The article provides a reconstruction of Eudoxus' approach to simultaneous risings and settings in his two works dedicated to the issue: the Phaenomena and the Enoptron. This reconstruction is based on the analysis of Eudoxus’ fragments transmitted by Hipparchus. These fragments are difficult and problematic, but a close analysis and a comparison with the corresponding passages in Aratus suggests a possible solution.

The debate over the value of pleasure among Eudoxus, Speusippus, and Aristotle is dramatically documented by the Nicomachean Ethics, particularly in the dialectical pros-and-cons concerning the so-called argument from contraries. Two similar versions of this argument are preserved at EN VII. 13, 1153b1–4, and X. 2, 1172b18–20. Many scholars believe that the argument at EN VII is either a report or an appropriation of the Eudoxean argument in EN X. This essay aims to revise this received view. It will (...) explain why these two arguments differ in premises, contents and purposes, and why these distinctions matter for a proper assessment of Aristotle’s understanding of pleasure and pain in general and his dialectical art in particular. (shrink)

A construction of the real number system based on almost homomorphisms of the integers $\mathbb {Z}$ was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On -saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently (...) by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG . In NBG , it can be obtained via a one-step construction by means of a definable ultrapower (modulo a suitable definable class ultrafilter). (shrink)

In the following essay I present a narrative of the development of astronomical models from Eudoxus to Kepler, as a case-study that vindicates an insightful and influential recent account of the concept of scientific understanding. Since this episode in the history of science and the concept of understanding are subjects to which Professor Roberto Torretti has dedicated two wonderful books—De Eudoxo a Newton: modelos matemáticos en la filosofía natural (2007), and Creative Understanding: philosophical reflections on physics (1990), respectively—this essay (...) is my contribution to celebrate his outstanding work and career in this volume. I dedicate this piece to Roberto, dear friend and mentor, in gratitude for all his inspirational work and personal support, which has greatly helped me, and many others, to better understand that human wonder we call scientific knowledge. (shrink)

THE book which Eudoxus of Cnidos was stated by some to have translated from the Egyptian is entitled in the manuscripts of Diog. Laert. 8. 89, a reading which R. D. Hicks retains in his Loeb edition. It was retained also in the edition of C. Gabr. Cobet and in the Tauchnitz edition ; so also H. S. Long in O.C.T.. Egyptian religion was richly theriolatrous. But does it proffer a suggestion of ‘Dialogues of Dogs’? The contrary belief is (...) suggested by the proposal of various emendations. (shrink)

The geometry of the alternative reconstruction of Eudoxan planetary theory is studied. It is shown that in this framework the hippopede acquires an analytical role, consolidating the theorys geometrical underpinnings. This removes the main point of incompatibility between the alternative reconstruction and Simpliciuss account of Eudoxan planetary astronomy. The analysis also suggests a compass and straight-edge procedure for drawing a point by point outline of the retrograde loop created by any given arrangement of the three inner spheres.

Dancy (philosophy, Florida State U.) presents two new interpretations of the evidence regarding the metaphysical ideas of two important figures in Plato's Academy, Eudoxus and Speusippus, and of Aristotle's reaction to those ideas.

In this paper, we examine a fundamental problem that appears in Greek philosophy: the paradoxes of self-reference of the type of “Third Man” that appears first in Plato’s 'Parmenides', and is further discussed in Aristotle and the Peripatetic commentators and Proclus. We show that the various versions are analysed using different language, reflecting different understandings by Plato and the Platonists, such as Proclus, on the one hand, and the Peripatetics (Aristotle, Alexander, Eudemus), on the other hand. We show that the (...) Peripatetic commentators do not focus on Plato’s solution but primarily on the formulation of the “Third Man” paradox. On the contrary, Proclus seems to be convinced that Plato suggests a sound solution to the paradox by defining the predicate of similarity (homogeneity) that demarcates two types of homogeneous entities – the eide and the participants in them in a way that their confusion would be inadmissible. We claim that Plato’s solution follows a sound line of reasoning that is formalisable in a language of Frege-Russell type; hence there exists a model in which Plato’s reasoning is valid. Furthermore, we notice that Plato’s definition of the second-order predicate of similarity is attained by resorting to first-order entities. In this sense, Plato’s definition is comparable to Eudoxus’ definition of ratio, which is also attained by resorting to first-order objects. Consequently, Plato seems to follow a logical practice established by the mathematicians of the 5th century, notably Eudoxus, in his solution to the paradox. (shrink)

