With mathematical reconstructions and philosophical arguments I show that Toomer's 1973 paper never contained any conclusive evidence for his claims that Hipparchus had a 3438'-based chord table, and that the Indians used that table to compute their sine tables. Recalculating Toomer's reconstructions with a 3600' radius -- i.e. the radius of the chord table in Ptolemy's Almagest, expressed in 'minutes' instead of 'degrees' -- generates Hipparchan-like ratios similar to those produced by a 3438' radius. It is therefore possible that (...) the radius of Hipparchus's chord table was 3600', and that the Indians independently constructed their 3438'-based sine table. (shrink)

Ptolemy reports three dated lunar eclipses observed by Hipparchus, and also refers to two more, without identifying them, which Hipparchus compared with two earlier counterparts (apparently, observed in Mesopotamia) to assess the validity of the Babylonian period relations of the lunar motion. Also, in Pliny the Elder’s Historia naturalis, we are told that a horizontal lunar eclipse (selenelion) at sunrise and moonset was reported (observed?) by Hipparchus. Reviewing a paper by G.J. Toomer in 1980, it is shown (...) that the pairs of the eclipses were, almost certainly, the ones occurring on “31 January 486 b.c. and 27 January 141 b.c.” and “19 November 502 b.c. and 14 November 157 b.c.”; and if Hipparchus observed from St. Stephen’s Hill in Rhodes, the most probable candidate for the selenelion at moonset was the lunar eclipse of 7 February 142 b.c., although he also had the chance to observe any of the four others, occurring on 3 July 150 b.c., 10 April 145 b.c., 26 November 139 b.c., and 15 November 138 b.c., on a sufficiently elevated mountain on the island. (shrink)

Following Schleiermacher, who was unable to account for several oddities in the dialogue, some scholars consider the Hipparchus a spurious Platonic work. This essay, by means of a dramatic re-enactment of the dialogue, accounts for those oddities. It demonstrates that the comrade is a recent immigrant to Athens who, having been deceived by a moneychanger in the agora, accuses ‘lovers of gain’ of being ‘profiteers’. Socrates exposes the comrade as fearful of risk-taking and then defends the reputation of (...) class='Hi'>Hipparchus, the Athenian King who encouraged commercial development. By further correcting the democratic account of his assassination, Socrates exposes the comrade’s envy, which hides his own profiteering ambitions. The discussion points to a series of dualities regarding commerce and philosophy, including both the generating and non-generating qualities of the philosophic life. It also reveals the rural origins of Socrates’ ‘new’ accusers, and the notion of justice held by them. (shrink)

The Hipparchus features a conversation between Socrates and an un-named companion, at an unknown time and place, about gain and whether we should in any way limit our pursuit of gain. Socrates argues intransigently that we should not place any limits on our pursuit of gain, while the companion, despite being unable to counter his arguments, is equally firm in his rejection of Socrates’ moral position. The dialogue thus shows the strength of the conviction, in the souls of decent (...) people, that unrestrained pursuit of the good things is not good for you. This conviction exists prior to calculation, and the action of the dialogue thereby testifies to the existence of man’s natural conscience. (shrink)

Attempts by twentieth-century historians to account for the successes and failures of the Hipparchian-Ptolemaic solar model provide valuable case studies for philosophers who are studying the relationship between observational data and theoretical constructs. A brief survey of recent literature on the solar model reveals that in some cases results which appear to be the product of highly accurate observation are, in fact, based on rather crude observations aided by a large measure of theoretical presupposition. On the other hand, mistaken results, (...) which have sometimes been explained as the outcome of theoretical bias or conscious deceit, can plausibly be accounted for as a result of honest mistakes at the observational level. (shrink)

