An Historical Inquiry on Geometry in Relativity: Reflections on Early

Transcription

D
History Research, ISSN 2159-550X
December 2011, Vol. 1, No. 1, 47-60
DAVID
PUBLISHING
An Historical Inquiry on Geometry in Relativity:
Reflections on Early Relationship Geometry-Physics
(Part One)
Raffaele Pisano
Cirphles-École Normale Supérieure Paris, France
Ferdinando Casolaro
University of Sannio, Italy

In order to present an historical discussion on spherical geometry in relativity between the 19th and 20th centuries,
the following research is offered in two parts. Part I: Reflections on Early Relationship Geometry-Physics is an
excursus focusing on the role played by geometry in history and its relationship with physics. Part II: Reflections
on Late Relationship Geometry-Physics is an epistemological investigation on the historical foundations of
geometry in special relativity focusing on a geometric model applied in S4 (O, x, y, z, t). In this sense, the two parts
are complementary.
Keywords: relationship geometry-physics-mathematics, foundations, historical epistemology of science.
A Very Short Introduction on Mathematics and Geometry
On Geometry and Mathematics
The reasoning ascribed to mathematics originated with men who were able, in a practical way, to
demonstrate the concept of one from two. They graphically described these cognitive processes by means of
illustrations: e.g., one animal is less dangerous than two animals. Each culture developed its calculus related to
their social typologies and demands. They used sexagesimal ratios associated with mythological and religious
cults. The Egyptians obtained several results in both arithmetic and geometry: they understood the basis of
multiplication and the technique of proofs. The Babylonians’ astronomy theories involved the use of primitive
reasoning by ratios. They used a simple triangle having sides 3, 4 and 5. It seems that this method was used for
buildings: e.g., a simple triangle, such as squadron, could be used. In order to calculate the equality between
two triangles the Egyptians measured first a side of one triangle, then those of the other triangle and
subsequently compared them. Moreover, by a degree of approximation, they could roughly measure a
circumference, some areas’ plane figures and the volume of solids. We should emphasize that these conclusions
were mostly conjecture and especially lacked the general rules applied in theory. Therefore, according to many
historians, the birth of a kind of science was delayed. It was therefore more difficult for these scientific ideas to
become widely accepted. Another reason however, could be related to social and cultural causes: new ideas and
theories were mainly broadcast among the more affluent and educated sections of society thus denying them
universal recognition. Nevertheless, the Egyptians applied this knowledge whereby geometry and the
observation of phenomena married well with special building techniques around the Nile river, where all the

Raffaele Pisano, Cirphles, Department of Philosophy, École Normale Supérieure Paris, France.
48
An Historical Inquiry On Geometry In Relativity:
Reflections On Early Relationship Geometry-Physics (Part One)
boundaries were destroyed by overflowing; for the re-building of planting lines, property ownership etc., it was
necessary to use agrimensuration techniques.
On the other hand, the Greeks could generalize a process of geometrical-mathematical knowledge,
considering it to be merely an intellectual instrument. The Greeks did not seem as interested in the applications
of geometry. They could show the equality between two triangles when only 3 data1 of each of them was
known. They obtained the other three elements, which should be naturally equal by logical reasoning. Thus,
geometry (Tannery, 1887; Dieudonne, 1985; Kvasz, 2006) progressed from a practical activity to an intellectual
one. Reasoning and physical phenomena formed a crucial combination: discovery-propriety-relationship.
Generally speaking historians of ancient science are unanimous in their claim that Pythagoras (fl. 575-490 B.C.)
and Pythagoreans’ reasoning constituted an important attempt at understanding the philosophical structure of
mathematics and its universe. Let us consider the discovery of musical harmonies2 in relation to mathematical
ratios:
Acoustical range
Eighth
Fifth
Fourth
Mathematical ratio
2:1
3:2
4:3
It is known that for Pythagoreans, a celestial body was represented by means of a range which
corresponded, in a reciprocal way to harmonies of acoustical chords. Later, they also assumed valid a
conjecture claiming that the motion of planets produced a sound called harmony of spheres. In effect, they
discovered that harmonic ratios remained valid in both musical notes which were produced by chord
perturbations, and these were produced by an air column in resonance. These aspects led them to consider the
musical harmonies in the universe and the motion of bodies related to a sound. The harmony of spheres was
obtained and the numbers were considered neutral. They were also able to transform a belief into early
scientific theory3. Briefly, we do not focus on Greek commensurability, a (Smith, 1958) problem largely dealt
with in secondary literature. In conclusion, all of these studies in geometry and mathematics contributed to
producing a specific field of applicability in both mythological questions and astronomical problems.
