said to hold for all mechanical instruments, 
that is, that whatever is gained in force by their means is lost in 
time and in speed. Perhaps to someone this may not appear so 
clearly in the present instance, and it may even seem that the 
force is multiplied without the mover traveling farther than the 
body moved. For let us suppose in the triangle ABC the line AB 
to be horizontal, AC to be the inclined plane whose height is 
measured by the perpendicular CB, and a movable body to be 
placed on the plane AC, and linked by the cord EDF to a force 

Fig. 23. 

or weight placed at F. Then if its proportion to the heaviness of 
the weight E is the same as that of the line BC to CA, from what 
has been proven, the weight F will descend, drawing the movable 
body E along the inclined plane without the weight F measuring 
a greater space in descending than the movable body E measures 
along the line AC. But here it should be noticed that although 
the movable body E will have passed over all the line AC in the 
same time that the other heavy body F will have fallen through 


an equal interval, nevertheless the heavy body E will not have 
been removed from the common center of heavy things more 
than the distance along the perpendicular CB, while the heavy 
body F descending perpendicularly will have dropped by a space 
equal to the whole line AC. And since heavy bodies do not have 
any resistance to transverse motions except in proportion to their 
removal from the center of the earth, then the movable body E 
not being raised more than the distance CB in the whole motion 
AC, while F has dropped perpendicularly as much as the whole 
length of AC, we may rightly say that the travel of the force F 
has the same ratio to the travel of the force E as the line AC has 
to the line CB, or as the weight E has to the weight F. Therefore 
it is very important to consider along what lines the motions are 
made, and especially of inanimate heavy bodies, whose moments 
have their whole power and their entire resistance in the line 
perpendicular to the horizon; and in other lines, transversely 
rising or falling, they have only that power, impetus, or resistance 
which is greater or lesser according as the inclinations approach 
more or less to the perpendicular elevation. 



(a) Scaling 

Galileo began his most significant book on pure science with a practical 
problem that arose out of his consultation services for the famous Venetian 
Arsenal with its dockyards and its galleys. It was a question of scaling; 
namely, why is the scaffolding and bracing for launching a large vessel 
greater than that for a small one? From purely geometrical considerations 
a similar large machine should not be proportionately stronger than a 
small one. For example, the law of the lever is the same for geometrically 
similar levers; if all distances are doubled or halved, the mechanical advan- 
tage remains unchanged. Evidently there is a missing physical factor, viz. 
the strength of materials. Geometry as pure mathematics is not sufficient 
for solving all mechanical problems; some kind of physical geometry is 
requisite. The first new science, strictly speaking, was truly the first engin- 
eering science, namely, the strength of materials. 

Dialogues Concerning Two New Sciences^ 

Salv. The constant activity which you Venetians display in 
your famous arsenal suggests to the studious mind a large field 
for investigation, especially that part of the work which involves 
mechanics; for in this department all types of instruments and 
machines are constantly being constructed by many artisans, 
among whom there must be some who, partly by inherited 
experience and partly by their own observations, have become 
highly expert and clever in explanation. 

Sagr. You are quite right. Indeed, I myself, being curious by 
nature, frequently visit this place for the mere pleasure of observ- 
ing the work of those who, on account of their superiority over 

t Ref. (9), pp. 1-3. 



other artisans, we call "first rank men." Conference with them 
has often helped me in the investigation of certain effects including 
not only those which are striking, but also those which are 
recondite and almost incredible. At times also I have been put to 
confusion and driven to despair of ever explaining something for 
which I could not account, but which my senses told me to be 
true. And notwithstanding the fact that what the old man told 
us a little while ago is proverbial and commonly accepted, yet it 
seemed to me altogether false, like many another saying which is 
current among the ignorant; for I think they introduce these 
expressions in order to give the appearance of knowing something 
about matters which they do not understand. 

Salv. You refer, perhaps, to that last remark of his when we 
asked the reason why they employed stocks, scaffolding and 
bracing of larger dimensions for launching a big vessel than they 
do for a small one ; and he answered that they did this in order to 
avoid the danger of the ship parting under its own heavy weight 
[vasta mole], a danger to which small boats are not subject? 

Sagr. Yes, that is what I mean; and I refer especially to his 
last assertion which I have always regarded as a false, though 
current, opinion; namely, that in speaking of these and other 
similar machines one cannot argue from the small to the large, 
because many devices which succeed on a small scale do not 
work on a large scale. Now, since mechanics has its foundation 
in geometry, where mere size cuts no figure, I do not see that the 
properties of circles, triangles, cylinders, cones and other solid 
figures will change with their size. If, therefore, a large machine 
be constructed in such a way that its parts bear to one another 
the same ratio as in a smaller one, and if the smaller is sufficiently 
strong for the purpose for which it was designed, I do not see 
why the larger also should not be able to withstand any severe 
and destructive tests to which it may be subjected. 

Salv. The common opinion is here absolutely wrong. Indeed, 
it is so far wrong that precisely the opposite is true, namely, 
that many machines can be constructed even more perfectly on a 
large scale than on a small; thus, for instance, a clock which 


indicates and strikes the hour can be made more accurate on a 
large scale than on a small. There are some intelligent people 
who maintain this same opinion, but on more reasonable grounds, 
when they cut loose from geometry and argue that the better 
performance of the large machine is owing to the imperfections 
and variations of the material. Here I trust you will not charge 
me with arrogance if I say that imperfections in the material, 
even those which are great enough to invalidate the clearest 
mathematical proof, are not sufficient to explain the deviations 
observed between machines in the concrete and in the abstract. 
Yet I shall say it and will affirm that, even if the imperfections 
did not exist and matter were absolutely perfect, unalterable and 
free from all accidental variations, still the mere fact that it is 
matter makes the larger machine, built of the same material and 
in the same proportion as the smaller, correspond with exactness 
to the smaller in every respect except that it will not be so strong 
or so resistant against violent treatment; the larger the machine, 
the greater its weakness. Since I assume matter to be unchange- 
able and always the same, it is clear that we are no less able to 
treat this constant and invariable property in a rigid manner than 
if it belonged to simple and pure mathematics. Therefore, 
Sagredo, you would do well to change the opinion which you, and 
perhaps also many other students of mechanics, have entertained 
concerning the ability of machines and structures to resist 
external disturbances, thinking that when they are built of the 
same material and maintain the same ratio between parts, they 
are able equally, or rather proportionally, to resist or yield to 
such external disturbances and blows. For we can demonstrate 
by geometry that the large machine is not proportionately 
stronger than the small. Finally, we may say that, for every 
machine and structure, whether artificial or natural, there is set 
a necessary limit beyond which neither art nor nature can pass ; 
it is here understood, of course, that the material is the same and 
the proportion preserved. 

Sagr. My brain already reels. My mind, like a cloud momen- 
tarily illuminated by a lightning-flash, is for an instant filled 


with an unusual light, which now beckons to me and which now 
suddenly mingles and obscures strange, crude ideas. From what 
you have said it appears to me impossible to build two similar 
structures of the same material, but of different sizes and have 
them proportionately strong; and if this were so, it would not be 
possible to find two single poles made of the same wood which 
shall be alike in strength and resistance but unlike in size. 

(b) Galileo's Problem 

What has become known as Galileo's problem concerns the resistance to 
fracture of a cantilever, i.e. a horizontal beam embedded in a wall (cf. Fig. 24). 
He assumed physically that the base of the fracture, where the beam is 
joined to the wall, is under a uniform tensile stress, which can be regarded 
as equivalent to a single resultant force acting towards the wall, at the center 
of the contact. He failed to realize that the fibers of the strained beam are 
not inextensible, and that there is a balance between forces of tension and 
those of compression (correctly noted by Edme Mariotte (c. 1620-84), who 
specified them in 1680 with the law of Robert Hooke (1635-1703) for the 
relation of elastic stress and equilibrium displacement). This false assump- 
tion, however, did not invalidate most of his conclusions inasmuch as they 
involved only ratios of the strengths of beams of similar cross-sections. 

Galileo's whole procedure was based on the law of the lever for equilibrium. 

Let a load W be suspended from the end of a beam of span L and depth 
D (or diameter). Then for a beam of neglible weight, with the lower edge 
of the beam along the wall as an axis (fulcrum), we have according to the 
law of the lever 


where R is the "resistance" of the beam. Hence 


W~ D ' 
Galileo indicated further how this conclusion has to be modified to include 
the weight of the beam. (He had previously derived the law of the lever for 
a beam having a significant weight.) 

Dialogues Concerning Two New Sciences^ 
Proposition I 

A prism or solid cylinder of glass, steel, wood or other break- 
able material which is capable of sustaining a very heavy weight 
fRef. (9), pp. 115-17. 


when applied longitudinally is, as previously remarked, easily 
broken by the transverse application of a weight which may be 
much smaller in proportion as the length of the cylinder exceeds 
its thickness. 

Let us imagine a solid prism ABCD fastened into a wall at 
the end AB, and supporting a weight E at the other end; under- 
stand also that the wall is vertical and that the prism or cylinder 
is fastened at right angles to the wall. It is clear that, if the 
cylinder breaks, fracture will occur at the point B where the edge 
of the mortise acts as a fulcrum for the lever BC, to which the 
force is applied; the thickness of the solid BA is the other arm of 
the lever along which is located the resistance. This resistance 
opposes the separation of the part BD, lying outside the wall, 
from that portion lying inside. From the preceding, it follows 
that the magnitude [momento] of the force applied at C bears to 
the magnitude [momento] of the resistance, found in the thickness 
of the prism, i.e., in the attachment of the base BA to its con- 
tiguous parts, the same ratio which the length CB bears to half 
the length BA; if now we define absolute resistance to fracture 
as that offered to a longitudinal pull (in which case the stretching 
force acts in the same direction as that through which the body 
is moved), then it follows that the absolute resistance of the 
prism BD is to the breaking load placed at the end of the lever 
BC in the same ratio as the length BC is to the half of AB in the 
case of a prism, or the semidiameter in the case of a cylinder. 
This is our first proposition. Observe that in what has here been 
said the weight of the solid BD itself has been left out of con- 
sideration, or rather, the prism has been assumed to be devoid 
of weight. But if the weight of the prism is to be taken account of 
in conjunction with the weight E, we must add to the weight E 
one half that of the prism BD : so that if, for example, the latter 
weighs two pounds and the weight E is ten pounds we must treat 
the weight E as if it were eleven pounds. 

Simp. Why not twelve ? 

Salv. The weight E, my dear Simplicio, hanging at the extreme 
end C acts upon the lever BC with its full moment of ten pounds : 


so also would the solid BD if suspended at the same point exert 
its full moment of two pounds ; but, as you know, this solid is 
uniformly distributed throughout its entire length, BC, so that 
the parts which lie near the end B are less effective than those 
more remote. 

Fig. 24. 

Accordingly if we strike a balance between the two, the weight 
of the entire prism may be considered as concentrated at its 
center of gravity which lies midway of the lever BC. But a weight 
hung at the extremity C exerts a moment twice as great as it would 
if suspended from the middle: therefore if we consider the 


moments of both as located at the end C we must add to the 
weight E one-half that of the prism. 

Simp. I understand perfectly; and moreover, if I mistake not, 
the force of the two weights BD and E, thus disposed, would 
exert the same moment as would the entire weight BD together 
with twice the weight E suspended at the middle of the lever BC. 

Salv. Precisely so, and a fact worth remembering. 

(c) Similar Beams 

The interesting question posed at the beginning of the Two New Sciences 
involved the relative diameters of two cylindrical beams where one given 
beam of span L\ and diameter D\ is just able to support its own weight W i 
without fracture, and a second beam of span L 2 is just able to support its 
weight W 2 , associated with an unknown diameter D 2 . Weight being propor- 
tional to volume, accordingly, 

W x cc L x Df- , 

W 2 oc L 2 D 2 i . 

Therefore, the corresponding bending moments 5j and B 2 are, respectively, 

Bi oc £i/>i2 - oc L<i D{2. , 


L 2 
B 2 oc L 2 D 2 2 — oc I 2 2£» 2 2 . 

Galileo had already proved (cf. Proposition IV, p. 119) that the maximum 
bending moment M of the resistance to fracture is proportional to the cube 
of the diameter of the base. 
If B = M max, then 

£> 3 oc 12 Z>2 , 

D oc U . 

But similar beams have DccL; hence they cannot also have similar strengths. 
A given form, therefore, cannot be physically magnified either in nature — 
or in art; its size is naturally limited. There is generally a right size! 

Dialogues Concerning Two New Sciences^ 

Sagr. The favor will be that much greater: nevertheless I hope 
you will oblige me by putting into written form the argument just 
given so that I may study it at my leisure. 

t Ref. (9), pp. 129-33. 


Salv. I shall gladly do so. Let A denote a cylinder of diameter 
DC and the largest capable of sustaining its own weight: the 
problem is to determine a larger cylinder which shall be at once 
the maximum and the unique one capable of sustaining its 
own weight. 

Let E be such a cylinder, similar to A, having the assigned 
length, and having a diameter KL. Let MN be a third propor- 
tional to the two lengths DC and KL: let MN also be the diameter 

Fig. 25. 

of another cylinder, X, having the same length as E: then, I say, 
X is the cylinder sought. Now since the resistance of the base DC 
is to the resistance of the base KL as the square of DC is to the 
square of KL, that is, as the square of KL is to the square of MN, 
or, as the cylinder E is to the cylinder X, that is, as the moment E 
is to the moment X; and since also the resistance [bending 
strength] of the base KL is to the resistance of the base MN as the 
cube of KL is to the cube of MN, that is, as the cube of DC is to 
the cube of KL, or, as the cylinder A is to the cylinder E, that is, 


as the moment of A is to the moment of E; hence it follows, ex 
aquali in proportion perturbata, that the moment of A is to'the 
moment of X as the resistance of the base DC is to the resistance 
of the base MN; therefore moment and resistance are related to 
each other in prism X precisely as they are in prism A. 
Let us now generalize the problem; then it will read as follows: 
Given a cylinder AC in which moment and resistance [ben- 
ding strength] are related in any manner whatsoever; let DE 
be the length of another cy Under; then determine what its 
thickness must be in order that the relation between its 
moment and resistance shall be identical with that of the 
cylinder AC. 
Using Fig. 25 in the same manner as above, we may say that, 
since the moment of the cylinder FE is to the moment of the 
portion DG as the square of ED is to the square of FG, that is, 
as the length DE is to I; and since the moment of the cylinder 
FG is to the moment of the cylinder AC as the square of FD is to 
the square of AB, or, as the square of ED is to the square of I, or, 
as the square of I is to the square of M, that is, as the length' I is 
to O; it follows, ex aquali, that the moment of the cylinder FE 
is to the moment of the cylinder AC as the length DE is to O, 
that is, as the cube of DE is to the cube of I, or, as the cube of FD 
is to the cube of AB, that is, as the resistance of the base FD is to 
the resistance of the base AB; which was to be proven. 

From what has already been demonstrated, you can plainly 
see the impossibility of increasing the size of structures to vast 
dimensions either in art or in nature; likewise the impossibility 
of building ships, palaces, or temples of enormous size in such a 
way that their oars, yards, beams, iron-bolts, and, in short, all 
their other parts will hold together; nor can nature produce 
trees of extraordinary size because the branches would break 
down under their own weight; so also it would be impossible to 
build up the bony structures of men, horses, or other animals so 
as to hold together and perform their normal functions if these 
animals were to be increased enormously in height; for this 
increase in height can be accomplished only by employing a 


material which is harder and stronger than usual, or by enlarging 
the size of the bones, thus changing their shape until the form 
and appearance of the animals suggest a monstrosity. This is 
perhaps what our wise Poet had in mind, when he says, in 
describing a huge giant : 

"Impossible it is to reckon his height 

"So beyond measure is his size." 

To illustrate briefly, I have sketched a bone whose natural 

length has been increased three times and whose thickness has 

been multiplied until, for a correspondingly large animal, it 

would perform the same function which the small bone performs 

Fig. 26. 

for its small animal. From the figures here shown you can see 
how out of proportion the enlarged bone appears. Clearly then 
if one wishes to maintain in a great giant the same proportion of 
limb as that found in an ordinary man he must either find a 
harder and stronger material for making the bones, or he must 
admit a diminution of strength in comparison with men of 
medium stature; for if his height be increased inordinately he will 
fall and be crushed under his own weight. Whereas, if the size of 


a body be diminished, the strength of that body is not diminished 
in the same proportion; indeed the smaller the body the greater 
its relative strength. Thus a small dog could probably carry on 
his back two or three dogs of his own size; but I believe that a 
horse could not carry even one of his own size. 

Simp. This may be so; but I am led to doubt it on account of 
the enormous size reached by certain fish, such as the whale 
which, I understand, is ten times as large as an elephant; yet 
they all support themselves. 

Salv. Your question, Simplicio, suggests another principle, 
one which had hitherto escaped my attention and which enables 
giants and other animals of vast size to support themselves and 
to move about as well as smaller animals do. This result may be 
secured either by increasing the strength of the bones and other 
parts intended to carry not only their weight but also the super- 
incumbent load; or, keeping the proportions of the bony structure 
constant, the skeleton will hold together in the same manner or 
even more easily, provided one diminishes, in the proper propor- 
tion, the weight of the bony material, of the flesh, and of anything 
else which the skeleton has to carry. It is this second principle 
which is employed by nature in the structure offish, making their 
bones and muscles not merely light but entirely devoid of weight. 

Simp. The trend of your argument, Salviati, is evident. Since 
fish live in water which on account of its density [corpulenza] or, 
as others would say, heaviness (gravitd) diminishes the weight 
[peso] of bodies immersed in it, you mean to say that, for this 
reason, the bodies of fish will be devoid of weight and will be 
supported without injury to their bones. But this is not all; for 
although the remainder of the body of the fish may be without 
weight, there can be no question but that their bones have weight. 
Take the case of a whale's rib, having the dimensions of a beam; 
who can deny its great weight or its tendency to go to the bottom 
when placed in water? One would, therefore, hardly expect these 
great masses to sustain themselves. 

Salv. A very shrewd objection! And now, in reply, tell me 
whether you have ever seen fish stand motionless at will under 


water, neither descending to the bottom nor rising to the top, 
without the exertion of force by swimming? 

Simp. This is a well-known phenomenon. 

Salv. The fact then that fish are able to remain motionless 
under water is a conclusive reason for thinking that the material 
of their bodies has the same specific gravity as that of water; 
accordingly, if in their make-up there are certain parts which 
are heavier than water there must be others which are lighter, 
for otherwise they would not produce equilibrium. 

Hence, if the bones are heavier, it is necessary that the muscles 
or other constituents of the body should be lighter in order that 
their buoyancy may counterbalance the weight of the bones. In 
aquatic animals therefore circumstances are just reversed from 
what they are with land animals inasmuch as, in the latter, the 
bones sustain not only their own weight but also that of the flesh, 
while in the former it is the flesh which supports not only its own 
weight but also that of the bones. We must therefore cease to 
wonder why these enormously large animals inhabit the water 
rather than the land, that is to say, the air. 

Simp. I am convinced and I only wish to add that what we call 
land animals ought really to be called air animals, seeing that they 
live in the air, are surrounded by air, and breathe air. 

Sagr. I have enjoyed Simplicio's discussion including both the 
question raised and its answer. Moreover I can easily understand 
that one of these giant fish, if pulled ashore, would not perhaps 
sustain itself for any great length of time, but would be crushed 
under its own mass as soon as the connections between the bones 
gave way. 

Salv. I am inclined to your opinion; and, indeed, I almost 
think that the same thing would happen in the case of a very big 
ship which floats on the sea without going to pieces under its 
load of merchandise and armament, but which on dry land and 
in air would probably fall apart. 


(d) A Cracked Column 

In the beginning Galileo told a strange anecdote about a large marble 
column resting horizontally on two supports at its ends. In order to prevent 
sagging and rupture at the middle a careful mechanic had placed an addi- 
tional support there. A few months later the column cracked — precisely 
in the middle. 

Suppose a cylinder supporting its own weight has the maximum length 
possible without fracture. If it is supported at both ends, it will have twice 
the length of such a beam supported by only at one of its ends in a wall. 
Consequently another support, say at the middle, produces an additional 
force and bending moment so that rupture occurs. 

Dialogues Concerning Two New Sciences^ 

Hitherto we have considered the moments and resistances of 
prisms and solid cylinders fixed at one end with a weight applied 
at the other end; three cases were discussed, namely, that in 
which the applied force was the only one acting, that in which the 
weight of the prism itself is also taken into consideration, and 


Fig. 27. 

that in which the weight of the prism alone is taken into con- 
sideration. Let us now consider these same prisms and cylinders 
when supported at both ends or at a single point placed some- 
where between the ends. In the first place, I remark that a 
t Ref. (9), pp. 134-8. 


cylinder carrying only its own weight and having the maximum 
length, beyond which it will break, will, when supported either 
in the middle or at both ends, have twice the length of one which 
is mortised into a wall and supported only at one end. This is 
very evident because, if we denote the cylinder by ABC and if we 
assume that one-half of it, AB, is the greatest possible length 
capable of supporting its own weight with one end fixed at B, 
then, for the same reason, if the cylinder is carried on the point G, 
the first half will be counterbalanced by the other half BC. So 
also in the case of the cylinder DEF, if its length be such that it 
will support only one-half this length when the end D is held fixed, 
or the other half when the end F is fixed, then it is evident that 
when supports, such as H and I, are placed under the ends D and 
F respectively the moment of any additional force or weight 
placed at E will produce fracture at this point. 

A more intricate and difficult problem is the following: neglect 
the weight of a solid such as the preceding and find whether 
the same force or weight which produces fracture when applied 
at the middle of a cylinder, supported at both ends, will also break 
the cylinder when applied at some other point nearer one end 
than the other. 

Thus, for example, if one wished to break a stick by holding 
it with one hand at each end and applying his knee at the middle, 
would the same force be required to break it in the same manner 
if the knee were applied, not at the middle, but at some point 
nearer to one end? 

Sagr. This problem, I believe, has been touched upon by 
Aristotle in his Questions in Mechanics. 

Salv. His inquiry however is not quite the same; for he seeks 
merely to discover why it is that a stick may be more easily 
broken by taking hold, one hand at each end of the stick, that 
is, far removed from the knee, than if the hands were closer 
together. He gives a general explanation, referring it to the 
lengthened lever arms which are secured by placing the hands at 
the ends of the stick. Our inquiry calls for something more: 
what we want to know is whether, when the hands are retained 


at the ends of the stick, the same force is required to break it 
wherever the knee be placed. 

Sagr. At first glance this would appear to be so, because the 
two lever arms exert, in a certain way, the same moment, seeing 
that as one grows shorter the other grows correspondingly longer. 

Salv. Now you see how readily one falls into error and what 
caution and circumspection are required to avoid it. What you 
have just said appears at first glance highly probable, but on 
closer examination it proves to be quite far from true; as will be 
seen from the fact that whether the knee — the fulcrum of the 
two levers — be placed in the middle or not makes such a dif- 
ference that, if fracture is to be produced at any other point than 
the middle, the breaking force at the middle, even when multiplied 
four, ten, a hundred, or a thousand times would not suffice. 
To begin with we shall offer some general considerations and then 
pass to the determination of the ratio in which the breaking force 
must change in order to produce fracture at one point rather 
than another. 

Let AB denote a wooden cylinder which is to be broken in 
the middle, over the supporting point C, and let DE represent 
an identical cylinder which is to be broken just over the support- 
ing point F which is not in the middle. First of all it is clear 
that, since the distances AC and CB are equal, the forces applied 
at the extremities B and A must also be equal. Secondly since 
the distance DF is less than the distance AC the moment of any 
force acting at D is less than the moment of the same force at A, 
that is, applied at the distance CA; and the moments are less in 
the ratio of the length DF to AC; consequently it is necessary to 
increase the force [momento] at D in order to overcome, or even 
to balance, the resistance at F; but in comparison with the length 
AC the distance DF can be diminished indefinitely: in order 
therefore to counterbalance the resistance at F it will be necessary 
to increase indefinitely the force [forza] applied at D. On the other 
hand, in proportion as we increase the distance FE over that of 
CB, we must diminish the force at E in order to counterbalance 
the resistance at F; but the distance FE, measured in terms of CB, 


cannot be increased indefinitely by sliding the fulcrum F toward 
the end D; indeed, it cannot even be made double the length CB. 
Therefore the force required at E to balance the resistance at F 
will always be more than half that required at B. It is clear then 
that, as the fulcrum F approaches the end D, we must of necessity 
indefinitely increase the sum of the forces applied at E and D in 
order to balance, or overcome, the resistance at F. 

Fig. 28. 

Sagr. What shall we say, Simplicio? Must we not confess 
that geometry is the most powerful of all instruments for sharpen- 
ing the wit and training the mind to think correctly? Was not 
Plato perfectly right when he wished that his pupils should be 
first of all well grounded in mathematics ? As for myself, I quite 
understood the property of the lever and how, by increasing or 
diminishing its length, one can increase or diminish the moment 
of force and of resistance; and yet, in the solution of the present 
problem I was not slightly, but greatly, deceived. 

Simp. Indeed I begin to understand that while logic is an 
excellent guide in discourse, it does not, as regards stimulation to 
discovery, compare with the power of sharp distinction which 
belongs to geometry. 


Sagr. Logic, it appears to me, teaches us how to test the 
conclusiveness of any argument or demonstration already dis- 
covered and completed; but I do not believe that it teaches us 
to discover correct arguments and demonstrations. But it would 
be better if Salviati were to show us in just what proportion the 
forces must be increased in order to produce fracture as the 
fulcrum is moved from one point to another along one and the 
same wooden rod. 

(e) Tubes 

The Second Day of the Two New Sciences concluded with a comparison 
of the strength of a hollow tube and that of a solid one (same material, 
volume, and length). Now in each case the resistance moment is proportional 
to the product of the cross-sectional area and the diameter. But the cross- 
sectional area of each column is the same, namely, volume/length. Therefore, 
the resistance moment must be proportional to the diameter, which, of 
course, is greater for the hollow cylinder. Hence, a hollow cylinder is more 
resistant to fracture. Although Galileo actually underestimated the relative 
strength of a hollow cylinder, his investigations marked the beginning of this 
unquestionably "new science", the very foundation of engineering theory. 
He pointed clearly to the direction requisite for further development. 

Dialogues Concerning Two New Sciences^ 

But, in order to bring our daily conference to an end, I wish 
to discuss the strength of hollow solids, which are employed in 
art — and still oftener in nature — in a thousand operations for 
the purpose of greatly increasing strength without adding to 
weight; examples of these are seen in the bones of birds and in 
many kinds of reeds which are light and highly resistant both to 
bending and breaking. For if a stem of straw which carries a 
head of wheat heavier than the entire stalk were made up of the 
same amount of material in solid form it would offer less resist- 
ance to bending and breaking. This is an experience which has 
been verified and confirmed in practice where it is found that a 

t Ref. (9), pp. 150-1. 


hollow lance or a tube of wood or metal is much stronger than 
would be a solid one of the same length and weight, one which 
would necessarily be thinner; men have discovered, therefore, 
that in order to make lances strong as well as light they must 
make them hollow. We shall now show that: 

In the case of two cylinders, one hollow the other solid but 

having equal volumes and equal lengths, their resistances 

[bending strengths] are to each other in the ratio of their 


Let AE denote a hollow cylinder and IN a solid one of the 

same weight and length; then, I say, that the resistance against 

fracture exhibited by the tube AE bears to that of the solid 

Fig. 29. 

cylinder IN the same ratio as the diameter AB to the diameter IL. 
This is very evident; for since the tube and the solid cylinder 
IN have the same volume and length, the area of the circular 
base IL will be equal to that of the annulus AB which is the base 
of the tube AE. (By annulus is here meant the area which lies 
between two concentric circles of different radii.) Hence their 
resistances to a straight-away pull are equal; but in producing 
fracture by a transverse pull we employ, in the case of the 
cylinder IN, the length LN as one lever arm, the point L as a 


fulcrum, and the diameter LI, or its half, as the opposing lever 
arm: while in the case of the tube, the length BE which plays 
the part of the first lever arm is equal to LN, the opposing lever 
arm beyond the fulcrum, B, is the diameter AB, or its half. 
Manifestly then the resistance [bending strength] of the tube 
exceeds that of the solid cylinder in the proportion in which the 
diameter AB exceeds the diameter IL, which is the desired result. 
Thus the strength of a hollow tube exceeds that of a solid 
cylinder in the ratio of their diameters whenever the two are 
made of the same material and have the same weight and length. 



