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151-560: The Platonic Year , which is also called the Great Year, has a different more ancient and mystical meaning. Plato hypothesized that winding the orbital motions of the Sun, Moon and naked eye planets forward or back in time would arrive at a point where they are in the same positions as they are today. He called this time period the Great Year and suggested that such a unified return would take place about every 36,000 years. By extension,

302-500: A cyclic quadrilateral , today called Ptolemy's theorem because its earliest extant source is a proof in the Almagest (I.10). The stereographic projection was ambiguously attributed to Hipparchus by Synesius (c. 400 AD), and on that basis Hipparchus is often credited with inventing it or at least knowing of it. However, some scholars believe this conclusion to be unjustified by available evidence. The oldest extant description of

453-547: A Great Year as 2484 years: but it has been argued that this is a miscopying of 2434, which represents 45 Exeligmos cycles. The origin of the Platonic Year would seem to have no connection with the precession of the equinoxes as this was unknown in Plato's time. Two centuries after Plato, Hipparchus is credited with discovering the period of equinox precession , and the term "Great Year" eventually came to be applied to

604-486: A Platonist or Pythagorean, in that such a one would have "the viewpoint of logical simplicity as an indispensable and effective tool of his research." British philosopher Alfred N. Whitehead is often misquoted of uttering the famous saying of "All of Western philosophy is a footnote to Plato." Many recent philosophers have also diverged from what some would describe as ideals characteristic of traditional Platonism. Friedrich Nietzsche notoriously attacked Plato's "idea of

755-493: A belief in the immortality of the soul, and several dialogues end with long speeches imagining the afterlife . In the Timaeus , Socrates locates the parts of the soul within the human body: Reason is located in the head, spirit in the top third of the torso , and the appetite in the middle third of the torso, down to the navel . Furthermore, Plato evinces a belief in the theory of reincarnation in multiple dialogues (such as

906-672: A corruption of another value attributed to a Babylonian source: 365 + ⁠ 1 / 4 ⁠ + ⁠ 1 / 144 ⁠ days (= 365.25694... days = 365 days 6 hours 10 min). It is not clear whether Hipparchus got the value from Babylonian astronomers or calculated by himself. Before Hipparchus, astronomers knew that the lengths of the seasons are not equal. Hipparchus made observations of equinox and solstice, and according to Ptolemy ( Almagest III.4) determined that spring (from spring equinox to summer solstice) lasted 94 1 ⁄ 2 days, and summer (from summer solstice to autumn equinox) 92 + 1 ⁄ 2 days. This

1057-400: A descendant of two kings, Codrus and Melanthus . His mother was Perictione , descendant of Solon , a statesman credited with laying the foundations of Athenian democracy . Plato had two brothers, Glaucon and Adeimantus , a sister, Potone , and a half brother, Antiphon. Plato may have travelled to Italy, Sicily , Egypt, and Cyrene . At 40, he founded a school of philosophy,

1208-460: A difference of approximately one day in approximately 300 years. So he set the length of the tropical year to 365 + 1 ⁄ 4 − 1 ⁄ 300 days (= 365.24666... days = 365 days 5 hours 55 min, which differs from the modern estimate of the value (including earth spin acceleration), in his time of approximately 365.2425 days, an error of approximately 6 min per year, an hour per decade, and ten hours per century. Between

1359-489: A different doctrine with respect to Forms to Plato and Socrates. Aristotle suggests that Socrates' idea of forms can be discovered through investigation of the natural world, unlike Plato's Forms that exist beyond and outside the ordinary range of human understanding. The Socratic problem concerns how to reconcile these various accounts. The precise relationship between Plato and Socrates remains an area of contention among scholars. Although Socrates influenced Plato directly,

1510-483: A fellow disciple of Plato. A variety of sources have given accounts of Plato's death. One story, based on a mutilated manuscript, suggests Plato died in his bed, whilst a young Thracian girl played the flute to him. Another tradition suggests Plato died at a wedding feast. The account is based on Diogenes Laërtius's reference to an account by Hermippus, a third-century Alexandrian. According to Tertullian , Plato simply died in his sleep. According to Philodemus, Plato

1661-571: A few people were capable or interested in following a reasoned philosophical discourse, but men in general are attracted by stories and tales. Consequently, then, he used the myth to convey the conclusions of the philosophical reasoning. Notable examples include the story of Atlantis , the Myth of Er , and the Allegory of the Cave . When considering the taxonomic definition of mankind , Plato proposed

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1812-491: A longer time of life on account of their virtue, and the good use they made of it in astronomical and geometrical discoveries, which would not have afforded the time of foretelling [the periods of the stars] unless they had lived six hundred years; for the great year is completed in that interval. It has been suggested that he obtained this value from Berossos (c. 3rd century BC) who reckoned time in intervals of 60, 600 and 3600 years. Isaac Newton (1642 – 1726/27 ) determined

