20th century cosmology. Translated from the French. Edited by Sergio Schiappori. With a preface by Vincenzo Fano. With an introduction by Giovanni Macchia (Q6488508)

It is a merit of the publishing house PGRECO to have reprinted the Italian translation of Jacques Merlau-Ponty's Cosmologie du XXe siècle. The original French text was published by Gallimard in 1965. The Italian translation dates to 1974 and was edited by Il Saggiatore. The volume under review is enriched by the Preface of Vincenzo Fano and by the Introduction of Giovanni Macchia. This book is important for various reasons: first of all, it was published shortly before the cosmic microwave background, discovered in 1965 itself, was interpreted as a residue of the Big Bang. This means that when Merleau-Ponty wrote, the diverse cosmological theories were on an equal footing and the Big Bang was not yet considered the standard theory. This makes his considerations particularly interesting. Furthermore, the phase of cosmology (1917--1961) considered by Merleau-Ponty is characterized by a highly hypothetical and deductive approach because the observations were insufficient to express a preference among the different theories. The main feature of this book is the following one: in the running text the use of mathematics is extremely restricted. The author speaks and comments on the themes he proposes expressing some mathematical concepts in words and giving for granted some other mathematical notions. Thus, at a certain level of reading, the text is understandable even without knowledge of all the mathematical apparatus it discusses. On the other hand, a full understanding needs such a knowledge. This work can be considered at least in two perspectives: 1) as a history of cosmologies from 1917 to 1961; 2) as a philosophical and epistemological commentary to such cosmologies. It seems to me that the first perspective is better developed and more original.NEWLINENEWLINEVincenzo Fano in his Preface (pp. I--V) gives a succinct, but useful picture of the main topics dealt with by Merleau-Ponty.NEWLINENEWLINEGiovanni Macchia (pp. VII--XXXVII), after having explained the main aspects of his biography and of his philosophy (pp. VIII--XII), draws a brief picture of cosmology in the epoch analysed by Merleau-Ponty. He points out that, in a sense, Newtonian physics had prevented a real cosmological debate (pp. XII--XVII). After that, Macchia (pp. XVIII--XXII) expounds the narratological aspect of the book, the way in which it was received by some cosmologists (he mentions the positive review by Sciama in 1968), and a comparison with David North's and Milton Karl Munitz's works on the history of cosmology. For, they were published in the same years as Merleau-Ponty's. In the final part of his Introduction Macchia faces some of the most important aspects of Merleau-Ponty's book, underlying that he worked in the period in which the vision of a static universe was overcome in favour of a dynamic one.NEWLINENEWLINEI will now summarize the main content of this so ponderous work: Einstein's Betrachtungen (1917) represent the birth of modern cosmology. From this work, the systematic research of Merleau-Ponty begins. However, the author provides an introductory chapter (pp. 15--34). After a brief outline on the most important opinions and discoveries in the cosmological field (Galileo, Herschel, Kant), he points out that at the beginning of the 20th century the structure of the universe was not clear. Merleau-Ponty recalls the opinion of the Swedish scientist Charlier who proposed a model of the universe based on two hypotheses. The first was that the observed spiral objects were not galaxies, but formations existing in the Milky Way (p. 17). The fundamental work of Hubble (which started in the 1920s), his collaborators and his successors is afterwards analysed. It established that the spirals are galaxies, they are distributed in the skies in an isotropic and uniform manner. Each galaxy can differ greatly from another. Hubble discovered the red-shift so that each galaxy seems to recede from the Earth with a speed proportional to its distance from our planet. (p. 19). Merleau-Ponty briefly recalls the initial interpretations of the red-shift, given by Eddington, Weyl and Wirtz. Some considerations on the problems of isotropy and homogeneity of the universe follow. These two features were accepted by most of the cosmologists, though there were some debates described by the author (pp. 23--28). As to the red-shift, it was immediately clear that: 1) it is similar to the Doppler-effect; 2) it is isotropic; 3) it is a direct function of the apparent magnitude, which, in its turn is a direct function of distance, so that the red-shift is a direct function of distance. The problem was that interpreting the red-shift as a Doppler effect means to recognize the universe's expansion, which was something new and unexpected. Some scientists (as Zwicky) refused this interpretation, but most of them adhered to it (pp. 29--30). Important hypotheses and discoveries partially connected to cosmology and astrophysics also took place in those years: il 1919 Perrin proposed that the transmutation of hydrogen in helium was the basis of the stars energy. Jeans, Bethe, Von Weizsäcker, Schatzman and Fowler worked on Perrin's hypothesis, which eventually was the right one (p. 33).NEWLINENEWLINEThe second chapter (pp. 35--69) concerns the relativistic cosmology. The new cosmology, Merleau-Ponty claims, begins in February 1917 with Einstein's ``Kosmologische Betrachtungen'': for the description of the universe given through Newton's law be coherent with the observed phenomena, it is necessary to add an additional term to the Poisson equation. A similar necessity also arises with Einstein's equation. With this modification, a universe in which time can go from \(-\infty\) to \(+\infty\), but space is finite, though unbounded, is produced (pp. 35--36). A first interesting consideration concerns inertia: according to Mach's principle, accepted by Einstein, the inertia of a body far away from any mass should tend to 0. This implies narrow conditions for the metric's coefficients of space-time. The calculations show that, under this condition for inertia, the space-coefficients of the metric tensor tend to 0, while the temporal one tends to infinity. This implies that potential energy tends to infinity. In turn, this has the consequence that the speed of the stars should be a non-negligible fraction of that of light, which is not the case (p. 37). The first thinkable solution is that the metric at infinity be the one connoting special relativity. But for various reasons, this supposition is not satisfying (p. 38). The second solution entails giving up the general determination of conditions to infinity, which is a view unacceptable for Einstein. Then, he decided to remove infinity and posed the universe to be finite (p. 39). Thus, space is closed. The author explains the implications of this idea (pp. 40--42). The main problem is the following one: Einstein was able to reduce the ten components of the energy tensor for cosmic matter to only one component, namely the mean density of matter-energy. A priori it can be a function of the coordinates, but if space is closed this hypothesis is not acceptable (p. 42). Therefore, a modification of the Einstein equations is necessary: the cosmological constant \(\Lambda\) is, thus, introduced. Merleau-Ponty describes the most relevant features of Einstein's universe, among which the following one is essential: ``In Einstein's universe space's curvature has no incidence on time. The intrinsic distinction between space and time called into question by the local geometry of special relativity, is, hence, restored on a cosmic level'' (p. 43). One of the problems of the first cosmological model devised by Einstein is that the modified field equations allow empty solutions and the hypothesis of a geometrically structured space-time independently of matter. This is in contradiction with one of the main tenets of Einstein's conceptions (p. 44). A further problem was that a satisfying mathematical expression for inertia was missing (p. 45). Finally, the discovery of the