(a) Simple Pendulum 

Galileo was fascinated throughout his life by all phenomena involving 
oscillations — apparently beginning with his (legendary) observations of the 
swinging chandelier in the Cathedral of Pisa. His substitution of a simple 
pendulum for this ornate lamp is typical of his seeking a simplified physical 
model to investigate a complex phenomenon. Despite the desire of some 
historians to regard Galileo as a mathematical Platonist, Galileo's investiga- 
tion of the simple pendulum appears to physicists to be a good example of 
the scientific method, namely, a search for related factors — not merely a 
simple linear proportion, not even a deduction from some a priori hypothesis, 
certainly not a so-called thought experiment. In the following passage he 
gave the basic law for comparing T\ and T 2 , the periods (time for a complete 
swing to and fro) of two pendula with lengths L\ and L 2 , respectively, 

T 2 J 

L 2 

The very fact that Galileo expressed this relation in terms of an irrational 
quantity, a square root, indicates that he could not have been a traditional 
Pythagorean. He himself noted that one might prefer to express the relation 

If =k 

T 2 2 L 2 ' 
Nowadays the period would be given absolutely by the formula 


= 2tr / — , 

where g is the acceleration due to gravity. It holds, however, only for small 
oscillations (for a displacement of 15° the error would be about 1.1%). 
Accordingly, there is some question as to whether Galileo's error (cf. ref. (7), 
p. 230) in assuming its correctness for any arc was due primarily to his poor 
time measurements or to his failure to examine carefully the periods for 
large arcs. 



Already in his earlier notes (cf. ref. (3), p. 108) he had reported his observa- 
tion that the oscillation of a lead bob lasts longer than that for a comparable 
wooden one (or cork; ref. (9), p. 85); he recognized also that the period is 
independent of the bob itself — except for any air resistance. He mentioned 
also (ref. (7), pp. 22, 26) that a bob would ascend to its initial height — 
were it not for this frictional loss. Here (ref. (7), p. 150), too, he illustrated 
the difference in persistence of motion for lead and for cotton. 

He considered also the case of a cannon ball dropped through the center 
of the earth as analogous to the action of a pendulum (ref. (7), p. 236). He 
likened the oscillations of water in a container to that of a pendulum (ref. (7), 
p. 428). The regulation of time in a wheel clock, he observed (ref. (7), p. 449X 
could be adjusted in accordance with the pendulum law. 

Dialogues Concerning Two New Sciences^ 

We come now to the other questions, relating to pendulums, 
a subject which may appear to many exceedingly arid, especially 
to those philosophers who are continually occupied with the 
more profound questions of nature. Nevertheless, the problem 
is one which I do not scorn. I am encouraged by the example of 
Aristotle whom I admire especially because he did not fail to 
discuss every subject which he thought in any degree worthy of 

Impelled by your queries I may give you some of my ideas 
concerning certain problems in music, a splendid subject, upon 
which so many eminent men have written: among these is 
Aristotle himself who has discussed numerous interesting acous- 
tical questions. Accordingly, if on the basis of some easy and 
tangible experiments, I shall explain some striking phenomena 
in the domain of sound, I trust my explanations will meet your 

Sagr. I shall receive them not only gratefully but eagerly. 
For, although I take pleasure in every kind of musical instrument 
and have paid considerable attention to harmony, I have never 
been able to fully understand why some combinations of tones are 
more pleasing than others, or why certain combinations not only 
fail to please but are even highly offensive. Then there is the old 

t Ref. (9), pp. 94-8. 


problem of two stretched strings in unison; when one of them is 
sounded, the other begins to vibrate and to emit its note; nor do 
I understand the different ratios of harmony [forme delle con- 
sonanze] and some other details. 

Salv. Let us see whether we cannot derive from the pendulum 
a satisfactory solution of all these difficulties. And first, as to 
the question whether one and the same pendulum really performs 
its vibrations, large, medium, and small, all in exactly the same 
time, I shall rely upon what I have already heard from our 
Academician. He has clearly shown that the time of descent is 
the same along all chords, whatever the arcs which subtend them, 
as well along an arc of 180° (i.e., the whole diameter) as along 
one of 100°, 60°, 10°, 2°, |°, or 4'. It is understood, of course, 
that these arcs all terminate at the lowest point of the circle, 
where it touches the horizontal plane. 

If now we consider descent along arcs instead of their chords 
then, provided these do not exceed 90°, experiment shows that 
they are all traversed in equal times; but these times are greater 
for the chord than for the arc, an effect which is all the more 
remarkable because at first glance one would think just the 
opposite to be true. For since the terminal points of the two 
motions are the same and since the straight line included between 
these two points is the shortest distance between them, it would 
seem reasonable that motion along this line should be executed 
in the shortest time; but this is not the case, for the shortest 
time — and therefore the most rapid motion — is that employed 
along the arc of which this straight line is the chord. 

As to the times of vibration of bodies suspended by threads of 
different lengths, they bear to each other the same proportion as 
the square roots of the lengths of the thread; or one might say 
the lengths are to each other as the squares of the times; so that 
if one wishes to make the vibration-time of one pendulum twice 
that of another, he must make its suspension four times as long. 
In like manner, if one pendulum has a suspension nine times as 
long as another, this second pendulum will execute three vibra- 
tions during each one of the first; from which it follows that the 


lengths of the suspending cords bear to each other the [inverse] 
ratio of the squares of the number of vibrations performed in the 
same time. 

Sagr. Then, if I understand you correctly, I can easily measure 
the length of a string whose upper end is attached at any height 
whatever even if this end were invisible and I could see only the 
lower extremity. For if I attach to the lower end of this string a 
rather heavy weight and give it a to-and-fro motion, and if I ask 
a friend to count a number of its vibrations, while I, during the 
same time-interval, count the number of vibrations of a pendulum 
which is exactly one cubit in length, then knowing the number of 
vibrations which each pendulum makes in the given interval of 
time one can determine the length of the string. Suppose, for 
example, that my friend counts 20 vibrations of the long cord 
during the same time in which I count 240 of my string which is 
one cubit in length; taking the squares of the two numbers, 20 
and 240, namely 400 and 57600, then, I say, the long string 
contains 57600 units of such length that my pendulum will 
contain 400 of them; and since the length of my string is one 
cubit, I shall divide 57600 by 400 and thus obtain 144. Accord- 
ingly I shall call the length of the string 144 cubits. 

Salv. Nor will you miss it by as much as a hand's breadth, 
especially if you observe a large number of vibrations. 

Sagr. You give me frequent occasion to admire the wealth 
and profusion of nature when, from such common and even 
trivial phenomena, you derive facts which are not only striking 
and new but which are often far removed from what we would 
have imagined. Thousands of times I have observed vibrations 
especially in churches where lamps, suspended by long cords, 
had been inadvertently set into motion; but the most which 
I could infer from these observations was that the view of those 
who think that such vibrations are maintained by the medium 
is highly improbable: for, in that case, the air must needs have 
considerable judgment and little else to do but kill time by 
pushing to and fro a pendent weight with perfect regularity. But 
I never dreamed of learning that one and the same body, when 


suspended from a string a hundred cubits long and pulled aside 
through an arc of 90° or even 1 ° or \°, would employ the same 
time in passing through the least as through the largest of these 
arcs ; and, indeed, it still strikes me as somewhat unlikely. Now 
I am waiting to hear how these same simple phenomena can 
furnish solutions for those acoustical problems — solutions which 
will be at least partly satisfactory. 

Salv. First of all one must observe that each pendulum has 
its own time of vibration so definite and determinate that it is 
not possible to make it move with any other period [altro periodo] 
than that which nature has given it. For let any one take in his 
hand the cord to which the weight is attached and try, as much 
as he pleases, to increase or diminish the frequency [frequenza] of 
its vibrations; it will be time wasted. On the other hand, one 
can confer motion upon even a heavy pendulum which is at rest 
by simply blowing against it; by repeating these blasts with a 
frequency which is the same as that of the pendulum one can 
impart considerable motion. Suppose that by the first puff we 
have displaced the pendulum from the vertical by, say, half an 
inch; then if, after the pendulum has returned and is about to 
begin the second vibration, we add a second puff, we shall impart 
additional motion; and so on with other blasts provided they are 
applied at the right instant, and not when the pendulum is coming 
toward us since in this case the blast would impede rather than 
aid the motion. Continuing thus with many impulses [impulsi] 
we impart to the pendulum such momentum [impeto] that a greater 
impulse [forza] than that of a single blast will be needed to stop it. 

Sagr. Even as a boy, I observed that one man alone by 
giving these impulses at the right instant was able to ring a bell 
so large that when four, or even six, men seized the rope and tried 
to stop it they were lifted from the ground, all of them together 
being unable to counterbalance the momentum which a single 
man, by properly-timed pulls, had given it. 


(b) Vibrating Freely 

When a one-dimensional string under tension T (gravity negligible) is 
plucked transversely, waves are propagated with a speed v 

V P 

where p is the mass per unit length. If the ends of the string are fixed, they 
reflect the progressive waves so that stationary wave patterns are set up 
corresponding to different frequencies of vibration. The fundamental (i e' 
the lowest frequency) occurs when the string (length L) vibrates as a whole 
(wavelength = 2L). Its period T x is the total distance over the speed i e 


Ti = 


Hence, the fundamental (lowest) frequency vi(=l/T{) is given by 


The higher frequencies are given generally by 
hi / T 

where n,- is an integer; these overtones are harmonics; for example, for 
n — 1 (corresponding to two vibrating segments) there occurs the octave 
i.e. twice the fundamental frequency. ' 

Evidently the frequency of a vibrating string may be raised by increasing 
the tension, or by decreasing the density (e.g. a thinner string for a given 
length), or by decreasing the length. 

P£re Marin Mersenne is usually credited with having found these laws of 
a vibrating string. Inasmuch as they were known to Galileo (cf. ref. (9), 
p. 100) even earlier, there is good reason to believe that he had already 
discovered them experientially — though not as precisely as Mersenne. 

His analysis of the musical notes of a vibrating plate in terms of the spacing 
of the stationary patterns of shavings on it is an excellent illustration how 
his innate curiosity about any natural phenomenon led him to a deeper 
understanding of it. This incident predates the later discovery of the now 
well-known Chladni figures (Ernst Chladni (1756-1827)) 


Dialogues Concerning Two New Sciences^ 

(c) Resonance 

Any body will vibrate freely with certain characteristic frequencies. If an 
external stimulus has the same frequency, the body will readily respond in 
unison. It was Galileo who first recognized this phenomenon of resonance. 
He noted that the corresponding strings of one instrument could be set in 
motion by the vibrating strings of a neighboring one via sound waves in 
the air. 

Dialogues Concerning Two New Sciences^ 

(d) Musical Intervals 

The Greeks had observed that harmonious notes from a given vibrating 
string are always in the ratio of the small integers, describing the number 
of its vibrating segments; hence inversely as the lengths of the smallest 
segments. It was Galileo, however, who identified the pitch of a sound with 
the frequency of its originating vibration. A musical octave, of course, 
occurs when the strings have two segments (half the string's length) each 
vibrating with double the frequency. 

From the law of vibrating strings we have 

t; fundamental 

and v octave 

- _L /— 

~ 2L V P ' 

~ 2L fj P 

Hence, the frequency of the octave is double that of the fundamental. Galileo, 
accordingly, proposed to measure the musical interval between two notes 
by the ratio of their frequencies. He applied his knowledge of the law of 
vibrating strings to the general case of musical intervals for strings of different 
materials and dimensions. His explanation of such consonance is basically 
acceptable even today. 

Galileo's investigations of such oscillations was the beginning of the 
modern science of acoustics. 

Dialogues Concerning Two New Sciences^ 

Salv. Your illustration makes my meaning clear and is quite 
as well fitted, as what I have just said, to explain the wonderful 
phenomenon of the strings of the cittern [cetera] or of the spinet 

t Ref. (9), pp. 98-104. 


[cimbalo], namely, the fact that a vibrating string will set another 
string in motion and cause it to sound not only when the latter 
is in unison but even when it differs from the former by an octave 
or a fifth. A string which has been struck begins to vibrate and 
continues the motion as long as one hears the sound [risonanza]; 
these vibrations cause the immediately surrounding air to vibrate 
and quiver; then these ripples in the air expand far into space and 
strike not only all the strings of the same instrument but even 
those of neighboring instruments. Since that string which is 
tuned to unison with the one plucked is capable of vibrating with 
the same frequency, it acquires, at the first impulse, a slight 
oscillation; after receiving two, three, twenty, or more impulses, 
delivered at proper intervals, it finally accumulates a vibratory 
motion equal to that of the plucked string, as is clearly shown by 
equality of amplitude in their vibrations. This undulation 
expands through the air and sets into vibration not only strings, 
but also any other body which happens to have the same period 
as that of the plucked string. Accordingly if we attach to the side 
of an instrument small pieces of bristle or other flexible bodies, 
we shall observe that, when a spinet is sounded, only those pieces 
respond that have the same period as the string which has been 
struck; the remaining pieces do not vibrate in response to this 
string, nor do the former pieces respond to any other tone. 

If one bows the base string on a viola rather smartly and 
brings near it a goblet of fine, thin glass having the same tone 
[tuono] as that of the string, this goblet will vibrate and audibly 
resound. That the undulations of the medium are widely dispersed 
about the sounding body is evinced by the fact that a glass of 
water may be made to emit a tone merely by the friction of the 
finger-tip upon the rim of the glass; for in this water is produced 
a series of regular waves. The same phenomenon is observed to 
better advantage by fixing the base of the goblet upon the bottom 
of a rather large vessel of water filled nearly to the edge of the 
goblet; for if, as before, we sound the glass by friction of the 
finger, we shall see ripples spreading with the utmost regularity 
and with high speed to large distances about the glass. I have 


often remarked, in thus sounding a rather large glass nearly full 
of water, that at first the waves are spaced with great uniformity, 
and when, as sometimes happens, the tone of the glass jumps an 
octave higher I have noted that at this moment each of the 
aforesaid waves divides into two; a phenomenon which shows 
clearly that the ratio involved in the octave [forma delV ottava] 
is two. 

Sagr. More than once have I observed this same thing, much 
to my delight and also to my profit. For a long time I have been 
perplexed about these different harmonies since the explanations 
hitherto given by those learned in music impress me as not 
sufficiently conclusive. They tell us that the diapason, i.e. the 
octave, involves the ratio of two, that the diapente which we call 
the fifth involves a ratio of 3 :2, etc. ; because if the open string of 
a monochord be sounded and afterwards a bridge be placed in 
the middle and the half length be sounded one hears the octave; 
and if the bridge be placed at 1/3 the length of the string, then on 
plucking first the open string and afterwards 2/3 of its length, the 
fifth is given; for this reason they say that the octave depends 
upon the ratio of two to one [contenuta trdldue e I'uno] and the 
fifth upon the ratio of three to two. This explanation does not 
impress me as sufficient to establish 2 and 3/2 as the natural 
ratios of the octave and the fifth; and my reason for thinking so 
is as follows. There are three different ways in which the tone of 
a string may be sharpened, namely, by shortening it, by stretching 
it and by making it thinner. If the tension and size of the string 
remain constant one obtains the octave by shortening it to one- 
half, i.e., by sounding first the open string and then one-half of it; 
but if length and size remain constant and one attempts to produce 
the octave by stretching he will find that it does not suffice to 
double the stretching weight; it must be quadrupled; so that, if 
the fundamental note is produced by a weight of one pound, four 
will be required to bring out the octave. 

And finally if the length and tension remain constant, while 
one changes the size of the string he will find that in order to 
produce the octave the size must be reduced to 1/4 that which 


gave the fundamental. And what I have said concerning the 
octave, namely, that its ratio as derived from the tension and 
size of the string is the square of that derived from the length, 
applies equally well to all other musical intervals [intervalli 
musici]. Thus if one wishes to produce a fifth by changing the 
length he finds that the ratio of the lengths must be sesquialteral, 
in other words he sounds first the open string, then two-thirds 
of it; but if he wishes to produce this same result by stretching or 
thinning the string, then it becomes necessary to square the ratio 
3/2 that is by taking 9/4 [dupla sesquiquarta]; accordingly, if the 
fundamental requires a weight of 4 pounds, the higher note will 
be produced not by 6, but by 9 pounds; the same is true in regard 
to size, the string which gives the fundamental is larger than that 
which yields the fifth in the ratio of 9 to 4. 

In view of these facts, I see no reason why those wise philo- 
sophers should adopt 2 rather than 4 as the ratio of the octave, 
or why in the case of the fifth they should employ the sesquialteral 
ratio, 3/2, rather than that of 9/4. Since it is impossible to count 
the vibrations of a sounding string on account of its high 
frequency, I should still have been in doubt as to whether a string, 
emitting the upper octave, made twice as many vibrations in the 
same time as one giving the fundamental, had it not been for the 
following fact, namely, that at the instant when the tone jumps 
to the octave, the waves which constantly accompany the 
vibrating glass divide up into smaller ones which are precisely 
half as long as the former. 

Salv. This is a beautiful experiment enabling us to distinguish 
individually the waves which are produced by the vibrations of a 
sonorous body, which spread through the air, bringing to the 
tympanum of the ear a stimulus which the mind translates into 
sound. But since these waves in the water last only so long as the 
friction of the finger continues and are, even then, not constant 
but are always forming and disappearing, would it not be a fine 
thing if one had the ability to produce waves which would persist 
for a long while, even months and years, so as to easily measure 
and count them ? 


Sagr. Such an invention would, I assure you, command my 

Salv. The device is one which I hit upon by accident; my part 
consists merely in the observation of it and in the appreciation of 
its value as a confirmation of something to which I had given 
profound consideration; and yet the device is, in itself, rather 
common. As I was scraping a brass plate with a sharp iron 
chisel in order to remove some spots from it and was running 
the chisel rather rapidly over it, I once or twice, during many 
strokes, heard the plate emit a rather strong and clear whistling 
sound; on looking at the plate more carefully, I noticed a long 
row of fine streaks parallel and equidistant from one another. 
Scraping with the chisel over and over again, I noticed that it 
was only when the plate emitted this hissing noise that any 
marks were left upon it; when the scraping was not accompanied 
by this sibilant note there was not the least trace of such marks. 
Repeating the trick several times and making the stroke, now 
with greater now with less speed, the whistling followed with a 
pitch which was correspondingly higher and lower. I noted also 
that the marks made when the tones were higher were closer 
together; but when the tones were deeper, they were farther apart. 
I also observed that when, during a single stroke, the speed 
increased toward the end the sound became sharper and the 
streaks grew closer together, but always in such a way as to 
remain sharply defined and equidistant. Besides whenever the 
stroke was accompanied by hissing I felt the chisel tremble in my 
grasp and a sort of shiver run through my hand. In short we see 
and hear in the case of the chisel precisely that which is seen and 
heard in the case of a whisper followed by a loud voice ; for, when 
the breath is emitted without the production of a tone, one does 
not feel either in the throat or mouth any motion to speak of in 
comparison with that which is felt in the larynx and upper part 
of the throat when the voice is used, especially when the tones 
employed are low and strong. 

At times I have also observed among the strings of the spinet 
two which were in unison with two of the tones produced by the 


aforesaid scraping; and among those which differed most in pitch 
I found two which were separated by an interval of a perfect 
fifth. Upon measuring the distance between the markings 
produced by the two scrapings it was found that the space which 
contained 45 of one contained 30 of the other, which is precisely 
the ratio assigned to the fifth. 

But now before proceeding any farther I want to call your 
attention to the fact that, of the three methods for sharpening a 
tone, the one which you refer to as the fineness of the string 
should be attributed to its weight. So long as the material of 
the string is unchanged, the size and weight vary in the same 
ratio. Thus in the case of gut-strings, we obtain the octave by 
making one string 4 times as large as the other; so also in the 
case of brass one wire must have 4 times the size of the other; 
but if now we wish to obtain the octave of a gut-string, by use of 
brass wire, we must make it, not four times as large, but four 
times as heavy as the gut-string: as regards size therefore the 
metal string is not four times as big but four times as heavy. 
The wire may therefore be even thinner than the gut notwith- 
standing the fact that the latter gives the higher note. Hence if 
two spinets are strung, one with gold wire the other with brass, 
and if the corresponding strings each have the same length, 
diameter, and tension it follows that the instrument strung with 
gold will have a pitch about one-fifth lower than the other 
because gold has a density almost twice that of brass. And here 
it is to be noted that it is the weight rather than the size of a 
moving body which offers resistance to change of motion [velocitd 
del moto] contrary to what one might at first glance think. For 
it seems reasonable to believe that a body which is large and light 
should suffer greater retardation of motion in thrusting aside the 
medium than would one which is thin and heavy; yet here 
exactly the opposite is true. 

Returning now to the original subject of discussion, I assert 
that the ratio of a musical interval is not immediately determined 
either by the length, size, or tension of the strings but rather by 
the ratio of their frequencies, that is, by the number of pulses of 


air waves which strike the tympanum of the ear, causing it also 
to vibrate with the same frequency. This fact established, we may 
possibly explain why certain pairs of notes, differing in pitch 
produce a pleasing sensation, others a less pleasant effect, and 
still others a disagreeable sensation. Such an explanation would 
be tantamount to an explanation of the more or less perfect 
consonances and of dissonances. The unpleasant sensation 
produced by the latter arises, I think, from the discordant 
vibrations of two different tones which strike the ear out of time 
[sproporzionatamente]. Especially harsh is the dissonance between 
notes whose frequencies are incommensurable; such a case occurs 
when one has two strings in unison and sounds one of them open, 
together with a part of the other which bears the same ratio to 
its whole length as the side of a square bears to the diagonal; this 
yields a dissonance similar to the augmented fourth or diminished 
fifth [tritono o semidiapente]. 

Agreeable consonances are pairs of tones which strike the ear 
with a certain regularity; this regularity consists in the fact that 
the pulses delivered by the two tones, in the same interval of time, 
shall be commensurable in number, so as not to keep the ear drum 
in perpetual torment, bending in two different directions in order 
to yield to the ever-discordant impulses. 

The first and most pleasing consonance is, therefore, the 
octave since, for every pulse given to the tympanum by the lower 
string, the sharp string delivers two; accordingly at every other 
vibration of the upper string both pulses are delivered simul- 
taneously so that one-half the entire number of pulses are 
delivered in unison. But when two strings are in unison their 
vibrations always coincide and the effect is that of a single string; 
hence we do not refer to it as consonance. The fifth is also a 
pleasing interval since for every two vibrations of the lower 
string the upper one gives three, so that considering the entire 
number of pulses from the upper string one-third of them will 
strike in unison, i.e., between each pair of concordant vibrations 
there intervene two single vibrations; and when the interval is 
a fourth, three single vibrations intervene. In case the interval is 


a second where the ratio is 9/8 it is only every ninth vibration of 
the upper string which reaches the ear simultaneously with one of 
the lower; all the others are discordant and produce a harsh 
effect upon the recipient ear which interprets them as dissonances. 



(a) How Fast? 

Although the ancients used the notion of speed, they were more concerned 
with the fact that a body went from one place to another — and why, rather 
than how and how much. Comparisons of speeds were made only in the 
form of ratios; there was little interest in speed per se; for example, in its 
variation along the path of the moving body. Galileo, however, carefully 
defined uniform motion, particularly with respect to all intervals of time, 
large or small. (He did not have a clear idea of so-called instantaneous speed, 
which is defined mathematically in terms of a derivative, i.e. a limit; he did 
regard a body as passing through all degrees of speed, including zero speed 
as one infinitely slow.) His analysis was based upon four axioms which 
guaranteed that the distance traversed is a monotonically increasing function 
of the speed and time — a linear proportionality is tacitly assumed. 

Galileo suggested a method for measuring the speed of light; namely, to 
determine the time for a signal to go from one spot to another several miles 
away. Despite the failure to record actually any time difference for a distance 
less than a mile, he believed from his observation of the spreading of a 
lightning flash among the clouds that a finite time is involved. Later the 
Accademia del Cimento repeated this experiment, again with a negative 
result. The speed of sound, however, was successfully determined by the 
Academy, viz. by noting the time for a discharged cannon at a known 
distance away to be heard after the flash had been observed. 

Dialogues Concerning Two New Sciences^ 


In dealing with steady or uniform motion, we need a single 
definition which I give as follows: 

t Ref. (9), pp. 154-5, 42-4. 


falling bodies 185 

By steady or uniform motion, I mean one in which the distances 
traversed by the moving particle during any equal intervals of 
time, are themselves equal. 


We must add to the old definition (which defined steady motion 
simply as one in which equal distances are traversed in equal 
times) the word "any," meaning by this, all equal intervals of 
time; for it may happen that the moving body will traverse equal 
distances during some equal intervals of time and yet the distances 
traversed during some small portion of these time-intervals may 
not be equal, even though the time-intervals be equal. 

From the above definition, four axioms follow, namely: 

Axiom I 
In the case of one and the same uniform motion, the distance 
traversed during a longer interval of time is greater than the 
distance traversed during a shorter interval of time. 

Axiom II 
In the case of one and the same uniform motion, the time 
required to traverse a greater distance is longer than the time 
required for a less distance. 

Axiom III 
In one and the same interval of time, the distance traversed at a 
greater speed is larger than the distance traversed at a less speed. 

Axiom IV 
The speed required to traverse a longer distance is greater than 
that required to traverse a shorter distance during the same 

Sagr. But of what kind and how great must we consider this 
speed of light to be? Is it instantaneous or momentary or does 
it like other motions require time? Can we not decide this by 
experiment ? 


Simp. Everyday experience shows that the propagation of light 
is instantaneous; for when we see a piece of artillery fired, at great 
distance, the flash reaches our eyes without lapse of time; but the 
sound reaches the ear only after a noticeable interval. 

Sagr. Well, Simplicio, the only thing I am able to infer from 
this familiar bit of experience is that sound, in reaching our 
ear, travels more slowly than light; it does not inform me whether 
the coming of the light is instantaneous or whether, although 
extremely rapid, it still occupies time. An observation of this 
kind tells us nothing more than one in which it is claimed that 
"As soon as the sun reaches the horizon its light reaches our 
eyes"; but who will assure me that these rays had not reached 
this limit earlier than they reached our vision? 

Salv. The small conclusiveness of these and other similar 
observations once led me to devise a method by which one might 
accurately ascertain whether illumination, i.e., the propagation 
of light, is really instantaneous. The fact that the speed of sound 
is as high as it is, assures us that the motion of light cannot fail 
to be extraordinarily swift. The experiment which I devised was 
as follows : 

Let each of two persons take a light contained in a lantern, or 
other receptacle, such that by the interposition of the hand, the 
one can shut off or admit the light to the vision of the other. 
Next let them stand opposite each other at a distance of a few 
cubits and practice until they acquire such skill in uncovering 
and occulting their lights that the instant one sees the light of his 
companion he will uncover his own. After a few trials the 
response will be so prompt that without sensible error [svario] 
the uncovering of one light is immediately followed by the 
uncovering of the other, so that as soon as one exposes his light he 
will instantly see that of the other. Having acquired skill at this 
short distance let the two experimenters, equipped as before, 
take up positions separated by a distance of two or three miles 
and let them perform the same experiment at night, noting care- 
fully whether the exposures and occultations occur in the same 
manner as at short distances; if they do, we may safely conclude 


that the propagation of light is instantaneous; but if time is 
required at a distance of three miles which, considering the going 
of one light and the coming of the other, really amounts to six, 
then the delay ought to be easily observable. If the experiment 
is to be made at still greater distances, say eight or ten miles, 
telescopes may be employed, each observer adjusting one for 
himself at the place where he is to make the experiment at night; 
then although the lights are not large and are therefore invisible 
to the naked eye at so great a distance, they can readily be 
covered and uncovered since by aid of the telescopes, once 
adjusted and fixed, they will become easily visible. 