1963-462: A more detailed discussion. Pliny ( Naturalis Historia II.X) tells us that Hipparchus demonstrated that lunar eclipses can occur five months apart, and solar eclipses seven months (instead of the usual six months); and the Sun can be hidden twice in thirty days, but as seen by different nations. Ptolemy discussed this a century later at length in Almagest VI.6. The geometry, and the limits of

2114-477: A participant in any of the dialogues, and with the exception of the Apology , there is no suggestion that he heard any of the dialogues firsthand. Some dialogues have no narrator but have a pure "dramatic" form, some dialogues are narrated by Socrates himself, who speaks in the first person. The Symposium is narrated by Apollodorus, a Socratic disciple, apparently to Glaucon. Apollodorus assures his listener that he

2265-513: A perfectly normal name, and "the common practice of naming a son after his grandfather was reserved for the eldest son", not Plato. According to Debra Nails, Plato's grandfather was the Aristocles who was archon in 605/4. Plato was born in Athens or Aegina , between 428 and 423 BC. He was a member of an aristocratic and influential family. His father was Ariston, who may have been

2416-439: A popular poem by Aratus based on the work by Eudoxus . Hipparchus also made a list of his major works that apparently mentioned about fourteen books, but which is only known from references by later authors. His famous star catalog was incorporated into the one by Ptolemy and may be almost perfectly reconstructed by subtraction of two and two-thirds degrees from the longitudes of Ptolemy's stars . The first trigonometric table

2567-505: A simpler sexagesimal system dividing a circle into 60 parts. Hipparchus also adopted the Babylonian astronomical cubit unit ( Akkadian ammatu , Greek πῆχυς pēchys ) that was equivalent to 2° or 2.5° ('large cubit'). Hipparchus probably compiled a list of Babylonian astronomical observations; Gerald J. Toomer , a historian of astronomy, has suggested that Ptolemy's knowledge of eclipse records and other Babylonian observations in

2718-540: A table giving the daily motion of the Moon according to the date within a long period. However, the Greeks preferred to think in geometrical models of the sky. At the end of the third century BC, Apollonius of Perga had proposed two models for lunar and planetary motion: Apollonius demonstrated that these two models were in fact mathematically equivalent. However, all this was theory and had not been put to practice. Hipparchus

2869-525: A tight range of only approximately ± 1 ⁄ 2 hour, guaranteeing (after division by 4,267) an estimate of the synodic month correct to one part in order of magnitude 10 million. Hipparchus could confirm his computations by comparing eclipses from his own time (presumably 27 January 141 BC and 26 November 139 BC according to Toomer ) with eclipses from Babylonian records 345 years earlier ( Almagest IV.2 ). Later al-Biruni ( Qanun VII.2.II) and Copernicus ( de revolutionibus IV.4) noted that

3020-453: A triangle formed by the two places and the Moon, and from simple geometry was able to establish a distance of the Moon, expressed in Earth radii. Because the eclipse occurred in the morning, the Moon was not in the meridian , and it has been proposed that as a consequence the distance found by Hipparchus was a lower limit. In any case, according to Pappus, Hipparchus found that the least distance

3171-419: A tripartite class structure corresponding to the appetite/spirit/reason structure of the individual soul. The appetite/spirit/reason are analogous to the castes of society. According to Socrates, a state made up of different kinds of souls will, overall, decline from an aristocracy (rule by the best) to a timocracy (rule by the honourable), then to an oligarchy (rule by the few), then to a democracy (rule by

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3322-485: Is 71 (from this eclipse), and the greatest 83 Earth radii. In the second book, Hipparchus starts from the opposite extreme assumption: he assigns a (minimum) distance to the Sun of 490 Earth radii. This would correspond to a parallax of 7′, which is apparently the greatest parallax that Hipparchus thought would not be noticed (for comparison: the typical resolution of the human eye is about 2′; Tycho Brahe made naked eye observation with an accuracy down to 1′). In this case,

3473-815: Is already implicitly known, or at exposing the contradictions and muddles of an opponent's position." Karl Popper , on the other hand, claims that dialectic is the art of intuition for "visualising the divine originals, the Forms or Ideas, of unveiling the Great Mystery behind the common man's everyday world of appearances". During the early Renaissance, the Greek language and, along with it, Plato's texts were reintroduced to Western Europe by Byzantine scholars. Some 250 known manuscripts of Plato survive. In September or October 1484 Filippo Valori and Francesco Berlinghieri printed 1025 copies of Ficino's translation, using