ABSTRACT: The 3rd BCE Stoic logician "Chrysippus says that the number of conjunctions constructible from ten propositions exceeds one million. Hipparchus refuted this, demonstrating that the affirmative encompasses 103,049 conjunctions and the negative 310,952." After laying dormant for over 2000 years, the numbers in this Plutarch passage were recently identified as the 10th (and a derivative of the 11th) Schröder number, and F. Acerbi showed how the 2nd BCE astronomer Hipparchus could have calculated them. What remained unexplained is (...) why Hipparchus’ logic differed from Stoic logic, and consequently, whether Hipparchus actually refuted Chrysippus. This paper closes these explanatory gaps. (1) I reconstruct Hipparchus’ notions of conjunction and negation, and show how they differ from Stoic logic. (2) Based on evidence from Stoic logic, I reconstruct Chrysippus’ calculations, thereby (a) showing that Chrysippus’ claim of over a million conjunctions was correct; and (b) shedding new light on Stoic logic and – possibly – on 3rd century BCE combinatorics. (3) Using evidence about the developments in logic from the 3rd to the 2nd centuries, including the amalgamation of Peripatetic and Stoic theories, I explain why Hipparchus, in his calculations, used the logical notions he did, and why he may have thought they were Stoic. OPEN ACCESS LINK. (shrink)

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.

Summary The Almagest of Ptolemy (mid-second century ad) contains eleven dated reports of observations of the positions of planets made during the third century bc in Babylon and Hellenistic Egypt. The present paper investigates the character, purpose, and conventions of the observational programmes from which these reports derive, the channels of their transmission to Ptolemy's time, and the fidelity of Ptolemy's presentation of them. Like the Babylonian observational programme, about which we have considerable knowledge through cuneiform documents, the Greco-Egyptian ones (...) were not directed towards the deduction of mathematical models of celestial motion but appear to have investigated patterns, correlations, and periodicities of phenomena. Ptolemy's immediate sources most likely were not the original series of observational records, but treatises by various astronomers of the intervening four centuries, including Hipparchus. While Ptolemy does not appear to have tampered with the wording of the reports, he faced difficulties and uncertainties in interpreting them; critically, he lacked sufficiently detailed information about the ancient calendars to be able to convert the reported dates accurately into his own chronological framework based on the Egyptian calendar. (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)

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)

English summary: The book is a translation with introduction and commentary of the Platonic dialogue Hipparchus. Based on the excursus, which has given the dialog its name and bestows him a special position within the Platonic oeuvre, a comparative analysis of the tradition concerning the tyrant Hipparchus, his erection of herms with epigrams in Attica and his murder by Harmodios and Aristogeiton will show that the excursus and the dialogue have the numerous references to the entire work of (...) Plato with regard to the topics of education, wisdom, friendship, love and deception. German description: Der kurze Dialog 'Hipparchos' ist ein Gesprach zwischen Sokrates und einem namenlosen Gesprachspartner. Heute im Allgemeinen fur unecht gehalten und daher in die Gruppe der Pseudoplatonica eingeordnet, ist er im Platonischen Corpus uberliefert und in der Antike ganz offensichtlich fur ein echtes Werk Platons gehalten worden. Der Kommentar analysiert den Gedankengang des Dialogs und ordnet ihn in einen plausiblen zeitgenossischen Kontext ein, der nicht so sehr die Bezuge auf das platonische Corpus in den Mittelpunkt stellt als vielmehr die Gemeinsamkeit mit der Entwicklung der griechischen Historiographie der spatklassischen und fruhhellenistischen Zeit. Die Appendizes vertiefen drei literarisch-historische Aspekte in der Darstellung der Figur des Hipparch: die uberlieferung zu dem Attentat, dem er zum Opfer fiel, sein Hermenprogramm sowie sein Bestreben, sich in Athen als ein Weiser darzustellen. Mit der Einordnung des Hipparch in den Umkreis der antiken Weisen knupft der Dialog an eine eher spezielle Tradition an, die unter die Weisen auch Tyrannen rechnet. Daher wird die Verbindung zwischen Tyrannis und griechischer Weisheitstradition im Rahmen der Appendizes ausfuhrlich erlautert. (shrink)