On Geometry and Astronomy
During the 7th and 8th centuries (Chaldean-Babylonian period) (Youschkevitch, 1976) scholars, fascinated
by the celestial bodies of the Upper Volta, attempted to closely investigate the properties of space and time in
3-dimensional geometry on spheres by means of formulas listed in tables.
The constant effort to coordinate the lunar and the tropical years seems to have been one of the important
stimuli in this area [astronomy and calendar] careful and continual observation of the apparent movements of
the moon and the sun. This activity became “scientifically significant” at least as early as 747 B.C., the date of
the first regularly repeated observations under the Assyrians. These observations were recorded in the form of
tables which were apparently used by later Greek astronomers and which a great tribute to the patience and skill
of Babylonian astronomers of Mesopotamia, although the length and completeness of these observations appear
to have been greatly exaggerated by modern authors. Many pertinent scientific data were organized in these
1
2
3
A side was one of a given triangle.
The numbers of harmonies’ ratios defines the lengths in which devices’ chords are divided.
Of course, the term scientific theory does not currently stand for scientific theory.
49
tables (Clagett, 2001).
Historical research into the evolution of mathematics in the past and its different fields of application
reveals a remarkable interest (Sarton, 1952) both in astronomy (Neugebauer, 1945) within their society or in the
astronomy–mathematics relationship. The Chaldeans, for example, went further than exploring metrics objects
related to lengths, areas, volumes, systems and sexagesimal degrees. Useful as these were in practice, the
Chaldeans also developed a kind of geometry, where they were able to conceive a curved space. The prediction
of the first possible sighting of celestial bodies each month and its impact on society was one of the main
problems (Cohen & Drabkin, 1948). In this sense, the difficulties were related to the moon and sun’s velocity,
the length of daylight and the ability of the observer. Although ancient Greek knowledge of astronomy is
incomplete, we can claim that the pseudo-scientific characters of the sun and moon (Kragh, 2008) were
discovered. Greek scholars, supported by the previous Babylonian culture, were able to conduct studies of
plane and spherical geometry. Euclidean geometry was considered useful for (ideally) describing observed
phenomena. Aristotle’s reasoning was aimed at organizing an axiomatic and encyclopaedically structure of
philosophical thought. The historian Hieronymus Georg Zeuthen (Zeuthen, 1902) (1839-1920) concluded that
Euclidean objects existed because a constructive representation by ruler–and–compass (Mascheroni, 1797;
Benjamin, 1969) was possible. Since Greek times, starting with the common belief that light travels in a
straight line, many theorems were produced (especially by Euclid). They are still considered to be among the
foundations of a mathematized light theory. Other studies concerned ambiguous speculations and were not
commonly accepted. They were a consequence of a principle (accepted by Plato but rejected as false by modern
physicists), that sight occurs through the effect of rays emanated–not by the viewed object but by the observer’s
eye. In Elements, Euclid (fl. 300 B.C.) briefly wrote about the geometry of the sphere, while in Phaenomena,
he dealt specifically with matters of spherical geometry. Astronomical tables were formulated based on
addition and subtraction formulae. The use of the chord (modern, sin) produced substantial results. Chords and
sin are essentially two equivalent functions:
 
chord    2 sin  
(1)
2
Therefore, according to modern trigonometry sin   is half of the chord subtended by a double angle.4
Claudius Ptolemaeus (fl. 90-168 B.C.) determined a geometrical relationship between chords-angles which he
listed in tables. We highlight his argument regarding Pythagoras’ theorem in which he was able to establish a
relationship:5


chord    chord 180     120 
(2)
From (2) and by means of a simple reasoning, which is not necessary to explain, one can show that
sin 2    cos 2    1
  2
On this subject one should also refer to Hipparchus’s (fl. 190-120 B.C.) On the Chords in a circle.
(3)
Autolycus from Pitane (Heath, 1931; Neugebauer, 1975) (fl. 360-290 B.C.) wrote On the Moving Sphere
(Pitane, 1971; Aujac, 1984, pp. 1-12; Pitane, 1979) and On Risings and Settings which were translated
(Autolycus, 1885) in the second half of the 19th century. On the Moving Sphere is part of a booklet called Little
Astronomy and was widely used by ancient astronomers. Some historians maintain that Autolycus preceded
4
5
We remark that several opinions concerning these aspects are different from one another. (Cfr: Geymonat L. 1973, I, 350-355.
We also specify that equation (2) cannot be entirely correct because the units of each side are different.
50
Euclid, others claimed that his works were derived from Eudoxus6 and from Cnidus’ (fl. 408-355 B.C.)
reasoning but it was generally concluded that these were the most ancient books on the matter. Greek scholars
already had an interest in both plane and spherical geometry. The abundance of reasoning on spherical
geometry theorems led historians to justifiably consider them to be known only by Greek scholars. Autolycus’
reasoning is surely more abstract than that of Euclid. Euclid’s style of presentation was not original. In On the
Moving Sphere, Autolycus deals mainly with the meridian circles, the maximum circles and parallels. Of course,
the lack of historical research in ancient times makes it difficult to propose adequate historical hypotheses.