In Nicomachean Ethics (= Eth. Nic.) 10.2, Aristotle addresses Eudoxus’ argument that pleasure is the chief good in his characteristically dialectical manner. The argument is that pleasure is the chief good, since all creatures, rational (ἔλλογα) and non-rational (ἄλογα) alike, are perceived to aim at pleasure (1172b9–11).1 At 1172b35–1173a5, Aristotle turns to an objection against Eudoxus’ argument. For some object (οἱ δ᾽ἐνιστάμενοι) to the argument by questioning one of its premisses, namely that what all creatures aim at is (...) the good (1172b12–15). Instead, they claim that what all creatures aim at is not good (ὡς οὐκ γαθὸν οὗ πάντ᾽ ἐφίεται, 1172b36). This claim is reasonably taken to mean that not everything that all creatures aim at is good. But, as we shall shortly see, Aristotle dismisses it in a way suggesting a less charitable interpretation. At any rate, the significance of this objection is that it challenges the strong claim that what all creatures aim at is the good with an argument against the weaker claim that what all creatures aim at is good (or a good). For if the weaker claim is refuted, then the strong claim is refuted as well. Aristotle takes issue with the argument against the weaker claim, but without committing himself to the strong claim. (shrink)

This paper discusses Aristotle’s statement (Metaph. A 9, 991a8-9) that both Anaxagoras and Eudoxus claimed that things are the result of a mixture of original elements, in relation to Plato’s metaphysics. Eudoxus used this immanentistic thesis to reform one central component of Plato’s Theory of Form, that is the “participation”. The first part of the paper analyzes some Anaxagorean aspects in Plato’s metaphysics, showing that Plato shares with Anaxagoras the “Transmission Theory of Causality” (as called by Dancy), but (...) he refuses its immanentistic version. The second part interprets Hip. Maj. 301b2-301c2 as a refusal of a immantestic interpretation of verbs like prosgivgnomai and kosmei'tai. It is also rejected Morgan’s thesis according to which Hippias supports an aware mereological metaphysical theory. The third part contests that Phaed. 100-106 is a defense of an immanentistic metaphysics abandoned by Plato in his later works. The meaning of the expression to; ejn hJmi'n does not include a mereological approach to the causality. In Plato’s metaphysics there is no strong contradiction between transcendence and immanence. The fourth part shows that the Parmenides refuses any immanentistic version of the relationship between Forms and things. Lastly, I will argue that from a Platonic point of view the only acceptable version is the separated interpretation of Transmission Theory of Causality. (shrink)

un essai de reconstitution Jean-Louis Gardies. n'avait, dans cette voie, cherché et réussi à en éviter qu'une seule. Le disciple ici va plus loin que le maître dans la voie que ce dernier avait ouverte, puisqu'il ne présuppose pas, mais ...

In Metaphysics M.2, 1077a9-14, Aristotle appears to argue against the existence of Platonic Forms on the basis of there being certain universal mathematical proofs which are about things that are ‘beyond’ the ordinary objects of mathematics and that cannot be identified with any of these. It is a very effective argument against Platonism, because it provides a counter-example to the core Platonic idea that there are Forms in order to serve as the object of scientific knowledge: the universal of which (...) theorems of universal mathematics are proven in Greek mathematics is neither Quantity in general nor any of the specific quantities, but Quantity-of-type-x. This universal cannot be a Platonic Form, for it is dependent on the types of quantity over which the variable ranges. Since for both Plato and Aristotle the object of scientific knowledge is that F which explains why G holds, as shown in a ‘direct’ proof about an arbitrary F (they merely disagree about the ontological status of this arbitrary F, whether a Form or a particular used in so far as it is F), Plato cannot maintain that Forms must be there as objects of scientific knowledge - unless the mathematics is changed. (shrink)