Sagr. This experiment strikes me as a clever and reliable 
invention. But tell us what you conclude from the results. 

Salv. In fact I have tried the experiment only at a short 
distance, less than a mile, from which I have not been able to 
ascertain with certainty whether the appearance of the opposite 
light was instantaneous or not; but if not instantaneous it is 
extraordinarily rapid — I should call it momentary; and for the 
present I should compare it to motion which we see in the 
lightning flash between clouds eight or ten miles distant from us. 
We see the beginning of this light — I might say its head and 
source — located at a particular place among the clouds; but it 
immediately spreads to the surrounding ones, which seems to be 
an argument that at least some time is required for propagation; 
for if the illumination were instantaneous and not gradual, we 
should not be able to distinguish its origin — its center, so to 
speak — from its outlying portions. What a sea we are gradually 
slipping into without knowing it! With vacua and infinities and 
indivisibles and instantaneous motions, shall we ever be able, 
even by means of a thousand discussions, to reach dry land? 

(b) Inertia 

There has been much controversy as to Galileo's contribution to our 
modern conception of inertia. On the one hand, he has been praised for 
having been solely responsible for its introduction; on the other hand, he 
has been accused of not even understanding it. The truth, I suppose, is 


somewhere between these extreme views. Without question Galileo did not 
conceive of the generalized axiom that was later proposed by Isaac Newton 
(1642-1727) as the first law of motion; nor did he apply any such inertial 
principle to celestial motions. 

Inklings of the inertial idea are found in his early notes "On Mechanics", 
where he associated rest and motion as two aspects of the same state — 
contrary to Aristotle's wide separation. In his notes "On Motion", he had 
previously suggested circular motion with constant speed as a neutral, 
third type of motion, neither natural nor violent. The first statement of the 
inertia principle was given (1607) in a letter to him; his first public reference 
in the second Sunspot letter (1613) — repeated (ref. (7), p. 147) in 1632, 
where he also specifically mentioned the forward lunges of a passenger in a 
boat that stops suddenly. It is in the Two New Sciences, however, that the 
physical concept of inertia is definitely formulated and practically utilized. 

Galileo had no clear conception of Newton's second law of motion. 
Nevertheless, he did recognize that the unbalanced force on a body is associated 
with its acceleration and, in the case of the free fall of bodies in a medium, 
that it is proportional to the acceleration (or resulting speed from rest). 

Dialogues Concerning Two New Sciences t 

Furthermore we may remark that any velocity once imparted 
to a moving body will be rigidly maintained as long as the 
external causes of acceleration or retardation are removed, a 
condition which is found only on horizontal planes; for in the 
case of planes which slope downwards there is already present a 
cause of acceleration, while on planes sloping upward there is 
retardation; from this it follows that motion along a horizontal 
plane is perpetual; for, if the velocity be uniform, it cannot be 
diminished or slackened, much less destroyed. Further, although 
any velocity which a body may have acquired through natural 
fall is permanently maintained so far as its own nature [suapte 
natura] is concerned, yet it must be remembered that if, after 
descent along a plane inclined downwards, the body is deflected 
to a plane inclined upward, there is already existing in this latter 
plane a cause of retardation; for in any such plane this same body 
is subject to a natural acceleration downwards. Accordingly we 
have here the superposition of two different states, namely, the 
velocity acquired during the preceding fall which if acting alone 

t Ref. (9), pp. 215-8, 244. 


would carry the body at a uniform rate to infinity, and the 
velocity which results from a natural acceleration downwards 
common to all bodies. It seems altogether reasonable, therefore, 
if we wish to trace the future history of a body which has 
descended along some inclined plane and has been deflected along 
some plane inclined upwards, for us to assume that the maximum 
speed acquired during descent is permanently maintained during 
the ascent. In the ascent, however, there supervenes a natural 
inclination downwards, namely, a motion which, starting from 
rest, is accelerated at the usual rate. If perhaps this discussion is 
a little obscure, the following figure will help to make it clearer. 

Let us suppose that the descent has been made along the 
downward sloping plane AB, from which the body is deflected 
so as to continue its motion along the upward sloping plane BC; 
and first let these planes be of equal length and placed so as to 
make equal angles with the horizontal line GH. Now it is well 
known that a body, starting from rest at A, and descending along 
AB, acquires a speed which is proportional to the time, which is 
a maximum at B, and which is maintained by the body so long as 
all causes of fresh acceleration or retardation are removed ; the 

Fig. 30. 

acceleration to which I refer is that to which the body would be 
subject if its motion were continued along the plane AB extended, 
while the retardation is that which the body would encounter if 
its motion were deflected along the plane BC inclined upwards; 
but, upon the horizontal plane GH, the body would maintain 
a uniform velocity equal to that which it had acquired at B after 
fall from A; moreover this velocity is such that, during an 


interval of time equal to the time of descent through AB, the body 
will traverse a horizontal distance equal to twice AB. Now let us 
imagine this same body to move with the same uniform speed 
along the plane BC so that here also during a time-interval equal 
to that of descent along AB, it will traverse along BC extended a 
distance twice AB ; but let us suppose that, at the very instant the 
body begins its ascent it is subjected, by its very nature, to the 
same influences which surrounded it during its descent from A 
along AB, namely, it descends from rest under the same accelera- 
tion as that which was effective in AB, and it traverses, during an 
equal interval of time, the same distance along this second plane 
as it did along AB ; it is clear that, by thus superposing upon the 
body a uniform motion of ascent and an accelerated motion of 
descent, it will be carried along the plane BC as far as the point 
C where these two velocities become equal. 

If now we assume any two points D and E, equally distant 
from the vertex B, we may then infer that the descent along BD 
takes place in the same time as the ascent along BE. Draw DF 

Fig. 31. 

parallel to BC; we know that, after descent along AD, the body 
will ascend along DF; or, if, on reaching D, the body is carried 
along the horizontal DE, it will reach E with the same momentum 
[impetus] with which it left D ; hence from E the body will ascend 
as far as C, proving that the velocity at E is the same as that at D. 
From this we may logically infer that a body which descends 
along any inclined plane and continues its motion along a plane 
inclined upwards will, on account of the momentum acquired, 
ascend to an equal height above the horizontal; so that if the 


descent is along AB the body will be carried up the plane BC as 
far as the horizontal line ACD: and this is true whether the 
inclinations of the planes are the same or different, as in the case 
of the planes AB and BD. But by a previous postulate the speeds 
acquired by fall along variously inclined planes having the same 
vertical height are the same. If therefore the planes EB and BD 
have the same slope, the descent along EB will be able to drive the 
body along BD as far as D; and since this propulsion comes from 
the speed acquired on reaching the point B, it follows that this 
speed at B is the same whether the body has made its descent 
along AB or EB. Evidently then the body will be carried up BD 
whether the descent has been made along AB or along EB. The 
time of ascent along BD is however greater than that along BC, 
just as the descent along EB occupies more time that that along 
AB; moreover it has been demonstrated that the ratio between 
the lengths of these times is the same as that between the lengths 
of the planes. 

Salviati. Once more, Simplicio is here on time; so let us 
without delay take up the question of motion. The text of our 
Author is as follows : 

The Motion of Projectiles 

In the preceding pages we have discussed the properties of 
uniform motion and of motion naturally accelerated along 
planes of all inclinations. I now propose to set forth those 
properties which belong to a body whose motion is compounded 
of two other motions, namely, one uniform and one naturally 
accelerated; these properties, well worth knowing, I propose to 
demonstrate in a rigid manner. This is the kind of motion seen in 
a moving projectile; its origin I conceive to be as follows: 

Imagine any particle projected along a horizontal plane without 
friction ; then we know, from what has been more fully explained 
in the preceding pages, that this particle will move along this 
same plane with a motion which is uniform and perpetual, 
provided the plane has no limits. But if the plane is limited and 


elevated, then the moving particle, which we imagine to be a 
heavy one, will on passing over the edge of the plane acquire, in 
addition to its previous uniform and perpetual motion, a down- 
ward propensity due to its own weight; so that the resulting 
motion which I call projection [projectio], is compounded of one 
which is uniform and horizontal and of another which is vertical 
and naturally accelerated. 

(c) A Thought Experiment 

Aristotle had argued that the speed of a freely falling body should be in 
proportion to its weight, the "cause" of its fall. To test this conclusion, 
Galileo proposed a so-called (critical) thought-experiment. Consider the 
combination of a large weight with a small one. On the one hand, the large 
one should be retarded by the slower, small weight; on the other hand, the 
combination, having a still greater weight, should move faster even than the 
large one alone. Obviously, these inconsistent deductions must be based 
upon a false assumption, namely, the proportion between speed and weight. 
Galileo indicated physically how any such logical inconsistencies disappear 
if all bodies are assumed to fall freely with the same speed. 

Dialogues Concerning Two New Sciences^ 

Salv. I greatly doubt that Aristotle ever tested by experiment 
whether it be true that two stones, one weighing ten times as much 
as the other, if allowed to fall, at the same instant, from a height 
of, say, 100 cubits, would so differ in speed that when the heavier 
had reached the ground, the other would not have fallen more 
than 10 cubits. 

Simp. His language would seem to indicate that he had tried 
the experiment, because he says: We see the heavier; now the 
word see shows that he had made the experiment. 

Sagr. But I, Simplicio, who have made the test can assure 
you that a cannon ball weighing one or two hundred pounds, or 
even more, will not reach the ground by as much as a span ahead 
of a musket ball weighing only half a pound, provided both are 
dropped from a height of 200 cubits. 

t Ref. (9), pp. 62-7. 


Salv. But, even without further experiment, it is possible to 
prove clearly, by means of a short and conclusive argument, that 
a heavier body does not move more rapidly than a lighter one 
provided both bodies are of the same material and in short such 
as those mentioned by Aristotle. But tell me, Simplicio, whether 
you admit that each falling body acquires a definite speed fixed 
by nature, a velocity which cannot be increased or diminished 
except by the use of force [violenza] or resistance. 

Simp. There can be no doubt but that one and the same body 
moving in a single medium has a fixed velocity which is deter- 
mined by nature and which cannot be increased except by the 
addition of momentum [impeto] or diminished except by some 
resistance which retards it. 

Salv. If then we take two bodies whose natural speeds are 
different, it is clear that on uniting the two, the more rapid one 
will be partly retarded by the slower, and the slower will be 
somewhat hastened by the swifter. Do you not agree with me in 
this opinion? 

Simp. You are unquestionably right. 

Salv. But if this is true, and if a large stone moves with a 
speed of, say, eight while a smaller moves with a speed of four, 
then when they are united, the system will move with a speed less 
than eight; but the two stones when tied together make a stone 
larger than that which before moved with a speed of eight. 
Hence the heavier body moves with less speed than the lighter; 
an effect which is contrary to your supposition. Thus you see 
how, from your assumption that the heavier body moves more 
rapidly than the lighter one, I infer that the heavier body moves 
more slowly. 

Simp. I am all at sea because it appears to me that the smaller 
stone when added to the larger increases its weight and by adding 
weight I do not see how it can fail to increase its speed or, at 
least, not to diminish it. 

Salv. Here again you are in error, Simplicio, because it is not 
true that the smaller stone adds weight to the larger. 

Simp. This is, indeed, quite beyond my comprehension. 


Salv. It will not be beyond you when I have once shown you 
the mistake under which you are laboring. Note that it is 
necessary to distinguish between heavy bodies in motion and the 
same bodies at rest. A large stone placed in a balance not only 
acquires additional weight by having another stone placed upon 
it, but even by the addition of a handful of hemp its weight is 
augmented six to ten ounces according to the quantity of hemp. 
But if you tie the hemp to the stone and allow them to fall freely 
from some height, do you believe that the hemp will press down 
upon the stone and thus accelerate its motion or do you think the 
motion will be retarded by a partial upward pressure ? One always 
feels the pressure upon his shoulders when he prevents the 
motion of a load resting upon him; but if one descends just as 
rapidly as the load would fall how can it gravitate or press upon 
him? Do you not see that this would be the same as trying to 
strike a man with a lance when he is running away from you with 
a speed which is equal to, or even greater, than that with which 
you are following him? You must therefore conclude that, 
during free and natural fall, the small stone does not press upon 
the larger and consequently does not increase its weight as it does 
when at rest. 

Simp. But what if we should place the larger stone upon the 

Salv. Its weight would be increased if the larger stone moved 
more rapidly; but we have already concluded that when the small 
stone moves more slowly it retards to some extent the speed of the 
larger, so that the combination of the two, which is a heavier 
body than the larger of the two stones, would move less rapidly, 
a conclusion which is contrary to your hypothesis. We infer 
therefore that large and small bodies move with the same speed 
provided they are of the same specific gravity. 

(d) The Medium's Role 

Nowadays in analyzing the free fall of a body we would begin by identifying 
less important influences, and then proceed to neglect them, if possible. 
Our everyday experience, for example, would lead us to regard the medium 


as having a less significant role than the attracting earth. Galileo, however, 
concentrated initially upon the medium itself, which seemed all-important in 
common phenomena such as the dropping of a heavy stone and the fluttering 
of a light leaf, or the fall of a solid gold metal and a thin gold leaf. He began, 
therefore, by investigating the behavior of bodies in different media. 

Aristotle had supposed that a body's speed in a medium varies inversely 
as the medium's density (regarded as its "resistance" to the motion). On 
this basis, the same body should move much more rapidly in air than in 
water. Inasmuch as Aristotle had supposed water to be only 10 times denser 
than air, in general, a body should fall with one-tenth the speed in water as 
compared with that in air. If, on the other hand, a material with a density 
one-tenth that of water, and consequently with one-tenth the speed in water, 
were to fall in air, its speed there should be equivalent to that of similarly 
dense wood, which, however, actually rises in water. A real dilemma! The 
assumed role of the medium appeared questionable. 

Let us consider the problem in modern terms. The unbalanced force F 
on an immersed body of density D and volume Fin a medium of density d is 

F={DV-dV)g , 
where g is the acceleration due to gravity. By Newton's second law of motion 

F = (DV)a , 
where DV is the mass of the body and a its resulting acceleration. 

For the same body falling from rest in two media of densities d and a", we 
have for their resulting speeds v and by v', respectively, 

_v .£_ — D ~ d 

v' ~ a' ~ D-d' ' 

For Galileo's particular example of a body (D = 1.2), in air (d = 0) and in 
water (d — 1), we have 

For two bodies (1 and 2) of the same volume in the same medium, we have 

CPi-<0 Diai 

(D 2 —d) D 2 a 2 

For free fall from rest, with negligible friction, the speed is proportional to 
the acceleration; hence in the same time 



a\ _ 
a 2 


v 2 



D 2 

V 2 

D 2 -d 

' D X 




l-d/D 2 


In the case of a medium of neglible density, say, air, the speeds of new bodies 

differing only in weight would be equal. 

Let us take Galileo's example of a marble egg and a hen's egg falling in water. 

The densities are approximately 2.7, 1.2, and 1.0, respectively, in cgs units. 


V 2 

Hence the marble egg would fall much faster than the hen's egg and their 
distance apart would further increase with time. On the other hand, if the 
density of the medium were gradually decreased, the relative speed would 
approach unity and the distance apart zero. On this basis, Galileo concluded 
that "in a vacuum all bodies would fall with the same speed" — not an 
a priori assumption, but an experimental extrapolation. 

Dialogues Concerning Two New Sciences^ 

Simp. Your discussion is really admirable; yet I do not find it 
easy to believe that a bird-shot falls as swiftly as a cannon ball. 

Salv. Why not say a grain of sand as rapidly as a grindstone? 
But, Simplicio, I trust you will not follow the example of many 
others who divert the discussion from its main intent and fasten 
upon some statement of mine which lacks a hair's-breadth of the 
truth and, under this hair, hide the fault of another which is as 
big as a ship's cable. Aristotle says that "an iron ball of one 
hundred pounds falling from a height of one hundred cubits 
reaches the ground before a one-pound ball has fallen a single 
cubit." I say that they arrive at the same time. You find, on 
making the experiment, that the larger outstrips the smaller by 
two finger-breadths, that is, when the larger has reached the 
ground, the other is short of it by two finger-breadths; now you 
would not hide behind these two fingers the ninety-nine cubits of 
Aristotle, nor would you mention my small error and at the same 
time pass over in silence his very large one. Aristotle declares 
that bodies of different weights, in the same medium, travel (in so 
far as their motion depends upon gravity) with speeds which are 
proportional to their weights; this he illustrates by use of bodies 
in which it is possible to perceive the pure and unadulterated 

t Ref. (9), pp. 64-8, 71-7. 


effect of gravity, eliminating other considerations, for example, 
figure as being of small importance [minimi momenti], influences 
which are greatly dependent upon the medium which modifies 
the single effect of gravity alone. Thus we observe that gold, the 
densest of all substances, when beaten out into a very thin leaf, 
goes floating through the air; the same thing happens with stone 
when ground into a very fine powder. But if you wish to maintain 
the general proposition you will have to show that the same ratio 
of speeds is preserved in the case of all heavy bodies, and that a 
stone of twenty pounds moves ten times as rapidly as one of two; 
but I claim that this is false and that, if they fall from a height of 
fifty or a hundred cubits, they will reach the earth at the same 

Simp. Perhaps the result would be different if the fall took 
place not from a few cubits but from some thousands of cubits. 

Salv. If this were what Aristotle meant you would burden 
him with another error which would amount to a falsehood; 
because, since there is no such sheer height available on earth, it 
is clear that Aristotle could not have made the experiment; yet 
he wishes to give us the impression of his having performed it 
when he speaks of such an effect as one which we see. 

Simp. In fact, Aristotle does not employ this principle, but 
uses the other one which is not, I believe, subject to these same 

Salv. But the one is as false as the other; and I am surprised 
that you yourself do not see the fallacy and that you do not 
perceive that if it were true that, in media of different densities 
and different resistances, such as water and air, one and the same 
body moved in air more rapidly than in water, in proportion as 
the density of water is greater than that of air, then it would 
follow that any body which falls through air ought also to fall 
through water. But this conclusion is false inasmuch as many 
bodies which descend in air not only do not descend in water, 
but actually rise. 

Simp. I do not understand the necessity of your inference; and 
in addition I will say that Aristotle discusses only those bodies 


which fall in both media, not those which fall in air but rise 
in water. 

Salv. The arguments which you advance for the Philosopher 
are such as he himself would have certainly avoided so as not to 
aggravate his first mistake. But tell me now whether the density 
[corpulenza] of the water, or whatever it may be that retards the 
motion, bears a definite ratio to the density of air which is less 
retardative; and if so fix a value for it at your pleasure. 

Simp. Such a ratio does exist; let us assume it to be ten; then, 
for a body which falls in both these media, the speed in water 
will be ten times slower than in air. 

Salv. I shall now take one of those bodies which fall in air 
but not in water, say a wooden ball, and I shall ask you to assign 
to it any speed you please for its descent through air. 

Simp. Let us suppose it moves with a speed of twenty. 

Salv. Very well. Then it is clear that this speed bears to some 
smaller speed the same ratio as the density of water bears to that 
of air; and the value of this smaller speed is two. So that really 
if we follow exactly the assumption of Aristotle we ought to infer 
that the wooden ball which falls in air, a substance ten times less- 
resisting than water, with a speed of twenty would fall in water 
with a speed of two, instead of coming to the surface from the 
bottom as it does; unless perhaps you wish to reply, which I do 
not believe you will, that the rising of the wood through the water 
is the same as its falling with a speed of two. But since the wooden 
ball does not go to the bottom, I think you will agree with me that 
we can find a ball of another material, not wood, which does fall 
in water with a speed of two. 

Simp. Undoubtedly we can; but it must be of a substance 
considerably heavier than wood. 

Salv. That is it exactly. But if this second ball falls in water 
with a speed of two, what will be its speed of descent in air ? 
If you hold to the rule of Aristotle you must reply that it will 
move at the rate of twenty; but twenty is the speed which you 
yourself have already assigned to the wooden ball; hence this 
and the other heavier ball will each move through air with the 


same speed. But now how does the Philosopher harmonize this 
result with his other, namely, that bodies of different weight move 
through the same medium with different speeds — speeds which 
are proportional to their weights? But without going into the 
matter more deeply, how have these common and obvious 
properties escaped your notice? Have you not observed that two 
bodies which fall in water, one with a speed a hundred times as 
great as that of the other, will fall in air with speeds so nearly 
equal that one will not surpass the other by as much as one 
hundredth part? Thus, for example, an egg made of marble will 
descend in water one hundred times more rapidly than a hen's 
egg, while in air falling from a height of twenty cubits the one will 
fall short of the other by less than four finger-breadths. In short, 
a heavy body which sinks through ten cubits of water in three 
hours will traverse ten cubits of air in one or two pulse-beats ; 
and if the heavy body be a ball of lead it will easily traverse the 
ten cubits of water in less than double the time required for ten 
cubits of air. And here, I am sure, Simplicio, you find no ground 
for difference or objection. We conclude, therefore, that the 
argument does not bear against the existence of a vacuum; but 
if it did, it would only do away with vacua of considerable size 
which neither I nor, in my opinion, the ancients ever believed to 
exist in nature, although they might possibly be produced by 
force [violenza] as may be gathered from various experiments 
whose description would here occupy too much time. 

Sagr. Seeing that Simplicio is silent, I will take the opportunity 
of saying something. Since you have clearly demonstrated that 
bodies of different weights do not move in one and the same 
medium with velocities proportional to their weights, but that 
they all move with the same speed, understanding of course that 
they are of the same substance or at least of the same specific 
gravity; certainly not of different specific gravities, for I hardly 
think you would have us believe a ball of cork moves with the 
same speed as one of lead; and again since you have clearly 
demonstrated that one and the same body moving through 
differently resisting media does not acquire speeds which are 


inversely proportional to the resistances, I am curious to learn 
what are the ratios actually observed in these cases. 

Salv. These are interesting questions and I have thought 
much concerning them. I will give you the method of approach 
and the result which I finally reached. Having once established 
the falsity of the proposition that one and the same body moving 
through differently resisting media acquires speeds which are 
inversely proportional to the resistances of these media, and 
having also disproved the statement that in the same medium 
bodies of different weight acquire velocities proportional to their 
weights (understanding that this applies also to bodies which 
differ merely in specific gravity), I then began to combine these 
two facts and to consider what would happen if bodies of different 
weight were placed in media of different resistances; and I found 
that the differences in speed were greater in those media which 
were more resistant, that is, less yielding. This difference was 
such that two bodies which differed scarcely at all in their speed 
through air would, in water, fall the one with a speed ten times as 
great as that of the other. Further, there are bodies which will 
fall rapidly in air, whereas if placed in water not only will not 
sink but will remain at rest or will even rise to the top : for it is 
possible to find some kinds of wood, such as knots and roots, 
which remain at rest in water but fall rapidly in air. 

Salv. Returning from this digression, let us again take up 
our problem. We have already seen that the difference of speed 
between bodies of different specific gravities is most marked in 
those media which are the most resistant: thus, in a medium of 
quicksilver, gold not merely sinks to the bottom more rapidly 
than lead but it is the only substance that will descend at all; all 
other metals and stones rise to the surface and float. On the other 
hand the variation of speed in air between balls of gold, lead, 
copper, porphyry, and other heavy materials is so slight that in a 
fall of 100 cubits a ball of gold would surely not outstrip one of 
copper by as much as four fingers. Having observed this I came 
to the conclusion that in a medium totally devoid of resistance 
all bodies would fall with the same speed. 


Simp. This is a remarkable statement, Salviati. But I shall 
never believe that even in a vacuum, if motion in such a place 
were possible, a lock of wool and a bit of lead can fall with the 
same velocity. 

Salv. A little more slowly, Simplicio. Your difficulty is not 
so recondite nor am I so imprudent as to warrant you in believing 
that I have not already considered this matter and found 
the proper solution. Hence for my justification and for your 
enlightenment hear what I have to say. Our problem is to find 
out what happens to bodies of different weight moving in a 
medium devoid of resistance, so that the only difference in speed 
is that which arises from inequality of weight. Since no medium 
except one entirely free from air and other bodies, be it ever so 
tenuous and yielding, can furnish our senses with the evidence 
we are looking for, and since such a medium is not available, we 
shall observe what happens in the rarest and least resistant media 
as compared with what happens in denser and more resistant 
media. Because if we find as a fact that the variation of speed 
among bodies of different specific gravities is less and less accord- 
ing as the medium becomes more and more yielding, and if 
finally in a medium of extreme tenuity, though not a perfect 
vacuum, we find that, in spite of great diversity of specific 
gravity [peso], the difference in speed is very small and almost 
inappreciable, then we are justified in believing it highly probable 
that in a vacuum all bodies would fall with the same speed. Let 
us, in view of this, consider what takes place in air, where for the 
sake of a definite figure and light material imagine an inflated 
bladder. The air in this bladder when surrounded by air will 
weigh little or nothing, since it can be only slightly compressed; 
its weight then is small being merely that of the skin which does 
not amount to the thousandth part of a mass of lead having the 
same size as the inflated bladder. Now, Simplicio, if we allow 
these two bodies to fall from a height of four or six cubits, by 
what distance do you imagine the lead will anticipate the bladder? 
You may be sure that the lead will not travel three times, or even 
twice, as swiftly as the bladder, although you would have made it 


move a thousand times as rapidly. 

Simp. It may be as you say during the first four or six cubits of 
the fall; but after the motion has continued a long while, I believe 
that the lead will have left the bladder behind not only six out of 
twelve parts of the distance but even eight or ten. 

Salv. I quite agree with you and doubt not that, in very long 
distances, the lead might cover one hundred miles while the 
bladder was traversing one; but, my dear Simplicio, this phenom- 
enon which you adduce against my proposition is precisely the 
one which confirms it. Let me once more explain that the 
variation of speed observed in bodies of different specific gravities 
is not caused by the difference of specific gravity but depends 
upon external circumstances and, in particular, upon the resist- 
ance of the medium, so that if this is removed all bodies would 
fall with the same velocity; and this result I deduce mainly from 
the fact which you have just admitted and which is very true, 
namely, that, in the case of bodies which differ widely in weight, 
their velocities differ more and more as the spaces traversed 
increase, something which would not occur if the effect depended 
upon differences of specific gravity. For since these specific 
gravities remain constant, the ratio between the distances 
traversed ought to remain constant whereas the fact is that this 
ratio keeps on increasing as the motion continues. Thus a very 
heavy body in a fall of one cubit will not anticipate a very light 
one by so much as the tenth part of this space; but in a fall of 
twelve cubits the heavy body would outstrip the other by one- 
third, and in a fall of one hundred cubits by 90/100, etc. 

Simp. Very well: but, following your own line of argument, 
if differences of weight in bodies of different specific gravities 
cannot produce a change in the ratio of their speeds, on the 
ground that their specific gravities do not change, how is it 
possible for the medium, which also we suppose to remain con- 
stant, to bring about any change in the ratio of these velocities? 

