ARCHIMEDEAN SCIENCE AND THE SCIENTIFIC
REVOLUTION
Agamenon R. E. Oliveira
Polytechnic School
Federal University of Rio de Janeiro
e-mail: [email protected]
ABSTRACT. According to Richard Westfall (Westfall, 1977) the
Scientific Revolution of the seventeenth century was dominated by two
themes: the Platonic-pythagorean tradition “which looked on nature in
geometric terms” and mechanical philosophy “which conceived of nature
as a huge machine”. This paper is an attempt to study the appropriation of
Archimedean science in the Scientific Revolution in Western Europe.
KEYWORDS. Archimedean Tradition, Scientific Revolution, History of
Mechanics, Galileo´s Scientific Method.
INTRODUCTION
The seventeenth century Scientific Revolution is a theme that continues
to attract the attention of many historians of science. These scientists
portray this process as occurring roughly in the following sequence:
Copernicus´ reformulation (1473-1543) of Ptolemy’s solution (100-170) of
the problem of planets with the need to restore their lost harmony; the
acceptance by Kepler (1571-1630) and Galileo (1564-1642) of its realistic
proposition; based on this perspective the development of mathematical
tools to study the heavens; the mathematization of free fall and projectile
motion to confirm the realistic basis of Copernicanism; and the
development of a new inertial conception of motion, associating an
abstract idealized concept of nature, linked to empirical and artificial
means of experiment (Cohen,1994).
The main objective of this paper is to present and discuss how the
Archimedean ‘legacy’ was received and transformed by the long process
that occurred during the revolution in science which culminated with
Galileo and Newton’s mechanicism (1642-1727) (Dugas,1988). The paper
pays special attention to the relationship between Archimedean techniques
and the scientific method presented mainly by Galileo (Cohen,1985). The
origins of integral calculus are also referred as an important part of
Archimedes’ ‘legacy’ (Urbaneja,2008).
A BIOGRAPHICAL NOTE ON ARCHIMEDES
Archimedes (see Fig. 1) was born in Syracuse, Sicily around the year of
287 B. C. His father was Phidias, an astronomer who investigated the size
of the sun and moon and their distance from the earth. In his youth
Archimedes seems to have spent some time in Egypt, where he invented
the water-screw as a means of pumping water out of the Nile to irrigate
fields (Hutchins, 1952).
Fig. 1. Marble Bust of Archimedes in Naple Museum.
He seems to have studied with Euclid’s pupils in Alexandria. It was
probably there that he became friends with Conon of Samos (280b.C.220b.C.) and Eratosthenes (285b.C.-194b.C.). He communicated his
discoveries to these two before they were made public and it was for
Eratosthenes that he wrote the ‘Method’. After the death of Conon,
Archimedes sent his discoveries to Conon’s friend and pupil, Dositheus of
Pelusium, to whom he dedicated other treatises.
Archimedes won great fame because of his mechanical inventions. At the
request of King Hiero he made catapults, battering rams, cranes and many
other engines and devices of war, which were later used with enormous
success in the defense of Syracuse against the Romans. Another military
story told by Lucian was that he used mirrors to set Roman ships on fire.
As described by Cicero (106b.C.-43b. C.) in his ‘Republic’ Archimedes
constructed an astronomical machine sufficiently accurate to show eclipses
of the sun and the moon. This apparatus consisted of concentric glass
spheres moved by water power and represents the Eudoxian system of the
world.
Archimedes only wrote about mathematical subjects, except in the lost
work ‘On Sphere-making’. His work dealt with arithmetic, geometry,
mechanics and hydrostatics (Heath,2002). He wrote no text books, unlike
Euclid and Apollonius (262b.C.-200b.C.). Although some of his writings
have been lost the most important have survived.
Archimedes’ concern with mathematics is considered to be the cause of
his death in the invasion that followed the capture of Syracuse by
Marcellus in 212 B. C. According to some historians Archimedes was so
intent on a mathematical diagram that when ordered by a soldier to attend
the victorious general he refused. He was then slain by the enraged soldier.
In accordance with Archimedes’ wishes, his family and friends inscribed
on his tomb the figure of his favorite theorem, a sphere and a
circumscribed cylinder and the ratio of the containing solid to the
contained (Arquimedes,2006).