3624-400: Is also close to an integer number of years (4,267 moons : 4,573 anomalistic periods : 4,630.53 nodal periods : 4,611.98 lunar orbits : 344.996 years : 344.982 solar orbits : 126,007.003 days : 126,351.985 rotations). What was so exceptional and useful about the cycle was that all 345-year-interval eclipse pairs occur slightly more than 126,007 days apart within

3775-482: Is also referenced by Jewish philosopher and Talmudic scholar Maimonides in his The Guide for the Perplexed . The works of Plato were again revived at the times of Islamic Golden ages with other Greek contents through their translation from Greek to Arabic. Neoplatonism was revived from its founding father, Plotinus. Neoplatonism, a philosophical current that permeated Islamic scholarship, accentuated one facet of

3926-502: Is by no means universally accepted, though Plato's works are still often characterized as falling at least roughly into three groups stylistically. Plato's unwritten doctrines are, according to some ancient sources, the most fundamental metaphysical teaching of Plato, which he disclosed only orally, and some say only to his most trusted fellows, and which he may have kept secret from the public, although many modern scholars doubt these claims. A reason for not revealing it to everyone

4077-401: Is consistent with 94 + 1 ⁄ 4 and 92 + 1 ⁄ 2 days, an improvement on the results ( 94 + 1 ⁄ 2 and 92 + 1 ⁄ 2 days) attributed to Hipparchus by Ptolemy. Ptolemy made no change three centuries later, and expressed lengths for the autumn and winter seasons which were already implicit (as shown, e.g., by A. Aaboe ). Hipparchus also undertook to find

4228-512: Is inconsistent with a premise of the Sun moving around the Earth in a circle at uniform speed. Hipparchus's solution was to place the Earth not at the center of the Sun's motion, but at some distance from the center. This model described the apparent motion of the Sun fairly well. It is known today that the planets , including the Earth, move in approximate ellipses around the Sun, but this was not discovered until Johannes Kepler published his first two laws of planetary motion in 1609. The value for

4379-478: Is none the less possible, however, to discern that the perfect number of time brings to completion the perfect year at that moment when the relative speeds of all eight periods have been completed together and, measured by the circle of the Same that moves uniformly, have achieved their consummation." In De Natura Deorum , Cicero wrote On the diverse motions of the planets the mathematicians have based what they call

4530-502: Is partially discussed in Phaedrus where Plato criticizes the written transmission of knowledge as faulty, favouring instead the spoken logos : "he who has knowledge of the just and the good and beautiful ... will not, when in earnest, write them in ink, sowing them through a pen with words, which cannot defend themselves by argument and cannot teach the truth effectually." It is, however, said that Plato once disclosed this knowledge to

4681-411: Is post-Hipparchus so the direction of transmission is not settled by the tablets. Hipparchus was recognized as the first mathematician known to have possessed a trigonometric table , which he needed when computing the eccentricity of the orbits of the Moon and Sun. He tabulated values for the chord function, which for a central angle in a circle gives the length of the straight line segment between

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4832-412: Is recounting the story, which took place when he himself was an infant, not from his own memory, but as remembered by Aristodemus, who told him the story years ago. The Theaetetus is also a peculiar case: a dialogue in dramatic form embedded within another dialogue in dramatic form. Some scholars take this as an indication that Plato had by this date wearied of the narrated form. In most of the dialogues,

4983-666: Is sometimes called the "father of astronomy", a title conferred on him by Jean Baptiste Joseph Delambre in 1817. Hipparchus was born in Nicaea ( ‹See Tfd› Greek : Νίκαια ), in Bithynia . The exact dates of his life are not known, but Ptolemy attributes astronomical observations to him in the period from 147 to 127 BC, and some of these are stated as made in Rhodes ; earlier observations since 162 BC might also have been made by him. His birth date ( c.  190  BC)

5134-414: Is the theory of forms (or ideas) , which has been interpreted as advancing a solution to what is now known as the problem of universals . He was decisively influenced by the pre-Socratic thinkers Pythagoras , Heraclitus , and Parmenides , although much of what is known about them is derived from Plato himself. Along with his teacher Socrates , and Aristotle , his student, Plato is a central figure in

5285-433: Is the Earth's orbital motion about the Sun which causes this apparent motion to occur. The Earth's axis of rotation is not set perpendicular to this plane but at a present angle of 23.5 degrees to the perpendicular. The alignment of the axis is maintained throughout the year so that the point of sky above the north or south poles remains unchanged throughout the Earth's annual rotation around the Sun. A slow conical motion of

5436-496: Is the first Greek credited with discovering axial precession roughly two hundred years after Plato's death (see below). Cicero (1st century BC) followed Plato in defining the Great Year as a combination of solar, lunar and planetary cycles. Plato's description of the perfect year is found in his dialogue Timaeus : And so people are all but ignorant of the fact that time really is the wanderings of these bodies, bewilderingly numerous as they are and astonishingly variegated. It