Salv. This objection with which you oppose my statement is 
clever; and I must meet it. I begin by saying that a heavy body 
has an inherent tendency to move with a constantly and uniformly 


accelerated motion toward the common center of gravity, that is, 
toward the center of our earth, so that during equal intervals of 
time it receives equal increments of momentum and velocity. 
This, you must understand, holds whenever all external and 
accidental hindrances have been removed; but of these there is 
one which we can never remove, namely, the medium which 
must be penetrated and thrust aside by the falling body. This 
quiet, yielding, fluid medium opposes motion through it with a 
resistance which is proportional to the rapidity with which the 
medium must give way to the passage of the body; which body, 
as I have said, is by nature continuously accelerated so that it 
meets with more and more resistance in the medium and hence 
a diminution in its rate of gain of speed until finally the speed 
reaches such a point and the resistance of the medium becomes 
so great that, balancing each other, they prevent any further 
acceleration and reduce the motion of the body to one which is 
uniform and which will thereafter maintain a constant value. 
There is, therefore, an increase in the resistance of the medium, 
not on account of any change in its essential properties, but on 
account of the change in rapidity with which it must yield and 
give way laterally to the passage of the falling body which is 
being constantly accelerated. 

Now seeing how great is the resistance which the air offers to 
the slight momentum [momento] of the bladder and how small 
that which it offers to the large weight [peso] of the lead, I am 
convinced that, if the medium were entirely removed, the advan- 
tage received by the bladder would be so great and that coming 
to the lead so small that their speeds would be equalized. 
Assuming this principle, that all falling bodies acquire equal 
speeds in a medium which, on account of a vacuum or something 
else, offers no resistance to the speed of the motion, we shall be 
able accordingly to determine the ratios of the speeds of both 
similar and dissimilar bodies moving either through one and the 
same medium or through different space-filling, and therefore 
resistant, media. This result we may obtain by observing how 
much the weight of the medium detracts from the weight of the 


moving body, which weight is the means employed by the falling 
body to open a path for itself and to push aside the parts of the 
medium, something which does not happen in a vacuum where, 
therefore, no difference [of speed] is to be expected from a 
difference of specific gravity. And since it is known that the 
effect of the medium is to diminish the weight of the body by the 
weight of the medium displaced, we may accomplish our purpose 
by diminishing in just this proportion the speeds of the falling 
bodies, which in a non-resisting medium we have assumed to 
be equal. 

Thus, for example, imagine lead to be ten thousand times as 
heavy as air while ebony is only one thousand times as heavy. 
Here we have two substances whose speeds of fall in a medium 
devoid of resistance are equal: but, when air is the medium, it 
will subtract from the speed of the lead one part in ten thousand, 
and from the speed of the ebony one part in one thousand, i.e. 
ten parts in ten thousand. While therefore lead and ebony 
would fall from any given height in the same interval of time, 
provided the retarding effect of the air were removed, the lead 
will, in air, lose in speed one part in ten thousand; and the ebony, 
ten parts in ten thousand. In other words, if the elevation from 
which the bodies start be divided into ten thousand parts, the 
lead will reach the ground leaving the ebony behind by as much 
as ten, or at least nine, of these parts. Is it not clear then that a 
leaden ball allowed to fall from a tower two hundred cubits 
high will outstrip an ebony ball by less than four inches? Now 
ebony weighs a thousand times as much as air but this inflated 
bladder only four times as much; therefore air diminishes the 
inherent and natural speed of ebony by one part in a thousand; 
while that of the bladder which, if free from hindrance, would 
be the same, experiences a diminution in air amounting to one 
part in four. So that when the ebony ball, falling from the tower, 
has reached the earth, the bladder will have traversed only 
three-quarters of this distance. Lead is twelve times as heavy as 
water; but ivory is only twice as heavy. The speeds of these two 
substances which, when entirely unhindered, are equal will be 


diminished in water, that of lead by one part in twelve, that of 
ivory by half. Accordingly when the lead has fallen through 
eleven cubits of water the ivory will have fallen through only six. 
Employing this principle we shall, I believe, find a much closer 
agreement of experiment with our computation than with that 
of Aristotle. 

In a similar manner we may find the ratio of the speeds of one 
and the same body in different fluid media, not by comparing the 
different resistances of the media, but by considering the excess 
of the specific gravity of the body above those of the media. Thus, 
for example, tin is one thousand times heavier than air and ten 
times heavier than water; hence, if we divide its unhindered speed 
into 1000 parts, air will rob it of one of these parts so that it will 
fall with a speed of 999, while in water its speed will be 900, 
seeing that water diminishes its weight by one part in ten while 
air by only one part in a thousand. 

Again take a solid a little heavier than water, such as oak, a 
ball of which will weigh let us say 1000 drachms; suppose an 
equal volume of water to weigh 950, and an equal volume of air, 
2; then it is clear that if the unhindered speed of the ball is 1000, 
its speed in air will be 998, but in water only 50, seeing that the 
water removes 950 of 1000 parts which the body weighs, leaving 
only 50. 

Such a solid would therefore move almost twenty times as 
fast in air as in water, since its specific gravity exceeds that of 
water by one part in twenty. And here we must consider the 
fact that only those substances which have a specific gravity 
greater than water can fall through it — substances which must, 
therefore, be hundreds of times heavier than air; hence when we 
try to obtain the ratio of the speed in air to that in water, we 
may, without appreciable error, assume that air does not, to any 
considerable extent, diminish the free weight [assoluta gravita], 
and consequently the unhindered speed [assoluta velocitd] of such 
substances. Having thus easily found the excess of the weight of 
these substances over that of water, we can say that their speed 
in air is to their speed in water as their free weight [totale gravita] 


is to the excess of this weight over that of water. For example, 
a ball of ivory weighs 20 ounces; an equal volume of water 
weighs 17 ounces; hence the speed of ivory in air bears to its 
speed in water the approximate ratio of 20: 3. 

Sagr. I have made a great step forward in this truly interesting 
subject upon which I have long labored in vain. 

(e) Changing Speed 

The quantitative definition of acceleration was original with Galileo. It 
grew out of his interest in a changing universe in contrast with the static 
conception of the Greeks, who were concerned with ends rather than with 
the means. He emphasized that his own concept was derived from experi- 
ence. In the case of uniformly accelerated motion he defined the acceleration 
of a moving body to be given by 


a — ~k — > 


where Av is the increment of the body's speed during a time interval At. 
Hence for a body starting from rest 

a = vt . 

He confessed at one time that he himself had supposed incorrectly that 
Av oc As (increment of distance s) rather than Av oc At. 

Having stated clearly his definitions, Galileo then proceeded in typical 
geometrical fashion to deduce consequential theorems. Old familiar geometry 
still afforded the most rigorous reasoning, and was practised more frequently 
than the new algebra. Nowadays we are more accustomed to the latter, and 
particularly to the later analytic geometry. 

His Theorem II, Proposition II (p. 174), related space intervals to the 
squares of the time involved, the so-called law of falling bodies. Starting 
with the definition of acceleration, we have for a body falling from rest 

s = vt , 
where v is the average speed, i.e. 

v = \ (gt-Q) = \gt . 
Hence, upon substitution, 

s = \gt* . 
Consider two successive distances, s\ and si, covered during times t\ and 
<2, respectively. Then, 

s\ = \ gti 2 , 


S2 = \ gti 2 ■ 
Asi = J 2 -ii=i^2 2 -J^i 2 , 



J*i = £*(fi+*2)0ri-f2) • 

fi-*2 = l , 

Asi=\g(ti+ti+i); ti = 1, 2, . . . 
Now inasmuch as *2 and t\ are successive times, the sum (/j+fj+i) must 
always be an odd number so that successive increments of distance, traversed 
in equal time intervals, increase as the odd numbers beginning with 1. 

Speed cannot be easily determined directly; it can be more precisely 
obtained indirectly from measurements of distance and of time. Even here 
time errors are considerable. Galileo ingeniously slowed down the motion 
by rolling balls down an inclined plane. In order to eliminate frictional 
contact between the ball and the plane, later he substituted pendula with 
bobs of lead and of cork. In this connection he was careful to use long, 
fine threads — again to reduce possible errors. Time measurement, however, 
presented a major difficulty; he determined time by weighing the water 
collected from a uniformly flowing source (probably with an accuracy of 
1 sec). 

In discovering that all bodies experience the same acceleration due to 
gravity Galileo did more than find a specific law for a particular body; he 
discovered a general law applicable to all astronomical bodies. 

Galileo continually strove to increase the accuracy of his investigations. 
His description of the precautions needed to reduce friction has become a 
classic — typical of an experientially minded investigator. He emphasized 
again and again the need for repetition of measurements. 

Galileo's formal deductive presentation with geometical reasoning, in the 
spirit of Archimedes, has caused many to regard his method of investigation 
as essentially a priori, i.e. hypothetico-deductive, with phenomena used 
primarily a posteriori for corroboration. The description of his inclined-plane 
experiment is prima facie evidence that experience was closely linked with 
theorizing in his procedure. In all his research Galileo exhibited a critical, 
informal interchange of theory and practice, of reasoning and experiment. 

Galileo surmised that the speed acquired by a body falling freely down 
an inclined plane from a given height is independent of the inclination of 
the plane. Later a theorem along this line was inserted in the Two New 
Sciences — elaborated by Viviani, following a suggestion of Galileo. 

Dialogues Concerning Two New Sciences^ 


The properties belonging to uniform motion have been dis- 
cussed in the preceding section; but accelerated motion remains 
to be considered. 

t Ref. (9), pp. 160-2, 167, 169-70, 174-9, 84-6, 183-4. 


And first of all it seems desirable to find and explain a defini- 
tion best fitting natural phenomena. For anyone may invent an 
arbitrary type of motion and discuss its properties; thus, for 
instance, some have imagined helices and conchoids as described 
by certain motions which are not met with in nature, and have 
very commendably established the properties which these curves 
possess in virtue of their definitions ; but we have decided to con- 
sider the phenomena of bodies falling with an acceleration such 
as actually occurs in nature and to make this definition of 
accelerated motion exhibit the essential features of observed 
accelerated motions. And this, at last, after repeated efforts we 
trust we have succeeded in doing. In this belief we are confirmed 
mainly by the consideration that experimental results are seen 
to agree with and exactly correspond with those properties 
which have been, one after another, demonstrated by us. Finally, 
in the investigation of naturally accelerated motion we were 
led, by hand as it were, in following the habit and custom of 
nature herself, in all her various other processes, to employ 
only those means which are most common, simple and easy. 

For I think no one believes that swimming or flying can be 
accomplished in a manner simpler or easier than that instinc- 
tively employed by fishes and birds. 

When, therefore, I observe a stone initially at rest falling 
from an elevated position and continually acquiring new incre- 
ments of speed, why should I not believe that such increases 
take place in a manner which is exceedingly simple and rather 
obvious to everybody? If now we examine the matter carefully 
we find no addition or increment more simple than that which 
repeats itself always in the same manner. This we readily under- 
stand when we consider the intimate relationship between 
time and motion; for just as uniformity of motion is defined by 
and conceived through equal times and equal spaces (thus we 
call a motion uniform when equal distances are traversed during 
equal time-intervals), so also we may, in a similar manner, 
through equal time-intervals, conceive additions of speed as 
taking place without complication; thus we may picture to our 


mind a motion as uniformly and continuously accelerated when, 
during any equal intervals of time whatever, equal increments 
of speed are given to it. Thus if any equal intervals of time 
whatever have elapsed, counting from the time at which the 
moving body left its position of rest and began to descend, the 
amount of speed acquired during the first two time-intervals 
will be double that acquired during the first time-interval alone; 
so the amount added during three of these time-intervals will 
be treble; and that in four, quadruple that of the first time- 
interval. To put the matter more clearly, if a body were to 
continue its motion with the same speed which it had acquired 
during the first time-interval and were to retain this same uni- 
form speed, then its motion would be twice as slow as that which 
it would have if its velocity had been acquired during two time- 

And thus, it seems, we shall not be far wrong if we put the 
increment of speed as proportional to the increment of time; 
hence the definition of motion which we are about to discuss may 
be stated as follows : A motion is said to be uniformly accelerated, 
when starting from rest, it acquires, during equal time-intervals, 
equal increments of speed. 

Sagr. Although I can offer no rational objection to this or 
indeed to any other definition, devised by any author whomso- 
ever, since all definitions are arbitrary, I may nevertheless 
without offense be allowed to doubt whether such a definition as 
the above, established in an abstract manner, corresponds to and 
describes that kind of accelerated motion which we meet in 
nature in the case of freely falling bodies. And since the Author 
apparently maintains that the motion described in his definition 
is that of freely falling bodies, I would like to clear my mind of 
certain difficulties in order that I may later apply myself more 
earnestly to the propositions and their demonstrations. 

Salv. It is well that you and Simplicio raise these difficulties. 
They are, I imagine, the same which occurred to me when I 
first saw this treatise, and which were removed either by discus- 
sion with the Author himself, or by turning the matter over in 


my own mind. 

Sagr. So far as I see at present, the definition might have 
been put a little more clearly perhaps without changing the 
fundamental idea, namely, uniformly accelerated motion is such 
that its speed increases in proportion to the space traversed; so 
that, for example, the speed acquired by a body in falling four 
cubits would be double that acquired in falling two cubits and 
this latter speed would be double that acquired in the first cubit. 
Because there is no doubt but that a heavy body falling from 
the height of six cubits has, and strikes with, a momentum 
[impeto] double that it had at the end of three cubits, triple that 
which it would have if it had fallen from two, and sextuple that 
which it would have had at the end of one. 

Salv. It is very comforting to me to have had such a com- 
panion in error; and moreover let me tell you that your proposi- 
tion seems so highly probable that our Author himself admitted, 
when I advanced this opinion to him, that he had for some time 
shared the same fallacy. But what most surprised me was to see 
two propositions so inherently probable that they commanded 
the assent of every one to whom they were presented, proven 
in a few simple words to be not only false, but impossible. 

Sagr. But now, continuing the thread of our talk, it would 

seem that up to the present we have established the definition of 

uniformly accelerated motion which is expressed as follows: 

A motion is said to be equally or uniformly accelerated 

when, starting from rest, its momentum (celeritatis momenta) 

receives equal increments in equal times. 

Salv. This definition established, the Author makes a single 
assumption, namely, 

The speeds acquired by one and the same body moving 
down planes of different inclinations are equal when the 
heights of these planes are equal. 

By the height of an inclined plane we mean the perpendicular 
let fall from the upper end of the plane upon the horizontal line 
drawn through the lower end of the same plane. Thus, to 
illustrate, let the line AB be horizontal, and let the planes CA 


and CD be inclined to it; then the Author calls the perpendicular 
CB the "height" of the planes CA and CD; he supposes that 
the speeds acquired by one and the same body, descending 
along the planes CA and CD to the terminal points A and D are 
equal since the heights of these planes are the same, CB; and 
also it must be understood that this speed is that which would 
be acquired by the same body falling from C to B. 

Sagr. Your assumption appears to me so reasonable that it 
ought to be conceded without question, provided of course there 
are no chance or outside resistances, and that the planes are 
hard and smooth, and that the figure of the moving body is 
perfectly round, so that neither plane nor moving body is rough. 

Fro. 32. 

All resistance and opposition having been removed, my reason 
tells me at once that a heavy and perfectly round ball descending 
along the lines CA, CD, CB would reach the terminal points A, 
D, B, with equal momenta [impeti eguali]. 

Salv. Your words are very plausible; but I hope by experi- 
ment to increase the probability to an extent which shall be little 
short of a rigid demonstration. 

Theorem II, Proposition II 

The spaces described by a body falling from rest with a uni- 
formly accelerated motion are to each other as the squares of 
the time-intervals employed in traversing these distances. 

Let the time beginning with any instant A be represented by 
the straight line AB in which are taken any two time-intervals 
AD and AE. Let HI represent the distance through which the 


body, starting from rest at H, falls with uniform acceleration. If 
HL represents the space traversed during the time-interval AD, 
and HM that covered during the interval AE, then the space 
MH stands to the space LH in a ratio which is the square of the 
ratio of the time AE to the time AD; or we may say simply that 
the distances HM and HL are related as the squares of AE 
and AD. 

Draw the line AC making any angle whatever with the line 
AB; and from the points D and E, draw the parallel lines DO 

Fig. 33. 

and EP; of these two lines, DO represents the greatest velocity 
attained during the interval AD, while EP represents the maxi- 
mum velocity acquired during the interval AE. But it has just 
been proved that so far as distances traversed are concerned 
it is precisely the same whether a body falls from rest with a 
uniform acceleration or whether it falls during an equal time- 
interval with a constant speed which is one-half the maximum 


speed attained during the accelerated motion. It follows therefore 
that the distances HM and HL are the same as would be trav- 
ersed, during the time-intervals AE and AD, by uniform velocities 
equal to one-half those represented by DO and EP respectively. 
If, therefore, one can show that the distances HM and HL are 
in the same ratio as the squares of the time intervals AE and 
AD, our proposition will be proven. 

But in the fourth proposition of the first book it has been 
shown that the spaces traversed by two particles in uniform 
motion bear to one another a ratio which is equal to the 
product of the ratio of the velocities by the ratio of the times. 
But in this case the ratio of the velocities is the same as the ratio 
of the time-intervals (for the ratio of AE to AD is the same as 
that of i EP to i DO or of EP to DO). Hence the ratio of the 
spaces traversed is the same as the squared ratio of the time- 
intervals. Q. E. D. 

Evidently then the ratio of the distances is the square of the 
ratio of the final velocities, that is, of the lines EP and DO, since 
these are to each other as AE to AD. 

Corollary I 

Hence it is clear that if we take any equal intervals of time 
whatever, counting from the beginning of the motion, such as 
AD, DE, EF, FG, in which the spaces HL, LM, MN, NI are 
traversed, these spaces will bear to one another the same ratio 
as the series of odd numbers, 1, 3, 5, 7; for this is the ratio of the 
differences of the squares of the lines [which represent time], 
differences which exceed one another by equal amounts, this 
excess being equal to the smallest line [viz. the one representing a 
single time-interval] : or we may say [that this is the ratio] of the 
differences of the squares of the natural numbers beginning with 

While, therefore, during equal intervals of time the velocities 
increase as the natural numbers, the increments in the distances 
traversed during these equal time-intervals are to one another as 
the odd numbers beginning with unity. 


Sagr. Please suspend the discussion for a moment since there 
just occurs to me an idea which I want to illustrate by means 
of a diagram in order that it may be clearer both to you and 
to me. 

Let the line AI represent the lapse of time measured from the 
initial instant A; through A draw the straight line AF making 
any angle whatever; join the terminal points I and F; divide the 
time AI in half at C; draw CB parallel to IF. Let us consider 
CB as the maximum value of the velocity which increases from 












Fig. 34. 

zero at the beginning, in simple proportionality to the intercepts 
on the triangle ABC of lines drawn parallel to BC; or what is 
the same thing, let us suppose the velocity to increase in proportion 
to the time; then I admit without question, in view of the preced- 
ing argument, that the space described by a body falling in the 
aforesaid manner will be equal to the space traversed by the same 
body during the same length of time travelling with a uniform 
speed equal to EC, the half of BC. Further let us imagine that 
the body has fallen with accelerated motion so that, at the in- 
stant C, it has the velocity BC. It is clear that if the body con- 


tinued to descend with the same speed BC, without acceleration' 
it would in the next time-interval CI traverse double the distance 
covered during the interval AC, with the uniform speed EC 
which is half of BC; but since the falling body acquires equal 
increments of time, it follows that the velocity BC, during the 
next time-interval CI will be increased by an amount represented 
by the parallels of the triangle BFG which is equal to the triangle 
ABC. If, then, one adds to the velocity GI half of the velocity 
FG, the highest speed acquired by the accelerated motion and 
determined by the parallels of the triangle BFG, he will have the 
uniform velocity with which the same space would have been 
described in the time CI; and since this speed IN is three times 
as great as EC it follows that the space described during the 
interval CI is three times as great as that described during the 
interval AC. Let us imagine the motion extended over another 
equal time-interval IO, and the triangle extended to APO; it is 
then evident that if the motion continues during the interval IO, at 
the constant rate IF acquired by acceleration during the time AI, 
the space traversed during the interval IO will be four times that 
traversed during the first interval AC, because the speed IF is 
four times the speed EC. But if we enlarge our triangle so as to 
include FPQ which is equal to ABC, still assuming the accelera- 
tion to be constant, we shall add to the uniform speed an incre- 
ment RQ, equal to EC; then the value of the equivalent uniform 
speed during the time-interval IO will be five times that during 
the first time-interval AC; therefore the space traversed will be 
quintuple that during the first interval AC. It is thus evident 
by simple computation that a moving body starting from rest 
and acquiring velocity at a rate proportional to the time, will, 
during equal intervals of time, traverse distances which are 
related to each other as the odd numbers beginning with unity, 
1, 3, 5;* or considering the total space traversed, that covered 
in double time will be quadruple that covered during unit time ; 
in triple time, the space is nine times as great as in unit time. 
And in general the spaces traversed are in the duplicate ratio of 
the times, i.e. in the ratio of the squares of the times. 


Simp. In truth, I find more pleasure in this simple and clear 
argument of Sagredo than in the Author's demonstration which 
to me appears rather obscure; so that I am convinced that 
matters are as described, once having accepted the definition of 
uniformly accelerated motion. But as to whether this accelera- 
tion is that which one meets in nature in the case of falling bodies, 
I am still doubtful; and it seems to me, not only for my own 
sake but also for all those who think as I do, that this would 
be the proper moment to introduce one of those experiments — 
and there are many of them, I understand — which illustrate in 
several ways the conclusions reached. 

Salv. The request which you, as a man of science, make, is a 
very reasonable one; for this is the custom — and properly so — 
in those sciences where mathematical demonstrations are applied 
to natural phenomena, as is seen in the case of perspective, 
astronomy, mechanics, music, and others where the principles, 
once established by well-chosen experiments, become the founda- 
tions of the entire superstructure. I hope therefore it will not 
appear to be a waste of time if we discuss at considerable length 
this first and most fundamental question upon which hinge 
numerous consequences of which we have in this book only a 
small number, placed there by the Author, who has done so 
much to open a pathway hitherto closed to minds of speculative 
turn. So far as experiments go they have not been neglected 
by the Author; and often, in his company, I have attempted in 
the following manner to assure myself that the acceleration 
actually experienced by falling bodies is that above described. 

A piece of wooden moulding or scantling, about 12 cubits 
long, half a cubit wide, and three finger-breadths thick, was 
taken; on its edge was cut a channel a little more than one 
finger in breadth; having made this groove very straight, smooth, 
and polished, and having lined it with parchment, also as smooth 
and polished as possible, we rolled along it a hard, smooth, and 
very round bronze ball. Having placed this board in a sloping 
position, by lifting one end some one or two cubits above the 
other, we rolled the ball, as I was just saying, along the channel, 


noting, in a manner presently to be described, the time required 
to make the descent. We repeated this experiment more than 
once in order to measure the time with an accuracy such that the 
deviation between two observations never exceeded one-tenth of 
a pulse-beat. Having performed this operation and having 
assured ourselves of its reliability, we now rolled the ball only 
one-quarter the length of the channel ; and having measured the 
time of its descent, we found it precisely one-half of the former. 
Next we tried other distances, comparing the time for the whole 
length with that for the half, or with that for two-thirds, or 
three-fourths, or indeed for any fraction; in such experiments, 
repeated a full hundred times, we always found that the spaces 
traversed were to each other as the squares of the times, and this 
was true for all inclinations of the plane, i.e. of the channel, along 
which we rolled the ball. We also observed that the times of 
descent, for various inclinations of the plane, bore to one another 
precisely that ratio which, as we shall see later, the Author had 
predicted and demonstrated for them. 

For the measurement of time, we employed a large vessel of 
water placed in an elevated position; to the bottom of this vessel 
was soldered a pipe of small diameter giving a thin jet of water, 
which we collected in a small glass during the time of each 
descent, whether for the whole length of the channel or for a part 
of its length ; the water thus collected was weighed, after each 
descent, on a very accurate balance; the differences and ratios of 
these weights gave us the differences and ratios of the times, and 
this with such accuracy that although the operation was repeated 
many, many times, there was no appreciable discrepancy in 
the results. 

Simp. I would like to have been present at these experiments ; 
but feeling confidence in the care with which you performed 
them, and in the fidelity with which you relate them, I am satis- 
fied and accept them as true and valid. 

Salv. The experiment made to ascertain whether two bodies, 
differing greatly in weight will fall from a given height with the 
same speed offers some difficulty; because, if the height is 


considerable, the retarding effect of the medium, which must be 
penetrated and thrust aside by the falling body, will be greater 
in the case of the small momentum of the very light body than 
in the case of the great force [violenza] of the heavy body; so 
that, in a long distance, the light body will be left behind; if the 
height be small, one may well doubt whether there is any differ- 
ence; and if there be a difference it will be inappreciable. 

It occurred to me therefore to repeat many times the fall 
through a small height in such a way that I might accumulate 
all those small intervals of time that elapse between the arrival 
of the heavy and light bodies respectively at their common 
terminus, so that this sum makes an interval of time which is 
not only observable, but easily observable. In order to employ 
the slowest speeds possible and thus reduce the change which the 
resisting medium produces upon the simple effect of gravity it 
occurred to me to allow the bodies to fall along a plane slightly 
inclined to the horizontal. For in such a plane, just as well as in 
a vertical plane, one may discover how bodies of different weight 
behave: and besides this, I also wished to rid myself of the 
resistance which might arise from contact of the moving body 
with the aforesaid inclined plane. Accordingly I took two balls, 
one of lead and one of cork, the former more than a hundred 
times heavier than the latter, and suspended them by means of 
two equal fine threads, each four or five cubits long. Pulling 
each ball aside from the perpendicular, I let them go at the same 
instant, and they, falling along the circumferences of circles 
having these equal strings for semi-diameters, passed beyond the 
perpendicular and returned along the same path. This free 
vibration [per lor medesime le andate e le tornate] repeated a 
hundred times showed clearly that the heavy body maintains so 
nearly the period of the light body that neither in a hundred 
swings nor even in a thousand will the former anticipate the 
latter by as much as a single moment [minimo momento], so 
perfectly do they keep step. We can also observe the effect of 
the medium which, by the resistance which it offers to motion, 
diminishes the vibration of the cork more than that of the lead, 


but without altering the frequency of either; even when the arc 
traversed by the cork did not exceed five or six degrees while that 
of the lead was fifty or sixty, the swings were performed in 
equal times. 

Simp. If this be so, why is not the speed of the lead greater 
than that of the cork, seeing that the former traverses sixty de- 
grees in the same interval in which the latter covers scarcely six ? 

Salv. But what would you say, Simplicio, if both covered 
their paths in the same time when the cork, drawn aside through 
thirty degrees, traverses an arc of sixty, while the lead pulled 
aside only two degrees traverses an arc of four? Would not 
then the cork be proportionately swifter? And yet such is the 
experimental fact. But observe this: having pulled aside the 
pendulum of lead, say through an arc of fifty degrees, and set it 
free, it swings beyond the perpendicular almost fifty degrees, 
thus describing an arc of nearly one hundred degrees; on the 
return swing it describes a little smaller arc; and after a large 
number of such vibrations it finally comes to rest. Each vibration, 
whether of ninety, fifty, twenty, ten, or four degrees occupies 
the same time: accordingly the speed of the moving body keeps 
on diminishing since in equal intervals of time, it traverses arcs 
which grow smaller and smaller. 

Precisely the same things happen with the pendulum of cork, 
suspended by a string of equal length, except that a smaller 
number of vibrations is required to bring it to rest, since on 
account of its lightness it is less able to overcome the resistance 
of the air; nevertheless the vibrations, whether large or small, are 
all performed in time-intervals which are not only equal among 
themselves, but also equal to the period of the lead pendulum. 
Hence it is true that, if while the lead is traversing an arc of 
fifty degrees the cork covers one of only ten, the cork moves 
more slowly than the lead; but on the other hand it is also true 
that the cork may cover an arc of fifty while the lead passes over 
one of only ten or six; thus, at different times, we have now 
the cork, now the lead, moving more rapidly. But if these same 
bodies traverse equal arcs in equal times we may rest assured 


that their speeds are equal. 