red-shift completely changed the perspectives of cosmology, so that, in many respects, this pioneering and fundamental work by Einstein became outdated within a relatively short time (pp. 46--47). An important aspect emphasised by Merleau-Ponty is that, with general relativity, the natural geometry becomes local and is suitable to include gravitation. But, if geometry and physics are local, how is it possible to speak of the universe? Einstein, who was convinced that it made perfect sense to speak of the universe as a whole, posed these problems and tried to give the first solutions. In this respect his Betrachtungen represent a fundamental contribution (pp. 48--51). The next step of the story told by the author regards De Sitter universe. As known, this is a universe without matter, but it is coherent with Einstein's equations. The discussions of the physical meaning of De Sitter universe engaged the community of physicists (Einstein, Friedmann, Eddington, Lemaître, Lanczos, Tolman, Robertson and Weyl) for several years (p. 52). The most important criticism addressed by De Sitter to Einstein is the following: through a stereographic transformation, it is possible to prove that the spatial part of the metric in Einstein's universe is null, whereas this is not the case for the time part. This implies the existence of a cosmic time, but appears not to be perfectly conform with the covariant character of general relativity. In contrast to this, all the coefficients of the metric are null at infinity in De Sitter universe, which can be interpreted as a limit case of a universe with very low density of matter (pp. 53-54). The author explains afterwards that De Sitter maintained the constant \(\Lambda\), but with a meaning different from Einstein's. This implied some advantages, but also some problems briefly explained by Merleau-Ponty (p. 55). He clarifies that, at the beginning, there were several misunderstandings with regard to De Sitter universe. Particularly, it was regarded as static. But, in fact, it is not: it is stationary, but not static, as Couderc and Robertson pointed out (pp. 56--57). As Merleau-Ponty remarks, Lemaître observed that, since De Sitter universe is empty, the distinction between space and time is unsubstantial. For, only the presence of matter determines a natural division in space and time (p. 57). In this case, the author also offers some mathematical details which allow to better understand the situation (pp. 57--59). A fundamental question is the following one: what does it mean that a universe devoid of matter is not static? Weyl answered that if such a little quantity of matter were introduced as not to modify the metric, this matter would tend to be dispersed, to diverge from the point where it was located (pp. 59--60). A fundamental contribution to the understanding and interpretation of De Sitter universe was given by Weyl in 1923 in his monograph ``Zur allgemeinen Relativitätstheorie'' where he posed the problem of finding the geometrical equivalent of a material system in this empty universe, so that a causal link was established among the points of the universe. Weyl was able to determine the geodetics on the De Sitter manifold and to determine the cosmic time as well as a universal parameter associated with the trajectories of each text-particle. He also found the red-shift, coherently with the recent discovery in Mount Palomar (pp. 61--62). The next scientist analysed by Merleau-Ponty is a milestone in modern cosmology: Aleksandr Friedmann. In his papers written in 1922 and 1924 a static universe is explicitly rejected. Friedmann poses physical and geometrical hypotheses. The former are those of Einstein and De Sitter. As to the latter, they are that: 1) space has constant curvature in any point, not in any time; 2) a system of coordinates can be chosen so that time is orthogonal to space (pp. 64--65). On the basis of these hypotheses, he obtains a differential equation in \(R,(R,)' R''\), being \(R\) the curvature radius of space (which is function only of time). It, on the basis of the value ascribed to \(\Lambda\), gives rise to three kinds of universe: two in eternal expansion (negative or null curvature) and one periodical (positive curvature). Merleau-Ponty explains time topology in the periodic universes (p. 65). In his second paper Friedmann deals with space with negative curvature, hence, an infinite space (p. 66). Merlau-Ponty remarks that Friedman was free of any philosophical presupposition with regard to the structure of the universe. This allowed him to analyse only the mathematical implications of his results. In this respect his cosmological view was more profound than those of all his contemporaries and he was open to accept the idea of an expanding universe (pp. 67--69).NEWLINENEWLINEThe third chapter (pp. 70--101) is dedicated to the development of relativistic cosmology. The two most important items concern Weyl's explanations concerning the geometry of De Sitter universe and Robertson's metric referred to the Friedmann equations. With regard to this latter question, the basic ideas of the relativistic cosmologies were that in the universe considered as a whole the local irregularities in the distribution of matter disappear, the local movements take place very slowly with respect to the light speed, and the density of matter is low. These presuppositions were empirically confirmed at the beginning of the 1930s (pp. 72--73). This granted, it is possible to construct the line element \(ds^2\) so that a positive real parameter \(R\) appears which measures the curvature of space (to be more precise: a kind of curvature), and which is a function of the cosmic time (p. 73). Under reasonable hypotheses on \(R(t)\) the existence of a universal Doppler effect is deduced (pp. 73--74). Robertson in 1929 had the merit to offer the expression of such a line element. However, significant uncertainties still existed. Particularly, is the notion of cosmic time the transcription of an empirical fact? (P. 75). Other classical problems concerned the finiteness or infinity of space and the nature of its expansion: is it followed by a contraction? Is the universe oscillating? Merleau-Ponty describes the progresses and the doubts of cosmology between the 1920s and the 1930s, as well as their connections with the value of the cosmic constant and with observations (pp. 76--78). The author considers some philosophical implications connected to this complex set of topics (pp. 79--81). Later on, he inserts a subsection (pp. 80--86) on Einstein's new cosmological views after Einstein thought Friedman to have been wrong, after, instead, having recognized his own error and after having repudiated the cosmological constant. Einstein recognized the possibility of an expanding universe and considered possible the idea that the structure of space was not completely determined by matter (p. 81). In 1932, jointly with De Sitter, he presents an expanding model of universe in which space is Euclidean. Merleau-Ponty remarks that the assumption \(\Lambda=0\) implies \(R''<0\). Consequently, the singularity \(R=0\) is in the past. At that epoch one thought that the Hubble time be \(1.8\cdot 10^9\) years, a period too short to explain the structure of the universe. Already in 1952 the Hubble time was considered equal to \(1.3\cdot 10^{10}\) years and this problem disappeared (p. 83). The author explains how, by posing \(\Lambda=0\), it is possible to determine the sign of the space's curvature only taking into account the speed of recession and the mean density, which are two observable magnitudes (p. 84). Afterwards, he introduces Lemaître's philosophy: space is finite according to Riemann elliptic geometry. This implies a return to the finitism that the Middle Ages had inherited from Aristotle (p. 87). Merleau-Ponty expounds Lemaître's philosophical reasoning against the possible infinity of the universe. As he correctly points out, this reasoning is not convincing (pp. 87--88). In the light of his finitism and of his admission of the cosmic time, Lemaître recognizes that the fate of the universe depends on the value of \(\Lambda\). An important aspect of his work is that it allows an