5587-532: Is the first astronomer known to attempt to determine the relative proportions and actual sizes of these orbits. Hipparchus devised a geometrical method to find the parameters from three positions of the Moon at particular phases of its anomaly. In fact, he did this separately for the eccentric and the epicycle model. Ptolemy describes the details in the Almagest IV.11. Hipparchus used two sets of three lunar eclipse observations that he carefully selected to satisfy

5738-482: Is virtue. In the Republic , Plato poses the question, "What is justice?" and by examining both individual justice and the justice that informs societies, Plato is able not only to inform metaphysics, but also ethics and politics with the question: "What is the basis of moral and social obligation?" Plato's well-known answer rests upon the fundamental responsibility to seek wisdom, wisdom which leads to an understanding of

5889-485: Is visible simultaneously on half of the Earth, and the difference in longitude between places can be computed from the difference in local time when the eclipse is observed. His approach would give accurate results if it were correctly carried out but the limitations of timekeeping accuracy in his era made this method impractical. Late in his career (possibly about 135 BC) Hipparchus compiled his star catalog. Scholars have been searching for it for centuries. In 2022, it

6040-535: The Gorgias and his ambivalence toward rhetoric expressed in the Phaedrus . But other contemporary researchers contest the idea that Plato despised rhetoric and instead view his dialogues as a dramatization of complex rhetorical principles. Plato made abundant use of mythological narratives in his own work; It is generally agreed that the main purpose for Plato in using myths was didactic. He considered that only

6191-538: The Laws features Socrates, although many dialogues, including the Timaeus and Statesman , feature him speaking only rarely. Leo Strauss notes that Socrates' reputation for irony casts doubt on whether Plato's Socrates is expressing sincere beliefs. Xenophon 's Memorabilia and Aristophanes 's The Clouds seem to present a somewhat different portrait of Socrates from the one Plato paints. Aristotle attributes

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6342-701: The Republic was prototypically totalitarian ; this has been disputed. Edmund Gettier famously demonstrated the Gettier problem for the justified true belief account of knowledge. That the modern theory of justified true belief as knowledge, which Gettier addresses, is equivalent to Plato's is, however, accepted only by some scholars but rejected by others. Primary sources (Greek and Roman) Secondary sources Hipparchus Hipparchus ( / h ɪ ˈ p ɑːr k ə s / ; Greek : Ἵππαρχος , Hípparkhos ; c.  190  – c.  120  BC)

6493-614: The Academy . It was located in Athens, on a plot of land in the Grove of Hecademus or Academus , named after an Attic hero in Greek mythology . The Academy operated until it was destroyed by Sulla in 84 BC. Many philosophers studied at the Academy, the most prominent being Aristotle. According to Diogenes Laërtius , throughout his later life, Plato became entangled with the politics of

6644-474: The Almagest came from a list made by Hipparchus. Hipparchus's use of Babylonian sources has always been known in a general way, because of Ptolemy's statements, but the only text by Hipparchus that survives does not provide sufficient information to decide whether Hipparchus's knowledge (such as his usage of the units cubit and finger, degrees and minutes, or the concept of hour stars) was based on Babylonian practice. However, Franz Xaver Kugler demonstrated that

6795-544: The Almagest . Some claim the table of Hipparchus may have survived in astronomical treatises in India, such as the Surya Siddhanta . Trigonometry was a significant innovation, because it allowed Greek astronomers to solve any triangle, and made it possible to make quantitative astronomical models and predictions using their preferred geometric techniques. Hipparchus must have used a better approximation for π than

6946-694: The Classical period who is considered a foundational thinker in Western philosophy and an innovator of the written dialogue and dialectic forms. He raised problems for what became all the major areas of both theoretical philosophy and practical philosophy , and was the founder of the Platonic Academy , a philosophical school in Athens where Plato taught the doctrines that would later become known as Platonism . Plato's most famous contribution

7097-522: The Herculaneum papyri , corroborates the claim that Plato was named for his "broad forehead". Seneca the Younger , writing hundreds of years after Plato's death, writes "His very name was given him because of his broad chest." According to the traditional story, Plato was originally named after his paternal grandfather, supposedly called Aristocles; the name "Plato" was only used as a nickname; and

7248-531: The Meno , Socrates uses a geometrical example to expound Plato's view that knowledge in this latter sense is acquired by recollection. Socrates elicits a fact concerning a geometrical construction from a slave boy, who could not have otherwise known the fact (due to the slave boy's lack of education). The knowledge must be of, Socrates concludes, an eternal, non-perceptible Form. Plato also discusses several aspects of epistemology . In several dialogues, Socrates inverts

7399-452: The Phaedo and Timaeus ). Scholars debate whether he intends the theory to be literally true, however. He uses this idea of reincarnation to introduce the concept that knowledge is a matter of recollection of things acquainted with before one is born, and not of observation or study. Keeping with the theme of admitting his own ignorance, Socrates regularly complains of his forgetfulness. In