Infinity is an intriguing topic, with connections to religion, philosophy, metaphysics, logic, and physics as well as mathematics. Its history goes back to ancient times, with especially important contributions from Euclid, Aristotle, Eudoxus, and Archimedes. The infinitely large is intimately related to the infinitely small. Cosmologists consider sweeping questions about whether space and time are infinite. Philosophers and mathematicians ranging from Zeno to Russell have posed numerous paradoxes about infinity and infinitesimals. Many vital areas of mathematics rest upon some (...) version of infinity. The most obvious, and the first context in which major new techniques depended on formulating infinite processes, is calculus. But there are many others, for example Fourier analysis and fractals.In this Very Short Introduction, Ian Stewart discusses infinity in mathematics while also drawing in the various other aspects of infinity and explaining some of the major problems and insights arising from this concept. He argues that working with infinity is not just an abstract, intellectual exercise but that it is instead a concept with important practical everyday applications, and considers how mathematicians use infinity and infinitesimals to answer questions or supply techniques that do not appear to involve the infinite.ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable. (shrink)

Why did Copernicus's research programme supersede Ptolemy's?’, Lakatos and Zahar argued that, on Zahar's criterion for ‘novel fact’, Copernican theory was objectively scientifically superior to Ptolemaic theory. They are mistaken, Lakatos and Zahar applied Zahar's criterion to ‘a historical thought-experiment’—fictional rather than real history. Further, in their fictional history, they compared Copernicus to Eudoxus rather than Ptolemy, ignored Tycho Brahe, and did not consider facts that would be novel for geostatic theories. When Zahar's criterion is applied to real history, (...) the results are distinctly different. Finally, most of the historical and conceptual problems in applying Zahar's criterion to the Copernican Revolution primarily arise from a deep difficulty in Lakatos's programme: the necessity of individuating research programmes and identifying their originators. 1 Working closely with David Dahl was crucial in developing this paper. Robert Westman's valiant effort to keep me on the historical straight and narrow drastically limited my tendency to a priori historical pronouncements. The Vassar Philosophy Department, John Tompsich, and Jean Sterling were also helpful. (shrink)

Glaucon in Plato's _Republic_ fails to grasp intermediates. He confuses pursuing a goal with achieving it, and so he adopts ‘mathematical platonism’. He says mathematical objects are eternal. Socrates urges a seriously debatable, and seriously defensible, alternative centered on the destruction of hypotheses. He offers his version of geometry and astronomy as refuting the charge that he impiously ‘ponders things up in the sky and investigates things under the earth and makes the weaker argument the stronger’. We relate his account (...) briefly to mathematical developments by Plato's associates Theaetetus and Eudoxus, and then to the past 200 years' developments in geometry. (shrink)

Aristotle asserts at 1073b10-13 that he intends to give in Metaphysics XII.8 a definite conception about the multitude of the divine transcendent entities, which function as the movers of the celestial spheres. In order to do so, he describes several celestial theories. First Eudoxus’s, then the modifications of this theory propounded by Callippus, and finally his own suggestion, the introduction of yet further spheres which integrate the celestial spheres into a single overarching scheme. For this, after explaining the spheres (...) providing the component motions of each planet, Aristotle introduces so-called rewinding spheres (anelittousai), which perform contrary revolutions to the ones performed by the spheres carrying the planet. (shrink)

The author describes the Cartesian way of solving the problem of the universal method in mathematics, in particular, the problem of applying algebra in geometry when it comes to the convergence of a discrete number and a continuous quantity. The article shows that the solution to this problem proposed by F. Viète is imperfect, since it introduces vague pseudo-geometric objects, and the geometric quantity is still far from an algebraic number. The author proves that Descartes' solution to this problem through (...) the use of Eudoxus proportions is based on such Cartesian epistemological principles as: the requirement of clarity and expressiveness of thinking; the idea of the central role of a holistic mathematical science; the idea of the existence of a simple and obvious nature of length as a basis for comparing all extended things; the elevation of the concept of ratio to the rank of a single subject of mathematical disciplines. (shrink)

(I) Aristotle of Stagira (384-322 BC) 0) A closed geocentric spherical cosmology. (Adopted from the great mathematician, Eudoxus, c. 400 to 347 BC; via Calippus; but Aristotle unifies their separate schemes for different heavenly bodies). (Aristotle cites mathematicians as estimating radius of earth: in fact 200% of correct figure. Eratosthenes ca. 250 BC estimates radius of earth as 120% of correct).