Nevertheless, one could claim that the propositions within On the Moving Sphere are arranged according to a
logical order: the propositions are first shown in a general form, and then they are repeated with a clear
reference to diagrams, and are proven. From an epistemological point of view, this aspect shows the Euclidean
style in Phaenomena (Berggren & Thomas, 1992), in which a spherical surface is initially defined as a surface
of revolution caused by a circumference around its own diameter. At that time, when geometry was applied to
cosmology (Kragh, 1991) questions also arose in astronomy such as determining the time of night by observing
the stars. The latter was one of the main problems in Greek astronomy. This question led astronomers to
improve methods used in spherical geometry.
Theodosius is quite interesting in Bitinia’s (fl. 379-395 B.C.) attempts. He collected his results in Spherica.
Unfortunately, at that time it did not help to solve the Greek problem mentioned above. Menelaus (fl. 98 B.C.),
in his studies on trigonometry, wrote Sphaerica collected from three books written in Arabic7 released and
later translated by Francesco Maurolycus (1494-1575). In Book I, the concepts of a spherical triangle are clearly
shown, e.g.: a figure composed of three arcs of the maximum circle belongs to a sphere and each of them is
lesser than a semi–circle. He aimed at showing that with spherical triangles; theorems were almost similar to
the ones shown by Euclid on the triangles in plane geometry. The sum of two sides of a spherical triangle is
greater than the third; the sum of the angles of a triangle is greater than two right angles and equal sides subtend
equal angles. Moreover, Menelaus demonstrated a singular theorem with no analogous theorems in the field of
the plane triangles: if the angles of a spherical triangle are respectively equal to those of another triangle, then
the two triangles are congruous. Other theorems of congruence for isosceles triangles completed the book.8
In the 2nd century Claudius Ptolomaeus (fl. 100-175 B.C.), faced with problems of spherical trigonometry,
calculated the chords of the arcs, and at the same time, introduced the theoretical basis of plane trigonometry.9
Ptolomaeus dealt with spherical triangles but in his manuscripts a systematic illustration is missing and the
theorems necessary to solve specific astronomical problems are the only ones to be proved. Later on, the
Arabians al’Battani (fl. 858-929) and Abu’l Wafa10 (fl. 940-998), to a greater extent than Persian Nasir–Ed din
(1201-1274), left their legacy of geometry and of spherical trigonometry. It seems that they were not known in
Europe until the 2nd half of the 15th century, when Georg von Peuerbach (1423-1461) and his pupil Johannes
Müller from Königsberg11 (called Regiomontanus: 1436-1476) began to translate Greek and Arabic works. In
6
Eudoxus studied homocentric and concentric systems. These constituted one of the main astronomical problems in antiquity. On
speed explained them, but is now lost.
7
A Grecian translation had been lost. See also a new interesting approach in a recent work by Szczeciniarz (Szczeciniarz).
8
Book II is mainly on astronomy and indirectly refers to spherical geometry. Book III deals with spherical trigonometry.
9
We should emphasise that trigonometry was mainly formulated in order to be applied to astronomy and since spherical
trigonometry proved more useful, it is understandable that it developed first.
10
He translated Euclid, Diophantus (fl. 284-298) and al–Khwarizmi’s (fl. 780-850) works adding his commentaries.
11
Daniel Santbech (fl. 1561) compiled a collection of De triangulis written in 1462-1463, Tabulae directionum: De triangulis
planis et sphaericis libri quinque (1533) and Compositio tabularum sinum recto.
51
particular, De triangulis Spaericis collected within an organic system all the information available on spherical
geometry and spherical trigonometry. Meanwhile, Johann Werner (1468-1522) improved Regiomontanus’s
ideas in De triangulis Sphaericis (1514) where spherical trigonometry was emphasized due to the need for
using a large amount of formulae as Regiomontanus and Werner (and later Copernicus) had used the functions
of sine and cosine only. George Joachim Rhaeticus (1514-1574), a pupil of Nicolaus Copernicus (1473-1543),
used all of the six trigonometric functions. Spherical trigonometry was further arranged in a system and partly
developed by Francois Viète (1540-1603), who obtained all of the necessary formulae to calculate a part of a
spherical rectangular triangle: two known parts and a practical rule known as Napier’s rule from John Napier
(1550-1617). In the 17th century, Edmund Halley (1656-1742) used spherical geometry (also reviewed in some
works). In the 18th century, thanks to Euler (1707-1783), a modern study of spherical trigonometry started to
define the whole of trigonometry by main principles. Carl Friedrich Gauss (1777-1855), who considered a
curved surface a space by itself, provided a generalization of spherical trigonometry.