Salv. Perfectly right. This point established, I pass to the 
demonstration of the following theorem : 

If a body falls freely along smooth planes inclined at any 
angle whatsoever, but of the same height, the speeds with 
which it reaches the bottom are the same. 

First we must recall the fact that on a plane of any inclination 
whatever a body starting from rest gains speed or momentum 
[la quantita delFimpeto] in direct proportion to the time, in agree- 
ment with the definition of naturally accelerated motion given by 
the Author. Hence, as he has shown in the preceding proposition, 
the distances traversed are proportional to the squares of the 
times and therefore to the squares of the speeds. The speed 
relations are here the same as in the motion first studied [i.e. 
vertical motion], since in each case the gain of speed is proportional 
to the time. 

Let AB be an inclined plane whose height above the level BC is 
AC. As we have seen above the force impelling [Fimpeto] a body 
to fall along the vertical AC is to the force which drives the same 

body along the inclined plane AB as AB is to AC. On the 
inclined AB, lay off AD a third proportional to AB and AC; 
then the force producing motion along AC is to that along AB 
(i.e. along AD) as the length AC is to the length AD. And 
therefore the body will traverse the space AD, along the incline 
AB, in the same time which it would occupy in falling the vertical 
distance AC, (since the forces [momenti] are in the same ratio 


as these distances) ; also the speed at C is to the speed at D as the 
distance AC is to the distance AD. But, according to the defini- 
tion of accelerated motion, the speed at B is to the speed of the 
same body at D as the time required to traverse AB is to the time 
required for AD ; and, according to the last corollary of the second 
proposition, the time of passing through the distance AB bears to 
the time of passing through AD the same ratio as the distance 
AC (a mean proportional between AB and AD) to AD. Accord- 
ingly the two speeds at B and C each bear to the speed at D the 
same ratio, namely, that of the distances AC and AD; hence 
they are equal. This is the theorem which I set out to prove. 

(f) Projectiles 

Undoubtedly the peak of Galileo's understanding of dynamics is found in 
his solution of the motion of a projectile — truly the climax of the Two 
New Sciences. Fundamental in this analysis was the new combination of a 
natural motion (free fall) with a violent one (projection). Although inde- 
pendent motions had previously been considered as acting simultaneously; 
for example, in the case of celestial orbits, they had always been of the same 
type, i.e. natural (circular) — or, at worse, merely hypothetical epicycles. 
On explaining the trajectory of a projectile, it had been customary to regard 
the occurrence of first one motion, say, the projection, and then that of, say, 
free fall — but never both together. Just as Galileo had refused to draw a 
line of demarcation between heavy and light bodies, between those at motion 
and those at rest, so here he considered both violent and natural motions 
occurring simultaneously. 

What is more, the total speed in this instance was viewed as continuously 
changing, so that Galileo had to use both the qualitative concept of inertia 
and the quantitative idea of acceleration. 

Let us consider a body projected upward with an initial speed v at an 
angle a with the surface. We must solve the following equations of motion: 



dfi g ' 

where the vertical y-axis is directed positively upward. Now suppose that 

dx dy 

at time / = 0, — = v cos a, and — — = v sin a. Upon integrating, we 

obtain at any future time 




= v cos a 


— - = v sin a—gt 

Also suppose at / = 0, x = y = 0. Then at any time t, 

x — (v cos) a t , 

y = (v sin a) t-\gt* . 

Eliminating t, we obtain the path, i.e. 

y — (tan a) x— - — - cos 2 a . x 2 ) 

which is the equation of a parabola. The range R (maximum horizontal 
displacement) is given by 

v 2 sm2a 
K = . 

Evidently the maximum range occurs when a = 45°, and the greatest height 
when a = 90°. The first fact had long been known empirically to gunners. 
Galileo predicted additionally, however, that the range for (45°+ /J) is the 
same as that for (45°— /?) — later confirmed. 

Galileo discussed practical objections to his obviously idealized representa- 
tion of a projectile, viz. convergence of the gravitational lines of force (not 
strictly parallel) and resistance (neglected) of the medium. His criterion for 
the use of such idealized models is still instructive: "to apply them with such 
limitations as experience will teach." 

Unfortunately practical gunners, who up to then had refused to consider 
theoretical speculations seriously, now failed to take into account the theo- 
retical limitations. For a century they dogmatically accepted Galileo's 
investigation without resistance in instances where the observed resistance 
was truly significant. Yet Galileo himself had already stressed the practical 
attainment of a terminal speed, which a freely falling body reaches in any 
resisting medium. 


Dialogues Concerning Two New Sciences f 

Theorem I, Proposition I 

A projectile which is carried by a uniform horizontal motion 
compounded with a naturally accelerated vertical motion 
describes a path which is a semi-parabola. 
Sagr. Here, Salviati, it will be necessary to stop a little while 
for my sake and, I believe, also for the benefit of Simplicio ; for 
it so happens that I have not gone very far in my study of Apollon- 
ius and am merely aware of the fact that he treats of the parabola 
and other conic sections, without an understanding of which I 
hardly think one will be able to follow the proof of other proposi- 
tions depending upon them. Since even in this first beautiful 
theorem the author finds it necessary to prove that the path of a 
projectile is a parabola, and since, as I imagine, we shall have to 
deal with only this kind of curves, it will be absolutely necessary 
to have a thorough acquaintance, if not with all the properties 
which Apollonius has demonstrated for these figures, at least 
with those which are needed for the present treatment. 

Salv. You are quite too modest, pretending ignorance of 
facts which not long ago you acknowledged as well known — I 
mean at the time when we were discussing the strength of materials 
and needed to use a certain theorem of Appollonius which gave 
you no trouble. 

Sagr. I may have chanced to know it or may possibly have 
assumed it, so long as needed, for that discussion ; but now when 
we have to follow all these demonstrations about such curves we 
ought not, as they say, to swallow it whole, and thus waste time 
and energy. 

Simp. Now even though Sagredo is, as I believe, well equipped 
for all his needs, I do not understand even the elementary terms; 
for although our philosophers have treated the motion of pro- 
jectiles, I do not recall their having described the path of a 
projectile except to state in a general way that it is always a 
curved line, unless the projection be vertically upwards. But 

t Ref. (9), pp. 245-57. 


if the little Euclid which I have learned since our previous dis- 
cussion does not enable me to understand the demonstrations 
which are to follow, then I shall be obliged to accept the theorems 
on faith without fully comprehending them. 

Salv. On the contrary, I desire that you should understand 
them from the Author himself, who, when he allowed me to see 
this work of his, was good enough to prove for me two of the 
principal properties of the parabola because I did not happen to 
have at hand the books of Apollonius. These properties, which 
are the only ones we shall need in the present discussion, he 
proved in such a way that no prerequisite knowledge was re- 

quired. These theorems are, indeed, given by Apollonius, but 
after many preceding ones, to follow which would take a long 
while. I wish to shorten our task by deriving the first property 
purely and simply from the mode of generation of the parabola 
and proving the second immediately from the first. 

Beginning now with the first, imagine a right cone, erected 
upon the circular base ibkc with apex at /. The section of this 
cone made by a plane drawn parallel to the side Ik is the curve 
which is called a parabola. The base of this parabola be cuts at 


right angles the diameter ik of the cirle ibkc, and the axis ad is 
parallel to the side Ik; now having taken any point /in the curve 
bfa draw the straight line/e parallel to bd; then, I say, the square 
of bd is to the square of fe in the same ratio as the axis ad is to 
the portion ae. Through the point e pass a plane parallel to the 
circle ibkc, producing in the cone a circular section whose 
diameter is the line geh. Since bd is at right angles to ik in the 
circle ibk, the square of bd is equal to the rectangle formed by id 
and dk; so also in the upper circle which passes through the 
points gfh the square of fe is equal to the rectangle formed by 
ge and eh; hence the square of bd is to the square of fe as the 
rectangule is to the rectangle And since the line ed is 
parallel to hk, the line eh, being parallel to dk, is equal to it; 
therefore the rectangle is to the rectangle as id is to 
ge, that is, as da is to ae; whence also the rectangle is to the 
rectangle, that is, the square of bd is to the square of fe, as 
the axis da is to the portion ae. q. e. d. 

The other proposition necessary for this discussion we demon- 
strate as follows. Let us draw a parabola whose axis ca is pro- 
longed upwards to a point d; from any point b draw the line be 
parallel to the base of the parabola; if now the point d is chosen 
so that da=ca, then, I say, the straight line drawn through the 
points b and d will be tangent to the parabola at b. For imagine, 
if possible, that this line cuts the parabola above or that its 
prolongation cuts it below, and through any point g in it draw 
the straight line/ge. And since the square of/e is greater than the 
square of ge, the square of/e will bear a greater ratio to the 
square of be than the square of ge to that of be; and since, by the 
preceding proposition, the square of/e is to that of be as the line 
ea is to ca, it follows that the line ea will bear to the line ca a 
greater ratio than the square of ge to that of be, or, than the 
square of ed to that of cd (the sides of the triangles deg and deb 
being proportional). But the line ea is to ca, or da, in the same 
ratio as four times the rectangle is to four times the square 
of ad, or, what is the same, the square of cd, since this is four 
times the square of ad; hence four times the rectangle bears 


to the square of cd a greater ratio than the square of ed to the 
square of cd; but that would make four times the rectangle 
greater than the square of ed; which is false, the fact being just 


/ ° \ 

I e V 

/ *\ 
J. £ 

ffi e 

Fig. 37. 

the opposite, because the two portions ea and ad of the line ed 
are not equal. Therefore the line db touches the parabola without 
cutting it. Q- E - D - 

Simp. Your demonstration proceeds too rapidly and, it seems 
to me, you keep on assuming that all of Euclid's theorems are 
as familiar and available to me as his first axioms, which is far 
from true. And now this fact which you spring upon us, that 
four times the rectangle is less than the square of de because 
the two portions ea and ad of the line de are not equal brings me 
little composure of mind, but rather leaves me in suspense. 

Salv. Indeed, all real mathematicians assume on the part of 
the reader perfect familiarity with at least the elements of Euclid; 
and here it is necessary in your case only to recall a proposition 
of the Second Book in which he proves that when a line is cut 


into equal and also into two unequal parts, the rectangle formed 
on the unequal parts is less than that formed on the equal (i.e. 
less than the square on half the line), by an amount which is the 
square of the difference between the equal and unequal segments. 
From this it is clear that the square of the whole line which is 
equal to four times the square of the half is greater than four 
times the rectangle of the unequal parts. In order to understand 
the following portions of this treatise it will be necessary to keep 
in mind the two elemental theorems from conic sections which 
we have just demonstrated; and these two theorems are indeed 
the only ones which the Author uses. We can now resume the 
text and see how he demonstrates his first proposition in which 
he shows that a body falling with a motion compounded of a 
uniform horizontal and a naturally accelerated [naturale descen- 
dente] one describes a semi-parabola. 

Let us imagine an elevated horizontal line or plane ab along 
which a body moves with uniform speed from a to b. Suppose 
this plane to end abruptly at b; then at this point the body will, 
on account of its weight, acquire also a natural motion down- 
wards along the perpendicular bn. Draw the line be along the 

e d c 




^-" 1 





Fig. 38. 

plane ba to represent the flow, or measure, of time; divide this 
line into a number of segments, be, cd, de, representing equal 
intervals of time; from the points b, c, d, e, let fall lines which are 
parallel to the perpendicular bn. On the first of these lay off 
any distance ci, on the second a distance four times as long, df; 


on the third, one nine times as long, eh; and so on, in proportion 
to the squares of cb, db, eb, or, we may say, in the squared ratio 
of these same lines. Accordingly we see that while the body 
moves from b to c with uniform speed, it also falls perpendicularly 
through the distance ci, and at the end of the time-interval be 
finds itself at the point i. In like manner at the end of the time- 
interval bd, which is the double of be, the vertical fall will be 
four times the first distance ci; for it has been shown in a previous 
discussion that the distance traversed by a freely falling body 
varies as the square of the time; in like manner the space eh 
traversed during the time be will be nine times ci; thus it is 
evident that the distances eh, df, ci will be to one another as the 
squares of the lines be, bd, be. Now from the points i,f, h draw 
the straight lines io, fg, hi parallel to be; these lines hi, fg, io are 
equal to eb, db and cb, respectively; so also are the lines bo, bg, bl 
respectively equal to ci, df, and eh. The square of hi is to that of 
fg as the line lb is to bg; and the square of fg is to that of io as 
gb is to bo; therefore the points /', /, h, lie on one and the same 
parabola. In like manner it may be shown that, if we take equal 
time-intervals of any size whatever, and if we imagine the particle 
to be carried by a similar compound motion, the positions of this 
particle, at the ends of these time-intervals, will lie on one and 
the same parabola. q. e. d. 

Salv. This conclusion follows from the converse of the first 
of the two propositions given above. For, having drawn a 
parabola through the points b and h, any other two points, /and 
/, not falling on the parabola must he either within or without; 
consequently the line/g is either longer or shorter than the line 
which terminates on the parabola. Therefore the square of hi 
will not bear to the square of fg the same ratio as the line lb to 
bg, but a greater or smaller; the fact is, however, that the square 
of hi does bear this same ratio to the square of fg. Hence the 
point / does lie on the parabola, and so do all the others. 

Sagr. One cannot deny that the argument is new, subtle and 
conclusive, resting as it does upon this hypothesis, namely, that 
the horizontal motion remains uniform, that the vertical motion 


continues to be accelerated downwards in proportion to the 
square of the time, and that such motions and velocities as these 
combine without altering, disturbing, or hindering each other, 
so that as the motion proceeds the path of the projectile does 
not change into a different curve: but this, in my opinion, is 
impossible. For the axis of the parabola along which we imagine 
the natural motion of a falling body to take place stands perpen- 
dicular to a horizontal surface and ends at the center of the earth; 
and since the parabola deviates more and more from its axis no 
projectile can ever reach the center of the earth or, if it does, as 
seems necessary, then the path of the projectile must transform 
itself into some other curve very different from the parabola. 

Simp. To these difficulties, I may add others. One of these is 
that we suppose the horizontal plane, which slopes neither up 
nor down, to be represented by a straight line as if each point on 
this line were equally distant from the center, which is not the 
case; for as one starts from the middle [of the line] and goes 
toward either end, he departs farther and farther from the center 
[of the earth] and is therefore constantly going uphill. Whence it 
follows that the motion cannot remain uniform through any 
distance whatever, but must continually diminish. Besides, I do 
not see how it is possible to avoid the resistance of the medium 
which must destroy the uniformity of the horizontal motion and 
change the law of acceleration of falling bodies. These various 
difficulties render it highly improbable that a result dervied from 
such unreliable hypotheses should hold true in practice. 

Salv. All these difficulties and objections which you urge are 
so well founded that it is impossible to remove them ; and, as for 
me, I am ready to admit them all, which indeed I think our 
Author would also do. I grant that these conclusions proved in 
the abstract will be different when applied in the concrete and 
will be fallacious to this extent, that neither will the horizontal 
motion be uniform nor the natural acceleration be in the ratio 
assumed, nor the path of the projectile a parabola, etc. But, on 
the other hand, I ask you not to begrudge our Author that which 
other eminent men have assumed even if not strictly true. The 


authority of Archimedes alone will satisfy everybody. In his 
Mechanics and in his first quadrature of the parabola he takes for 
granted that the beam of a balance or steelyard is a straight line, 
every point of which is equidistant from the common center of 
all heavy bodies, and that the cords by which heavy bodies are 
suspended are parallel to each other. 

Some consider this assumption permissible because, in practice, 
our instruments and the distances involved are so small in com- 
parison with the enormous distance from the center of the earth 
that we may consider a minute of arc on a great circle as a straight 
line, and may regard the perpendiculars let fall from its two 
extremities as parallel. For if in actual practice one had to 
consider such small quantities, it would be necessary first of all 
to criticise the architects who presume, by use of a plumbline, to 
erect high towers with parallel sides. I may add that, in all their 
discussions, Archimedes and the others considered themselves as 
located at an infinite distance from the center of the earth, in 
which case their assumptions were not false, and therefore their 
conclusions were absolutely correct. When we wish to apply 
our proven conclusions to distances which, though finite, are very 
large, it is necessary for us to infer, on the basis of demonstrated 
truth, what correction is to be made for the fact that our distance 
from the center of the earth is not really infinite, but merely very 
great in comparison with the small dimensions of our apparatus. 
The largest of these will be the range of our projectiles — and 
even here we need consider only the artillery — which, however 
great, will never exceed four of those miles of which as many 
thousand separate us from the center of the earth; and since these 
paths terminate upon the surface of the earth only very slight 
changes can take place in their parabolic figure which, it is 
conceded, would be greatly altered if they terminated at the 
center of the earth. 

As to the perturbation arising from the resistance of the 
medium this is more considerable and does not, on account of its 
manifold forms, submit to fixed laws and exact description. 
Thus if we consider only the resistance which the air offers to the 


motions studied by us, we shall see that it disturbs them all and 
disturbs them in an infinite variety of ways corresponding to the 
infinite variety in the form, weight, and velocity of the projectiles. 
For as to velocity, the greater this is, the greater will be the 
resistance offered by the air; a resistance which will be greater as 
the moving bodies become less dense [men gravi]. So that although 
the falling body ought to be displaced [andare accelerandosi] in 
proportion to the square of the duration of its motion, yet no 
matter how heavy the body, if it falls from a very considerable 
height, the resistance of the air will be such as to prevent any 
increase in speed and will render the motion uniform; and in 
proportion as the moving body is less dense [men grave] this 
uniformity will be so much the more quickly attained and after 
a shorter fall. Even horizontal motion which, if no impediment 
were offered, would be uniform and constant is altered by the 
resistance of the air and finally ceases ; and here again the less 
dense [piu leggiero] the body the quicker the process. Of these 
properties [accidentia of weight, of velocity, and also of form 
[figura], infinite in number, it is not possible to give any exact 
description; hence, in order to handle this matter in a scientific 
way, it is necessary to cut loose from these difficulties; and having 
discovered and demonstrated the theorems, in the case of no 
resistance, to use them and apply them with such limitations as 
experience will teach. And the advantage of this method will 
not be small; for the material and shape of the projectile may be 
chosen, as dense and round as possible, so that it will encounter 
the least resistance in the medium. Nor will the spaces and 
velocities in general be so great but that we shall be easily able 
to correct them with precision. 

In the case of those projectiles which we use, made of dense 
[grave] material and round in shape, or of lighter material and 
cylindrical in shape, such as arrows, thrown from a sling or 
crossbow, the deviation from an exact parabolic path is quite 
insensible. Indeed, if you will allow me a little greater liberty, 
I can show you, by two experiments, that the dimensions of our 
apparatus are so small that these external and incidental 


resistances, among which that of the medium is the most con- 
siderable, are scarcely observable. 

I now proceed to the consideration of motions through the 
air, since it is with these that we are now especially concerned; 
the resistance of the air exhibits itself in two ways: first by 
offering greater impedance to less dense than to very dense 
bodies, and secondly by offering greater resistance to a body in 
rapid motion than to the same body in slow motion. 

Regarding the first of these, consider the case of two balls 
having the same dimensions, but one weighing ten or twelve 
times as much as the other; one, say, of lead, the other of oak, 
both allowed to fall from an elevation of 150 or 200 cubits. 

Experiment shows that they will reach the earth with slight 
difference in speed, showing us that in both cases the retardation 
caused by the air is small; for if both balls start at the same 
moment and at the same elevation, and if the leaden one be 
slightly retarded and the wooden one greatly retarded, then the 
former ought to reach the earth a considerable distance in 
advance of the latter, since it is ten times as heavy. But this 
does not happen; indeed, the gain in distance of one over the 
other does not amount to the hundredth part of the entire fall. 
And in the case of a ball of stone weighing only a third or half as 
much as one of lead, the difference in their times of reaching the 
earth will be scarcely noticeable. Now since the speed [impeto] 
acquired by a leaden ball in falling from a height of 200 cubits is 
so great that if the motion remained uniform the ball would, in 
an interval of time equal to that of the fall, traverse 400 cubits, 
and since this speed is so considerable in comparison with those 
which, by use of bows or other machines except fire arms, we are 
able to give to our projectiles, it follows that we may, without 
sensible error, regard as absolutely true those propositions which 
we are about to prove without considering the resistance of the 

Passing now to the second case, where we have to show that 
the resistance of the air for a rapidly moving body is not very 
much greater than for one moving slowly, ample proof is given 


by the following experiment. Attach to two threads of equal 
length — say four or five yard's — two equal leaden balls and 
suspend them from the ceiling; now pull them aside from the 
perpendicular, the one through 80 or more degrees, the other 
through not more than four or five degrees; so that, when set 
free, the one falls, passes through the perpendicular, and 
describes large but slowly decreasing arcs of 160, 150, 140 degrees 
etc. ; the other swinging through small and also slowly diminishing 
arcs of 10, 8, 6, degrees, etc. 

In the first place it must be remarked that one pendulum passes 
through its arcs of 180°, 160°, etc., in the same time that the 
other swings through its 10°, 8°, etc., from which it follows that 
the speed of the first ball is 16 and 18 times greater than that of 
the second. Accordingly, if the air offers more resistance to the 
high speed than to the low, the frequency of vibration in the large 
arcs of 180° or 160°, etc., ought to be less than in the small arcs 
of 10°, 8°, 4°, etc., and even less than in arcs of 2°, or 1°; but this 
prediction is not verified by experiment; because if two persons 
start to count the vibrations, the one the large, the other the 
small, they will discover that after counting tens and even hun- 
dreds they will not differ by a single vibration, not even by a 
fraction of one. 

This observation justifies the two following propositions, 
namely, that vibrations of very large and very small amplitude 
all occupy the same time and that the resistance of the air does 
not affect motions of high speed more than those of low speed, 
contrary to the opinion hitherto generally entertained. 

Sagr. On the contrary, since we cannot deny that the air 
hinders both of these motions, both becoming slower and finally 
vanishing, we have to admit that the retardation occurs in the 
same proportion in each case. But how? How, indeed, could 
the resistance offered to the one body be greater than that offered 
to the other except by the impartation of more momentum and 
speed [impeto e velocita] to the fast body than to the slow? And 
if this is so the speed with which a body moves is at once the 
cause and measure [cagione e misura] of the resistance which it 


meets. Therefore, all motions, fast or slow, are hindered and 
diminished in the same proportion; a result, it seems to me, of 
no small importance. 

Salv. We are able, therefore, in this second case to say that 
the errors, neglecting those which are accidental, in the results 
which we are about to demonstrate are small in the case of our 
machines where the velocities employed are mostly very great and 
the distances negligible in comparison with the semi-diameter of 
the earth or one of its great circles. 

Simp. I would like to hear your reason for putting the pro- 
jectiles of fire arms, i.e., those using powder, in a different class 
from the projectiles employed in bows, slings, and crossbows, on 
the ground of their not being equally subject to change and 
resistance from the air. 

Salv. I am led to this view by the excessive and, so to speak, 
supernatural violence with which such projectiles are launched; 
for, indeed, it appears to me that without exaggeration one might 
say that the speed of a ball fired either from a musket or from 
a piece of ordnance is supernatural. For if such a ball be allowed 
to fall from some great elevation its speed will, owing to the 
resistance of the air, not go on increasing indefinitely; that which 
happens to bodies of small density in falling through short 
distances — I mean the reduction of their motion to uniformity — 
will also happen to a ball of iron or lead after it has fallen a few 
thousand cubits; this terminal or final speed [terminata velocita] 
is the maximum which such a heavy body can naturally acquire 
in falling through the air. This speed I estimate to be much smaller 
than that impressed upon the ball by the burning powder. 

An appropriate experiment will serve to demonstrate this fact. 
From a height of one hundred or more cubits fire a gun [archibuso] 
loaded with a lead bullet, vertically downwards upon a stone 
pavement; with the same gun shoot against a similar stone from 
a distance of one or two cubits, and observe which of the two balls 
is the more flattened. Now if the ball which has come from the 
greater elevation is found to be the less flattened of the two, this 
will show that the air has hindered and diminished the speed 


initially imparted to the bullet by the powder, and that the air 
will not permit a bullet to acquire so great a speed, no matter 
from what height it falls; for if the speed impressed upon the ball 
by the fire does not exceed that acquired by it in falling freely 
[naturalmente] then its downward blow ought to be greater 
rather than less. 

This experiment I have not performed, but I am of the opinion 
that a musket-ball or cannon-shot, falling from a height as great 
as you please, will not deliver so strong a blow as it would if 
fired into a wall only a few cubits distant, i.e., at such a short 
range that the splitting or rending of the air will not be sufficient 
to rob the shot of that excess of supernatural violence given it by 
the powder. 

The enormous momentum [impeto] of these violent shots may 
cause some deformation of the trajectory, making the beginning 
of the parabola flatter and less curved than the end; but, so far as 
our Author is concerned, this is a matter of small consequence in 
practical operations, the main one of which is the preparation of 
a table of ranges for shots of high elevation, giving the distance 
attained by the ball as a function of the angle of elevation; and 
since shots of this kind are fired from mortars [mortari] using 
small charges and imparting no supernatural momentum [impeto 
sopranaturale] they follow their prescribed paths very exactly. 

(g) Galileian Relativity 

Newton's second law of motion states that an unbalanced force Fona 
body with mass M will be equal to the time-rate of change of its momentum 
Mv, where v is the velocity of the body. Let A {Mv) be the change of 
momentum of the body in a time interval At. Then 

F= A(Mv) 

At ' 

where all measurements are made relative to a given frame of reference 5. 
Suppose we measure also the velocity vi in a different frame of reference 5" 
which has a constant velocity v relative (horizontally) to S. 

v = v'+v . 



Av = Av" . 


F = 


i.e. the mathematical form of the describing equation is the same for both 
systems. The solutions, accordingly, will also be formally the same. For 
example, a body falling freely (vertically) in the first frame S will have identi- 
cally the same path and speed in the second S". In general, the motion will 
be described similarly in all frames of reference that differ only in constant 
velocity — later called Galileian relativity by physicists. 

The Galileian transformation equations between measurements in system 
S" and those in system 5 can be shown to be 

x' — x+v t , 

y = y , 

z' =Z , 
t' = t . 

Dialogue Concerning Two Chief World Systems — 

Ptolemaic and Copernican f 

For a final indication of the nullity of the experiments brought 
forth, this seems to me the place to show you a way to test them 
all very easily. Shut yourself up with some friend in the main 
cabin below decks on some large ship, and have with you there 
some flies, butterflies, and other small flying animals. Have a 
large bowl of water with some fish in it; hang up a bottle that 
empties drop by drop into a wide vessel beneath it. With the 
ship standing still, observe carefully how the little animals fly 
with equal speed to all sides of the cabin. The fish swim indif- 
ferently in all directions; the drops fall into the vessel beneath; 
and, in throwing something to your friend, you need throw it 
no more strongly in one direction than another, the distances 
being equal; jumping with your feet together, you pass equal 
spaces in every direction. When you have observed all these 
things carefully (though there is no doubt that when the ship is 
standing still everything must happen in this way), have the ship 
proceed with any speed you like, so long as the motion is uniform 

t Ref. (7), pp. 186-7, 250. 


and not fluctuating this way and that. You will discover not the 
least change in all the effects named, nor could you tell from any 
of them whether the ship was moving or standing still. In jump- 
ing, you will pass on the floor the same spaces as before, nor will 
you make larger jumps toward the stern than toward the prow 
even though the ship is moving quite rapidly, despite the fact 
that during the time that you are in the air the floor under you 
will be going in a direction opposite to your jump. In throwing 
something to your companion, you will need no more force to 
get it to him whether he is in the direction of the bow or the stern, 
with yourself situated opposite. The droplets will fall as before 
into the vessel beneath without dropping toward the stern, 
although while the drops are in the air the ship runs many spans. 
The fish in their water will swim toward the front of their bowl 
with no more effort than toward the back, and will go with equal 
ease to bait placed anywhere around the edges of the bowl. 
Finally the butterflies and flies will continue their flights indif- 
ferently toward every side, nor will it ever happen that they are 
concentrated toward the stern, as if tired out from keeping up 
with the course of the ship, from which they will have been 
separated during long intervals by keeping themselves in the air. 
And if smoke is made by burning some incense, it will be seen 
going up in the form of a little cloud, remaining still and moving 
no more toward one side than the other. The cause of all these 
correspondences of effects is the fact that the ship's motion is 
common to all the things contained in it, and to the air also. 