intuitive and realistic interpretation of \(\Lambda\): it is a repulsive cosmic force which, in a static condition, counterbalances gravity (pp. 89--90). Merleau-Ponty clarifies the merits and the problems of such an interpretation (p. 90). The next section (pp. 91--95) is dedicated to Eddington's interpretation of the cosmological constant: as Lemaître, Eddington thought the universe to be finite, but he did not believe either in the existence of a singular state or in the fact that the cosmological constant could be considered as a force (pp. 91--92). Eddington did not appreciate Einstein's idea that the metric was determined by matter. For him, this was too a materialistic view. In 1927 Eddington interprets \(\Lambda\) as a sample of length and does not regard the red-shift as a litmus paper of the expanding universe. Merleau-Ponty stresses the main merit of finitism: it offers a satisfying interpretation of the gravity law because Einstein's equations with \(\Lambda\) allow to define a constant curvature radius in every point and direction. This implies a satisfying situation with regard to the metrical problems (pp. 93--94). In 1933, in his work ``The expanding Universe'' Eddington claims that \(\Lambda\) must be positive because this is the only condition which permits the existence of a length sample (p. 94). Finally, Merleau-Ponty refers to empiricist cosmology of Tolman: strictly speaking cosmology is a science which cannot be admitted by an empiricist. The author expounds the attempt developed by Tolman to make his cosmological interests coherent with his empiricist convictions. The resulting picture is hardly satisfying (pp. 97--99). While photographing the situation in the middle of the 1930s, the observations revealed an expanding universe and the theory oriented towards the plausibility of the existence of a singular state in the past.NEWLINENEWLINEThe fourth chapter (pp. 105--125) concerns the ideas for a new cosmology. Rather than of ``new cosmology'', it would be more appropriate to speak of ``new cosmologies'' because Merleau-Ponty refers to those scientists who denied general relativity be necessarily the foundation of cosmology. Basically, the author. identifies three lines of thought: 1) Milne's kinematic relativity; 2) Hoyle, Bondi and Gold's theory of steady state; 3) Dirac and Jordan's ideas on the possible relations between the values assumed by the cosmological constants (pp. 105--108). Though it is difficult to find a unified matrix for the philosophy of these cosmologists, they refuse empiricism, positivism and materialism in favour of a purely deductive approach (pp. 108--111). From the technical standpoint, the main doubts of these scientists concerned three aspects: 1) while dealing with the entire cosmos, the operation of measuring a distance and the concept of distance itself are not clear and the global geometry of space is involved. General relativity inherits from classical science the idea that a distance is measurable by a rigid ruler and a time by a clock. This is not appropriate on a cosmic scale; 2) the problem of cosmic time; 3) the relations among the basic cosmical constants were not explained (pp. 108--113). The new cosmologists think that the laws of the universe cannot be induced by extending laws which are valid in a local context. They have to be established a priori and verified. Metric must be deduced as an answer to the question: ``what is universe's metrical structure for this structure to be communicated between two observers independently of their position in space and time?'' In their perspective dynamics has to be deduced from cosmology and not the opposite (pp. 114--117). The two most important principles introduced in this context are the cosmological principle and the perfect cosmological principle: the former claims that on a large scale the universe looks the same to all observers. The latter adds this to be true at any time. These scientists, and particularly Milne, offer an inter-subjectivist interpretation of these principles, not a realist one (pp. 117--119). The first result obtained by Milne in 1934 is that an expanding Newtonian universe can be conceived. He proved this assertion for the Einstein-De Sitter universe. Milne tried to extend the equivalence between a Newtonian universe and a relativistic one to the relativistic models compatible with the cosmic principle. The problem was that space was not Euclidean. However, Milne and McCrea, independently the one from the other, solved this problem. Milne considered his proof as an epistemological manifest: nothing as the structure of real space exists. The reason to choose a structure is related to the possibility to transform the world as a subjective and internal fact in a system of well-founded appearances. Milne considers that the equations of general relativity ought not be extended to the universe. Thus, he assumes only special relativity (pp. 119--125).NEWLINENEWLINEThe fifth chapter is dedicated to Milne's kinematic relativity (pp. 126--198). He developed his theory between 1935 and 1948. According to Milne, the world is a system of appearances. Anything beyond them is an intellectual construction. Therefore, physical space is not real, and it is possible to measure only distances, not lengths. Space and time are separated because time is the subject of an immediate perception, while this is not true for space (pp. 126--130). Milne founds the initial steps of his reasoning on the concept of observer. He is an ego who, at the beginning, has only the consciousness of the passing of time, of simultaneity and of the relation ``before and after''. This given, how is it possible that two observers read the same time on their clocks or that their clocks are congruent? Milne solved this complicated problem. Even more difficult was the extension to the solution to three observers, which was solved by Milne and Whitrow through the continuous groups. The next step concerned spatial coordinates: Milne proved that it is possible to determine: 1) a function epoch-distance which, measured by A, supplies the distance of B from A as a function of the A coordinate time; 2) the function ``running-clock'' which provides the relation in A between the coordinate time in A and the coordinate time in B (pp. 131--132). Within his theory Milne is able to reconstruct special relativity. A non-intuitive fact which strongly characterizes Milne's theory is the existence of two temporal scales: there is a temporal scale \(t\) in which all the observers have a common time-origin. In proximity of \(t=0\) the observers (Merleau-Ponty speaks of the members of a linear equivalence) are joined together, after that they diverge. The coordinates transform according to Lorentz transformations. There is, however, another time-scale \(\tau\) which derives from a different description of the equivalence. The two scales are related by the equation \(\tau=t_0+t_0\log t/t_0\), where \(t_0\) is a constant in which the two scales coincide, as it is apparent posing \(t=t_0\). In the scale \(\tau\) all the observers are relatively at rest, there is no natural origin of times and simultaneity regains its absolute meaning (pp. 133--134). While using the scale \(t\) the situation is analogous to that typical of special relativity. In contrast to this, in the scale \(\tau\), since the simultaneity is absolute, time has a public character (pp. 136--137). Milne claims that his two temporal scales are nothing odd because classical physics has referred to time in two different meaning: the time of mechanics is different from that of Maxwell equations in electromagnetism: his \(\tau\) represents the former, his \(t\) the latter (p. 137). Milne's universe is homogeneous, but in this case, too, it is necessary to define homogeneity in the two temporal scales. This homogeneous substratum expands like a sphere with a radius \(r=ct\). Milne proves that his universe has the following feature: each observer is at the centre of an expanding sphere. He does not measure a uniform density. The density of matter increases departing from the centre and tends to infinity on the sphere's surface, which, thence, is like an extended singularity. Despite its