In a later period all of the studies mentioned produced a generalization of geometry on curved surfaces
which became known as non–Euclidean geometries. In order to have a global view of scientific works12
contested in the 19th century we present an incomplete excursus of the most relevant contributions to geometry
and mathematics in that short revolutionary period.
Table 1
A partial excursus on geometry and physics–mathematics manuscripts
Year
1799
1799
1805
1820
1821
1822
1823
1826
1827
1829(30)
1829
1831
1832
1833
1835(38)
12
13
Author and manuscript
Theory
C.F. Gauss (177-1855): Demonstratio nova theorematis omnem functionem algebraicam
Geometry
rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse.
G. Monge (1746-1818): Traité de Géométrie descriptive
Geometry
P. S. Laplace (1749-1827): Traité de Mécanique céleste
Physics–mathematics
J.B.J. Fourier (1768-1830): Mémoire d'analyse sur le mouvement de la chaleur dans les
fluides
A. L. Cauchy (1789-1857): Cours d'analyse de l‘École Polytechnique
Mathematics
J. V. Poncelet (1788-1867): Traité des propriétés projectives des figures.
Geometry
A. L. Cauchy (1789-1857): Résumé des leçons sur le calcul différentiel
Mathematics
N.I. Lobacevskij13 (1793-1856): Exposition succincte des principes de la géométrie, avec
Geometry
une démonstration rigoureuse de la Théorie des Paralléles
C. F. Gauss (177-1855): Disquisitiones generales circa superfice curvas
Geometry
N.I Lobacevskij (1793-1856): “O Nachalakh Geometrii” in Kazanskij Vestnik
Geometry
A. L. Cauchy (1789-1857): Leçons sur le calcul différentiel
Mathematics
E. Galois (1811-1832): Deux mémoires d’analyse in Azra J. P., Bourgne R.
Mathematics(-geometry)
Gauthier–Villars, 1862, Paris.
J. Bolyai (1802-1860): “Appendix” in scientiam spatii absolute Veram exhibens: a veritate
aut iaisitate Axiomatis XI Euctidei (a priori haud unquam decidcnda)
independentem;adjeeta,ad casum faisitatis, quadratura circuti geometrica. Auctore Geometry
Johanne Bolyai de Bolya, Geometrarum in Exercïtu Caesareo Regio Austriaco Castrensium
Capitaneo. Maros-Vasarhelyini.
J. Bolyai (1802-1860): Tentamen iuventutem studiosam in elementa matheseos puræ
elementaris ac sublimioris methodo intuitiva evidentiaque huic propria introducendi: cum
Geometry
appendice triplic. Auctore Professare Mathescos et Physices Chemiaeque publico
ordinario. Toums Primus, umptibus Academiæ scientiarum hungaricæ.
N.I Lobacevskij (1793-1856): Novye nacala geometrij s polnoj teoriej parallel'nyh
Geometry
Pisano & Capecchi, 2009, 83-90.
For the history of geometries: Boris & Rosenfeld, 1998.
52
1836
1836
1837
1840
1868
1878
1885
1887
P. S. Laplace (1749-1827): Exposition du système du monde
N. I. Lobacevskij (1793-1856): “Primeniene voobrazaemoj geometrii k nekotorym
Geometry
integrlam”, in Kazanskij Vestnik
N.I Lobacevskij (1793-1856): “Géométrie imaginaire” in Journal für die reine und
Geometry
angewandte Mathematik, XVII.
N. I. Lobacevskij (1793-1856): Geometrische Untersuchungen zur Theorie der
Geometry
Parallellinien
N. I. Lobacevskij (1793-1856): La Science Absolue de l’Espace indépendante de vérité ou
Geometry
de la fausseté de l’Axiome XI d’Euclide (que l'on ne pourra jamais établir a priori)
J. B. J. Fourier (1768-1830): Analytical theory of heat
Autolycus from Pitane. (fl. 360-290). De sphaera quae movetur liber de ortibus et
Geometry
occasibus libri duo, transl. by. Hultsch, Teubneri, Lipsiae.
J. H. Poincaré, “Sur les hypothèses fondamentales de la géométrie” in Bulletin de la
Geometry
Société Mathématique de France, XV
Although the chronology is not very detailed, as in Table 1 above, basically, there is enough evidence to
suggest the significant role and influence played by geometry and mathematics in future physics:
physics-mathematics or rational theory.