Plato began it. After thinking about the nature of argument he concluded that the correct way of reasoning was the axiomatic way, and formulated the programme of axiomatization that Eudoxus and Euclid subsequently carried out. Since then the axiomatic method has been firmly established, not only as the method for mathematics, but as a paradigm to which all other disciplines should strive to be assimilated; and in this present century not only has axiomatization been carried through as completely as (...) it can be, but the most determined efforts have been made to wish hypotheticodeductive schemata on to biology, economics, and even history. (shrink)

The assumption that Ptolemy adopted star coordinates from a star catalog by Hipparchus is investigated based on Hipparchus’ equatorial star coordinates in his Commentary on the phenomena of Aratus and Eudoxus. Since Hipparchus’ catalog was presumably based on an equatorial coordinate system, his star positions must have been converted into the ecliptical system of Ptolemy’s catalog in his Almagest. By means of a statistical analysis method, data groups consistent with this conversion of coordinates are identified. The found groups show (...) a high degree of agreement between Hipparchus’ and Ptolemy’s data. The value of the obliquity of the ecliptic underlying the conversion is estimated by adjustment and statistically agrees with Ptolemy’s value of this parameter. The results allow the assumption that Ptolemy’s coordinates were determined from Hipparchus’ coordinates by an accurate star globe or even by calculation. For a calculative derivation of ecliptical coordinates from equatorial ones, possible calculation methods are discussed considering the mathematics of the Almagest. (shrink)

The aim of this text is to present the evolution of the relation between the concept of number and magnitude in ancient Greek mathematics. We will briefly revise the Pythagorean program and its crisis with the discovery of incommensurable magnitudes. Next, we move to the work of Eudoxus and present its advances. He improved the Pythagorean theory of proportions, so that it could also treat incommensurable magnitudes. We will see that, as the time passed by, the existence of incommensurable (...) magnitudes was no longer something strange. Already in the period of Plato and Aristotle, their existence was common place: up to the point of being considered absurd that all magnitudes were commensurable. Aristotle criticized the Pythagorean program and defended that, though belonging to the same category, number and magnitude are of distinct species: number is discrete and magnitude is continuous. We finish by presenting briefly how the concept of number was amplified throughout the centuries until it came also to include the notion of continuity. (shrink)

The figure of the cordial host of the Academy, who invited the most gifted mathematicians and cultivated pure research, whose keen intellect was able if not to solve the particular problem then at least to show the method for its solution: this figure is quite familiar to students of Greek science. But was the Academy as such a center of scientific research, and did Plato really set for mathematicians and astronomers the problems they should study and methods they should use? (...) Our sources tell about Plato's friendship or at least acquaintance with many brilliant mathematicians of his day (Theodorus, Archytas, Theaetetus), but they were never his pupils, rather vice versa -- he learned much from them and actively used this knowledge in developing his philosophy. There is no reliable evidence that Eudoxus, Menaechmus, Dinostratus, Theudius, and others, whom many scholars unite into the group of so-called "Academic mathematicians," ever were his pupils or close associates. Our analysis of the relevant passages (Eratosthenes' Platonicus, Sosigenes ap. Simplicius, Proclus' "Catalogue of geometers", and Philodemus' "History of the Academy", etc.) shows that the very tendency of portraying Plato as the architect of science goes back to the early Academy and is born out of interpretations of his dialogues. (shrink)

The derivative is a basic concept of differential calculus. However, if we calculate the derivative as change in distance over change in time, the result at any instant is 0/0, which seems meaningless. Hence, Newton and Leibniz used the limit to determine the derivative. Their method is valid in practice, but it is not easy to intuitively accept. Thus, this article describes the novel method of differential calculus based on the double contradiction, which is easier to accept intuitively. Next, the (...) geometrical meaning of the double contradiction is considered as follows. A tangent at a point on a convex curve is iterated. Then, the slope of the tangent at the point is sandwiched by two kinds of lines. The first kind of line crosses the curve at the original point and a point to the right of it. The second kind of line crosses the curve at the original point and a point to the left of it. Then, the double contradiction can be applied, and the slope of the tangent is determined as a single value. Finally, the meaning of this method for the foundation of mathematics is considered. We reflect on Dehaene’s notion that the foundation of mathematics is based on the intuitions, which evolve independently. Hence, there may be gaps between intuitions. In fact, the Ancient Greeks identified inconsistency between arithmetic and geometry. However, Eudoxus developed the theory of proportion, which is equivalent to the Dedekind Cut. This allows the iteration of an irrational number by rational numbers as precisely as desired. Simultaneously, we can define the irrational number by the double contradiction, although its existence is not guaranteed. Further, an area of a curved figure is iterated and defined by rectilinear figures using the double contradiction. (shrink)