Now transfer this argument to the whirling of the earth and to 
the rock placed on top of the tower, whose motion you cannot 
discern because in common with the rock you possess from the 
earth that motion which is required for following the tower; you 
do not need to move your eyes. Next, if you add to the rock a 
downward motion which is peculiar to it and not shared by you, 
and which is mixed with this circular motion, the circular portion 
of the motion which is common to the stone and the eye continues 
to be imperceptible. The straight motion alone is sensible, for to 
follow that you must move your eyes downward. 


I wish I could tell this philosopher, in order to remove him 
from error, to take with him a very deep vase filled with water 
some time when he goes sailing, having prepared in advance a 
ball of wax or some other material which would descend very 
slowly to the bottom — so that in a minute it would scarcely 
sink a yard. Then, making the boat go as fast as he could, so that 
it might travel more than a hundred yards in a minute, he should 
gently immerse this ball in the water and let it descend freely, 
carefully observing its motion. And from the first, he would see 
it going straight toward that point on the bottom of the vase to 
which it would tend if the boat were standing still. To his eye 
and in relation to the vase its motion would appear perfectly 
straight and perpendicular, and yet no one could deny that it 
was a compound of straight (down) and circular (around the 
watery element). 

Now these things take place in motion which is not natural, 
and in materials with which we can experiment also in a state of 
rest or moving in the opposite direction, yet we can discover no 
difference in the appearances, and it seems that our senses are 
deceived. Then what can we be expected to detect as to the earth, 
which, whether it is in motion or at rest, has always been in the 
same state? And when is it that we are supposed to test by 
experiment whether there is any difference to be discovered among 
these events of local motion in their different states of motion and 
of rest, if the earth remains forever in one or the other of these 
two states ? 



Galileo's unique contribution to the understanding of sunspot phenomena 
was his identification of their motion with the dark spots as rotation of the 
sun. He regarded them (wrongly) as clouds rather than depressions. The 
existence of these changing imperfections in a heavenly body was prima 
facie evidence of the falsity of the Greek notion of a fixed heaven made up 
possibly of some unchangeable quintessence. 

History and Demonstrations Concerning Sunspots and 
their Phenomena^ 

I have remained silent also until I might hope to give some 
satisfaction to your inquiry about the solar spots, concerning 
which you have sent me some brief essays by the mysterious 
"Apelles." The difficulty of this matter, combined with my 
inability to make many continued observations, has kept (and 
still keeps) my judgment in suspense. And I, indeed, must be 
more cautious and circumspect than most other people in pro- 
nouncing upon anything new. As Your Excellency well knows, 
certain recent discoveries that depart from common and popular 
opinions have been noisily denied and impugned, obliging me to 
hide in silence every new idea of mine until I have more than 
proved it. Even the most trivial error is charged to me as a capital 
fault by the enemies of innovation, making it seem better to 
remain with the herd in error than to stand alone in reasoning 
correctly. I might add that I am quite content to be last and to 
come forth with a correct idea, rather than to get ahead of other 

t Ref. (4), pp. 90-1, 98-9, 106-9. 



people and later be compelled to retract what might be said 
sooner, indeed, but with less consideration. 

These considerations have made me slow to respond to Your 
Excellency's requests and still make me hesitate to do more than 
advance a rather negative case by appearing to know rather 
what sunspots are not than what they really are, it being much 
harder for me to discover the truth than to refute what is false. 
But in order to satisfy Your Excellency's wishes in part at least, 
I shall consider those things which seem to me worthy of notice 
in the three letters of this man Apelles, as you require, and in 
particular what he has to say with regard to determining the 
essence, the location, and the motion of these spots. 

First of all, I have no doubt whatever that they are real objects 
and not mere appearances or illusions of the eye or of the lenses 
of the telescope, as Your Excellency's friend well establishes in 
his first letter. I have observed them for about eighteen months, 
having shown them to various friends of mine, and at this time 
last year I had many prelates and other gentlemen at Rome 
observe them there. It is also true that the spots do not remain 
stationary upon the body of the sun, but appear to move in 
relation to it with regular motions, as your author has noted 
in that same letter. Yet to me it appears that this motion is in 
the opposite direction from what Apelles says — that is, they 
move from west to east, slanting from south to north, and not 
from east to west and north to south. This may be clearly 
perceived in the observations he himself describes, which compare 
in this regard with my own observations and with what I have 
seen of those made by other people. The spots seen at sunset are 
observed to change place from one evening to the next, descending 
from the part of the sun then uppermost, and the morning spots 
ascend from the part then below; and they appear first in the 
more southerly parts of the sun's body and disappear or separate 
from it in the more northerly regions. Thus the spots describe 
lines on the face of the sun similar to those along which Venus 
and Mercury proceed when those planets come between the sun 
and our eyes. Hence they move with respect to the sun as do 


Venus and Mercury and the other planets, which motion is from 
west to east and obliquely to the horizon from south to north. 
If Apelles assumes that the spots do not revolve about the sun, 
but merely pass beneath it, then their motion may be properly 
called "from east to west." But assuming that the spots circle 
about the sun, being now beyond it and now this side of it, their 
rotation should be said to be from west to east, since that is the 
direction in which they move when they are in the more distant 
portions of their orbits. 

It now remains for us to consider the judgment of Apelles 
concerning the essence and substance of these spots, which in 
sum is that they are neither clouds nor comets, but stars that 
go circling about the sun. I confess to Your Excellency that I am 
not yet sufficiently certain to affirm any positive conclusion about 
their nature. The substance of the spots might even be any of a 
thousand things unknown and unimaginable to us, while the 
phenomena commonly observed in them — their shapes, their 
opacity, and their movement — may lie partly or wholly outside 
the realm of our general knowledge. Therefore I see nothing 
discreditable to any philosopher in confessing that he does not 
know, and cannot know, what the material of the solar spots may 
be. But if, proceeding on a basis of analogy with materials known 
and familiar to us, one may suggest something that they may be 
from their appearance, my view would be exactly opposite to that 
of Apelles. To me it seems that none of the essentials belonging 
to stars are in any way adapted to the spots, while on the other 
hand I find in them nothing at all which does not resemble our 
own clouds. This may be seen by arguing as follows. 

Sunspots are generated and decay in longer and shorter periods ; 
some condense and others greatly expand from day to day; they 
change their shapes, and some of these are most irregular ; here 
their obscurity is greater and there less. They must be simply 
enormous in bulk, being either on the sun or very close to it. 
By their uneven opacity they are capable of impeding the sun- 


light in differing degrees; and sometimes many spots are produced, 
sometimes few, sometimes none at all. 

Now of all the things found with us, only clouds are vast and 
immense, are produced and dissolved in brief times, endure for 
long or short periods, expand and contract, easily change shape, 
and are more dense and opaque in some places and less so in 
others. Indeed, all other materials not only lack these properties 
but are far from having them. Moreover there is no doubt that 
if the earth shone with its own light and not by that of the sun, 
then to anyone who looked at it from afar it would exhibit 
congruent appearances. For as now this country and now that 
was covered by clouds, it would appear to be strewn with dark 
spots that would impede the terrestrial splendor more or less 
according to the greater or less density of their parts. These spots 
would be seen darker here and less dark there, now more 
numerous and again less so, now spread out and now restricted; 
and if the earth revolved upon an axis, they would follow its 
motion. And since clouds are of no great depth with respect to 
the breadth in which they are normally extended, those seen at 
the center of the visible hemisphere would appear quite broad, 
while those toward the edges would look narrower. In a word, no 
phenomena would be perceived that are not likewise seen in 

I therefore repeat and more positively confirm to Your Excel- 
lency that the dark spots seen in the solar disk by means of the 
telescope are not at all distant from its surface, but are either 
contiguous to it or separated by an interval so small as to be 
quite imperceptible. Nor are they stars or other permanent 
bodies, but some are always being produced and others dissolved. 
They vary in duration from one or two days to thirty or forty. 
For the most part they are of most irregular shape, and their 
shapes continually change, some quickly and violently, others 
more slowly and moderately. They also vary in darkness, appear- 
ing sometimes to condense and sometimes to spread out and 


rarefy. In addition to changing shape, some of them divide into 
three or four, and often several unite into one; this happens less 
near the edge of the sun's disk than in its central parts. Besides 
all these disordered movements they have in common a general 
uniform motion across the face of the sun in parallel lines. From 
special characteristics of this motion one may learn that the sun 
is absolutely spherical, that it rotates from west to east around its 
own center, carries the spots along with it in parallel circles, and 
completes an entire revolution in about one lunar month. Also 
worth noting is the fact that the spots always fall in one zone of 
the solar body, lying between the two circles which bound the 
declinations of the planets — that is, they fall within 28° or 29° 
of the sun's equator. 

The diiferent densities and degrees of darkness of the spots, 
their changes of shape, and their collecting and separating are 
evident directly to our sight, without any need of reasoning, as a 
glance at the diagrams which I am enclosing will show. But that 
the spots are contiguous to the sun and are carried around by its 
rotation can only be deduced and concluded by reasoning from 
certain particular events which our observations yield. 

First, to see twenty or thirty spots at a time move with one 
common movement is a strong reason for believing that each 
does not go wandering about by itself, in the manner of the 
planets going around the sun. In order to explain this, let us 
define the poles in the solar globe and its circles of longitude and 
latitude as we do in the celestial sphere. If the sun is spherical 
and rotates, there will be two points at rest called the poles, and 
all other points on its surface will describe parallel circles which 
are larger or smaller according to their distance from the poles. 
The largest of all will be the central circle, equally distant from 
the two poles. The dimension of the spots along these circles 
will be called their breadth, and by their length we shall mean 
their dimension extending toward the poles and determined by a 
line perpendicular to that which determines their breadth. 

These terms defined, let us consider the specific events observed 
in the sunspots from which one may arrive at a knowledge of their 


positions and movements. To begin with, the spots at their first 
appearance and final disappearance near the edges of the sun 
generally seem to have very little breadth, but to have the same 
length that they show in the central parts of the sun's disk. Those 
who understand what is meant by foreshortening on a spherical 
surface will see this to be a manifest argument that the sun is a 
globe, that the spots are close to its surface, and that as they are 
carried on that surface toward the center they will always grow 
in breadth while preserving the same length. All of them do not 
thin out equally to a hairsbreadth when close to the circumference, 
but this is because they are not all simple spots on the surface, but 
also have a certain height. Some have more thickness and some 
have less, just as our clouds, which may spread out for tens or 
hundreds of miles in length and breadth and may have greater or 
less thickness; yet these are not more than a few hundred or 
perhaps a thousand yards thick. And the thickness of the sun- 
spots, though small in comparison with their other two dimen- 
sions, may be much greater in one spot than another, so that the 
thinnest spots when close to the edge of the sun look extremely 
slender — especially as the inner part of this edge is brightly 
lighted — while the thicker spots appear broader. But many of 
them are reduced to a threadlike thinness, and this could not 
happen at all if their motion across the face of the sun took place 
at even a short distance from the solar globe. For this maximum 
thinning takes place at the point of greatest foreshortening, and 
it would occur outside the face of the sun if the spots were any 
perceptible distance away from its surface. 

In the second place, one must observe the apparent travel of 
the spots day by day. The spaces passed by the same spot in 
equal times become always less as the spot is situated nearer the 
edge of the sun. Careful observation shows also that these 
increases and decreases of travel are quite in proportion to the 
versed sines of equal arcs, as would happen only in circular 
motion contiguous to the sun itself. In circles even slightly 
distant from it, the spaces passed in equal times would appear to 
differ very little against the sun's surface. 


A third thing which strongly confirms this conclusion may be 
deduced from the spaces between one spot and another. Some of 
these separations remain constant, others greatly increase toward 
the center of the solar disk, being quite narrow elsewhere, and 
insensible near the edge; still others show extreme variability. 
The events are such that they could be met with only in circular 
motion made by different points on a rotating globe. Spots 
located close together along the same parallel of solar latitude 
seem almost to touch each other at their first emergence; if 
farther apart, they will at any rate be much closer near the edge 
than near the center of the sun. As they move away from the 
edge, they are seen to separate more and more; at the center, they 
have their maximum separation; and as they move on from there 
they approach each other again. Accurate observation of the 
ratios of these separations and approaches shows that they can 
occur only upon the very surface of the solar globe. 

That the spots are very thin in comparison with their length 
and breadth may be deduced from the gaps between them, for 
they are often distinct all the way out to the very limb of the sun. 
This would not happen if they were very high and thick, especially 
when quite close together. Likewise separations among groups 
of very small spots have been seen all the way to the edge, though 
much foreshortened by the curvature of the surface. Some may 
say from this that such spots must be surfaces of little or no 
thickness, since when close to the edge of the disk the bright 
spaces between them are not foreshortened more than their own 
breadths are diminished, which it seems could not happen if their 
height were appreciable. But I say this is not a necessary conse- 
quence, because one must consider also the brilliance of the 
sunlight which illuminates the spots edgewise ... I could give 
many examples, but in order to avoid prolixity I shall save this to 
write of in another place. 

I should be mentioned that the spots are not completely fixed 
and motionless on the face of the sun, but continually change in 


shape, collect together, and disperse. But this variation is small 
in relation to the general rotation of the sun, and should not 
trouble anyone who will judiciously weigh the general movement 
against the small accidental variation. And just as all the 
phenomena in these observations agree exactly with the spots' 
being contiguous to the surface of the sun, and with this surface 
being spherical rather than any other shape, and with their being 
carried around by the rotation of the sun itself, so the same 
phenomena are opposed to every other theory that may be 
proposed to explain them. 



There has been considerable controversy among modern historians as to 
who should be credited with the invention of the telescope. Galileo made no 
such claim for himself. His own technical skill, however, was evident in his 
ability to design and construct a telescope merely on hearsay. What is more, 
he continually strove to improve the instrument, first by making use of 
specially ground lenses (not merely available spectacle lenses) and later by 
increasing the magnification. His curiosity turned his terrestrial spy-glass 
toward heaven (if not the first to do so, then at least the first to appreciate 
scientifically the view). The earth's moon was revealed as having a rough 
surface, contrary to Aristotelian notions. He accounted for "the new moon 
in the old moon's arms" as earthshine, and gave a reasonable estimate of 
height (4 miles) of the mountains on the moon — quantitative evidence that 
the moon is not a perfect sphere. 

His discovery of bright planetary moons about Jupiter is a fascinating 
example of a scientific investigation that began with his casually noting an 
interesting phenomenon, but then continued with his determinedly seeking 
to understand it. The later discovery of moonlike phases of the planet Venus 
was even less a matter of luck. Galileo's work with the telescope belies a 
modern portraiture of him as a Platonic dreamer or even as a pure mathe- 
matician. His whole research approach was essentially experiential. 

Such discoveries, to be sure, did not confirm the Copernican hypothesis. 
They did, however, give credence to the existence of a universe, i.e. not a 
duoverse with a perfect heaven and an imperfect earth, and to the probability 
that the earth is not the center of motion for all heavenly bodies. 

The Starry Messenger^ 

About ten months ago a report reached my ears that a certain 
Fleming had constructed a spyglass by means of which visible 
objects, though very distant from the eye of the observer, were 
distinctly seen as if nearby. Of this truly remarkable effect several 

t Ref. (4), pp. 28-33, 40-1, 51-7. 



experiences were related, to which some persons gave credence 
while others denied them. A few days later the report was con- 
firmed to me in a letter from a noble Frenchman at Paris, Jacques 
Badovere, which caused me to apply myself wholeheartedly to 
inquire into the means by which I might arrive at the invention 
of a similar instrument. This I did shortly afterwards, my basis 
being the theory of refraction. First I prepared a tube of lead, at 
the ends of which I fitted two glass lenses, both plane on one side 
while on the other side one was spherically convex and the other 
concave. Then placing my eye near the concave lens I perceived 
objects satisfactorily large and near, for they appeared three times 
closer and nine times larger than when seen with the naked eye 
alone. Next I constructed another one, more accurate, which 
represented objects as enlarged more than sixty times. Finally, 
sparing neither labor nor expense, I succeeded in constructing for 
myself so excellent an instrument that objects seen by means of 
it appeared nearly one thousand times larger and over thirty 
times closer than when regarded with our natural vision. 

It would be superfluous to enumerate the number and impor- 
tance of the advantages of such an instrument at sea as well as on 
land. But forsaking terrestrial observations, I turned to celestial 
ones, and first I saw the moon from as near at hand as if it were 
scarcely two terrestrial radii away. After that I observed often 
with wondering delight both the planets and the fixed stars, and 
since I saw these latter to be very crowded, I began to seek (and 
eventually found) a method by which I might measure their 
distances apart. 

Here it is appropriate to convey certain cautions to all who 
intend to undertake observations of this sort, for in the first place 
it is necessary to prepare quite a perfect telescope, which will show 
all objects bright, distinct, and free from any haziness, while 
magnifying them at least four hundred times and thus showing 
them twenty times closer. Unless the instrument is of this kind 
it will be vain to attempt to observe all the things which I have 
seen in the heavens, and which will presently be set forth. Now 
in order to determine without much trouble the magnifying power 


of an instrument, trace on paper the contour of two circles or 
two squares of which one is four hundred times as large as the 
other, as it will be when the diameter of one is twenty times that 
of the other. Then, with both these figures attached to the same 
wall, observe them simultaneously from a distance, looking at the 
smaller one through the telescope and at the larger one with the 
other eye unaided. This may be done without inconvenience while 
holding both eyes open at the same time; the two figures will 
appear to be of the same size if the instrument magnifies objects 
in the desired proportion. 

Such an instrument having been prepared, we seek a method 
of measuring distances apart. This we shall accomplish by the 
following contrivance. 

Fig. 39. 

Let ABCD be the tube and E be the eye of the observer. Then if 
there were no lenses in the tube, the rays would reach the object 
FG along the straight lines ECF and EDG. But when the lenses 
have been inserted, the rays go along the refracted lines ECH and 
EDI; thus they are brought closer together, and those which were 
previously directed freely to the object FG now include only the 
portion of it HI. The ratio of the distance EH to the line HI then 
being found, one may by means of a table of sines determine the 
size of the angle formed at the eye by the object HI, which we shall 
find to be but a few minutes of arc. Now, if to the lens CD we 
fit thin plates, some pierced with larger and some with smaller 
apertures, putting now one plate and now another over the lens 
as required, we may form at pleasure different angles subtending 
more or fewer minutes of arc, and by this means we may easily 
measure the intervals between stars which are but a few minutes 
apart, with no greater error than one or two minutes. And for 


the present let it suffice that we have touched lightly on these 
matters and scarcely more than mentioned them, as on some other 
occasion we shall explain the entire theory of this instrument. 

Now let us review the observations made during the past two 
months, once more inviting the attention of all who are eager for 
true philosophy to the first steps of such important contem- 
plations. Let us speak first of that surface of the moon which 
faces us. For greater clarity I distinguish two parts of this 
surface, a lighter and a darker; the lighter part seems to surround 
and to pervade the whole hemisphere, while the darker part 
discolors the moon's surface like a kind of cloud, and makes it 
appear covered with spots. Now those spots which are fairly 
dark and rather large are plain to everyone and have been seen 
throughout the ages; these I shall call the "large" or "ancient" 
spots, distinguishing them from others that are smaller in size but 
so numerous as to occur all over the lunar surface, and especially 
the lighter part. The latter spots had never been seen by anyone 
before me. From observations of these spots repeated many 
times I have been led to the opinion and conviction that the 
surface of the moon is not smooth, uniform, and precisely 
spherical as a great number of philosophers believe it (and the 
other heavenly bodies) to be, but is uneven, rough, and full of 
cavities and prominences, being not unlike the face of the earth, 
relieved by chains of mountains and deep valleys. The things 
I have seen by which I was enabled to draw this conclusion are 
as follows. 

On the fourth or fifth day after new moon, when the moon is 
seen with brilliant horns, the boundary which divides the dark 
part from the light does not extend uniformly in an oval line as 
would happen on a perfectly spherical solid, but traces out an 
uneven, rough, and very wavy line as shown in the figure below. 
Indeed, many luminous excrescences extend beyond the boundary 
into the darker portion, while on the other hand some dark 
patches invade the illuminated part. Moreover a great quantity 
of small blackish spots, entirely separated from the dark region, 
are scattered almost all over the area illuminated by the sun with 


the exception only of that part which is occupied by the large and 
ancient spots. Let us note, however, that the said small spots 
always agree in having their blackened parts directed toward the 
sun, while on the side opposite the sun they are crowned with 
bright contours like shining summits. There is a similar sight on 
earth about sunrise, when we behold the valleys not yet flooded 
with light though the mountains surrounding them are already 
ablaze with glowing splendor on the side opposite the sun. And 
just as the shadows in the hollows on earth diminish in size as 
the sun rises higher, so these spots on the moon lose their black- 
ness as the illuminated region grows larger and larger. 

Fig. 40. 

Again, not only are the boundaries of shadow and light in the 
moon seen to be uneven and wavy, but still more astonishingly 
many bright points appear within the darkened portion of the 
moon, completely divided and separated from the illuminated 


part and at a considerable distance from it. After a time these 
gradually increase in size and brightness, and an hour or two later 
they become joined with the rest of the lighted part which has 
now increased in size. Meanwhile more and more peaks shoot 
up as if sprouting now here, now there, lighting up within the 
shadowed portion; these become larger, and finally they too are 
united with that same luminous surface which extends ever 
further. An illustration of this is to be seen in the figure above. 
And on the earth, before the rising of the sun, are not the highest 
peaks of the mountains illuminated by the sun's rays while the 
plains remain in shadow? Does not the light go on spreading 
while the larger central parts of those mountains are becoming 
illuminated? And when the sun has finally risen, does not the 
illumination of plains and hills finally become one? But on the 
moon the variety of elevations and depressions appears to surpass 
in every way the roughness of the terrestrial surface, as we shall 
demonstrate further on. 

That the lighter surface of the moon is everywhere dotted with 
protuberances and gaps has, I think, been made sufficiently clear 
from the appearances already explained. It remains for me to 
speak of their dimensions, and to show that the earth's irregu- 
larities are far less than those of the moon. I mean that they are 
absolutely less, and not merely in relation to the sizes of the 
respective globes. This is plainly demonstrated as follows. 

I had often observed, in various situations of the moon with 
respect to the sun, that some summits within the shadowy portion 
appeared lighted, though lying some distance from the boundary 
of the light. By comparing this separation to the whole diameter 
of the moon, I found that it sometimes exceeded one-twentieth of 
the diameter. Accordingly, let CAF be a great circle of the lunar 
body, E its center, and CF a diameter, which is to the diameter 
of the earth as two is to seven. 

Since according to very precise observations the diameter of 
the earth is seven thousand miles, CF will be two thousand, CE 
one thousand, and one-twentieth of CF will be one hundred miles. 
Now let CF be the diameter of the great circle which divides the 


light part of the moon from the dark part (for because of the very 
great distance of the sun from the moon, this does not differ 
appreciably from a great circle), and let A be distant from C by 
one-twentieth of this. Draw the radius EA, which, when pro- 
duced, cuts the tangent line GCD (representing the illuminating 
ray) in the point D. Then the arc CA, or rather the straight line 
CD, will consist of one hundred units whereof CE contains one 
thousand, and the sum of the squares of DC and CE will be 
1,010,000 This is equal to the square of DE; hence ED will 
exceed 1,004, and AD will be more than four of those units of 
which CE contains one thousand. Therefore the altitude AD on 
the moon, which represents a summit reaching up to the solar 
ray GCD and standing at the distance CD from C, exceeds four 
miles. But on the earth we have no mountains which reach to a 
perpendicular height of even one mile. Hence it is quite clear that 
the prominences on the moon are loftier than those on the earth. 

Fig. 41. 

On the seventh day of January in this present year 1610, at the 
first hour of night, when I was viewing the heavenly bodies with 
a telescope, Jupiter presented itself to me; and because I had 
prepared a very excellent instrument for myself, I perceived (as 
I had not before, on account of the weakness of my previous 
instrument) that beside the planet there were three starlets, small 


indeed, but very bright. Though I believed them to be among the 
host of fixed stars, they aroused my curiosity somewhat by 
appearing to lie in an exact straight line parallel to the ecliptic, 
and by their being more splendid than others of their size. Their 
arrangement with respect to Jupiter and each other was the 

East *• * O * West 

that is, there were two stars on the eastern side and one to the 
west. The most easterly star and the western one appeared larger 
than the other. I paid no attention to the distances between them 
and Jupiter, for at the outset I thought them to be fixed stars, as 
I have said. But returning to the same investigation on January 
eighth — led by what, I do not know — I found a very different 
arrangement. The three starlets were now all to the west of 
Jupiter, closer together, and at equal intervals from one another 
as shown in the following sketch: 

East O * * * West 

At this time, though I did not yet turn my attention to the way 
the stars had come together, I began to concern myself with the 
question how Jupiter could be east of all these stars when on the 
previous day it had been west of two of them. I commenced to 
wonder whether Jupiter was not moving eastward at that time, 
contrary to the computations of the astronomers, and had got in 
front of them by that motion. Hence it was with great interest 
that I awaited the next night. But I was disappointed in my hopes, 
for the sky was then covered with clouds everywhere. 

On the tenth of January, however, the stars appeared in this 
position with respect to Jupiter: 

East *■ * O West 

that is, there were but two of them, both easterly, the third (as 
I supposed) being hidden behind Jupiter. As at first, they were 
in the same straight line with Jupiter and were arranged precisely 
in the line of the zodiac. Noticing this, and knowing that there 


was no way in which such alterations could be attributed to 
Jupiter's motion, yet being certain that these were still the same 
stars I had observed (in fact no other was to be found along the 
line of the zodiac for a long way on either side of Jupiter), my 
perplexity was now transformed into amazement. I was sure that 
the apparent changes belonged not to Jupiter but to the observed 
stars, and I resolved to pursue this investigation with greater care 
and attention. 

And thus, on the eleventh of January, I saw the following 
disposition : 

East •*■ * O West 

There were two stars, both to the east, the central one being three 
times as far from Jupiter as from the one farther east. The latter 
star was nearly double the size of the former, whereas on the night 
before they had appeared approximately equal. 

I had now decided beyond all question that there existed in the 
heavens three stars wandering about Jupiter as do Venus and 
Mercury about the sun, and this became plainer than daylight 
from observations on similar occasions which followed. Nor 
were there just three such stars; four wanderers complete their 
revolutions about Jupiter, and of their alterations as observed 
more precisely later on we shall give a description here. Also 
I measured the distances between them by means of the telescope, 
using the method explained before. Moreover I recorded the 
times of the observations, especially when more than one was 
made during the same night — for the revolutions of these planets 
are so speedily completed that it is usually possible to take even 
their hourly variations. 

Thus on the twelfth of January at the first hour of night I saw 
the stars arranged in this way: 

East * *<3 * West 

The most easterly star was larger than the western one, though 
both were easily visible and quite bright. Each was about two 
minutes of arc distant from Jupiter. The third star was invisible 


at first, but commenced to appear after two hours; it almost 
touched Jupiter on the east, and was quite small. All were on the 
same straight line directed along the ecliptic. 