On Geometry and Mathematics Applied to Mechanics
It is likely that the Greeks would have been the first to attempt to solve some problems of representation
and it is not improbable that in the classical age a system of representation similar to the one developed during
the Renaissance had already been developed. There is, however, no evidence that a mathematical and unique
definition of space can be found in the pre-historical representations.
At the beginning of the Renaissance, architects (Briggs, 1927; Pisano, 2009; Pisano & Gaudiello, 2009)
were interested in re-evaluating14 De architettura (Vitruvius, 1990) by Vitruvius (fl. 80 (70?)-23 B.C.). The
application of geometry to early mechanics became the first guide15 for both design (practical) (Casolaro, 2004)
and building. Its translation also included figures by Daniele Barbaro (1513-1570) published in Venice in 1567
and it made an important contribution to the popularization of Roman works (Reymond, 1927) during the
Renaissance.
Figure 1. Tower section–Book I16
14
De architectura was probably written between 27 and 23 B.C. In 1414, the manuscript was re-translated by Poggio Bracciolini
(1380-1459), historian and Italian humanist. In regard to the originality of the manuscript, George Sarton (1884-1956) and John
Louis Emil Dreyer (1852-1926) thought that De architectura could also be post and written by a “pseudo-Vitruvius”. (Cfr.: Stahl,
1972, 123).
15
It seems that Theophrastus (fl. 372-287 B.C.) was also a guide for the architects at that time.
16
Vitruvius, 1567. I dieci libri dell'architettura, by Daniele Barbaro, Libro I, cap.V, 49. (The two images present the importance
of geometry both as pure discipline and applied to architecture). On the history of science and architecture, please see recently:
Capecchi & Pisano, 2010.
53
Figure 2. Plant and wall–Book I17
In particular, if we refer to mechanics (Clagett, 1959), and more specifically to geometry in statics
(Capecchi & Pisano, 2007), we will notice many Renaissance building problems (Pisano, 2008) already
discussed in De architectura. For example, Vitruvius’ reasoning on Gerone II’s (fl. 308-216. A.C.) golden
crown and Book IX completed by Archimedes (fl. 287-212) are very well–known. De architectura consists of
ten Books which correspond with the reasoning put forward by Heron from Alexandria (fl. I B.C.? II A.C.?).
Without a doubt, Vitruvius would also have been familiar with the theories of Marcus Terentius Varro Lucullus
(fl. 116-27 B.C.) as found in Disciplinarum. In Book I, Vitruvius suggested (sometimes ambiguously) that both
mathematical and geometrical knowledge were necessary for architects and builders and applied geometry was
useful in astronomy (Book IX). The 12th century brought dramatic changes in style and in Gothic architecture.
One of the main problems was to achieve maximum light and space by minimizing the least clutter of walls and
structures. Generally speaking, one of the first physical sciences to be influenced by mathematics was René
Descartes’18 (1625-1637) Optics in which any physical law was followed by a mathematical interpretation. The
methods of perspective, in geometry, can be considered to be among those used to represent figures of space on
a plane. Gothic architecture lengthened vertical lines and increased the geometric complexity of the plants
producing more aisles. Thus, Gothic with radial chapels, perspectives continually changing, according to the
dynamic and ever changing interplay of light, often filtered through the coloured stained glass. Therefore, the
Gothic cathedral around which generations of builders and craftsmen flourished is in fact an extremely logical
and stable building. The representation of the technical details of a figure is achieved through the methods of
descriptive geometry (Monge, 1811), while projective geometry (Poncelet, 1865-1866) serves to produce a
representation through a transformation caused by the use of projections and sections.
An Epistemological Excursus on Astronomy
The history of astronomy (as cited in Dumas, 1957; as cited in Taton, 1961, pp. 123-126), (including
cosmology) was born at least two thousand five hundred years ago (Duhem, 1958): Comparing such a long
17
Vitruvius, 1567, Libro I, V, 52.
Descartes, 1897-1913. See Discours de la méthode et Essais, Specimina philosophiae Vol. VI; Physico–mathematica Vol. X,
Le Monde ou Traité de la lumière, Vol. XI (Id. 1964-1974).