spherical appearance, Milnes' universe is Euclidean and infinite (pp. 138--141). Milne's universe is, thus, populated with observers-particles, but they are not the only objects admitted in the substratum. He admits that a free particle, a test particle, can be introduced. To this particle, no clock connected to those of the observers is associated. This free particle allows Milne to pass from kinematics to dynamics. An a priori dynamics can be founded by defining the movement of a free particle without referring to physical experience (p. 141). The fundamental problem is to determine the vector acceleration of the free particle as a combination of the kinematical elements connoting its position and movement at a given instant, if a fundamental observer of the substratum is assumed as the spatial reference centre and as a time-measurer (pp. 141--142). Milne shows that an acceleration vector can be determined, given two series of conditions and a function \(G(\xi)\), where \(\xi\) is a parameter without any physical dimension (p. 142). On this basis, he develops his dynamics whose most important results are: 1) the inertial mass of a free particle obeys neither to the transformation law of special relativity nor to conservation law of classical mechanics; 2) the fundamental particles have the same mass in all the fundamental systems of reference; 3) Milne's dynamics avoids the need to establish an inertial system (p. 143). The time which compares in Milne's dynamics is \(\tau\), not \(t\). His equations, transformed in the scale \(\tau\), coincide with the classical acceleration for the free particle. However, dynamics in this scale is different from Newton's because the public space is hyperbolic and the free particle's trajectory is a geodetic in this space (p. 144). Milne develops his dynamics both in the scales \(\tau\), and \(t\). All the concepts assume a different meaning in the two scales and, it is necessary to clarify when to use one or the other. For example, in the scale \(t\) everything works as if the substratum exerts an action like that of a couple. Universe has two aspects: the planetary aspect with a centre and with rotations around such a centre and its extragalactic aspect where rectilinear movements of recession prevail. According to Milne, within kinematic relativity, these two aspects are explained thanks to the use of the two temporal scales (p. 145). Merleau-Ponty explains further properties of the two scales. Milne has a preference for the scale \(t\). In this scale the inverse square law holds, but the constant of gravitation is not anymore a constant, but it is proportional to \(t\). The author then specifies further mathematical features of Milne's universe (pp. 148--151). The red-shift represents a problem for Milne's theory because in the scale \(t\), it is easily deduced; whereas in the scale \(\tau\) the particles of the substratum, which are the centres of galaxies, are relatively at rest. Thence, the red-shift cannot be explained kinematically (p. 152). How is it possible to explain the red-shift in the scale \(\tau\)? Milne starts from Bohr's formula connecting the energy of a photon of atomic origin, its frequency, and Planck constant \(h\). A fundamental conclusion is that in the scale \(t\) the value of \(h\) is a function of \(t\), whereas in the scale \(\tau\), it is a true constant. These reasonings allow Milne to reach the conclusion that in the equation \(\tau=t_0+t_0\log(t/t_0)\), the parameter \(t_0\) is not the same when a photon is emitted and when it is received. Through an argumentation, which Merleau-Ponty does not consider completely convincing, Milne proves how the red-shift can be obtained in the scale $\tau$, although he himself had some doubts that through the scale \(\tau\) only is it possible to have a good explanation of the red-shift (pp. 152--154). From an astronomical standpoint, the galaxies' centres are singularities. The system of galaxies seen as substratum is infinite, in expansion, and fulfills the cosmological principle. The universe is contained within the sphere \(R=ct\). Everything works in the scale \(t\), but it is not easy to determine which scale is useful to take measures in astronomy. Galaxies exchange energy through the following mechanism: a particle recedes from the nucleus to which it is associated accelerating until it reaches the speed of light and afterwards it begins to decelerate and approaches another nucleus to which is gets associated (pp. 155--157). From a philosophical standpoint Milne had the intention to construct a priori a model of universe on the basis of epistemological and methodological axioms (p. 157). Milne's doctrine is connected to theology as he wants to conciliate natural philosophy with Christian faith. Two fundamental items of Milne's philosophy of physics are: 1) the metrical knowledge is grounded on time. This contradicts a tradition which goes from Augustin to Bergson; 2) after Leibniz, he is the first philosopher-scientist who defines universe as an ideal community of monads-observers. Metric intersubjectivity precedes the description of physical objectivity. Milne criticizes empiricism and formalism because they do not face the question of ``why'' but only that of ``how''. He can be regarded as a metaphysicist of intellectual intuition: there is no physical space, but only the abstract space in which the thinker arranges phenomena (pp. 158--161). Afterwards Merleau-Ponty dedicates some considerations to the link between science and theology in Milne, the most important of which is the following one: in the scale \(t\) there is an origin, the temporal zero, which is an absolute instant, despite time's relativity. It represents creation. It makes no sense to ask what there was before creation. Instead, the state of the universe must be perfectly determinable after creation (pp. 163--165). The author stresses some other aspects of Milnes' cosmo-theology, also pointing out his anachronistic elements (pp. 165--166). The other significant author who adhered to kinematic relativity is Whitrow. His philosophy was different from Milne's. He was a Kantian, at least from a methodological point of view. He was interested in the theory of knowledge: how is it possible to make the scientific method uniform and communicable? In particular: is a system of dynamics possible which is epistemologically as founded as Euclidean geometry? (pp. 166--169). In this context, the main problem consists in giving a satisfying treatment of inertia. According to Whitrow, it is necessary to find a definition of inertia which is coherent with Leibniz's relativity of time and with Mach's principle. He analyses why the theory of relativity is not completely satisfying in this respect and, after a philosophical digression which, admittedly, is not adamant, he concludes that kinematic relativity offers the best picture of inertia since, in scale \(\tau\), which corresponds to classical mechanics, the movement of the observers in respect to the substratum is rectilinear and uniform. In this way, Mach's principle is fulfilled, and the inertia law is epistemologically founded. Merleau-Ponty highlights that a conspicuous problem in kinematic relativity is that the concept of mass is not defined (pp. 170--174). Later on, he develops some considerations on the philosophy of cosmology: kinematic relativity can be seen as the cosmological transfiguration of Leibniz's monadology, but monads are communicating. The relativistic principle of covariance has a monadological flavour: it is necessary only insofar as the universe can be regarded as a set of different metrical appearances. Kinematic relativity goes beyond Einstein as it considers the universe as a multiplicity of egos, and not as a material system. In conclusion of his philosophical considerations, Merleau-Ponty draws a moral (which he seems to share) from kinematic relativity: physics is not the representation of the universe in itself, but the description of the procedures through which the monads conciliate their measurements (pp. 175--182). The author claims that the most original aspect of kinematic relativity is the attempt to assume the measure of time as a foundation for the