The present study aims at giving factual support to the thesis that the Parmenides is serious in intention, rigorous in logical demonstration, and stylistically meticulous in its original composition. While this consideration may be tedious, still it is useful. Against a past history which has claimed to find the tone hilarious, the logic fallacious, the work inauthentic, the text in need of bracketing by divination, the whole incoherent— against these eccentricities a certain firm sobriety seems called for. I hope that (...) my conclusions find support and persist through fluctuations of philosophic and philological styles.The main difficulty with my thesis is that the text as we now have it (in Burnet's and Dies' editions) shows exceptions to every rule that might apply to style and even to logical structure. Thus it is almost but not quite uniformly true that each theorem opens with a theorem statement; that each is marked by a "questioning" response; that each deduction is valid when formalized in propositional calculus; and so on. Are the exceptions the result of careless composition; are they deliberate warnings not to take the proofs too seriously; or are they the result of errors in transmission? One way to test this is to reconstruct early versions of the text: if these show more logical rigor than the later versions the notion of a wholly valid original is supported; if they do not, the result may point toward the need for a less serious interpretation. A second thing to look for is the possibility that, here and there, parts of a coherent original text are uniquely conserved in sources other than the principal mss, B,T, and W. This assumes, of course, that the "original" text in question is the one with the best logical form, and that assumption seems justified. As a matter of fact, later copyings almost universally show deterioration, not improvement in style and logic.My textual findings are more compatible with some lines of interpretation than with others. Thus I offer some reasons for not accepting treatments of the work as a joke, mystical revelation, or abrupt metaphysical revision. The structure of the dialogue is best explained, I think, by reading it as an indirect proof that some non-Platonic interpretations of the theory of forms are unsatisfactory. In particular, these are the Megarian version (represented in the first part of the dialogue by Zeno, in the second by the First Hypothesis), and the Anaxagorean version put forward by Eudoxus (represented in the first part by Cephalus and his friends from Clazomenae, in the second by the Second Hypothesis). At the same time, the dialogue shows the need for a philosophic method other than the "hypothetical— deductive" way of dianoia; presumably this other method is the "dialectic" discussed in Republic VII. It also follows that any interpretation of the Phaedo which falls into either the Eudoxian or Megarian difficulties was not— at least at the date of the Parmenides—Plato's intended reading.Findings concerning the relations and reliability of the manuscripts of this dialogue also apply to the texts of the other Platonic dialogues which these mss contain.Further, the Parmenides is such an important work both historically and intrinsically that any insight which textual study can bring to its interpretation is a contribution to Western philosophy. (shrink)

(I) Aristotle of Stagira (384-322 BC) 0) A closed geocentric spherical cosmology. (Adopted from the great mathematician, Eudoxus, c. 400 to 347 BC; via Calippus; but Aristotle unifies their separate schemes for different heavenly bodies). (Aristotle cites mathematicians as estimating radius of earth: in fact 200% of correct figure. Eratosthenes ca. 250 BC estimates radius of earth as 120% of correct).

Aristotle takes his ethics to be an inquiry into the ultimate good of human life. In the course of his criticism of Plato and Eudoxus, Aristotle formulates two general conditions on the concept of the ultimate good. Firstly, the ultimate good has to be something prakton. The primary sense of prakton is not, as it is often taken to be, of something that is "realizable" in human action, but of something that is, or can be, aimed at in human (...) action. Secondly, in addition to being based on an appropriate moral psychology and theory of action, an adequate ethical theory has to be non-reductivist--it has to give its due weight to the plurality and variety of the goodness of particular intrinsic goods. I examine why Aristotle's criticism of Eudoxus' hedonism as a reductive ethical theory is ineffective, and point out some deficiencies in Aristotle's own account of pleasure. ;Aristotle takes the view that every desire--whether rational or non-rational--is for something that appears good to the agent. This view is compared with what seems to be the currently accepted view, that some of our desires do not involve a valuation of any sort. Although I do not claim that the current view is wrong, I do argue that Aristotle's position may provide an alternative, and plausible, way of thinking about the nature of human motivation. I discuss next Aristotle's concept of telos. A telos has a dual character: it is something an action is aimed at, and also something attained in human action. Although Aristotle's ethics is not teleological, outcomes are important for the ethical evaluation of actions. In order to explicate how outcomes matter, I introduce the concepts of an Aristotelian outcome and of a good Aristotelian outcome. The notion of a third actualization is introduced to throw some light on the sort of success in the conduct of life which Aristotle takes eudaimonia to be. ;Finally, I provide an account of Aristotle's theory of practical reason. I argue that his ethical theory demands a very wide interpretation of the concept of deliberation. An analysis of Aristotle's concept of choice is given, and an attempt made to specify the conditions under which an action counts as chosen. Deliberation is ethical if it is guided by the deliberator's conception of eudaimonia. I explicate briefly what it is for a person to have such a conception, what such a conception is like, and in what way a person can be thought to be guided by it. Finally, the concept of practical reason is linked with the concept of the ultimate good. I propose an interpretation of the three conditions Aristotle imposes on the ultimate good in EN I.7--finality, self-sufficiency and choiceworthiness which requires that we initially construe the concepts of finality and of self-sufficiency in a narrow sense, and take the concept of finality as a comparative notion. The three conditions can then be regarded as specifying how deliberation ought to be conducted if it is to lead to eudaimonia. (shrink)