18
54
period with nearly four centuries of physics, from the Renaissance until today, seems odd. The history of
astronomy seems much older than the history of physics. Nonetheless, at least with the work of Alexandre
Koyré (Koyrè, 1957) (1892-1864) and Thomas Kuhn (Kuhn, 1962, 1978) (1922-1996), it has been transformed
from a descriptive history (results, devices, scientists and institutions) to an interpretative one. This
transformation did not occur in the history of astronomy, although it had been initiated during the Copernicus
revolution which was extremely important both from an intellectual and also foundational point of view. This
historical deficiency may depend on many different factors, one being a history too strictly tied to the division
between two cosmological theories, so confining the interpretative data to just a few hypotheses. Yet in the
history of astronomy a further historiographical problem exists, that is the question of the temporal aspect:
Table 2
A temporal excursus for the history of astronomy (adapted by Drago, 1991)
Items
Astronomy
Pre–Hellenism
Clergies
Objects
Cosmos, Sun and
Stars
Theories
Trigonometry
Type of Knowledge
of science
Isolated and
non–systematic
Hellenism
1540
1687
1820
Copernicus,
Ptolomaeus
Newton
Laplace
Kepler, Galilei
Cosmos, Sun and Sun, Earth and Solar Sun, Earth and Solar Solar system and
Stars
system
system
Galaxies
Trigonometry,
Trigonometry,
Infinitesimal
Infinitesimal
geometry
geometry
analysis
analysis, mechanics
The only science for A new science for a A physical theory to
a (mathematical)
general
interpret terrestrial
A mathematicaldescription of nature understanding of the
and celestial
pattern theory
(cosmos)
World
phenomena
It is worth noticing that, according to the previous Table 1, the ancient history of astronomy cannot be
fully examined, due to its insufficient theorization, since it is largely rooted in the cultural geographical and
temporal context. As a matter of fact, its interpretation mainly depends on its social and cultural background.
Subsequently, using a uniquely conservative approach (religious), the next step was the study (laically) of the
universe. That is why this period seems, from a temporal point of view, very short. Only two millennia later did
astronomers reconsider the conjecture of a cosmological view founded on a curve basis and only when it was
necessary to face the problem of determining the most suitable geometry for the space ruling laws. The
scientific research at that time seems to echo (or be centered around) the scientific programmes in Traité de
Mécanique céleste and Exposition du système du monde (Laplace, 1836). Both are the crucial programmes
proposed by Pierre Simon Laplace (Gillispie, 1997) (1749-1827) and are focused on central forces offering the
2
possibility of realizing differential equations d x  f ( r  , v, t ) by  as an exponent different even from 2.
dt
Man being induced, by the illusions of his senses, to consider himself as the centre of the universe, it was
easy to persuade him, that the stars influenced the events of his life, and could prognosticate to him his future
destiny.19
Simeon-Denis Poisson (1781-1840), fully respecting the Newtonian paradigm (Newton I., 1730)20 in his
interpretation of all types of celestial and earthly phenomena through cause–forces, that is, typically, the central
forces, applied the scheme to many other cases, in particular to heat phenomena (Lavoisier & Laplace, 1784),
19
Laplace 1809, 248-249, line 23. (In French: “L’homme, porté par les illusions des sens à se regarder comme le centre de
l'univers, se persuade facilement que les astres influent sur sa destinée, et qu'il est possible de la prévoir par l'observation de leurs
aspects au moment de la naissance.” (Laplace 1836, Tome II, 373, line 3).
20
Please, see the Newtonian claim on last queries in: Newton I. 1730, 388-389.
55
discovering some laws of gases, e.g., the law of adiabatics (Poisson, 1823).
Table 3
Main Newtonian paradigm
Items in theory
Space
Time
Basic concepts
Inertia
Atom
Fluid
Mass
Interaction
Newtonian Mechanics (1687)
Absolute and mathematical.
Absolute and mathematical.
Accelerations.
Perpetual.
Infinitesimal part of matter.
Phlogiston (corporeal)
Inertial.
Force–cause.
Problem of the theory


F  ma
Arguing techniques
Solutions
Differential equations.
Any possible solution, for a given force.
Charles Singer (1876-1960) (Singer, 1959), referring to the Newtonian paradigm and the development of
astronomy, proposed the following classification:
(a) Experimental astronomy: observation of celestial bodies by telescope
(b) Dynamical astronomy: Laplacian program
(c) Astrophysics: physical and chemical studies of celestial bodies.
With regard to astronomy, theories which were included in mechanics became rational ones, that is to say,
a physical-mathematical (Paty, 2001; Paty, 1994) theory in which, in some cases, the physical principles (and
experimental phenomena) could also be considered.
These aspects developed mainly due to Jean Baptiste Joseph Fourier (Grattan-Guinness & Ravetz, 1972; as
cited in Gillispie & American Council of Learned Societies, 1970-1990) (1768-1830) who included numerous
mathematical calculations resulting in the theory appearing to be a completely mathematical one: Théorie
analytique de la chaleur (Fourier, 1822). Let us examine some historical examples which occurred in the 19th
century.