metric description of the universe (p. 182). Milne thinks that the objections against the measure of time derive from the fact that they would be against a measure of time uniformity. However, this does not exclude that the measure of time be based on a more immediate and original experience (pp. 183--184). Dingle moves the following objection to Milne's conception: there are two kinds of facts: 1) those which only belong to a temporal sequence, as the tic-tac of a clock; 2) those which can be distinguished only in reference to another quantity, as the successive positions of a mobile. According to Dingle, Milne's mistake is that he founds the time-measure only in the first class of facts. Merleau-Ponty claims that the fundamental question on the metric nature of time is to understand whether the measure of time can be formed and can represent the basis for other metrical operations without presupposing an order of the world with which our instruments can agree. The author develops a series of considerations on this topic, and concludes that, probably, Milne's project was too ambitious and that the empiricist principle according to which the cosmic regularities are observed and not constructed is truthful (pp. 185--189). Finally, in the scale \(\tau\), namely the mechanical one, time is irreversible, while in the cosmic scale \(t\), it is reversible (pp. 190--191). Robertson analysed the power and the limit of the deductive method in cosmology. He focused only on the mathematical apparatus of the theory. He identifies three steps: 1) to define the idealised space-time, namely Milne's substratum; 2) to determine the movement of a free particle; 3) to define statistical systems of particles connected with diverse theories of gravitation (pp. 192--193). Robertson proves that: a) Milne's operative methodology and the cosmological principle leave the kinematics of the substratum more indeterminate than Milne himself thought; b) if the operative method is applied in all its generality, space-time has a Riemannian metric which has the same form and level of generality as in the relativistic cosmology; c) the law of movement of a test-particle is less determined than Milne thought and it is consistent with every theory of gravitation compatible with the cosmological principle. But if the gravitational law is not connected with the substratum, the description of cosmic structure and the determination of the laws of the universe are irreducibly different (pp. 194--198).NEWLINENEWLINEThe sixth chapter (pp. 199--237) is dedicated to the cosmology of steady state. Bondi and Gold, the first theoreticians of the steady state theory, introduce the perfect cosmological principle. They accept the universe's expansion. The universe as a whole does not have an age; only the local processes have. Steady state theory breaks with general relativity: no dynamics is proposed, neither any equations for the gravity field. The continuous creation of matter ex nihilo is admitted (pp. 200--201). From an epistemological standpoint, as in Milne, the deductive method is used. Cosmology cannot be constructed by an extrapolation from a local theory. Bondi claims that the deductive theories are criticized because of the introduction of ad hoc hypotheses. This is an error: the hypotheses must be clearly stated and be as few as possible. The creators of this theory claim that, on a cosmic scale, laws and descriptive enunciations tend to coincide (pp. 201--202). From a gnoseological point of view, these authors think that a series of facts and properties exist whose meaning is cosmological. The most important one is inertia, which is considered as a result of interactions between the text-body and the rest of universe; the second example of a fact of this kind is the darkness of the night sky, which can be explained thinking of the recession of stars and galaxies (pp. 204--206). In 1948, Bondi and Gold introduce the perfect cosmological principle and claim that, while dealing with the universe, it is impossible to separate what is accidental from what is essential. This is because there is only one universe (pp. 207--208). It is, thence, necessary to introduce a priori hypotheses which allow to overcome this difficulty. Bondi and Gold postulate that the interaction between the ordinary physical laws and the cosmic structure implies a stationary situation for the universe. However, given the existence of thermodynamical imbalance, the universe cannot be static (pp. 208--209). Red-shift and Olbers' paradox show that the universe is expanding. The two authors renounce the hydrodynamic principle of continuity and postulate that matter be created ex nihilo: the mass of a proton per litre per \(10^9\) years. Strictly speaking the principle of mass-energy conservation is not violated because the mass-energy in the observable universe remains constant (p. 210). Special relativity and Milne's concept of substratum, jointly with the perfect cosmological principle (p.c.p. hereafter) permit: 1) to delete cosmic time; 2) to determine the \(ds^2\) of the substratum, up to a constant (p. 211). Merleau-Ponty clarifies that the steady state theory is involved in the problem of cosmic time less than all the other theories (pp. 212--213). This notwithstanding, a problem with cosmic time still exists: it is possible to calculate the time interval beyond which the signals from a galaxy will not reach anymore an observer. There is, thence, a relation between galaxies' statistical distribution, depending on their age as measured in the proper time of the observer and the characteristic time of expansion as measured in the scale of substratum's cosmic time. The identification of the two scales relies upon the hypothesis the real world be identified with substratum (p. 213--214). In the steady state theory, the curvature \(k\) of the tri-space is null and \(R'/R\) is constant. This determines \(R\) up to a constant, which is Hubble constant whose value is determined empirically (p. 215). The metric element corresponds to that of De Sitter's metric in relativistic cosmology, but its interpretation is different because in the steady state theory no necessary connection between geometric form and mass-energy density exists (pp. 215--216). The expression of density in function of time ought offer an observational discriminant between the steady state theory and the relativistic theories, but at that time no instrument was suitable to detect such a difference (p. 216). Some considerations follow on the matter creation, a process that raised many doubts and on which several hypotheses were developed (pp. 216--219). One of the major limitations of the steady state theory is that, after having highlighted the dependence of physics on cosmology, their creators do not realise this idea. Fred Hoyle, an astronomer who strongly contributed to the edification of this theory, claimed this to be due to two facts: 1) the strong connection between local physics and remote universe prevents to conceive a completely controlled experience; 2) the p.c.p. is satisfied only at a large scale, not locally (pp. 219--220). As a final commentary, Merleau-Ponty asserts that the steady state theory highlights the contrast between the cosmological and the local standpoint as well as the lack of similarity between cosmological determinism and physical determinism (pp. 222--223). The next sub-chapter concerns Sciama's inertia theory, which can be included in the context of the steady state theory: Sciama wants to construct an effective theory of inertia compatible with Mach's principle. He starts from the assumptions: 1) the universe exerts appreciable forces on local matter; 2) the local irreversible processes are connected to the irreversible expansion of the universe; 3) the content of the universe is the same as its laws. Sciama proposes the law of inertia be \(1/r\). In this assumption the far matter's action prevails on the near one and the action of an additional close body is also negligible. Sciama, by means of an analogy with electromagnetism, adds a vector potential to the classical gravitational scalar potential. In the expression of the gravitational field, a term proportional to the