7550-665: The Renaissance , George Gemistos Plethon brought Plato's original writings to Florence from Constantinople in the century of its fall. Many of the greatest early modern scientists and artists who broke with Scholasticism , with the support of the Plato-inspired Lorenzo (grandson of Cosimo), saw Plato's philosophy as the basis for progress in the arts and sciences. The 17th century Cambridge Platonists , sought to reconcile Plato's more problematic beliefs, such as metempsychosis and polyamory, with Christianity. By

7701-773: The Scholastic philosophers referred to Aristotle as "the Philosopher". The only Platonic work known to western scholarship was Timaeus , until translations were made after the fall of Constantinople , which occurred during 1453. However, the study of Plato continued in the Byzantine Empire , the Caliphates during the Islamic Golden Age , and Spain during the Golden age of Jewish culture . Plato

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7852-572: The eccentricity attributed to Hipparchus by Ptolemy is that the offset is 1 ⁄ 24 of the radius of the orbit (which is a little too large), and the direction of the apogee would be at longitude 65.5° from the vernal equinox . Hipparchus may also have used other sets of observations, which would lead to different values. One of his two eclipse trios' solar longitudes are consistent with his having initially adopted inaccurate lengths for spring and summer of 95 + 3 ⁄ 4 and 91 + 1 ⁄ 4 days. His other triplet of solar positions

8003-627: The justified true belief definition in the Theaetetus , concluding that justification (or an "account") would require knowledge of difference , meaning that the definition of knowledge is circular . In the Sophist , Statesman , Republic , Timaeus , and the Parmenides , Plato associates knowledge with the apprehension of unchanging Forms and their relationships to one another (which he calls "expertise" in dialectic), including through

8154-457: The metaphysical tradition that strongly influenced Plato and continues today. Heraclitus viewed all things as continuously changing , that one cannot "step into the same river twice" due to the ever-changing waters flowing through it, and all things exist as a contraposition of opposites. According to Diogenes Laërtius, Plato received these ideas through Heraclitus' disciple Cratylus . Parmenides adopted an altogether contrary vision, arguing for

8305-435: The "twin pillars of Platonism" as the theory of Forms, on the one hand, and, on the other hand, the doctrine of immortality of the soul. In the dialogues Socrates regularly asks for the meaning of a general term (e. g. justice, truth, beauty), and criticizes those who instead give him particular examples, rather than the quality shared by all examples. "Platonism" and its theory of Forms (also known as 'theory of Ideas') denies

8456-464: The 19th century, Plato's reputation was restored, and at least on par with Aristotle's. Plato's influence has been especially strong in mathematics and the sciences. Plato's resurgence further inspired some of the greatest advances in logic since Aristotle, primarily through Gottlob Frege . Albert Einstein suggested that the scientist who takes philosophy seriously would have to avoid systematization and take on many different roles, and possibly appear as

8607-571: The 1st century AD: Axiochus , Definitions , Demodocus , Epigrams , Eryxias , Halcyon , On Justice , On Virtue , Sisyphus . No one knows the exact order Plato's dialogues were written in, nor the extent to which some might have been later revised and rewritten. The works are usually grouped into Early (sometimes by some into Transitional ), Middle , and Late period; The following represents one relatively common division amongst developmentalist scholars. Whereas those classified as "early dialogues" often conclude in aporia ,

8758-482: The 4th century BC and Timocharis and Aristillus in the 3rd century BC already divided the ecliptic in 360 parts (our degrees , Greek: moira) of 60 arcminutes and Hipparchus continued this tradition. It was only in Hipparchus's time (2nd century BC) when this division was introduced (probably by Hipparchus's contemporary Hypsikles) for all circles in mathematics. Eratosthenes (3rd century BC), in contrast, used

8909-407: The Earth's polar axis about its normal to the plane of the ecliptic is caused by the attractive force of the other heavenly bodies on the equatorial protuberance of the Earth. A similar conical motion can also be observed in a gyroscope that is subjected to lateral forces. The resultant motion of the Earth's axis is called general precession and the equinox points in the ecliptic move westward along

9060-460: The European philosophical tradition is that it consists of a series of footnotes to Plato." There is a traditional story that Plato ( ‹See Tfd› Greek : Πλάτων , Plátōn , from Ancient Greek : πλατύς , romanized :  platys , lit.   'broad') is a nickname . According to Diogenes Laërtius, writing hundreds of years after Plato's death, his birth name