On the thirteenth of January four stars were seen by me for the 
first time, in this situation relative to Jupiter: 

East * O*** West 

Three were westerly and one was to the east; they formed a 
straight line except that the middle western star departed slightly 
toward the north. The eastern star was two minutes of arc away 
from Jupiter, and the intervals of the rest from one another and 
from Jupiter were about one minute. All the stars appeared to 
be of the same magnitude, and though small were very bright, 
much brighter than fixed stars of the same size. 

On the twenty-sixth of February, midway in the first hour of 
night, there were only two stars : 

East * O * West 

One was to the east, ten minutes from Jupiter; the other to the 
west, six minutes away. The eastern one was somewhat smaller 
than the western. But at the fifth hour three stars were seen: 

East * O * * West 

In addition to the two already noticed, a third was discovered to 
the west near Jupiter; it had at first been hidden behind Jupiter 
and was now one minute away. The eastern one appeared farther 
away than before, being eleven minutes from Jupiter. 

This night for the first time I wanted to observe the progress of 
Jupiter and its accompanying planets along the line of the zodiac 
in relation to some fixed star, and such a star was seen to the east, 
eleven minutes distant from the easterly starlet and a little 
removed toward the south, in the following manner: 

East * O * * West 


On the twenty-seventh of February, four minutes after the first 
hour, the stars appeared in this configuration : 

East * * O * * West 


The most easterly was ten minutes from Jupiter; the next, thirty 
seconds; the next to the west was two minutes thirty seconds from 
Jupiter, and the most westerly was one minute from that. Those 
nearest Jupiter appeared very small, while the end ones were 
plainly visible, especially the westernmost. They marked out an 
exactly straight line along the course of the ecliptic. The progress 
of these planets toward the east is seen quite clearly by reference 
to the fixed star mentioned, since Jupiter and its accompanying 
planets were closer to it, as may be seen in the figure above. 
At the fifth hour, the eastern star closer to Jupiter was one 
minute away. 

At the first hour on February twenty-eighth, two stars only 
were seen; one easterly, distant nine minutes from Jupiter, and 
one to the west, two minutes away. They were easily visible and 
on the same straight line. The fixed star, perpendicular to this 
line, now fell under the eastern planet as in this figure : 

East # O * West 


At the fifth hour a third star, two minutes east of Jupiter, was 
seen in this position : 

East # * O * West 

On the first of March, forty minutes after sunset, four stars all 
to the east were seen, of which the nearest to Jupiter was two 
minutes away, the next was one minute from this, the third two 
seconds from that and brighter than any of the others ; from this 
in turn the most easterly was four minutes distant, and it was 
smaller than the rest. They marked out almost a straight line, but 


the third one counting from Jupiter was a little to the north. The 
fixed star formed an equilateral triangle with Jupiter and the 
most easterly star, as in this figure : 

East * * * * O West 


On March second, half an hour after sunset, there were three 
planets, two to the east and one to the west, in this configuration : 

East ** O * West 

The most easterly was seven minutes from Jupiter and thirty 
seconds from its neighbor; the western one was two minutes 
away from Jupiter. The end stars were very bright and were 
larger than that in the middle, which appeared very small. The 
most easterly star appeared a little elevated toward the north 
from the straight line through the other planets and Jupiter. The 
fixed star previously mentioned was eight minutes from the 
western planet along the line drawn from it perpendicularly to 
the straight line through all the planets, as shown above. 

I have reported these relations of Jupiter and its companions 
with the fixed star so that anyone may comprehend that the 
progress of those planets, both in longitude and latitude, agrees 
exactly with the movements derived from planetary tables. 

Such are the observations concerning the four Medicean planets 
recently first discovered by me, and although from these data 
their periods have not yet been reconstructed in numerical form, 
it is legitimate at least to put in evidence some facts worthy of 
note. Above all, since they sometimes follow and sometimes 
precede Jupiter by the same intervals, and they remain within very 


limited distances either to east or west of Jupiter, accompany- 
ing that planet in both its retrograde and direct movements in 
a constant manner, no one can doubt that they complete their 
revolutions about Jupiter and at the same time effect all together 
a twelve-year period about the center of the universe. That they 
also revolve in unequal circles is manifestly deduced from the 
fact that at the greatest elongation from Jupiter it is never possible 
to see two of these planets in conjunction, whereas in the vicinity 
of Jupiter they are found united two, three, and sometimes all 
four together. It is also observed that the revolutions are swifter 
in those planets which describe smaller circles about Jupiter, since 
the stars closest to Jupiter are usually seen to the east when on the 
previous day they appeared to the west, and vice versa, while the 
planet which traces the largest orbit appears upon accurate 
observation of its returns to have a semimonthly period. 

Here we have a fine and elegant argument for quieting the 
doubts of those who, while accepting with tranquil mind the 
revolutions of the planets about the sun in the Copernican 
system, are mightily disturbed to have the moon alone revolve 
about the earth and accompany it in an annual rotation about the 
sun. Some have believed that this structure of the universe should 
be rejected as impossible. But now we have not just one planet 
rotating about another while both run through a great orbit 
around the sun; our own eyes show us four stars which wander 
around Jupiter as does the moon around the earth, while all 
together trace out a grand revolution about the sun in the space 
of twelve years. 



Looking out a window we note that a tree hides something behind it farther 
away. If we shift to the right or to the left, the background seems to also 
move to the right or left, respectively. The actual displacement depends 
upon how far we move, and how close the tree is. If, say, the tree is just 
in front of a house, the displacement will be relatively small. 

Imagine a star nearer to the earth than others behind it. As the earth 
revolves annually about the sun, we move now to its right, now to its left. 
The near star, therefore, should appear periodically displaced relative to the 
others — it should show parallax ("change beside"). Astronomers, however, 
failed for centuries to detect any such displacement due to the earth's motion 
— owing to the great distance of all stars. Stellar parallax was not detected 
until 1837^10, when Bessel repeated accurate measurements of 61 Cygni, a 
fifth magnitude star which had been found to have a very large proper motion 
(indicative of proximity to the earth). For this star, his observations yielded 
a parallax of only 0.35. 

Dialogue Concerning Two Chief World Systems — 
Ptolemaic and Copernican^ 

Sagr. Getting back to the point, I invite Simplicio to consider 
how the approach and retreat which the earth makes with respect 
to some fixed star near the pole may be made as if by a straight 
line, for such is the diameter of the earth's orbit. Hence the 
attempt to compare the rising and falling of the polestar due to 
motion along such a diameter with that due to motion over the 
small circle of the earth strongly indicates a lack of understanding. 

Simp. But we are still in the same difficulty, since not even the 
small variation which ought to exist is to be found, and if the 
variation is null, then the annual motion attributed to the earth 
along its orbit must also be admitted to be null. 

tRef. (7), pp. 377-9, 382-6. 



Sagr. Now I shall let Salviati resume, who I believe would not 
shrug off as nonexistent the rising or dropping of the polestar or 
of some other fixed star. I say this even though such events 
may not be known to anyone, and were assumed by Copernicus 
himself to be, I shall not say null, but unobservable because of 
their smallness. 

Salv. I said earlier that I do not believe anyone has set himself 
the task of observing whether variations which might depend 
upon an annual movement of the earth are to be perceived in any 
fixed star at the various seasons of the year, and I added that 
I doubt whether anyone has very clearly understood just what 
variations should appear, or among what stars. Therefore it will 
be good for us to examine this point carefully. 

I have indeed found authors writing in general terms that the 
annual motion of the earth should not be admitted because it is 
improbable that visible changes would not then be seen in the 
fixed stars. Not having heard anyone go on to say what, in 
particular, these visible changes ought to be, and in what stars, 
I think it quite reasonable to suppose that those who say generally 
that the fixed stars remain unchanged have not understood (and 
perhaps have not even tried to find out) the nature of these 
alterations, or what it is that they mean ought to be seen. In 
making this judgment I have been influenced by knowing that 
the annual movement attributed to the earth by Copernicus, if 
made perceptible in the steller sphere, would not produce visible 
alterations equally among all stars, but would necessarily make 
great changes in some, less in others, still less in yet others, and 
finally none in some stars, however great the size of the circle 
assumed for this annual motion. The alterations which should 
be seen, then, are of two sorts; one is an apparent change in size 
of these stars, and the other is a variation in their altitudes at the 
meridian, which implies as a consequence the varying of places 
of rising and setting, of distances from the zenith, etc. 

Sagr. I think that what I see coming is like a ball of string 
so snarled that without God's help I may never manage to 
disentangle it; for to confess my deficiencies to Salviati, I have 


often thought about this without ever getting hold of the loose 
end of it. I say this not so much in reference to things pertaining 
to the fixed stars as to an even more terrifying task that you 
have brought to my mind by mentioning these meridian altitudes, 
latitudes of rising, distances from the zenith, etc. The reeling of 
my brain has its origin in what I shall now tell you. 

Copernicus assumes the stellar sphere to be motionless, with 
the sun likewise motionless in the center of it. Therefore all 
alterations in the sun or in the fixed stars which may appear to 
us must necessarily belong to the earth; that is, be ours. But the 
sun rises and sets along a very great arc on our meridian — 
almost forty-seven degrees — and its deviations in rising and 
setting vary by still greater arcs along the oblique horizons. Now 
how can the earth be so remarkably tilted and elevated with 
respect to the sun, and not at all so with regard to the fixed stars — 
or so little as to be imperceptible? This is the knot which has 
never passed through my comb, and if you untie it for me I shall 
consider you greater than an Alexander. 

Salv. These difficulties do credit to Sagredo's ingenuity; the 
question is one which Copernicus himself despaired of explaining 
in such a way as to make it intelligible, as will be seen both from 
his own admission of its obscurity and from his setting out twice 
to explain it, in two different ways. And without affectation 
I admit not having understood his explanation myself, until I had 
made it intelligible in still another way which is quite plain and 
clear, and this only after a long and laborious application of 
my mind. 

Simp. Aristotle saw the same objection, and made use of it to 
disprove some of the ancients who would have had it that the 
earth was a planet. Against them he reasoned that if it were, it 
would be necessary for it, like the other planets, to have more 
than one movement to produce these variations in the risings and 
settings of the fixed stars as well as in their meridian altitudes. 
And since he raised the difficulty without solving it, it must 
necessarily be very difficult of solution, if not entirely impossible. 

Salv. The strength and force of the knotting make the untying 


the more beautiful and admirable, but this I do not promise you 
today; you must excuse me until tomorrow. For the present, let 
us go on considering and explaining these alterations and dif- 
ferences which ought to be perceived in the fixed stars on account 
of the annual movement, as we were just saying. 

Sagr. I see it clearly now, thanks to your having awakened 
my mind, first by telling me positively that a fallacy existed, and 
next by commencing to interrogate me in general about the means 
of my recognizing the stoppings and retrograde motions of the 
planets. Now, this is known by comparing the planets with the 
fixed stars, in relation to which they are seen to vary their move- 
ments now westward, now eastward, and sometimes to remain 
practically motionless. But beyond the stellar sphere there is not 
another sphere, immensely more remote and visible to us, with 
which we might compare the fixed stars. Hence not a trace could 
we discover in them of anything corresponding to what appears 
among the planets. I believe that this is what you were so anxious 
to draw from my mouth. 

Salv. And there it is, with the addition of your most subtle 
insight to boot. And if I, with my little joke, opened your mind, 
you with yours have reminded me that it is not entirely impossible 
for something some time to become observable among the fixed 
stars by which it might be discovered what the annual motion 
does reside in. Then they, too, no less than the planets and the 
sun itself, would appear in court to give witness to such motion 
in favor of the earth. For I do not believe that the stars are 
spread over a spherical surface at equal distances from one center; 
I suppose their distances from us to vary so much that some are 
two or three times as remote as others. Thus if some tiny star 
were found by the telescope quite close to some of the larger ones, 
and if that one were therefore very very remote, it might happen 
that some sensible alterations would take place in it corresponding 
to those of the outer planets. 

So much for the moment with regard to the special case of 
stars placed in the ecliptic. Let us now go to the fixed stars 
outside the ecliptic, and assume a great circle vertical to its plane, 


for example a circle that would correspond in the stellar sphere 
to the solstitial colure. This we shall mark CEH, and it will be 
a meridian at the same time. Let us take in it a star outside the 
ecliptic, which can be E here. Now this will indeed vary its 
elevation with the movement of the earth, because from the 
earth at A it will be seen along the ray AE, with the elevation of 
the angle EAC, but from the earth at B it will be seen along the 
ray BE, with an angle of elevation EBC. This is greater than 
EAC, on account of its being an exterior angle of the triangle 
EAB, while the other is the opposite interior angle. Hence the 
distance of the star E from the ecliptic would be seen to be 
changed, and also its meridian altitude would be greater in 
position B than in the place A, in proportion as the angle EBC 
exceeds EAC; that is, by the angle AEB. For the side AB of the 
triangle EAB being produced to C, the exterior angle EBC (being 
equal to the two opposite interior angles E and A) exceeds A by 
the size of the angle E. And if we take another star in the same 
meridian farther from the ecliptic — let this be the star H — then 
this will be even greater in variation when seen from the two 
positions A and B, according as the angle AHB becomes greater 
than the angle E. This angle will continue to increase in propor- 
tion as the star observed gets farther from the ecliptic, until 
finally the maximum alteration will appear in that star which is 
placed at the very pole of the ecliptic. For a complete under- 
standing, this may be demonstrated as follows : 

Let the diameter of the earth's orbit be AB, whose center is G, 
and assume it to be extended out to the stellar sphere in the 
points D and C. From the center G, let the axis GF of the ecliptic 
be erected as far as the same sphere, in which a meridian DFC 
vertical to the plane of the ecliptic is assumed to be described. 
Taking, in the arc FC, any points H and E as places of fixed stars, 
add the lines FA, FB, AH, HG, HB, AE, GE, and BE. Then 
AFB is the angle of difference (or we may say the parallax) of the 
star placed at the pole F; that of the star at H is the angle AHB, 
and for the star at E it is the angle AEB. I say that the angle of 
difference of the polestar F is the maximum; of the others, those 


closest to this maximum are larger than those more distant from 
it. That is, the angle F is greater than the angle H, and this is 
greater than the angle E. 

Suppose a circle described about the triangle FAB. Since the 
angle F is acute, its base AB being less than the diameter DC of 
the semicircle DFC, it will fall in the larger portion of the circum- 
scribed circle cut by the base AB. And since AB is divided in the 
center and at right angles to FG, the center of the circumscribed 
circle will be in the line FG. Let this be the point I. Now of all 
the lines drawn to the circumference of the circumscribed circle 
from the point G, which is not its center, the greatest is that which 

passes through the center. Hence FG will be greater than any 
other line drawn through G to the circumference of the same 
circle, and therefore this circumference will cut the line GH, which 
is equal to the line GF, and cutting GH it will also cut AH. Let it 
cut that in L, and add the line LB. Then the two angles AFB and 
ALB will be equal, being included in the same portion of the 
circumscribed circle. But ALB, an exterior angle, is greater than 
the interior angle H; therefore angle F is greater than angle H. 

By the same method we may show that the angle H is greater 
than the angle E, because the center of the circle described about 
the triangle AHB is on the perpendicular GF, to which the line 
GH is closer than the line GE; hence its circumference cuts GE 
and also AE, from which the proposition is obvious. 

From this we conclude that the alteration of appearance (which, 
using the proper technical term, we may call the parallax of the 
fixed stars) is greater or less according as the stars observed are 


more or less close to the pole of the ecliptic, and that finally for 
stars on the ecliptic itself the alteration is reduced to nothing. 
Next, as to the earth approaching and retreating from the stars 
by its motion, those stars which are on the ecliptic are made 
nearer or farther by the entire diameter of the earth's orbit, as we 
have already seen. For those which lie near the pole of the 
ecliptic, this approach and retreat is almost nothing, while for 
others the alteration is made greater as the stars become closer 
to the ecliptic. 

In the third place we may see that this alteration of appearance 
is greater or less according as the observed star is closer to or 
more remote from us. For if we draw another meridian less 
distant from the earth (which shall be DFI here), a star placed at 
F and seen along the same ray AFE with the earth at A, when it 
is later observed from the earth at B will be seen along the ray 
BF, and will make the angle of difference BFA greater than the 
first one, AEB, being exterior to the triangle BFE. 

Sagr. I have listened to your discourse with great pleasure, 
and with profit too; now, to make sure that I have understood 
everything, I shall state briefly the heart of your conclusions. 

It seems to me that you have explained to us two sorts of differing 
appearances as being those which because of the annual motion 
of the earth we might observe in the fixed stars. One is their 
variation in apparent size as we, carried by the earth, approach 


them or recede from them; the other (which likewise depends 
upon this same approach and retreat) is their appearing to us to 
be now more elevated and now less so on the same meridian. 
Besides this you tell us (and I thoroughly understand) that these 
two alterations do not occur equally in all stars, but to a greater 
extent in some, to a lesser in others, and not at all in still others. 
The approach and retreat by which the same star ought to appear 
larger at one time and smaller at another is imperceptible and 
practically nonexistent for stars which are close to the pole of the 
ecliptic, but it is great for the stars placed in the ecliptic itself, 
being intermediate for those in between. The reverse is true of 
the other alteration; that is, the elevation or lowering is nil for 
stars along the ecliptic and large for those encircling the pole of 
the ecliptic, being intermediate for those in the middle. 

Furthermore, both these alterations are more perceptible in 
the closest stars, less sensible in those more distant, and would 
ultimately vanish for those extremely remote. 

So much for my part. The next thing, so far as I can see, is to 
convince Simplicio. I think he will not easily be reconciled to 
admitting such alterations as these to be imperceptible, stemming 
as they do from such a vast movement of the earth and from a 
change that carries the earth to places twice as far apart as our 
distance from the sun. 

Simp. Really, to be quite frank, I do feel a great repugnance 
against having to concede the distance of the fixed stars to be so 
great that the alterations just explained would have to remain 
entirely imperceptible in them. 

Salv. Do not completely despair, Simplicio; perhaps there is 
yet some way of tempering your difficulties. First of all, that the 
apparent size of the stars is not seen to alter visibly need not 
appear entirely improbable to you when you see that men's 
estimates in such a matter may be so grossly in error, particularly 
when looking at brilliant objects. Looking, for example, at a 
burning torch from a distance of two hundred paces, and then 
coming closer by three or four yards, do you believe that you 
yourself would perceive it as larger? For my part, I should 


certainly not discover this even if I approached by twenty or thirty 
paces; sometimes I have even happened to see such a light at a 
distance, and been unable to decide whether it was coming 
toward me or going away, when in fact it was approaching. Now 
what of this? If the same approach and retreat of Saturn (I mean 
double the distance from the sun to us) is almost entirely imper- 
ceptible, and if it is scarcely noticeable in Jupiter, what could it 
amount to in the fixed stars, which I believe you would not 
hesitate to place twice as far away as Saturn? In Mars, which 
while approaching us . . . 

Simp. Please do not labor this point, for I am indeed convinced 
that what you have said about the unaltered appearance of the 
apparent sizes of the fixed stars may very well be the case. But 
what shall we say to that other difficulty which arises from no 
variation at all being seen in their changing aspects ? 



Although Galileo was not deeply spiritual, he was sincerely religious and 
a loyal churchman. The celebrated Galileo case, therefore, does not truly 
revolve about the perennial issues of science versus religion per se. On the 
contrary, he himself showed genuine understanding of this problem in his 
own attempt to reconcile public natural science with his personal religious 
faith. Believing in God as the Author of both nature and the Bible, he refused 
to regard any apparent disagreement as more than a natural human short- 
coming due to necessarily incomplete human understanding. His attitude, 
I believe, would be generally acceptable to many modern Christians. 

What took place historically was an inevitable clash between the limited 
private opinion of an individual and the limited accepted judgment of a 
social group, of which he was a member. The controversy was theological 
only in the sense that the orthodox theology of that period was couched in 
Aristotelian philosophy, which was characterized by such universal breadth 
and organic coherence that the disturbance of any part was propagated 
throughout the whole. Furthermore, the entire matter was complicated by 
the friction of narrow personalities: Peripatetics and Platonists, Dominicans 
and Jesuits, the Pope and even Galileo himself. Neither "truth" nor academic 
freedom was basically at stake; the affair was primarily sociological. The 
history of science itself, as well as the history of the Church, has had similar 
incidents when an individual's idea has been lost in a social morass. 

Galileo's personal views on the relative roles of science and of the Bible 
are clearly stated in his letter to the dowager Grand Duchess. He looked 
upon scientific evidence as one aspect of divine wisdom, but at the same 
time he warned of human fallibility in interpreting ancient biblical writings. 
He urged tolerant confidence in the fact that "two truths cannot contradict 
each other". His presentation revealed a deep understanding of the Scrip- 
tures and a broad reading of the Church Fathers. Individuals today still 
have to resolve their own continually changing conflicts that inevitably 
arise because of having to incorporate new experiences in necessarily incom- 
plete old understandings. Clues can be obtained from studying the behavior 
of other religious men of science like Galileo. 



Letter to Madame Christine of Lorraine, Grand Duchess 
of Tuscany, Concerning the Use of Biblical Quotation in 
Matters of Science^ 

The reason produced for condemning the opinion that the 
earth moves and the sun stands still is that in many places in the 
Bible one may read that the sun moves and the earth stands still. 
Since the Bible cannot err, it follows as a necessary consequence 
that anyone takes an erroneous and heretical position who 
maintains that the sun is inherently motionless and the earth 

With regard to this argument, I think in the first place that it is 
very pious to say and prudent to affirm that the holy Bible can 
never speak untruth — whenever its true meaning is understood. 
But I believe nobody will deny that it is often very abstruse, and 
may say things which are quite different from what its bare words 
signify. Hence in expounding the Bible if one were always to 
confine oneself to the unadorned grammatical meaning, one might 
fall into error. Not only contradictions and propositions far 
from true might thus be made to appear in the Bible, but even 
grave heresies and follies. Thus it would be necessary to assign 
to God feet, hands, and eyes, as well as corporeal and human 
affections, such as anger, repentance, hatred, and sometimes even 
the forgetting of things past and ignorance of those to come. 
These propositions uttered by the Holy Ghost were set down in 
that manner by the sacred scribes in order to accommodate them 
to the capacities of the common people, who are rude and 
unlearned. For the sake of those who deserve to be separated 
from the herd, it is necessary that wise expositors should produce 
the true senses of such passages, together with the special reasons 
for which they were set down in these words. This doctrine is so 
widespread and so definite with all theologians that it would be 
superfluous to adduce evidence for it. 

Hence I think that I may reasonably conclude that whenever 
the Bible has occasion to speak of any physical conclusion 

f Ref. (4), pp. 181-3, 185-7. 


(especially those which are very abstruse and hard to understand), 
the rule has been observed of avoiding confusion in the minds of 
the common people which would render them contumacious 
toward the higher mysteries. Now the Bible, merely to conde- 
scend to popular capacity, has not hesitated to obscure some very 
important pronouncements, attributing to God himself some 
qualities extremely remote from (and even contrary to) His 
essence. Who, then, would positively declare that this principle 
has been set aside, and the Bible has confined itself rigorously to 
the bare and restricted sense of its words, when speaking but 
casually of the earth, of water, of the sun, or of any other created 
thing? Especially in view of the fact that these things in no way 
concern the primary purpose of the sacred writings, which is the 
service of God and the salvation of souls — matters infinitely 
beyond the comprehension of the common people. 

This being granted, I think that in discussions of physical 
problems we ought to begin not from the authority of scriptural 
passages, but from sense-experiences and necessary demon- 
strations; for the holy Bible and the phenomena of nature proceed 
alike from the divine Word, the former as the dictate of the Holy 
Ghost and the latter as the observant executrix of God's com- 
mands. It is necessary for the Bible, in order to be accommodated 
to the understanding of every man, to speak many things which 
appear to differ from the absolute truth so far as the bare meaning 
of the words is concerned. But Nature, on the other hand, is 
inexorable and immutable; she never transgresses the laws 
imposed upon her, or cares a whit whether her abstruse reasons 
and methods of operation are understandable to men. For that 
reason it appears that nothing physical which sense-experience 
sets before our eyes, or which necessary demonstrations prove to 
us, ought to be called in question (much less condemned) upon 
the testimony of biblical passages which may have some different 
meaning beneath their words. For the Bible is not chained in 
every expression to conditions as strict as those which govern all 
physical effects; nor is God any less excellently revealed in Nature's 
actions than in the sacred statements of the Bible. Perhaps this 


is what Tertullian meant by these words: 

"We conclude that God is known first through Nature, and 
then again, more particularly, by doctrine; by Nature in His 
works, and by doctrine in His revealed word." 

From this I do not mean to infer that we need not have an 
extraordinary esteem for the passages of holy Scripture. On the 
contrary, having arrived at any certainties in physics, we ought to 
utilize these as the most appropriate aids in the true exposition of 
the Bible and in the investigation of those meanings which are 
necessarily contained therein, for these must be concordant with 
demonstrated truths. I should judge that the authority of the 
Bible was designed to persuade men of those articles and propo- 
sitions which, surpassing all human reasoning, could not be made 
credible by science, or by any other means than through the very 
mouth of the Holy Spirit. 

Yet even in those propositions which are not matters of faith, 
this authority ought to be preferred over that of all human 
writings which are supported only by bare assertions or probable 
arguments, and not set forth in a demonstrative way. This I hold 
to be necessary and proper to the same extent that divine wisdom 
surpasses all human judgment and conjecture. 

But I do not feel obliged to believe that that same God who has 
endowed us with senses, reason, and intellect has intended to 
forgo their use and by some other means to give us knowledge 
which we can attain by them. 

The same disregard of these sacred authors toward beliefs about 
the phenomena of the celestial bodies is repeated to us by 
St. Augustine in his next chapter. On the question whether we 
are to believe that the heaven moves or stands still, he writes thus : 

"Some of the brethren raise a question concerning the motion 
of heaven, whether it is fixed or moved. If it is moved, they say, 
how is it a firmament? If it stands still, how do these stars which 
are held fixed in it go round from east to west, the more northerly 
performing shorter circuits near the pole, so that heaven (if there 
is another pole unknown to us) may seem to revolve upon some 
axis, or (if there is no other pole) may be thought to move as a 


discus ? To these men I reply that it would require many subtle 
and profound reasonings to find out which of these things is 
actually so; but to undertake this and discuss it is consistent 
neither with my leisure nor with the duty of those whom I desire 
to instruct in essential matters more directly conducing to their 
salvation and to the benefit of the holy Church." 

From these things it follows as a necessary consequence that, 
since the Holy Ghost did not intend to teach us whether heaven 
moves or stands still, whether its shape is spherical or like a 
discus or extended in a plane, nor whether the earth is located 
at its center or off to one side, then so much the less was it 
intended to settle for us any other conclusion of the same kind. 
And the motion or rest of the earth and the sun is so closely linked 
with the things just named, that without a determination of the 
one, neither side can be taken in the other matters. Now if the 
Holy Spirit has purposely neglected to teach us propositions of 
this sort as irrelevant to the highest goal (that is, to our salvation), 
how can anyone affirm that it is obligatory to take sides on them, 
and that one belief is required by faith, while the other side is 
erroneous ? Can an opinion be heretical and yet have no concern 
with the salvation of souls ? Can the Holy Ghost be asserted not 
to have intended teaching us something that does concern our 
salvation ? I would say here something that was heard from an 
ecclesiastic of the most eminent degree : "That the intention of 
the Holy Ghost is to teach us how one goes to heaven, not how 
heaven goes." 