temporal derivative of the vector potential appears. Inertia derives from this term. Merleau-Ponty adds some other brief technical specifications (pp. 223--227). The last sub-chapter is dedicated to Fred Hoyle, probably the most well-known theoretician of the steady state theory: unlike Bondi and Gold, he emphasises the principle of the creation of matter rather than the p.c.p. He postulates the existence of a specific creational field, besides the traditional physical fields. He accepts Einstein equations and makes a change that has effects on a cosmic scale (pp. 228--231). Once again, unlike Bondi and Gold, Hoyle considers cosmology to be the result of an induction starting from local knowledges. He is not a monist. In the course of the years, Hoyle tried to better specify the physical features of the creational field and attempted to make it consistent with Einstein's general principle of covariance. In Hoyle's intention, the steady state universe should result from equations similar to Einstein's (pp. 232--235). Merlau-Ponty offers a sketch of the basic technical developments of Hoyle's theory (pp. 236--237).NEWLINENEWLINEThe relativistic cosmologies, kinematic relativity and steady state theory are the most important cosmological approaches between 1917 and 1960s, but there were also others, of which the author speaks in the 7th chapter (pp. 238--274). The first of these alternative cosmologies focuses on the values of the universal constants. The first who thought of a relation among the different cosmical constants was Eddington in 1933. He believed that \(\Lambda\) had to define an absolute length unit and that two independent numerical relations existed: the cosmic radius \(R\) and the number \(N\) of protons. Starting from supposed relations among these constants, he deduced other 27 constants (pp. 239--242). Dirac in 1937--1938, though influenced by Eddington, offers a different interpretation of the constants: those close to 1 or which differ from 1 by little powers of 10 are true constants. Instead, the others are not constant, but magnitudes proportional to the age of the universe or to its square. He argues like this: Eddington did not explain the big numbers, therefore, they are inexplicable. Thence, they indicate the age of the universe. Merleau-Ponty defines this reasoning ``bizarre'' (p. 243). Dirac stresses the contrast between the principle of covariance and the existence of a time direction in the history of cosmology. Hence, general relativity cannot be the basis of cosmology (p. 243). There is a history of the universe, to which the value of the atomic constants is connected (p. 244). This is the fundamental principle, with whose help Dirac draws several conclusions, for example, the fact that the universe is infinite and Euclidean (p. 245). In deducing these results, Dirac does not rely on any physical theory. He arrives at a double temporal scale, as Milne, but in the scale \(t\) of Milne the gravitational ``constant'' is an increasing function of time. In contrast to this, in Dirac's scale equivalent to \(t\), it is a decreasing function (p. 246). Merleau-Ponty briefly focuses on the merits and shortcomings of the Dirac approach (pp. 246--247). The third theoretician of the universal constants was Pascual Jordan. He first offered a theory in which the creation of matter becomes an ordinary physical process. His cosmology is syncretistic and has several obscurities. It is based on: 1) the Dirac principle; 2) the creation of matter. Apropos, he claims that some supernovae are generated by the creation of very condensed and expanding matter; 3) projective relativity. This is a theory due, from a mathematical point of view, to Lichnérowicz and Thiry, in which the four usual coordinates are replaced by five homogeneous coordinates. In this way Maxwell and Einstein equations are covariant only for a particular group of transformations. Merleau-Ponty gives a brief explanation of how Jordan used this theory in a cosmological context (pp. 248--253). The second sub-chapter is dedicated to those cosmologists who renounced cosmological principle and cosmic time as well as to homogeneity and isotropy of the universe. Kurt Gödel is the most celebrated of these theoreticians: he is convinced that the most profound consequence of the relativity of simultaneity is that the becoming is only an appearance. He has an idealistic conception of time and wants to prove that the idea of cosmic time is extraneous to relativity. In order to prove his thesis, Gödel found a solution of Einstein equation in which no cosmical time exists. Gödel's universe has a constant, negative curvature, the metric is static, the matter's lines of the universe are infinite timelike lines, no structural red-shift exists. It is anisotropic because in any point a rotation is observable, which, hence, indicates a privileged direction. Most surprisingly: closed time-lines exist (pp. 256--258). The lack of the red-shift makes Gödel's universe implausible as a model of our universe. However, to this objection Gödel answered that his model proves that the passing of time depends on a particular configuration of matter. One might say that it is a contingent property (pp. 259--262). After a summary of the theories so far analysed (pp. 262--265) Merleau-Ponty deals with other cosmical models. The first one is due to the Indian cosmologist Amalkumar Raychaudhury, who, besides expansion, realised that the cosmic fluid can have a movement of rotation and fringing. The former acts against gravity, whereas the latter deforms the cosmic fluid transversely and acts in the sense of condensation. This model of universe does not attenuate the singularity problem. However, Schücking and Heckmann proved that such a singularity is not reduced to a material point, but loses one or two of its dimensions, not all three. B. B. Robinson continued these studies and analysed a series of models where the fringing is only a function of time, and space degenerates into a line or a surface, but never in a point (pp. 265--268). Lindquist and Wheeler reached an isotropic, but not homogeneous model of universe: the lattice universe. The universe is considered like a sort of crystal composed of cells. The solutions of cosmological equations in the single cells must be considered in order to obtain a global solution (p. 269). Jaroslav Pachner, probably influenced by Infeld, developed the ideas of Lindquist and Wheeler. Following Einstein's opinion, he thought that the material tensor should not be present in the cosmological equations. On these bases, Merleau-Ponty offers a sketch of Pachner's universe (pp. 270--274).NEWLINENEWLINEThe 8th chapter (pp. 277--323) concerns the cosmic becoming. The author points out that the new cosmology has introduced two new elements in the classical discussions on becoming: 1) the problem of singularity; 2) the fact that the cosmic becoming does not concern anymore the modifications of things in space, but the becoming of space-time itself (pp. 277--279). The first sub-chapter is dedicated to irreversibility and cosmology and is opened by a fascinating question: suppose the concept of entropy and the second principle of thermodynamics be applicable to the universe as a whole. Time direction is given by the increment of entropy. But when the universe will have reached the complete equilibrium, how will it be possible to consider time? At a microscopic level something will continue to happen, but how to regard time? (pp. 279--281). A further problem concerns the way in which the statistical laws of thermodynamics have to be applied to the notion of time (pp. 282--283). Merleau-Ponty refers to the answers of several scientists. Some deal with the question in an epistemological perspective, others in a cosmological one; but, admittedly, none of the proposed answers is satisfying (pp. 284--286). Given the difficulty to answer these so general questions, the author poses four other more specific problems connected to them: 1) Is the universe evolving? The answer seems sure: yes. For the universe to be in equilibrium it would be necessary, at least that: a) the process of dematerialization of stars be compensated by an inverse process of energy materialization. No indication of such a process exists; b) energy be uniformly distributed between mass and radiation. This is not the case: almost all the energy of the universe is in the form of mass. Furthermore, the red-shift is a further clue for the universe not to be in equilibrium. Thence, it is evolving (pp. 287--290). As Merleau-Ponty clarifies the reasons given by scientists as Dauvillier and Zwicky against the equilibrium of the universe are weak (pp. 291--293). 