Table 4
Examples of the roles played by mechanics in the theory (adapted by Drago, 1991)
Celeste
Planetary
MECHANICS
(Mathematics) Engineering
Corporeal
Molecular
Stars, comets, planets satellites, Stability of planetarium system
Geodesy, cartography
Mechanics, instruments, frictions, structures,
analytical theory of heat
Statics, dynamics, hydrodynamics, crystallography
Elasticity
In 181121, on the occasion of a competition, whose subject was the propagation of heat in solid bodies,
Fourier improved his work, presented again in Mémoire su la théorie analytique de la chaleur (Fourier, 1829). In
21
On 21 December 1807, he presented his work on heat, Théorie de la propagation de la chaleur dans les solides, (Fourier, 1807.
Tome I, 112-116. (n. 6, mars 1808); Id., 1888-1890.Œuvres de Fourier) at the Académie des Sciences de Paris. He focused on
heat transmission between discrete masses and special case-studies of continuous bodies (e.g. ring, sphere, cylinder, rectangular
prism and cube). The prize was conferred only in 1812.
56
1822 Fourier (Grattan-Guinness 1970-1990; as cited in Gillispie & American Council of Learned Societies,
1970-1990) published a monographic work, the Théorie analytique de la chaleur (Fourier, 1822) which
contained (in its first part) the whole of his Mémoire of 1811. The basic proprieties claimed by Fourier concern
the capacity of bodies to absorb and transfer heat. However, a small group of primeval facts deduct the principles
of the theory. As a matter of fact, one can see this aspect in rational mechanics, where Fourier claimed that
The principles of the theory are derived, as are those of rational mechanics, from a very small number of primary
facts, the causes of which are not considered by geometers, but which they admit as the results of common
observations confirmed by all experiment. The differential equations of the propagation of heat express the most
general conditions, and reduce the physical questions to problems of pure analysis, and this is the proper object of
theory. They are not less rigorously established than the general equations of equilibrium and motion. 22
For his Habilitationsschrift (probationary essay), Bernhard Riemann (1826-1866) (Freudenthal, 1970-1980) studied Fouriers’ series and presented the essay in 1853. Although trigonometric series were long used
in astronomy, Fourier made frequent use of them in solving the heat equation but did little to resolve the
fundamental issues. Riemann’s essay made considerable progress towards solving this problem, first by giving
a criterion for a function to be integrable and then by obtaining a necessary condition for a Riemann integrable
function to be representable by Fourier’s series. Nevertheless, at that time, Fourier’s analytic theory of heat was
too under-developed to be included in a definitive mathematical (-mechanical) scheme (Grattan-Guinness,
1969), in fact, “part of its impact was that it did not fit into the scheme of rational and celestial mechanics”23.
Fourier based his scientific thought firmly on the role played by mathematics with regard to physical
phenomena, which guided his thinking towards a rational physics. Many physical concepts (e.g., velocity,
propagation) were supported by means of differential equations and integral calculus. This mathematical aspect
gave these concepts rationality thereby producing a physical-mathematical theory. Therefore it can be
concluded that a theoretical physics theory becomes more significant when it is included in a larger and
consolidated theory and strongly interpreted by mathematics.
Following Fourier’s approach, in 1861 Gabriel Lamé (1795-1870) wrote Leçons sur la théorie analytique
de la chaleur (Lamé, 1861), focusing on the mathematical role to be played in the transmission of heat,
independently, both of physical law phenomena and the nature of heat. Like Fourier (Pisano & Capecchi, 2009),
Lame was not interested in experiments and measuring procedures, hypotheses on the nature of heat, the
physical theory of heat, or specific heat, as one can read in his theoretical premise in his Discours
préliminaire24 where he establishes that his course on the analytical theory of heat does not presuppose the
laws of heat-exchange, or radiation.
22
Fourier 1878, 6, line 11. (In French: “Les principes de cette théorie sont déduits, comme ceux de la mécanique rationnelle, d'un
très–petit nombre de faits primordiaux, dont les géomètres ne considèrent point la cause, mais qu'ils admettent comme résultant
des observations communes et confirmés par toutes les expériences. Les équations différentielles de la propagation de la chaleur
expriment les conditions les plus générales, et ramènent les questions physiques à des problèmes d'analyse pure, ce qui est
proprement l'objet de la théorie. Elles ne sont pas moins rigoureusement démontrées que les équations générales de l'équilibre et
du movement.” (Fourier 1822, xj, line 4).
23
Keston D. “Joseph Fourier, Politician & Scientist”, http://www.todayinsci.com/F/Fourier_JBJ/FourierPoliticianScientistBio.htm
(Quotations marks from the author).
24
“Le Cours que j’entreprends aujourd’hui a pour objet principal d’établir la Théorie analytique de la Chaleur, sans partir
d’aucun principe hypothétique relatif à la constitution intérieure des milieux solides, sans présupposer les lois de l’échange
calorifique, ou du rayonnement particulaire, sans adopter aucune restriction pour les variations de la conductibilité autour d’un
même point. Je suis de plus en convaincu qu’en évitant, de cette manière, toute idée préconçue sur les lois naturelles, […]. En
effet, la Théorie de l’Élasticité, complètement dégagée de tout principe hypothétique, peut démontrer rigoureusement, en
s’appuyant sur les faits, que dans les milieux diaphanes, les particuliers pondérables vibrent lumineusement” (Lamé, 1861. V).