THERE WAS A TIME, EONS AGO, when philosophy as the love of wisdom could lay claim to all knowledge. Aristotle’s corpus of writings covered all the main areas of inquiry then known, including an original organon on syllogistic logic and scientific method. But this hegemony over knowledge was soon challenged by separatist disciplines forming their own research strategies. As early as the third century B.C.E., following the deaths of Alexander and Aristotle, the ruling Ptolemies created in Alexandria two centers of (...) scientific research, the Library and the Museum, rivaling the Academy and Lyceum of Athens. In those state-supported institutions specialized research by grammarians, mathematicians, astronomers, physicists, biologists, and physiologists replaced the more speculative syntheses of knowledge of the previous Hellenic period. The resident scholars and those affiliated by correspondence or as legatees were the originators of axiomatic geometry and what we now refer to as protoscience: for example, Euclid, Herophilus, Erasistratus, Apollonius, Heraclides, Hipparchus, and Aristarchus. Each of the major founders of modern classical science acknowledged their indebtedness to these extraordinary ancient Greeks who initiated the scientific exploration of the universe. (shrink)

The gravitational influence of Jupiter on Saturn produces, among other things, non-negligible changes in the eccentricity of Saturn that affect the magnitude of error of Ptolemaic astronomy. The value that Ptolemy obtained for the eccentricity of Saturn is a good approximation of the real eccentricity—including the perturbation of Jupiter—that Saturn had during the time of Ptolemy's planetary observations or a bit earlier. Therefore, it seems more probable that the observations used for obtaining the eccentricity of Saturn were done near Ptolemy’s (...) time, and rather unlikely earlier than the first century AD. Even if this is not quite a demonstration that Ptolemy used observations of his own, my argument increases its probability and practically discards the idea that Ptolemy borrowed values or observations from astronomers further back than the first century AD, such as Hipparchus or the Babylonians. (shrink)

The understanding of the soul in the West has been profoundly shaped by Christianity, and its influence can be seen in certain assumptions often made about the soul: that, for example, if it does exist, it is separable from the body, free, immortal, and potentially pure. The ancient Greeks, however, conceived of the soul quite differently. In this ambitious new work, Michael Davis analyzes works by Homer, Herodotus, Euripides, Plato, and Aristotle to reveal how the ancient Greeks portrayed and understood (...) what he calls “the fully human soul.” Beginning with Homer’s _Iliad_, Davis lays out the tension within the soul of Achilles between immortality and life. He then turns to Aristotle’s _De Anima_ and _Nicomachean Ethics_ to explore the consequences of the problem of Achilles across the whole range of the soul’s activity. Moving to Herodotus and Euripides, Davis considers the former’s portrayal of the two extremes of culture—one rooted in stability and tradition, the other in freedom and motion—and explores how they mark the limits of character. Davis then shows how _Helen_ and _Iphigeneia among the Taurians_ serve to provide dramatic examples of Herodotus’s extreme cultures and their consequences for the soul. The book returns to philosophy in the final part, plumbing several Platonic dialogues—the _Republic_, _Cleitophon_, _Hipparchus_, _Phaedrus_, _Euthyphro_, and _Symposium_—to understand the soul’s imperfection in relation to law, justice, tyranny, eros, the gods, and philosophy itself. Davis concludes with Plato’s presentation of the soul of Socrates as self-aware and nontragic, even if it is necessarily alienated and divided against itself. _The Soul of the Greeks_ thus begins with the imperfect soul as it is manifested in Achilles’ heroic, but tragic, longing and concludes with its nontragic and fuller philosophic expression in the soul of Socrates. But, far from being a historical survey, it is instead a brilliant meditation on what lies at the heart of being human. (shrink)