2) Is the direction of cosmic evolution determined without any ambiguity? Once again, the answer is in the affirmative: the second principle of thermodynamics defines a privileged direction; the red-shift defines a direction of cosmic evolution. The red-shift can be compared with a local process, the Doppler effect, which can be interpreted thermodynamically. Hence, the thermodynamic arrow and the cosmic arrow are identified, and no ambiguity arises (pp. 293--294). Gold and Hogarth add other considerations on the non-ambiguity of the arrow time (pp. 295--297). This subject is, nonetheless, full of difficulties: Tolman argues that if the universe is considered as a system subject to an adiabatic and isentropic transformation, the cosmological time direction cannot be the same as the thermodynamic one. On the other hand, as it seems to be the case, if the adiabatic transformation is irreversible, the energetic content of every volume-element can increase indefinitely, which implies an indefinite increment of entropy in a series of universe-oscillations having increasing amplitude and period (pp. 298--300). 3) What is the meaning to ascribe to the possible final state of universe? First of all, the author analyses the problem of stellar death. From a historical point of view, this is an interesting document because the described picture is coherent with what we nowadays know of the stellar evolution, but also shows remarkable uncertainties that now have been overcome (pp. 302--304). A very brief discussion on the end of the universe is proposed, but it adds nothing to what has already been told apropos of entropy and destiny of the universe (pp. 305--306). 4) Had the evolution of the universe a beginning instant and is it possible to date such instant? Merleau-Ponty analyses both the affirmative and the negative answers. In the former case, he stresses the difficulty to construct a projection between a segment of real numbers and the time-instants. Walker and Whitrow, for example, think that the ordered set of instants does not have an initial element. Alternatively, one can think time to be open, but limited in the past. There is no initial instant, but the temporal series converges to a limit, which is not the beginning of the universe in time, but the beginning of time. Lemaître and Milne, granted the enormous differences of their theories, propend for this opinion (pp. 309--310). Finally, one can think that the series of the cosmic events is projected on the totality of real numbers, but it has a discontinuity in the past. It is a singularity. This is Gamow's, and nowadays we should add also our, conception of the universe's origin. If, in contrast to the expounded opinions one thinks that no origin of the universe exists, it is possible to refer to a periodic universe. Here Merleau-Ponty develops an interesting consideration: if periodicity is complete, namely if at the time \(t\) of a cycle, exactly the same events as those which happens at the time \(t\) of another cycle take place, then our universe would be atemporal. Otherwise, the possibility -- regarded more plausible by the author -- that the events at the time \(t\) of two cycles be different can be considered (pp. 311--312). There are other possible options for those who deny that the universe had a birth, but Merleau-Ponty dedicates few considerations to them (pp. 312--313). While commenting on the various hypotheses, the author claims that several clues, and particularly the red-shift, are favourable evidences for the existence of a singular state at the beginning of the universe's history (pp. 314--316). However, this singular state is problematic: how is it causally connected to the following structure of the universe, taking into account that it should be a condition with minimal entropy? No shared answer to these questions exists (pp. 316-317). Furthermore, if the expansion-contraction model is accepted, how could the contracting phase be consistent with the laws of thermodynamics? (pp. 318-320). After a series of long analyses concerning these issues, Merleau-Ponty declares that: ``[\dots] preference should be given to the steady state theory'' (p. 322).NEWLINENEWLINEThe 9th chapter (pp. 324--377) regards the cosmogonical theories. The author points out that modern cosmology has brought down one of the pillars of classical science: the principle of conservation of mass. Matter lost its supremacy: the matter of a star is transformed into energy, in the steady state theory, matter is created ex nihilo and so on (pp. 324--327). Afterwards, the author analyses the dichotomy condensation/explosion. At least from Newton's time, gravity was considered the most important force for the structure of the universe. However, recently studied phenomena such as the supernovae and the exploding nuclei of some galaxies show that explosion has a crucial role within the universe. The red-shift also suggests that explosion could have a significant role in the genesis of the universe itself (pp. 330--331). All cosmogonies present some problem. They have to answer three questions: 1) the genesis of the atomic elements from hydrogen; 2) the genesis of the stars; 3) the genesis of galaxies and galaxy clusters (pp. 332--333). The first cosmogony considered by the author is that of Lemaître. The first solution he gave is the so called universe of Lemaître-Eddington. It is a closed, elliptic, Einstein universe whose indefinitely accelerated expansion begins infinitely far in the past. Asymptotically far in the past this universe tended to coincide with Einstein's static universe. Lemaître proposed a mechanism for the equilibrium-rupture, which was very refined from a mathematical standpoint, but was not completely satisfactory from a physical one (pp. 335--336). To overcome the problem of the equilibrium-rupture, in 1931 Lemaître conceived the theory of the primeval atom. The enormous energy of the cosmic rays was interpreted by him as a clue of the primeval explosion which generated the universe. The existence of heavy elements was seen as another proof in favour of this idea (pp. 337--338). The universe has an initial phase of exploding acceleration, the expansion afterwards decelerates and the universe reaches Einstein's equilibrium state. Later on, it accelerates once again (pp. 338--339). Merleau-Ponty explains how the diverse structures of universe (stars, galaxies, etc.) were created in the various phases of the universe evolution (pp. 340--431). The author underlines that only the geometrical foundations are solid in Lemaître's construction. In a physical perspective, it is weak: the phasis immediately after the explosion is not well described and the recent studies on the cosmic rays have shown that, very probably, they are local and recent phenomena (pp. 342--343). The next analysed cosmogony is Gamow's: he started with an initial condition and has three phases similar to those of Lemaître. However, Gamow's examination of the initial phases is more profound than in Lemaître: starting from the hypothesis of an expanding gas of neutrons, Gamow tried to reconstruct the genesis of the elements. The cosmological constant is null, space is open and negatively curved (pp. 344--345). Merleau-Ponty recalls that, although in the time he was writing it was already known that the genesis of the heavy elements is not cosmic, the contributions by Gamow, Alpher and Hermann on this topic were a remarkable progress on any other theory conceived before them. I add that, with regard to the light elements until lithium and beryllium, this theory is essentially still accepted. The author enters many useful details of Gamow's theory of elements (pp. 346--349). He describes the different phases of