ARCHIMEDES´ NEW SCIENCE
Statics and Hydrostatics – Aristotle’s mechanics is integrated in a theory
of physics which is part of a system of the world. Archimedes made statics
an autonomous theoretical science, based on postulates of an experimental
origin and supported by mathematical demonstrations (Serres,1997).
In Book One of the treatise ‘On the Equilibrium of Planes’ Archimedes
developed the principle of the lever. He enunciated eight postulates which
provided the foundation for fifteenth propositions. In Book One of his
treatise ‘On Floating Bodies’ Archimedes developed his famous Principle.
It originally appears in Proposition 3 which states: “Of solids those which,
size for size, are of equal weight with a fluid will, if let down into the fluid,
be immersed so that they do not project above the surface but do not sink
lower”. In Book Two of the same treatise, Archimedes modified the
principle which is the subject of Proposition 5, Book one, to the following
form: “ Any solid lighter than a fluid will, if placed in the fluid, be so far
immersed that the weight of the solid will be equal to the weight of the
fluid displaced” (Archimedes,1952).
Both principles became fundamental principles of mechanics. The
principle of the lever gave rise to the principle of virtual velocities which is
equivalent to the equilibrium conditions for a static problem. Also, Galileo
uses the principle of the lever and quotes Archimedes in the Second Day of
the Discorsi. Archimedes’ Principle is the origin of the Galilean theory of
motion, as we will see below (Archimedes,1952).
Historically it was perhaps Leonardo da Vinci (1452-1519) who
recognized the importance of the general concept of moments in a static
sense (Dugas,1954). Galileo Galilei asserts that the moment was the
inclination of the same body considered in the situation which it occupied
on the arm of a lever or a balance (Drake,1995).
The principle of virtual works was formalized by Jean Baptiste Fourier
(1768-1830), but the letter sent by Johann Bernoulli (1667-1748) to Pierre
Varignon (1654-1772), written in January 26, 1717, needs to be
acknowledged as a historical fact of great importance in the history of the
principle (Fourier,1798).
The Geometrization of Mechanics and the Mechanization of Geometry –
‘The Method Treating of Mechanical Problems’, is a treatise addressed to
Eratosthenes in which Archimedes reveals the method used in his
mathematical discoveries. In fact, the same method was applied in other
treatises, such as ‘On the Equilibrium of Planes’, ‘Quadrature of the
Parabola’ (Fig. 2) and ‘On Floating Bodies’.
Fig. 2 – Excerpt of propositions 16 and 17 of “ On the Parabola
Quadrature "
The essence of ‘The Method’ can be deduced by analyzing any problem
solved by Archimedes. It is possible to identify three steps in its
development. The first step is a pure geometrical approach, where the
objects are selected in order to compare unknown quantities with known
ones using the properties of the lever. In the second step, a mechanical
approach is applied. The equilibrium laws of the lever regarding the
fulcrum are used to make comparisons between geometrical quantities. In
the final step the operations made in the second step are repeated but now
in relation to the entire figures under consideration. This means, if the
positions of the center of gravity of figures and the volume of one of them
are known, the equilibrium conditions permit the volume of the other to be
found (Archimedes, 1952).
´The Method` only survived on the Palimpsest discovered in 1906, as
shown in Fig. 3. The most significant contribution made by the Palimpsest
appears in proposition 14 of ´The Method` where Archimedes is
measuring the volume of a cylindrical segment. In that proposition
Archimedes pushes the discussion of the mathematical use of actual
infinity approximately about 2000 years back in time.
Fig. 3 – Two Views of The Palimpsest.
The Origin of Integral Calculus – It is possible to see in EudoxusArchimedes’ method of exhaustion the origin of integral calculus. This
method is based on the theory of proportions presented by Eudoxus (408355a. C.) of Cnidus. It consists of finding the area of a shape by inscribing
inside it a sequence of polygons whose areas converge to the area of the
containing shape. If the sequence is correctly constructed, the difference in
area between the nth polygon and the containing shape will become
arbitrarily small as n becomes large. As this difference becomes arbitrarily
small, the possible values for the area of the shape are systematically
‘exhausted’ by the lower bound areas successively established by the
sequence members (Eves, 1997).