9211-755: The Form of the Good. Plato views "The Good" as the supreme Form, somehow existing even "beyond being". In this manner, justice is obtained when knowledge of how to fulfill one's moral and political function in society is put into practice. The dialogues also discuss politics. Some of Plato's most famous doctrines are contained in the Republic as well as in the Laws and the Statesman . Because these opinions are not spoken directly by Plato and vary between dialogues, they cannot be straightforwardly assumed as representing Plato's own views. Socrates asserts that societies have

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9362-564: The Forms are the causes of everything else, he [i.e. Plato] supposed that their elements are the elements of all things. Accordingly, the material principle is the Great and Small [i.e. the Dyad], and the essence is the One ( τὸ ἕν ), since the numbers are derived from the Great and Small by participation in the One". "From this account it is clear that he only employed two causes: that of the essence, and

9513-626: The Forms were the truths of geometry , such as the Pythagorean theorem . The theory of Forms is first introduced in the Phaedo dialogue (also known as On the Soul ), wherein Socrates disputes the pluralism of Anaxagoras , then the most popular response to Heraclitus and Parmenides. For Plato, as was characteristic of ancient Greek philosophy, the soul was that which gave life. Plato advocates

9664-533: The Geography of Eratosthenes"). It is known to us from Strabo of Amaseia, who in his turn criticised Hipparchus in his own Geographia . Hipparchus apparently made many detailed corrections to the locations and distances mentioned by Eratosthenes. It seems he did not introduce many improvements in methods, but he did propose a means to determine the geographical longitudes of different cities at lunar eclipses (Strabo Geographia 1 January 2012). A lunar eclipse

9815-499: The Great Year, "which is completed when the sun, moon and five planets having all finished their courses have returned to the same positions relative to one another. The length of this period is hotly debated, but it must necessarily be a fixed and definite time." Macrobius (early fifth century AD) in his commentary on Cicero's Somnium Scipionis states that 'the philosophers' reckon the Great Year as 15,000 years. Censorinus (3rd century AD) wrote that Aristarchus of Samos reckoned

9966-549: The Greek. Prediction of a solar eclipse, i.e., exactly when and where it will be visible, requires a solid lunar theory and proper treatment of the lunar parallax. Hipparchus must have been the first to be able to do this. A rigorous treatment requires spherical trigonometry , thus those who remain certain that Hipparchus lacked it must speculate that he may have made do with planar approximations. He may have discussed these things in Perí tēs katá plátos mēniaías tēs selēnēs kinēseōs ("On

10117-584: The Hellespont and are thought by many to be more likely possibilities for the eclipse Hipparchus used for his computations.) Ptolemy later measured the lunar parallax directly ( Almagest V.13), and used the second method of Hipparchus with lunar eclipses to compute the distance of the Sun ( Almagest V.15). He criticizes Hipparchus for making contradictory assumptions, and obtaining conflicting results ( Almagest V.11): but apparently he failed to understand Hipparchus's strategy to establish limits consistent with

10268-542: The Islamic context, Neoplatonism facilitated the integration of Platonic philosophy with mystical Islamic thought, fostering a synthesis of ancient philosophical wisdom and religious insight. Inspired by Plato's Republic, Al-Farabi extended his inquiry beyond mere political theory, proposing an ideal city governed by philosopher-kings . Many of these commentaries on Plato were translated from Arabic into Latin and as such influenced Medieval scholastic philosophers. During

10419-409: The Moon eclipsed while apparently it was not in exact opposition to the Sun. Parallax lowers the altitude of the luminaries; refraction raises them, and from a high point of view the horizon is lowered. Hipparchus and his predecessors used various instruments for astronomical calculations and observations, such as the gnomon , the astrolabe , and the armillary sphere . Hipparchus is credited with

10570-458: The Moon's equation of the center in the Hipparchan model.) Before Hipparchus, Meton , Euctemon , and their pupils at Athens had made a solstice observation (i.e., timed the moment of the summer solstice ) on 27 June 432 BC ( proleptic Julian calendar ). Aristarchus of Samos is said to have done so in 280 BC, and Hipparchus also had an observation by Archimedes . He observed

10721-451: The Qur’anic conception of God—the transcendent—while seemingly neglecting another—the creative. This philosophical tradition, introduced by Al-Farabi and subsequently elaborated upon by figures such as Avicenna , postulated that all phenomena emanated from the divine source. It functioned as a conduit, bridging the transcendental nature of the divine with the tangible reality of creation. In

10872-576: The Sun is on the equator (i.e., in one of the equinoctial points on the ecliptic ), but the shadow falls above or below the opposite side of the ring when the Sun is south or north of the equator. Ptolemy quotes (in Almagest III.1 (H195)) a description by Hipparchus of an equatorial ring in Alexandria; a little further he describes two such instruments present in Alexandria in his own time. Hipparchus applied his knowledge of spherical angles to