The climata played an important role in Greek geography. As used in the mathematical geography of Hipparchus and Ptolemy the word denotes a narrow belt or strip of land, 400 stades wide, on each side of a parallel of latitude; inhabitants of the same clitma were assumed to be situated in the same geographical latitude, since, for practical purposes, the celestial phenomena, lengths of the longest and shortest days, and general climatic conditions did not change appreciably within this distance. (...) We may compare the modern conception of arbitrary zones each possessing its own standard time; but whereas these are belts of longitude contiguous to each other, the climata were belts of latitude not necessarily contiguous. (shrink)

Τ τε 'Ομρον ξ ποβολς γγραφε [σόλων] αψδεσθαι, οον που ό πρτος ληξεν κεθεν ρχεσθαι τòν χ;óμεν㦿ν . It would be tedious to trace the course of the controversy about this phrase. Since Hermann showed that ύποβάλλειν means ‘subiicere alteri quod recordetur uel dicat’ , the many conflicting interpretations have been obliged to resort to some device, more or less strained, in order to reconcile the natural sense of ύποβολή with what at least appears to be the explanatory clause added (...) in the text of Diogenes, and with the parallel phrase in the Hipparchus, 228d: ήνάγκασε τούς αΨδος Παναθηναίοις ξ ύπολψεως φεξης ατδιιναι. We should expect some term meaning performance by turns, and sufficiently technical to need the explanation given by Diogenes. Mr. Pallis has cut the knot in the Classical Quarterly by suggesting ξ ποβς on the model of the pseudo-Platonic passage. There is, however, no classical authority for this word, and a change as small will restore a known technical word with the required sense. (shrink)