The idea behind ‘the Method’ is attributed to Antiphon, but the theory
was given greater rigor by Eudoxus. The first use of the term exhaustion
appeared in 1647 by George de Saint-Vincent in his ‘Opus geometricum
quadraturae circuli et sectionum coni’. Archimedes used this method for
several geometric purposes, such as the length, area, volume and center of
gravity of many geometric figures. However, the method of exhaustion is
unable to efficiently solve the problems which it suggests. Only after the
development of analytic geometry by Fermat (1601-1665) and Descartes
(1596-1650), as well as the appearance of the concept of limits, was it
possible to construct automatic algebraic operations to solve them
(Descartes, 1991).
THE RECEPTION OF ARCHIMEDES´ WORK IN WESTERN EUROPE
Archimedes’ works were translated directly from Greek by William of
Moerbeke (1215-1286), a Flemish Dominican. He was made Latin bishop
of Corinth in Greece about 1286. At the request of Thomas Aquinas (12251274) he undertook a complete translation of the works of Aristotle. The
reason for the request was the concern that by the thirteenth century Arabic
versions had distorted the original meaning of Aristotle (384b. C.-322b.
C.). Another concern was that the influence of the rationalist Averroes
(1126-1198) could be a source of philosophical and theological error.
Thus, Moerbeke was the first to translate the ‘Politics’ into Latin, which
unlike other parts of Aristotelian corpus had not been translated into
Arabic. The translations of Moerbeke (Fig. 4) were already standard
classics by the 14th century. He also translated mathematical treatises by
Hero of Alexandria and Archimedes. Then, he translated almost all of
Archimedes works except ‘The Method’ and ‘Stomachion’. In the early
1450s Pope Nicholas V (1397-1455) commissioned Jacobus de Sancto
Cassiano Cremonensis to make a new translation of Archimedes with
commentaries of Eutocius. This translation became the standard version
and was printed in 1544 (Arquimedes, 2007).
Fig. 4 - Moerbeke’s Translation to Latin of Archimedes Works.
Another translation of Archimedes was made by Niccolo Tartaglia (15001557) in 1543 in Venice. Tartaglia belongs to an important Italian school
of mechanics. He published ‘Nova Scientia’ in 1537 and ‘Quesiti et
Inventioni Diversi’ in 1546, where dynamical problems were solved. In
1544 the Editio Princeps of Greek and Latin versions of Archimedes
appeared in Basel.
Two new translations of Archimedes appeared in order to reconstruct his
work. The authors of these translations were Frederico Commandino
(1506-1575) and Francesco Maurolico (1494-1575). Because of the fruitful
consequences of the translations by the former we will look at some of its
details and implications.
Commandino studied philosophy and medicine at Padua from 1534-1544.
He dedicated himself to studying the mathematical classics and he had the
mathematical knowledge, as well as the language skills, to edit and
translate these classics. He worked on the classic texts of Archimedes,
Ptolemy, Euclid, Aristarchus, Pappus, Apollonius, Eutocius, Hero and
Serenus.
The first translation of his published was an edition of Archimedes in
1558. In 1565 Commandino published his original work ‘De Centro
Gravitatis’, described by Stillman Drake (Drake,1981), Galileo´s
biographer, as a “pioneer treatise on centers of gravity in the Archimedean
tradition”. In the preface of this work Commandino refers to his edition of
Archimedes´ ‘On Floating Bodies’.
Commandino also has influenced his pupil Guidobaldo del Monte (15451607), an important mathematician who was part of Galileo´s circle.
Guidobaldo studied at the University of Padua in 1564 and helped Galileo
in his academic career. Under his patronage Galileo was appointed to a
professorship of mathematics at the University of Pisa in 1589.
Guidobaldo helped Galileo again in 1592 when he had to apply for the
chair of mathematics at the University of Padua. He was a critic of
Galileo´s principle of isochronisms of the pendulum, one of Galileo´s
major discoveries which Guidobaldo believed was impossible.
THE MATHEMATIZATION OF NATURE IN THE SCIENTIFIC
REVOLUTION
The supreme instance of abstraction in the scientific method is the use of
mathematics, especially geometry, for the study of physical problems.