57
Table 5
Basic concepts in Fourier’s theory
Items in theory
Space
Time
Basic concepts
Heat
Problem of the theory
Arguing techniques
Solutions
Fourier’s Théorie analytique de la chaleur (1822)
Mathematical for calculus.
Mathematical for calculus.
Temperature as a mathematical and continuous function. Velocity of heat transmission.
Declared doubts on the nature of heat overcame by trust in mathematics laws25.
To calculate the temperature T= f(x,y,z,t) for an isotropic and homogènes solid bodies–system which
expresses the value of v (Fourier, 1822).
Differential calculus and Integral calculus.
Differential equations applied to particular case-study and integral calculus applied to general cases
to obtain (only) mathematical results. Series expansions.
Ultimately, the recognition of the importance of mathematics in a theory occurs when it is included in a
new theory: e.g., propagation, velocity etc. mathematically applied to physics. The theory was promoted by
means of an advanced mathematics (differential calculus by partial derivates, integral calculus, series, etc.) to
interpret each field of phenomena. In this sense, a theory is absorbed by mathematics, producing crucial
consequences: that is the birth of new theories, e.g., Théorie analytique de la chaleur, mécanique céleste and
Exposition du système du monde (Laplace, 1805, 1836). Nevertheless, there is the risk that the original theory
can become devalued (Pisano & Capecchi, 2009).
Some Final Remarks
Greek geometry as a proved science is essentially considered from 500 to 300 B.C. (Archimedean era).
One has to wait for a more thorough application of geometry to machines (Knobloch, 2004).
We do not have to wait until medieval or early modern times for the application of geometry to the
investigation of nature, for it began in both physics and astronomy in the fourth century and matured in the
Hellenistic and Greek–Roman periods (Clagett, 2001).
Thus, even if it is not formalized and is incomplete, the Greek astronomers already had enough knowledge
to apply plane and spherical geometry in the interpretation of natural phenomena and astronomy, e.g., for the
motions of the moon and sun. Let us consider the Chaldean–Babylonian era. This part of the history of science
is fragmented with regard to philosophical and device studies. This science in antiquity had above all crucial
theoretical significance. It was oriented towards astronomy, geography and mathematics. In fact, practical
studies, useful to astronomers and geographers (tables, formulas etc.) were formulated. Nevertheless, an
intrinsic improvement in spherical theoretical geometry, e.g., by Autolycus and Menelaus, was developed. Let
us also recall the spherical triangles theory developed at the same time as the plane theory (Elements, Book I).
Theodosius also disputed spherics theorems, even if he presented them like theorems on solids. A collection of
interesting results on spherical geometry was developed up to the time of Ptolomaeus who gave us one of the
first trigonometric rules for finding the chord as the sum of two angles. Since the 1950s, all of these studies
have not always been historically recorded as scientific ones by historians. On this point Clagett claimed:
25
“15. Il restait encore à comparer les faits avec la Théorie. […]. Ces expériences confirment le principe dont on est parti, et qui
est adopté de tous les physiciens, malgré la diversité de leurs hypothèses sur la nature de la chaleur. (Fourier, 1822. 12, line 16).
“Section II. […] 22. On ne pourrait former que des hypothèses incertaines sur la nature de la chaleur, mais la connaissance de lois
mathématique auxquelles ses effets sont assujettis est Independent da toute hypothèse; elle exige seulement l’examen attentif des
faits principaux que les observations communes ont indiquées, […]” (Fourier, 1822, 18, line 7).
58
One of my colleagues has raised a question as to whether the title [of the book:] Greek Science in Antiquity is a
tautology. My answer is no for two important reasons. [1] In the first place there is other than Greek Science in
antiquity (e.g., Babylonian Science or Indian Science). [2] Secondly and more important, I have stressed throughout
the volume that the Greek scientific corpus has had a long life in antiquity, in the Middle Ages, and in early modern
times (Clagett, 2001).
The basis of Science was essentially empirical in nature (Preti, 1957), without a theoretical approach and
generalized with the exception of late geometry and astronomy after the Chaldean–Babylonian era. Science was
largely related to social problems and the difference between pure and practical science was almost
indistinguishable. We have to wait for the Greek period to have an organized and generalized collection of
mathematical and geometrical studies. But epistemological questions remain.26
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An Historical Inquiry on Geometry in Relativity: Reflections on Early

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