Τ⋯ τε 'Ομ⋯ρον ⋯ξ ὑποβολ⋯ς γ⋯γραφε [σόλων] ῥαψῳδεῖσθαι, οἷον ὅπου ό πρ⋯τος ἒληξεν ⋯κεῖθεν ἂρχεσθαι τòν ⋯χ;óμεν㦿ν.It would be tedious to trace the course of the controversy about this phrase. Since Hermann showed that ύποβάλλειν means ‘subiicere alteri quod recordetur uel dicat’, the many conflicting interpretations have been obliged to resort to some device, more or less strained, in order to reconcile the natural sense of ύποβολή with what at least appears to be the explanatory clause added in the text (...) of Diogenes, and with the parallel phrase in the Hipparchus, 228d: ήνάγκασε τούς ῥαΨῳδοὺς Παναθηναίοις ⋯ξ ύπολ⋯ψεως ⋯φεξης α⋯τ⋯ διι⋯ναι. We should expect some term meaning performance by turns, and sufficiently technical to need the explanation given by Diogenes. Mr. Pallis has cut the knot in the Classical Quarterly by suggesting ⋯ξ ὺποβ⋯ς on the model of the pseudo-Platonic passage. There is, however, no classical authority for this word, and a change as small will restore a known technical word with the required sense. (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: Classical Art: A Life History DAVID CAST This is a wonderful book, rich in its purposes, wide in its range and, thanks to the author’s home institution, Christ’s College, Cambridge, lavishly illustrated with images of objects, many familiar, some less so. And it is written with an elegance and clarity that belies the depths of scholarship in its history. The first letter of the subtitle suggests the tenor (...) of Vout’s account, that since every age has conjured up a different idea of the classical, what is here is yet one more such history, a modest if particular account of what was and is still there in the records made of the Greek and Roman past. The basic metaphor Vout uses to define the focus of this history is that of conversation, whether such conversing is seen to exist between the objects themselves or between the past and the present, between those objects that have outgrown their intended functions to become part of a canon that has dictated taste over the ages, as she puts it, shaping culture and culture’s questions. Thus conversation is here, plus also what Vout is not afraid, when necessary, to call the misappropriations and translocations of the Greek and Roman objects she takes note of. Yet another contribution she adds to the idea of the classical comes from what she calls, perhaps a little awkwardly, the long durée approach to its subjects, seen first when these objects were produced, then as they survived Christianity and the fall of Rome and Byzantium, the French Revolution, modernism and fascism, to live Caroline Vout, Classical Art: A Life History from Antiquity to the Present, Princeton University Press, 2018. xi + 376 pages; 80 color + 132 b/w illustrations. $39.50. ISBN 9780691177038; E-book ISBN9781400890279. arion 27.1 spring/summer 2019 on in galleries and auction rooms, making the heart soar and the blood boil, as Vout claims, even today. It is a complex story she has to tell, difficult yet valuable, and its method is nicely exemplified by the first example considered, the two statues, known as the Tyrannicides, dated to 477–476 BCE, and located now in the National Archaeological Museum in Naples, attributed on the museum caption to the sculptors Kritios and Nesiotes, and dated to 477–476 BCE. The details here are crucial for, on investigation, it is clear these are not the statues that were erected in the Athenian agora to honor the men, Harmodius and Aristogeiton, who, by tradition, had killed the tyrant’s brother Hipparchus. Yet even this second group, as she notes, had been a stand-in, since the first piece to honor these men, made by the sculptor Antenor, had been stolen by the Persians in the sack of Athens in 480–479 BCE. Nor is this group here the actual second group but rather a copy, made by an unknown copyist working under the Roman Empire. What we have here are what might be called pretenders—yet ones, despite doubts about their authenticity, that have long been taken to mark a particular moment in the formal history of Greek art, the advent of the so-called Severe style in Athens and the birth of the classical moment of sculpture. Any full account of these statues depends then on a rich history, plus a range of other events; the existence of ancient casts of the Tyrannicides during the Roman Empire, marking out such statues as art, rather than having still their first meaning; the prior history in the Renaissance of the acquisition of ancient sculpture as a sign of power; the sight of the figure of Aristogeiton, then in the Palazzo Farnese in Rome, by Johann Joachim Winckelmann; the building up of these statues from fragments of other statues; in 1859 the identification of these figures as the Tyrannicides; and finally a reading of classical texts, those of Pausanias, Pliny and Lucian, that spoke, even if in contradictory ways, of all the actors here, Antenor, Kritios and Aristogeiton. To which should be added the recent accounts of these pieces, provided by schol172 classical art: a life history ars of classical art like Brunilde Ridgway, R. T... (shrink)

Herodotus' extension of tisis from a merely ethical principle to an encompassing law of nature is now widely recognized. The unjust expulsion of Demaratus from the Spartan kingship obtains its clear revenge from both Leotychidas and Cleomenes . Hipparchus' vision of a giant prophet who announces the universal penalty for human injustice embodies a statement of the ethical law which Herodotus sees operating in the realm of animals as well as of men: for any act of injustice one must (...) pay the penalty, . Herodotus similarly analyses the Persians' aim at Salamis: they sought to force the Greeks to ‘pay t n e penalty’ for the Greek success at Artemision. (shrink)

Herodotus' extension of tisis from a merely ethical principle to an encompassing law of nature is now widely recognized. The unjust expulsion of Demaratus from the Spartan kingship obtains its clear revenge from both Leotychidas and Cleomenes. Hipparchus' vision of a giant prophet who announces the universal penalty for human injustice embodies a statement of the ethical law which Herodotus sees operating in the realm of animals as well as of men: for any act of injustice one must pay (...) the penalty,. Herodotus similarly analyses the Persians' aim at Salamis: they sought to force the Greeks to ‘pay t n e penalty’ for the Greek success at Artemision. (shrink)