Nature as mathematized by Galileo finds itself represented at a different
level of abstraction than nature realized in daily experience. This
difference appears to Galileo in a form of a problem of how to ensure that
the mathematically expressed laws he found were valid in some way at the
level of experience too. The necessary means to establish this were
discovered by Galileo in experiments. He had already used ‘mental
experiments’ to heuristically explain mathematical regularities that are
behind free fall and projectile motion. Experiments also appeared to
Galileo to provide a means by which to bridge the gap between
phenomena represented at an idealized and at an empirical level (Galileu,
1988).
Kepler, with the fundamental help of Ticho Brahe’s accurate (1546-1601)
observations, mathematizes nature in the sense of creating a mathematical
physics of the heavens, completely distinct from the model on which many
astronomers’ best efforts had been expended since the time of Ptolemy,
including Copernicus himself. Galileo realized the impossibility of
‘Archimedenizing’ the Aristotelian concept of motion and saw also that
free fall and projectile motion were key phenomena and could be the
cornerstones of a new science of motion. In addition, Galileo was
convinced that this new science could support the heliocentric doctrine and
to provide its realistic basis. Bringing Archimedean techniques to study
these phenomena, he mathematized nature in a modest range of terrestrial
phenomena (Hall, 1981). Both Galileo and Kepler established the universe
of precision. The Scientific Revolution had begun.
To study how Archimedean science was firstly used by Galileo, we have
to focus our analysis on the return of Galileo to Pisa in 1589. In Pisa at that
time philosophical discussions were greatly concern with motion.
Francesco Buonamici (1533-1575), while he was one of Galileo´s
professors completed an important work, ‘De Motu’, published in Florence
in 1591. The text is directly influenced by Archimedean themes. He
discussed the problem of Hiero´s crown and used Archimedes´ treatise ‘On
Floating Bodies’.
Despite this influence Buonamici states that the Archimedean analysis on
the decrease of the weight of bodies immersed in water did not have a
universal character because it was restricted as a mathematical
explanation. The progress Galileo made by developing a theory of motion
supported by natural causes is remarkable. Here he found in Archimedes
the foundation on which to construct a realistic theory of motion. It seems
that it happens in the period spent in Pisa, where he wrote the work known
as the manuscripts of ‘De Motu Antiquora’, with the objective of
describing the natural motion of bodies. Galileo never published this work.
However, its subject appears later in his ‘Discorsi’, published in 1638.
The development of Galileo´s ideas about motion considers gravity as the
unique cause of motion (free fall and projectile motion) and the
explanation of the upright motion of light bodies immersed in water is due
to Archimedes´ principle. The fact that fluid is heavier than the bodies is
the cause of upright motion. While Buonamici saw only a mathematical
explanation to this phenomenon, Galileo inverted the approach completely,
renewing the conceptual basis of the theory of motion and thus creating a
new science (Geymonat,1997).
Looking at the history of mechanics during the above mentioned period,
it is important to note the contribution of Giovanni Benedetti (1530-1590)
because he anticipates some of Galileo´s ideas (Koyré,1973). He stated
that the speed of a falling body would depend on its surface because of
friction with the air, and only in a vacuum would bodies of different sizes
fall at the same speed. He also stated that bodies composed of the same
material fall at the same speed regardless of their weight. He justified his
claim with an argument using Archimedes´ results on bodies in a fluid.
Benedetti’s ‘Diversarum Speculationum’, which appeared in 1585,
contains a section on mechanics, and in it circular motion is studied. He
wrote that if a body is released from circular motion it will travel in a
straight line which is tangent to the original circle of motion. This result
also anticipates some of the achievements of Huygens (1629-1695) and
Newton.
CONCLUSION
Archimedes was the first Greek mathematician to put into evidence the
addition of terms of an infinite series. This result appears when he
calculated the area under a parabola segment using the method of
exhaustion. A good approximation to the ∏ number is obtained with this
method (Heath, 2003). Archimedes also anticipates by several centuries
the modern concepts of power series developed by Taylor (1685-1731) and
MacLaurin (1698-1746).
The questions that arise out of Archimedean problems show the
impossibility of Greek mathematics being able to solve these problems
with the classical theory of proportions. Neither the tools nor the general
methods were available for this. Only the Scientific Revolution would
provide the solution.
After the appearance of a new mathematical language of analytic
geometry with the pioneering work of François Viète (1540-1603) and
Descartes the necessary innovation was introduced into the mathematical
corpus creating new conditions and other possibilities for science.
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