But let us again consider the degree to which necessary demon- 
strations and sense experiences ought to be respected in physical 
conclusions, and the authority they have enjoyed at the hands of 
holy and learned theologians. From among a hundred attesta- 
tions I have selected the following: 

"We must also take heed, in handling the doctrine of Moses, 
that we altogether avoid saying positively and confidently any- 
thing which contradicts manifest experiences and the reasoning 
of philosophy or the other sciences. For since every truth is in 
agreement with all other truth, the truth of Holy Writ cannot 


be contrary to the solid reasons and experiences of human 

And in St. Augustine we read: "If anyone shall set the authority 
of Holy Writ against clear and manifest reason, he who does this 
knows not what he has undertaken; for he opposes to the truth 
not the meaning of the Bible, which is beyond his comprehension, 
but rather his own interpretation; not what is in the Bible, but 
what he has found in himself and imagines to be there." 

This granted, and it being true that two truths cannot contra- 
dict one another, it is the function of wise expositors to seek out 
the true senses of scriptural texts. These will unquestionably 
accord with the physical conclusions which manifest sense and 
necessary demonstrations have previously made certain to us. 
Now the Bible, as has been remarked, admits in many places 
expositions that are remote from the signification of the words 
for reasons we have already given. Moreover, we are unable to 
affirm that all interpreters of the Bible speak by divine inspiration, 
for if that were so there would exist no differences between them 
about the sense of a given passage. Hence I should think it would 
be the part of prudence not to permit anyone to usurp scriptural 
texts and force them in some way to maintain any physical 
conclusion to be true, when at some future time the senses and 
demonstrative or necessary reasons may show the contrary. Who 
indeed will set bounds to human ingenuity ? Who will assert that 
everything in the universe capable of being perceived is already 
discovered and known? Let us rather confess quite truly that 
"Those truths which we know are very few in comparison with 
those which we do not know." 

We have it from the very mouth of the Holy Ghost that God 
delivered up the world to disputations, so that man cannot find out 
the work that God hath done from the beginning even to the end. 
In my opinion no one, in contradiction to that dictum, should 
close the road to free philosophizing about mundane and physical 
things, as if everything had already been discovered and revealed 
with certainty. Nor should it be considered rash not to be 
satisfied with those opinions which have become common. No 


one should be scorned in physical disputes for not holding to the 
opinions which happen to please other people best, especially 
concerning problems which have been debated among the 
greatest philosophers for thousands of years. 


1564 Born February 15th in Pisa, Italy (father a cloth merchant of lower 

1574 Moved to Florence, Italy. 

1579 Attended monastery school of Sta Maria di Vallombrosa (about 
25 miles east of Florence) for one year. 

1581 Matriculated at the University of Pisa. 

1583 Discovered the isochronism of the swinging chandelier in the Cathe- 
dral of Pisa. 

1585 Returned with family to Florence. 

1585 Investigated the concept of center of gravity. 

1586 Constructed a hydrostatic balance and wrote a paper on The Little 
Balance {La Bilancetta) and its use (published 1644). 

1587 First visit (educational) to Rome (met Jesuit astronomer, Father 
Christopher Clavius, at the Romano Collegio). 

1589 Appointed to the Chair of Mathematics at the University of Pisa. 

1590 Discovered the cycloid. 

1590-1 Experimented from the Leaning Tower of Pisa (legendary). Wrote 
lecture notes, On Motion (De Motu), applying Archimedes' principle 
to motion in a medium; adhered to Aristotle's doctrine of natural 
places (and the medieval notion of impetus) although criticizing some 
of his other ideas (published in 1883). 

1591 Death of father, Vincenzio Galilei. 

1592 Appointed to the Chair of Mathematics at the University of Padua. 
1594 Caught chill resulting in an arthritic condition during his whole life. 
1597 Designed and constructed a popular "geometric and military com- 
pass" (like a modern proportional compass or divider). 

1600 Wrote lecture notes, On Mechanics (Le Meccaniche), a systematic 
summary about the statics of simple machines as then known (early 
manuscript versions of 1593 and 1594; published in French in 1634, 
by M. Mersenne, and in 1649 in Italian). 

1600 Favorite daughter, Virginia (later Sister Maria Celeste of San Matteo 
convent in Arcetri), born of Venetian mistress, Marina Gamba, 
who he kept in Padua. 

1601 Daughter Livia (later Sister Archangela of San Matteo convent) born. 

1602 Investigated magnetism. 

1604 Wrote letter to Fra Paolo Sarpi, stating correctly the law of falling 

bodies, but giving an incorrect proof. 
1604 Lectured on the (super) nova, noting its appearance among the fixed 




1605 Tutored Cosimo de' Medici, son of Ferdinand I, Grand Duke of 
Tuscany, during summer vacation at Florence. 

1606 Constructed a thermoscope. 
1606 Son, Vincenzio, born. 

1606 Published his first book, Instructions on the Use of the Geometric and 
Military Compass (Le Operazioni del Compasso Geometrico et Militare). 

1607 Published a polemic Defense against the Calumnies and Impostures of 
Baldesar Capra (Difesa Contro alle Calunnie e imposture di Baldesar 
Capra), a Padua student from Milan, who plagiarized Galileo's work 
and was later condemned for doing so. 

1609 Designed and constructed a telescope (Galileian, opera glass). 

1610 Used the telescope to view the sky, thereby discovering the myriads 
of stars in the Milky Way, the moon's mountains, Jupiter's moons 
(four), Venus phases, and Saturn's ring (indefinite). 

1610 Published The Sidereal Messenger (Sidereus Nuncius), describing his 
astronomical investigations (good reading for all students today). 

1610 Appointed Chief Mathematician and Philosopher to Cosimo II, 
Grand Duke of Tuscany, at Florence. 

1611 Second visit (triumphant) to Rome; observed sun spot; elected sixth 
member of the Accademia dei Lincei (L. — comparable to later 
English FRS). 

1612 Published Discourse on Bodies in Water (Discorso . . . intorno alle 
cose, che stanno in su aequo), describing experimental investigations 
on floating bodies (ice, ebony chips, wax models, etc.) — first public 
dispute on natural philosophy (physics) with an informal "league" of 
peripatetic philosophers — worth reading by physics teachers. 

1613 Published, under Lincean auspices, three Letters on Sunspots to 
Mark Welser at Augsburg in opposition to the views of the Jesuit 
Father Christopher Scheiner — the beginning of a bitter controversy 
involving priority of sun-spot discovery (neither was rightfully first); 
contained his first (1612) public endorsement of the Copernican 

1613 Wrote a letter to a former pupil the Benedictine, Benedetto Castelli, 
holder of the Chair of Mathematics at the University of Pisa, about 
his personal views as to the relation of science and religion; copied 
by someone for a Dominican priest, Niccolo Lorini, of the Convent 
of San Marco in Florence (famous for Savonarola and Fra Angelico). 

1614-31 Dwelt at the Villa L'Ombrellino at Bellosguardo (west across the 
Arno) as a guest of Lorenzo Segni. 

1614 Denounced as anti-Christian by the Dominican Fra Thomaso Caccini 
from the pulpit of the Convent of Santa Maria Novella — the begin- 
ning of a planned attack by this arch-villain of the celebrated Galileo 

1615 Denounced as a heretic to the Holy Office at Rome by Father Lorini 
on the basis of the letter to Castelli. 


1615 Revised formally Castelli letter as Letter to Madame Christina of 
Lorraine, Grand Duchess of Tuscany (Lettero alia Granduchessa di 
Toscano, Crestina di Lorena) giving his personal views (generally 
acceptable— and worth reading today) of apparent conflicts between 
Biblical statements and scientific findings (published 1636). 

1615-16 Third visit (promotional) to Rome, to win friends for theCopernican 
theory; matter referred officially by Pope Paul V to the Qualifiers 
of the Congregation of the Index, who ruled that the idea of a central 
sun was philosophically absurd and formally heretical, and that the 
idea of a moving earth was censurable in philosophy and erroneous 
in faith; through Robert Cardinal Bellarmine, Galileo was forbidden, 
at least, to hold or teach the Copernican theory (the record is not 
clear as to the specific injunction). 

1623 Death of Cosimo II; accession of Ferdinand II. 

1623 Maffeo Cardinale Berberini (a friend of Galileo) became Pope 
Urban VIII. 

1623 Published The Assayer ill Saggiatore), an outstanding polemic against 
the opinions of an anonymous Jesuit (Father Horatio Grassi) about 
the nature of comets (three had appeared in 1618), and a philosophical 
defense (manifesto) of science (e.g. need for intellectual freedom, 
distinction of primary and secondary qualities, use of mathematics, 

1624 Fourth visit (friendly) to Rome — six audiences with Pope Urban 
VIII; designed a compound microscope (?). 

1626 Investigated magnetism. 

1630 Fifth visit (license) to Rome to obtain permission to publish "The 
Two Chief World Systems"; death of Galileo's influential friend 
Prince Federico Cesi, head of the Lincean Academy. 

1631 Moved to Martellini's villa, "II Giojello" (east across the Arno), near 
the San Matteo convent. 

1632 Published The Two Chief World Systems {Dialogo . . . sopra i due 
Massimi Sistemi del Mondo, Tolemaico, e Copernicano), a clever 
literary dialog by an Aristotelean philosopher, Simplicio, and a 
scientific Florentine, Filippo Salviati, who compare the relative 
arguments for the Ptolemaic and Copernican theories in an effort to 
win the neutral, urbane Venetian, Giovanni Francesco Sagredo 
(discussed telescope data, the rotation and revolution of the earth 
stellar parallax). The Copernican case, based chiefly upon an explana- 
tion (false) of tidal phenomena, is obviously presented more con- 
vincingly even than the Pope's unanswerable logic (told by Simplicio 
at the end) — the obiter dicta include magnetism, relativity of motion, 
errors in measurement, etc. 

1632 Ordered by the Inquisition to stand trial at Rome (sale of "Dialogue" 

1633 Sixth visit (trial) to Rome; examined three times by Inquisition; 
sentenced (22 June), largely on basis of his failure to comply with 
1616 injunction (record not clear); confessed and abjured in Domini- 
can Convent of Santa Maria sopra la Minerva. 


1633 Lived at Siena in custody of a former pupil, Archbishop Ascanio 
Piccolomini; returned to Arcetri home under house arrest. 

1634 Death of daughter Virginia. 

1637 Blinded in both eyes (glaucoma?). 

1638 Wrote letter to Daniello Antonini on his observation of lunar 

1638 Published, at Leyden, the Two New Sciences (Discorsi a dimostrazioni 
matematiche, intorno a due nuove scienze Attenenti alia meccanica e i 
movimenti locali, completed 1636); viz., a novel approach to the 
strength of materials (including a discussion of resistance to fracture, 
the weight of air, the existence of a vacuum, resistance of a medium, 
terminal speed, acoustics, speed of light, etc.), and his mature view 
of terrestrial motions (i.e., velocity, acceleration, inertia, falling 
bodies, motion on an inclined plane, projectiles, etc.) — discussed 
by the same characters as in The Two Chief World Systems in a more 
scientific (quite clear) though less literary manner — should be read 
by all physics students. 

1638 Visited by the 29-year-old John Milton. 

1639 Refused freedom by Pope. 

1639 Received the 18-year-old Vincenzio Viviani as his "last disciple". 
1641 Welcomed Evangelista Torricelli (a student of Castelli) as a col- 
laborator (his successor). 

1641 Investigated applicability of pendulum motion to a clock. 

1642 Died (8 January); buried in Cappella del Companile del Noviziato 
in Santa Croce. 

1736 Transferred to present tomb in Santa Croce (funds left by Viviani), 
under authorization of Pope Clement XII (Florentine). 

1835 Works removed from the Index by Pope Gregory XVI. 

1893 Encyclical Providentissimus Deus of Pope Leo XIII states the official 
position of the Roman Church on the relations of science and religion 
— little different from Galileo's posture. 

1965 Mentioned graciously by Pope Paul VI on visit to Santa Croce. 


Accademia dei Lincei 17, 23, 278 

Accademia del Cimento 11,184 

Accademia della Crusca 6 

Acceleration 34, 195, 206, 207, 209, 

Acoustics 34, 171, 174, 176, 182 

Air, weight 74, 115 ff. 

"Apelles" 239,240,241 

Apollonius 223, 224 

Aquinas, Thomas 19 

Arcetrei see Florence 

Archangela, Sister see Galilei Livia, 

Archimedes 5, 58, 120, 122, 123, 
125, 126, 133, 135, 136, 140, 
207, 230 
principle 76, 133, 140 

Ariosto, Ludovico 4 

Aristotle 7, 8, 14, 17, 18, 19, 22, 27, 
28, 50, 64, 67, 76, 77, 85, 86, 
88, 89, 90, 93, 94, 95, 96, 
106 ff., 114, 115, 121, 125, 
127, 164, 171, 188, 192, 195, 
196, 197, 198, 199, 205, 247, 
262, 269 

Armature 11, 63, 69 ff. 

Assayer, The 23, 34, 44 ff., 50 ff. 

Astronomy 36 

Atomism 34, 52 

Authority 50, 132 

Balance 83, 102 ff., 133 ff. 
"Balance, Little" 5, 135 
Antonio Cardinal 30 
Francesco Cardinal 30 
Maffeo Cardinal 16, 18, 23, 26, 
see also Pope Urban VIII 
Barometer 74 

Beams 154 

similar 157 
Bellarmine, Robert Cardinal 16,21, 

22, 29, 31, 32, 279 
Benedetti, Giovanni Battista 7 
Berkeley, George 23 
Bernoulli, Jean 10 
Bessel, Friedrich Wilhelm 28, 260 
Bible 19, 20, 269 ff., 274 

Alessandra 35 

Sestilia 24, 35 
Bodies in Water 18, 38, 121 ff. 
Borghini, Jacopo 3 
Boscalgia, Cosimo 19 
Brahe, Tycho 12, 19, 23, 50, 51 
Bruno, Giordano 18 
Buonamico, Francesco 7 
Buonarrati, Jr., Michelangelo 16, 

Caccini, Fra Thomaso 19, 278 

Cantilever 154 ff. 

Capponi, Giovanni Vincenzio 36 

Capra, Baldassare 12 

Castelli, Father Benedetto 19,35,278 

Cavalieri, Bonaventura 35 

Celeste, Sister Maria see Galilei, 

Cesarini, Virginio 23 
Cesi, Prince Federico 17, 26, 279 
Chandelier, swinging 5, 170, 173 
Chladni, Ernst 175 
Chladni figures 175 
Christina, Grand Duchess of Tuscany 

13, 19, 20, 269 
Church fathers 269 
Church, Roman Catholic 30, 31, 



282 INDEX 

Ciampoli, Monsignor Giovanni 26 
Cigoli, Ludovico Cardi da 4 
Clavius, Father Christopher 6, 16 
Cocchi, Antonio 36 
Colombe, Ludovico delle 18, 19 
Comets 22, 38 
Comets 22 
dip 68 
needle 68 
Continuum 34, 52 ff., 54 
Copernican system 19, 21, 22, 24, 

26, 27, 28, 29, 31, 32, 33, 247, 

Copernicus, Nicolaus 14, 29, 51, 

261, 262 
Cremonini, Cesare 9, 14 
Cycloid 7 
Cygni, 61 28, 260 

Dante Alighieri 4 

Demonstration 78 ff., 81 ff., 99 ff., 

102 ff., 123, 153, 216, 271 
Density 76, 120, 134, 181, 195, 196, 

197, 198 
Descartes, Rene 35 
"Divino Afflante Spiritu" 20 
Dominicans 19, 20, 31, 38, 269 
Donati, Doge Leonardo 13 

magnetism 63 
motion 29 

Engineering science 151,167 

"Eppur si muove" 32 

Euclid 5,226 

Experiment 18, 63, 68, 70, 73, 91, 
128, 172, 179, 185, 186 ff., 
192, 193, 196, 197, 207, 216 ff., 
217, 231, 238 

Fall, free 
different materials 89, 91, 92, 

192, 196, 199, 200, 201, 205, 

different materials and media 

196, 199, 200, 202, 204 
different media 77 ff., 84 ff., 

86 ff, 88 ff., 94, 97 ff., 101, 

109 ff, 127, 194 ff., 197 ff., 

law 11, 206,210, 21 Iff., 213, 215 
leaf 197 
"resistance" of medium 203, 

218ff., 222, 229, 230, 232ff. 
ship 236 ff, 238 
terminal speed 203, 222, 234 
tower 204, 237 
two stones together 193 
vacuum 77, 106 ff, 111 ff., 196, 201 
wax model 18, 34, 91 ff., 124, 

128 ff. 
Floating 17, 34, 81 ff, 120 
ebony 18, 120 ff, 127, 204 
ice 18, 121 ff. 
kettle 120 
wood 103 
Florence 4, 9, 15, 16, 17, 22, 27, 33, 

Arcetri 27 
San Marco 20 
San Matteo 24 
Santa Croce 3, 36 
Santa Maria del Fiore 19 
Santa Maria Novella 19 
buoyant 76, 78, 120 
unbalanced 188 
see also Motion, Newton's 

second law 
Fracture 154 ff, 167 

column 163 ff. 
Freedom of thought 38 

Fabricius, Johannes 17 
Fabrizio d'Acquapendente, Gero- 
nimo 9 

Carlo 24 
Cosimo 24 
Galileo 3 ff, 276 ff. 

INDEX 283 

Galileo (ancestor) 3 

Galileo (grandson) 24 

Livia 12, 24, 277 

Michelangelo 12 

Vincenzio (father) 3, 9, 277 

Vincenzio (nephew) 24 

Vincenzio (son) 12, 24 

Virginia (daughter) 12, 24, 33, 38, 

Virginia (sister) 9 

Tribuna di 36 
Galileo case 17, 18, 21, 30 31, 32, 

36, 269 
Galileo's illnesses 

arthritis 10 

blindness 34, 35 

hernia 24 

hypochondria 22 
Galileo's lamp 5 
Galileo's problem 154 ff. 
Galletti, Taddeo 12 
Gamba, Marina 12, 277 
Gassendi, Pierre 35 
"Geometric and Military Compass" 

10, 11 
Geometry 36, 151, 153, 166, 206 
Gilbert, William 11, 63 ff. 
Giucciardini, Piero 22 
God 269, 270 

Gorri, Antonio Francesco 36 
Grassi, Father Orazia 22, 50 
Great World Systems 39 

see also Two Chief World Systems 
Grienberger, Father Christopher 1 6 
Guidobaldo, Marquis de Monte 6 
Guiducci, Maria 22 

Heat 44 ff., 48 ff. 

Heaviness 77, 105 

Hiero 133, 135 

Holy Office, Congregation of the 

("Inquisition") 20, 21, 30, 

31, 32, 33, 34 
Hooke, Robert 154 

Impetus 8, 126, 192, 210, 220, 233 
Imprimatur 29, 31 

Inclined plane 10, 140, 210, 211, 
law 140, 148 
Index Expurgatorius 32 
Inertia see Motion, Newton's first 

Infinite 52, 59, 61 
Injunction (1616) 21, 29, 31, 32 
Instruments 10, 230, 247, 248 

Jesuits 6, 16, 17, 19, 22, 38, 269 
Jordan, Nemorius 140 
Jupiter, moons 13, 14, 15, 22, 28, 
247, 254 ff. 

Kepler, Johannes 12, 15, 16, 29 

Lagrange, Joseph Louis Comte de 34 
Landini 27 

Landucci, Benedetta di Luca 12, 24 
Lens 13, 15, 240, 247 
Leo XIII, Pope 20 
"Letter to Grand Duchess Christ- 
ina" 270 ff. 
Lever 10, 140 

arm 133 

bent 140 

law 133, 140, 151, 154 
Libri, Giulio 14 
Light, speed 34, 184, 186 
Lightness 76 

see also Heaviness 
Line 53, 55, 59, 61 
Lippershey, Johannes 13 
Locke, John 23, 44 
Lodestone 63 ff., 66, 67, 69 

see also Magnetism 
Logic 167 
Longitude 34 
Lorini, Father Niccolo 20, 278 

fundamental 140 
law 141, 149 
screw 140 ff., 148 
simple 140 

284 INDEX 

Maculano, Fra Vincenzio 31 
Magnetism 63 ff. 

pole 68 

terella 63 
Maraffi, Fra Luigi 20 
Mariotte, Edmfe 154 
Martyr 32 

strength 151 ff. 

strength of 52 
cohesion 52 
tubes 167 ff. 
Mathematics 18, 34, 50, 51, 70, 73, 

151, 153, 166, 216 
Measurement 173, 207, 217 

time 170, 207, 217 
"Mechanics" 9, 38, 141, 188 
Medicean planets 13, 258 

Cosimo de' the Great 3 

Cosimo I de' the Elder 3 

Cosimo II de' 11, 13, 15, 17, 18, 
22, 23, 24, 26 

Ferdinand I de' 6, 13 

Ferdinand II de' 11,23 

Giovanni de' 9 

Leopold Cardinal de' 1 1 

Leopold II de' 36 
Mersenne, Pere Marin 9, 175 
Metaphysics 7, 20 
Methodology 18, 23, 37, 38, 43, 73, 
76, 123, 135, 170, 207, 208 

"manifesto of science" 23 
Micanzio, Fra Fulgenzio 33, 35 
Milky Way 13 
Milton, John 35 
Moment (of force) 144 ff. 

bending 157, 163 

resistance 167 
Momentum 210, 218, 233, 235 
Moon-surface 13, 247, 250 ff., 258 
"Motion" 8, 38, 77, 188 

circular 188, 237 

compound 189, 191 ff., 237 

inclined plane 142, 143, 172, 
188 ff., 191,207,211, 216 ff. 

natural 8, 34, 77, 81 ff., 85, 86 ff., 
188, 221 

Newton's first law 34, 142, 171, 
187 ff., 191, 221 

Newton's second law 76, 188, 
195, 235 

"resistance" of medium 195,222 

uniform 184 ff. 

uniformly accelerated 210 

violent, see Projectile 

consonance 176, 182 

pitch 176 

tone 181 
Musical interval 34, 176, 179, 182 

fifth 178, 179, 181, 182 

fourth 182 

harmonic 175 

octave 176, 178 

Natural phenomena 7, 10, 20, 208, 

Nature 7, 50, 173, 269 ff. 
"New science" 167 
Newton, Sir Isaac 34, 36, 188 
Niccolini, Marquis Francesco 26, 

Noaillis, Comte de 33 
Nova (super) 12, 13 
odd 215 
square 52, 60 

center of earth 171 
natural 170 ff. 
water 171 

Osiander, Andreas 29 

Padua, University 9, 36 

Pappus 143 

Parallax, stellar 28, 260 ff., 265 

Pascal, Blaise 32 

Paul III, Pope 13 

INDEX 285 

Paul V, Pope 16 
Paul VI, Pope 36 

period 170, 171 

resistance 171, 207, 218 ff., 233 

simple 8, 170 
law 170, 171, 172 
Peripatetics 269 
Petrarca, Francesco 4 
Philosophy, natural see Physics 
Physics 37, 50, 51, 63 
Piccolomini, Archbishop Ascanio 

33, 280 
"Pigeon League" 19 
Pisa 3, 7, 19 

Leaning Tower 8 

Ponti de Mezzo 7 

University 4, 6, 9, 15 
Pius XIII, Pope 20 
Place, natural 8, 76 
Plato 18, 50, 166, 170, 269 
Projectile 34, 188, 191 ff., 221 ff., 

bows 231,234 

firearms 234 

horizontal plane 229 

impact 234 ff. 

lance 194 

parabolic trajectory 222 ff., 229, 

range 222, 230 

resistance of medium 229 ff. 
"Providentissimus Deus" 20, 280 
Ptolemaic system 26 
Ptolemy 50, 51 
Pump lift 74 ff. 
Pythagoras 170 

Quality 23, 44 
Quantity 44, 50 
Quintessence 239 

Galileian 28, 235 ff. 
transformation equations 236, 

Religion 20, 31, 32, 269 

Research 7 

Resonance 174, 176, 177 
goblet 177 

Riccardi, Father Niccolo 26 

Ricci, Ostilio 4 

Ripples 177 

Rome 4, 6, 16, 21, 23, 26, 30, 31, 
Santa Maria Maggiore 4 
Santa Maria sopra Minerva 32 

Sagredo, Giovanni Francesco 27, 

33, 52, 279 
St. Augustine 272, 274 
Salviati, Filippo 17, 27, 33, 52, 279 
Sarpi, Fra Paolo 11,277 
Sarsi, Lothario 23, 51 
Saturn, rings 14 
Scaling 151 ff. 

Scheiner, Father Christopher 17, 278 
Scientific revolution 37, 38 
Screw 140 ff. 

Sense experience 44 ff., 271 
Siena 33 

Simplicio 27, 30, 33, 52 
Size 161 

bones 160 

fish 161 

right 157 
Sound 34, 47 

speed 184 
Specific gravity 124, 133 ff., 202, 

air 115 ff. 

alloy 133 ff. 
Speed 184, 207 

instantaneous 184 

measurement by impact 234 
Starry Messenger 13, 26, 247 ff. 
Statics 9 
Stevin, Simon 140 
Storia delle Scienze di Firenze, 

Istituto e Museo di 36 
Structure 33 
Sun 243 

rotation 246 

286 INDEX 

Sunspots 17, 21, 28, 239 ff. 
Sunspots and their Phenomena 17, 

239 ff. 
Surface tension 120 
Susterman, Giusto 27 
Symmetry 140 

Tasso, Torquato 4 

Telescope 13, 15, 240, 242, 247, 
248 ff., 250 

Tertullian 272 

Thales 63 

Thermoscope 11 

Thought experiment 170, 192 ff. 

Tide 26,28,29 

Torricelli, Evangelista 10, 35, 74 

Truth 20,269,274 

Tuscan waters 33 

Two Chief World Systems 11, 26, 
27, 33, 34, 38, 39, 63, 236 ff., 
260 ff. 

Two New Sciences 6, 8, 11, 33, 39, 
52,74, 115 ff., 151 ff., 154 ff., 
157 ff., 163 ff., 167 ff., 171 ff., 
176 ff., 184 ff., 188 ff., 192 ff., 
196 ff., 207 ff, 221, 223 ff. 

Tychonic system 27, 28 

Universe 247 

Urban VIII, Pope 16, 18, 23, 27 
30, 31, 32, 269 
"unanswerable argument" 30 
see also Barberini, Maffeo 

Vacuum 54, 75, 77, 115, 196, 201 
Vallombrosa, Santa Maria di 4, 277 
Vatican Council II 36 
Venice 15, 16 

arsenal 11, 151 

San Marco Campanile 13 
Venus, phases 16, 28, 247 

fundamental 175 

law 175 

plate 175, 180 

string 175, 176 
Vinta, Belisario 15 
Virtual velocities, principle 10 
Viviani, Vincencio 35, 36, 207, 280 

Wave 175, 179, 182 

speed 175 

stationary 175 
Weight in vacuum 105 ff., 115, 

118 ff. 
Welser, Mark 17, 278 

, 1TW r ; UVEW* 001 


There is a tendency nowadays for undergraduates to learn their 
physics completely from textbooks, not becoming acquainted 
with the original literature and therefore not realising how the 
subject grew and developed. The purpose of the series in which 
this volume is published is to present a set of reasonably priced 
books which give, for a particular subject or a particular 
physicist, reprints of those papers which record the development 
of new ideas, preceded by a careful introduction which places 
the papers in the context of present-day physics. 




jus Student 
Research Professor 
Clever Courtier 

Popular Author 


rpreting Sense 

Primary Qualities 

Mathematical Language 
Continue — Mathematical and Physical 
The Pump that Failed 
Apparent Lightness 
Weighing Air 
Floating Ebony 
Analyzing an Alloy 
The Screw as a Machine 
Strength of Materials 


Galileo's Problem 

Similar Beams 

A Cracked Column 

Natural Oscillations 

Simple Pendulum 

Vibrating Freely 


Musical Intervals 
Falling Bodies 

How Fast? 


A Thought Experiment 

The Medium's Role 

Changing Speed 


Galileian Relativity 
Spots on the Sun 
New Moons 
Parallax of a Star 
Nature — God's Handiwork 

=n O 





I O