the universe's evolution in Gamow's theory (pp. 350--351). In this context, a great difficulty is to determine the processes through which condensation takes place, namely the formation of stars, planets and galaxies. The novelty of Gamow's approach is that he introduced the properties of turbulence in the study of the stars', planets' and proto-galaxies' origin (pp. 353--355). Gamow finds the traces of the original turbulence in the galaxies' distribution, which are irregularly concentrated in clusters in a way compatible with a compression level sufficient to explain condensations in the primitive gas (p. 355). Afterwards, the author analyses Hoyle's cosmogony: this scientist thinks than in an infinite universe the concept of total matter and energy is meaningless. Therefore, given this feature and the creation of matter ex nihilo, it is necessary to wonder what quantities are conserved and what the meaning of the conservation principles is. He thinks of the quantity of matter-energy in the proper volume unity of any observer (p. 357). Hoyle developed a theory of the matter creation: it is created in the forms of proton and electrons. Afterwards the galaxies are formed starting from the original hydrogen. Only later the elements different from hydrogen are generated (pp. 358--359). A profound problem faced by Hoyle regards the condensation of matter. The author refers to the solutions proposed for this difficult issue (pp. 360--362). Hoyle thought that the galaxies become radio sources after a certain period from their creation. From the point of view of a human observer, there will be a zone of the universe particularly rich of radio galaxies. In a general computation of the age of galaxies, Hoyle individuates several generations (pp. 363--364). The next section concerns the cosmological ideas of the Soviet scientist Victor Ambarzumian. Before dealing with this topic, following Mikulak, the author recalls the leitmotiv of Soviet cosmology: 1) the universe is infinite in time and space; 2) matter and energy are uncreated, indestructible, eternal; 3) no decadence exists in the universe; 4) it is not possible to extend local laws to the universe. As a matter of fact, these diktats posed strong limitation to Soviet cosmology. Things began to change after 1958 (pp. 368--370). Ambarzumian thinks the cosmological unity to be a myth and a simplification of reality. Therefore, it makes no sense to speak of expansion in general. Furthermore, the universe is not homogeneous. However, while referring to the metagalaxies, namely to the zone of universe (relatively) near us, we can say that this part of the universe is homogeneous and in expansion (pp. 368--371). Ambarzumian thinks the universe's expansion to be quick and almost explosive (p. 371). He is profoundly empiricist and certainly does not deny the red-shift. However, he is not convinced that it can be interpreted in a cosmological perspective. This scientist highlighted the concept of ``stellar association'', which, intuitively, is a set of similar stars (for example blue giant stars). The original idea of Ambarzumian is that the associations are composed of stars having the same origin and produced by a single process. The associations are formed and dispersed quickly. Afterwards Ambarzumian, mutatis mutandis, applies his ideas to the galaxies. All in all, Ambarzumian's cosmological ideas are not far from those of the non-stationary relativistic cosmology. The problem of a singularity exists at the beginning of the relativistic universe. In Ambarzumian's explosive cosmology it is moved to the genesis of stars and galaxies (pp. 372--377).NEWLINENEWLINEThe 10th chapter (pp. 381--402) is entitled ``The theoretical cosmology and the observed Universe''. Merleau-Ponty stresses the difficulties of cosmological observation, despite the improvements due to the Hale and Schmidt telescopes. The radio astronomy, he claims, is still in an initial phase. What can be said is that the isotropy in the distribution of galaxies and the features of the red-shift confirm the idea of a cosmic time, of a space with constant curvature and of an homogeneous substratum of galaxies (pp. 381--385). An important result would be to determine the Robertson metric observationally because this could indicate whether the theories which chose \(k\) and \(R(t)\) a priori are acceptable or instead those -- as the steady state theory and the kinematic relativity -- which deny this possibility. The idea is to deduce some relations between measurable quantities and unknowns in the Robertson metric. There are some quantities -- as the galaxies' magnitude or their apparent diameters and angular distances -- which can connect observations with theory, but the former are still not adequate (pp. 386--387). Two great difficulties concern the concept of distance and that of physical evolution of the galaxies as it is hard to link them to observable aspects of the universe (pp. 388--389). In 1961 Allan Sandage thought of three relations between significant cosmological magnitudes which could be verified by observation. The only really utilizable is the relation between the red-shift of a galaxy and its apparent magnitude, which can measure the Hubble parameter \(H_0\) at the first order, and the deceleration parameter \(q_0\) at the second order. The author briefly expounds Sandage's methods which he considers empirical (pp. 389--390). In the kinematic relativity \(q_0\) is equal to 0, in the steady state theory it is \(-1\), in the relativistic cosmology it is negative for all the models in which the universe's expansion is accelerating, and it is positive in those where the universe's expansion is decelerating. In the universe of Einstein-De Sitter it is \(\frac{1}{2}\). According to the calculation of Sandage it is equal to about \(+0.2\) (today it is estimated to be equal to \(-0.6\pm 0.2\)). Therefore, the universe's expansion ought decelerate; but, the author claims that this calculation is very uncertain (p. 392). Curvature is even more difficult to be empirically determined and for \(\Lambda\) it is impossible. Therefore, at the beginning of the 1960s the observations seem to indicate: 1) the red-shift excludes static metrics; 2) the deceleration parameter has a small positive value. Hence, expansion is decelerating; 3) curvature is very difficult to determine. The indications seem in favour of a hyperbolic space (pp. 394--395). In the rest of the chapter, Merleau-Ponty analyses the potential contribution of radio astronomy to cosmology. He claims that this branch of astronomy is potentially powerful for cosmology, but for the moment it is still in an embryonic phase. According to the Cambridge catalogue of 1961, it seems that the distribution of galaxies is not coherent with the steady state theory. But this conclusion is highly hypothetical and based on several axioms refused by the theoreticians of the steady state theory (pp. 395--402).NEWLINENEWLINEThe 11th and last chapter (pp. 403--427) is a summary of what expounded in the book and an anticipation of future research. One of the most interesting aspects of this chapter is the attempt to show that general relativity is not indispensable in cosmology (pp. 408--412). With regard to classical problems, as that of infinity of space, the author claims that, in the light of the recent physical and cosmological research, this problem is not dialectic in the Kantian sense: observation could decide whether space is finite or infinite. Simply, we have not yet suitable observational means to solve the question. But it is, however, an observational problem (pp. 413--416). With regard to the nature of space-time, the author wonders if it must be considered as a mathematical device which joins two different aspects of reality, or if it is the fundamental aspect of physical reality. He is favourable to the first option, a position that today, and in the epoch in which the book was written as well, few physicists would share. In his opinion, the existence of a cosmic time independent of space is a clue in favour of his idea (pp. 417--423).NEWLINENEWLINEA mathematical appendix follows where several details missing in the book are explained (pp. 449--505).