THERE WAS A TIME, EONS AGO, when philosophy as the love of wisdom could lay claim to all knowledge. Aristotle’s corpus of writings covered all the main areas of inquiry then known, including an original organon on syllogistic logic and scientific method. But this hegemony over knowledge was soon challenged by separatist disciplines forming their own research strategies. As early as the third century B.C.E., following the deaths of Alexander and Aristotle, the ruling Ptolemies created in Alexandria two centers of (...) scientific research, the Library and the Museum, rivaling the Academy and Lyceum of Athens. In those state-supported institutions specialized research by grammarians, mathematicians, astronomers, physicists, biologists, and physiologists replaced the more speculative syntheses of knowledge of the previous Hellenic period. The resident scholars and those affiliated by correspondence or as legatees were the originators of axiomatic geometry and what we now refer to as protoscience: for example, Euclid, Herophilus, Erasistratus, Apollonius, Heraclides, Hipparchus, and Aristarchus. Each of the major founders of modern classical science acknowledged their indebtedness to these extraordinary ancient Greeks who initiated the scientific exploration of the universe. (shrink)

In a recent paper I discussed the origin of the concept of the climata in Greek geography, and adduced reasons for attributing the formulation and elaboration of the concept to Hipparchus . The above passage in Strabo was naturally mentioned in the course of the argument, and I drew attention in a footnote to the unsatisfactory nature of the account given by him of the climata. I now propose to examine the passage in more detail.

Recent investigations have thrown new light on such topics as the early Greek belief in heliocentricity, the relation between Greek and Babylonian astronomy, the reliability of Ptolemy's Syntaxis, Hipparchus's theory of motion for the sun, Hipparchus's value for the obliquity of the ecliptic, and Eratosthenes' estimate of the size of the earth. Some claims resulting from these investigations are controversial, especially the reevaluation of Ptolemy (though it is notable that no one any longer uses data from the Syntaxis (...) for investigating such things as the spin of the earth). This essay presents the evidence for these claims; it makes no pretense of presenting the evidence against them. (shrink)

This volume contains translations of the following dialogues: Hipparchus, Minos, Lovers, Cleitophon, Theages, Alcibiades I, Laches, Lesser Hippias, Greater Hippias, and Ion. Each translation is accompanied by a short interpretive study, some previously published, designed to serve as an introduction and stimulus to further reading and studying of the respective dialogues. Contributors, both translators and interpreters, are mostly professors of Political Science, and they include Allan Bloom and the late Leo Strauss. In his introduction, the editor discusses the status (...) of these neglected texts and argues for an unprejudiced reconsideration of these fascinating Platonic dramas and important works of political philosophy. (shrink)

The latest book of Reviel Netz presents a highly erudite analysis of the style of Hellenistic mathematics. Besides the Introduction and Conclusion the book is composed of four chapters. Before turning to more general remarks I would like first to outline the contents of the book.The Introduction starts with the presentation of Archimedes’ Spiral lines. In a condensed form, the author outlines his approach and calls attention to the stylistic peculiarities of that work. Most of the themes discussed in more (...) detail in the following chapters, such as the analysis of the narrative structure of the treatise, its working with surprise, the exposition of the rhetorical devices, the discussion of technical tools, and the breaking of genre-boundaries emerge here side by side in a single treatise.The first chapter, called The carnival of calculation, starts with another work of Archimedes, the Stomachion, for which Netz offers a new interpretation as a treatise in geometrical combinatorics. Then he turns to Archimedes’ On the measurement of the circle, where, in a detailed analysis of the calculations made by Archimedes, Netz presents a pattern that he identifies in several other works of Hellenistic mathematics, a pattern he calls ‘asymptotic calculation’. It consists in a combination of strict geometric arguments with calculations of proportions, some precise, some only approximate, which together with rather arbitrary simplifications lead to a result that is achieved after an abrupt and unexpected halt. Netz shows that, according to the surviving testimonies, a similar pattern can be found in many Hellenistic authors, such as Aristarchus, Hipparchus, and Apollonius of Perga. "[I]n many Hellenistic mathematical works we see calculations that give rise to values that are big and unwieldy, or else are only asymptotically found. Such calculations end up not with the satisfaction of having simplified … ". (shrink)