11023-448: The apparent diameter of the Sun and Moon. Pappus of Alexandria described it (in his commentary on the Almagest of that chapter), as did Proclus ( Hypotyposis IV). It was a four-foot rod with a scale, a sighting hole at one end, and a wedge that could be moved along the rod to exactly obscure the disk of Sun or Moon. Hipparchus also observed solar equinoxes , which may be done with an equatorial ring : its shadow falls on itself when

11174-407: The apparent diameters of the Sun and Moon with his diopter . Like others before and after him, he found that the Moon's size varies as it moves on its (eccentric) orbit, but he found no perceptible variation in the apparent diameter of the Sun. He found that at the mean distance of the Moon, the Sun and Moon had the same apparent diameter; at that distance, the Moon's diameter fits 650 times into

11325-486: The causation of good and of evil". The most important aspect of this interpretation of Plato's metaphysics is the continuity between his teaching and the Neoplatonic interpretation of Plotinus or Ficino which has been considered erroneous by many but may in fact have been directly influenced by oral transmission of Plato's doctrine. A modern scholar who recognized the importance of the unwritten doctrine of Plato

11476-417: The cause of precession and established the rate of precession at 1 degree per 72 years, very close to the best value measured today, thus demonstrating the magnitude of the error in the earlier value of 1 degree per century. Plato Plato ( / ˈ p l eɪ t oʊ / PLAY -toe ; Greek : Πλάτων, Plátōn , born c.  428-423 BC, died 348 BC), was an ancient Greek philosopher of

11627-413: The center of the Earth, but the observer is at the surface—the Moon, Earth and observer form a triangle with a sharp angle that changes all the time. From the size of this parallax, the distance of the Moon as measured in Earth radii can be determined. For the Sun however, there was no observable parallax (we now know that it is about 8.8", several times smaller than the resolution of the unaided eye). In

11778-467: The change in the length of the day (see ΔT ) we estimate that the error in the assumed length of the synodic month was less than 0.2 second in the fourth century BC and less than 0.1 second in Hipparchus's time. It had been known for a long time that the motion of the Moon is not uniform: its speed varies. This is called its anomaly and it repeats with its own period; the anomalistic month . The Chaldeans took account of this arithmetically, and used

11929-491: The character of a writer were attributed to that writer even when the actual author was unknown. The works taken as genuine in antiquity but are now doubted by at least some modern scholars are: Alcibiades I (*), Alcibiades II (‡), Clitophon (*), Epinomis (‡), Letters (*), Hipparchus (‡), Menexenus (*), Minos (‡), Lovers (‡), Theages (‡) The following works were transmitted under Plato's name in antiquity, but were already considered spurious by

12080-466: The chords for angles with increments of 7.5°. In modern terms, the chord subtended by a central angle in a circle of given radius R equals R times twice the sine of half of the angle, i.e.: The now-lost work in which Hipparchus is said to have developed his chord table, is called Tōn en kuklōi eutheiōn ( Of Lines Inside a Circle ) in Theon of Alexandria 's fourth-century commentary on section I.10 of

12231-403: The circle, i.e., the mean apparent diameters are 360 ⁄ 650 = 0°33′14″. Like others before and after him, he also noticed that the Moon has a noticeable parallax , i.e., that it appears displaced from its calculated position (compared to the Sun or stars ), and the difference is greater when closer to the horizon. He knew that this is because in the then-current models the Moon circles

12382-420: The city of Syracuse , where he attempted to replace the tyrant Dionysius , with Dionysius's brother-in-law, Dion of Syracuse , whom Plato had recruited as one of his followers, but the tyrant himself turned against Plato. Plato almost faced death, but was sold into slavery. Anniceris , a Cyrenaic philosopher, bought Plato's freedom for twenty minas , and sent him home. Philodemus however states that Plato

12533-484: The common man's intuition about what is knowable and what is real. Reality is unavailable to those who use their senses. Socrates says that he who sees with his eyes is blind. While most people take the objects of their senses to be real if anything is, Socrates is contemptuous of people who think that something has to be graspable in the hands to be real. In the Theaetetus , he says such people are eu amousoi (εὖ ἄμουσοι), an expression that means literally, "happily without

12684-547: The complete written philosophical work of Plato, based on the first century AD arrangement of Thrasyllus of Mendes . The modern standard complete English edition is the 1997 Hackett Plato: Complete Works , edited by John M. Cooper. Thirty-five dialogues and thirteen letters (the Epistles ) have traditionally been ascribed to Plato, though modern scholarship doubts the authenticity of at least some of these. Jowett mentions in his Appendix to Menexenus, that works which bore

12835-410: The concept of form as distinct from matter, and that the physical world is an imitation of an eternal mathematical world. These ideas were very influential on Heraclitus, Parmenides and Plato. The two philosophers Heraclitus and Parmenides , influenced by earlier pre-Socratic Greek philosophers such as Pythagoras and Xenophanes , departed from mythological explanations for the universe and began