The article includes a short biography of Hermann of Dalmatia and gives an account of his translations and philosophical and scientific work. In order to have a better understanding of Hermann’s philosophy, a reminder of Greek and Arabic philosophy of nature, on which he relies in his interpretation of the world picture, needs to be presented. Cosmological models by Plato, Aristotle, Eudoxus, Heraclides of Pont, Apollonius of Perga, Hipparchus, Ptolemy, and the Arab scientist Abu Ma’shar, are presented. The main (...) focus of interest is on Hermann’s translations. The immense importance of his translations from Greek and Arabic into Latin is due to the fact that some of the seminal Greek and Arabic works became known in Western Europe in the middle of the twelfth century. Hermann is also important as the author of the original work De essentiis, which presented a blend of Platonian and Aristotelian as well as Western European and Arab traditions. (shrink)

Although Aristotle’sMetaphysicsreceived much attention in the nineteenth and the twentieth centuries, scholars and historians of science were not particularly interested in clarifying the aim of Aristotle’s appeal to astronomy in Λ 8. Read with monotheistic prejudices, this chapter was quickly abandoned by Aristotelian scholars as a gratuitous insertion, which downgrades Aristotle’s God for the sake of some supplementary principles, whose existence was dictated by celestial mechanics. On the other hand, historians of astronomy read the astronomical excursus as providing a picture (...) of Aristotle as an able astronomer, who made an important contribution to the theory of concentric celestial spheres of Eudoxus and Callippus by adding the counteracting spheres. The present article argues (a) that Aristotle purposefully turned to astronomy as the only mathematical science whose objects were correlative to the immaterial first substances or gods, the number of which had to be precisely determined by his own project of first philosophy; and (b) that he had no aspiration of improving the astronomical theory of his peers, contrary to an interpretation that first emerged with the Peripatetic exegete Sosigenes in the second century A.D. (shrink)

In Virgil's third eclogue, the goatherd Menalcas responds to his challenger Damoetas by offering as his wager in their contest of song a pair of embossed cups,caelatum diuini opus Alcimedontis, decorated with a pattern of vine and ivy. In the middle of this design, he says, are two figures. One is the astronomer Conon, and the other—at this point Menalcas, afflicted with a sudden loss of memory, professes to have forgotten the name of the second figure, and breaks off into (...) a question :quis fuit alter, | descripsit radio totum qui gentibus orbem, | tempora quae messor, quae curuus arator haberet?Various candidates for the identity of this second astronomer have been suggested, one of the most favoured being Eudoxus of Cnidus, whosePhaenomenahad been versified by the Hellenistic poet Aratus. In 1930 it was proposed by Léon Herrmann that Menalcas in fact answers his own question in his reference tocuruUS ARAToratEcl.3.42: the solution to Virgil's riddle is already written into the question, in the form of the anagram ofAratusconcealed within these two words. Strictly speaking, Hermann notes only that ‘Aratorau v. 42 évoque le nom d’Aratos’; he and later exponents of the theory tend to ignore the preceding ‘-us’—but there seems no reason to exclude it if an allusion to the Greek poet is to be seen in the two following syllables. (shrink)

The article includes a short biography of Hermann of Dalmatia and gives an account of his translations and philosophical and scientific work. In order to have a better understanding of Hermann’s philosophy, a reminder of Greek and Arabic philosophy of nature, on which he relies in his interpretation of the world picture, needs to be presented. Cosmological models by Plato, Aristotle, Eudoxus, Heraclides of Pont, Apollonius of Perga, Hipparchus, Ptolemy, and the Arab scientist Abu Ma’shar, are presented. The main (...) focus of interest is on Hermann’s translations. The immense importance of his translations from Greek and Arabic into Latin is due to the fact that some of the seminal Greek and Arabic works became known in Western Europe in the middle of the twelfth century. Hermann is also important as the author of the original work De essentiis, which presented a blend of Platonian and Aristotelian as well as Western European and Arab traditions. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 43.4 (2005) 490-491 [Access article in PDF] Husain Sarkar. Descartes' Cogito: Saved from the Great Shipwreck. New York: Cambridge University Press, 2003. Pp. xviii + 305. Cloth, $65.00. Descartes's first critics attacked his cogito, ergo sum as deficient; his present critics attack it as excessive. Either way, it is an Archimedean point in Descartes's world and merits a book-length study. In this book, (...) Husain Sarkar makes a powerful case that the cogito is not an argument, but rather an "experiment," generating an intuition enabling Descartes to retrieve a timber from what Bourdin called the "great shipwreck" of the hyperbolic doubt.Sarkar places the experiment within his own fresh readings of Descartes's philosophy. They will be illuminating to new Descartes scholars. Some readings will seem perverse or cryptic. Constructive passages illuminate more, polemic passages less, as when Sarkar finds seven flaws in Hintikka's cogito interpretation. Despite the polemic excesses, Sarkar writes with modesty, epitomized by his slogan, "Reason with them in the most courteous manner."With verve and imagination Sarkar presents the cogito as an experiment which can be carried out in any world at all and yields an intuition that qualifies as a Cartesian first principle. He suggests various implications of his interpretation—for example, for the epistemic role of will: "The will initially learns the limits that it must recognize only at the foot of the cogito... It is in the state of the cogito that the will learns what it means to be ineluctably inclined toward affirming a proposition, and what it is to be free" (256).What is the content of the intuited 'I exist'? Sarkar calls upon Robert Adams's notion of a "primitive thisness," entirely distinct from a substance's properties or "suchnesses," and upon David Lewis's thesis that besides knowing propositions a person can know a self-ascription of a property. Still, he makes no headway at all. True, the question is extraordinarily [End Page 490] difficult. It has not yielded appreciably to brilliant phenomenologists or brilliant students of direct reference and indexicality or, for that matter, me. The content of 'I' statements is a topic for another book, and no one I know of is writing it.In the book's centerpiece, Sarkar marshals a series of attacks on the cogito as argument. Sometimes they miss—more polemic overkill. Usually they hit home. Their cumulative effect is persuasive.Descartes says that "I think, therefore I exist" is the first principle of his philosophy (VI, 32; VIIIA, 7; V, 147: references are to the Adam and Tannery edition). But if the cogito were an argument, one of its premises would become his first principle.An argument requires movement of thought and that requires reliance on memory (X, 370 and 408), and in the cogito Descartes strips memory of its credentials (VII, 24, superseding the assurances of X, 368).An argument relies on logical principles, but Descartes suspects these and reasoning generally. In the First Meditation he calls into doubt the simplest mathematics; elsewhere he generalizes from mathematics to "self-evident principles" and to "all the arguments I had previously taken as demonstrative proofs" (VI, 32; IXB, 6; similar aspersions are cast in V, 137 and 139). (But see also Mark Olson, "Descartes' First Meditation: Mathematics and the Laws of Logic" JHP 26 [1988]: 407–438.)I'll add a wrinkle. Descartes's mouthpiece Eudoxus says you can use your own doubt to deduce facts that are even more certain than those commonly built upon the great principle that it is impossible that a thing should both exist and at the same time not exist (X, 522). The conclusion of any argument rests on such logical principles. If the cogito were an argument, its certainty could not exceed theirs. Descartes suggests here that it does.Sarkar also cites Descartes's denigrations of "logic," but these often pertain not to reasoning generally but to syllogistic logic. Thus Margaret Wilson took the cogito to be a nonsyllogistic argument. Still, Sarkar effectively deploys... (shrink)