12986-476: The distances and sizes of the Sun and the Moon, in the now-lost work On Sizes and Distances ( ‹See Tfd› Greek : Περὶ μεγεθῶν καὶ ἀποστημάτων Peri megethon kai apostematon ). His work is mentioned in Ptolemy's Almagest V.11, and in a commentary thereon by Pappus ; Theon of Smyrna (2nd century) also mentions the work, under the title On Sizes and Distances of the Sun and Moon . Hipparchus measured

13137-420: The ecliptic at the rate of about 50.3 seconds of arc per year as a result. In 25,772 years, the points are once again at the same point in the sky where observations began. In addition the tilt, or obliquity , of the Earth's axis is not constant but changes in a cycle of its own. During a cycle that averages about 40,000 years, the tilt of the axis varies between 22.1 and 24.5 degrees. Plato (c. 360 BC) used

13288-459: The famous Euthyphro dilemma in the dialogue of the same name: "Is the pious ( τὸ ὅσιον ) loved by the gods because it is pious, or is it pious because it is loved by the gods?" ( 10a ) In the Protagoras dialogue it is argued through Socrates that virtue is innate and cannot be learned, that no one does bad on purpose, and to know what is good results in doing what is good; that knowledge

13439-573: The first book, Hipparchus assumes that the parallax of the Sun is 0, as if it is at infinite distance. He then analyzed a solar eclipse, which Toomer presumes to be the eclipse of 14 March 190 BC. It was total in the region of the Hellespont (and in his birthplace, Nicaea); at the time Toomer proposes the Romans were preparing for war with Antiochus III in the area, and the eclipse is mentioned by Livy in his Ab Urbe Condita Libri VIII.2. It

13590-481: The first century; Ptolemy's second-century Almagest ; and additional references to him in the fourth century by Pappus and Theon of Alexandria in their commentaries on the Almagest . Hipparchus's only preserved work is Commentary on the Phaenomena of Eudoxus and Aratus ( ‹See Tfd› Greek : Τῶν Ἀράτου καὶ Εὐδόξου φαινομένων ἐξήγησις ). This is a highly critical commentary in the form of two books on

13741-419: The first method is very sensitive to the accuracy of the observations and parameters. (In fact, modern calculations show that the size of the 189 BC solar eclipse at Alexandria must have been closer to 9 ⁄ 10 ths and not the reported 4 ⁄ 5 ths, a fraction more closely matched by the degree of totality at Alexandria of eclipses occurring in 310 and 129 BC which were also nearly total in

13892-429: The first surviving text discussing it is by Menelaus of Alexandria in the first century, who now, on that basis, commonly is credited with its discovery. (Previous to the finding of the proofs of Menelaus a century ago, Ptolemy was credited with the invention of spherical trigonometry.) Ptolemy later used spherical trigonometry to compute things such as the rising and setting points of the ecliptic , or to take account of

14043-400: The first to develop a reliable method to predict solar eclipses . His other reputed achievements include the discovery and measurement of Earth's precession, the compilation of the first known comprehensive star catalog from the western world, and possibly the invention of the astrolabe , as well as of the armillary sphere that he may have used in creating the star catalogue. Hipparchus

14194-440: The geometry of book 2 it follows that the Sun is at 2,550 Earth radii, and the mean distance of the Moon is 60 + 1 ⁄ 2 radii. Similarly, Cleomedes quotes Hipparchus for the sizes of the Sun and Earth as 1050:1; this leads to a mean lunar distance of 61 radii. Apparently Hipparchus later refined his computations, and derived accurate single values that he could use for predictions of solar eclipses. See Toomer (1974) for

14345-544: The good itself" along with many fundamentals of Christian morality, which he interpreted as "Platonism for the masses" in Beyond Good and Evil (1886). Martin Heidegger argued against Plato's alleged obfuscation of Being in his incomplete tome, Being and Time (1927). Karl Popper argued in the first volume of The Open Society and Its Enemies (1945) that Plato's proposal for a " utopian " political regime in

14496-734: The greatest overall astronomer of antiquity . He was the first whose quantitative and accurate models for the motion of the Sun and Moon survive. For this he certainly made use of the observations and perhaps the mathematical techniques accumulated over centuries by the Babylonians and by Meton of Athens (fifth century BC), Timocharis , Aristyllus , Aristarchus of Samos , and Eratosthenes , among others. He developed trigonometry and constructed trigonometric tables , and he solved several problems of spherical trigonometry . With his solar and lunar theories and his trigonometry, he may have been

14647-470: The history of Western philosophy . Plato's entire body of work is believed to have survived intact for over 2,400 years—unlike that of nearly all of his contemporaries. Although their popularity has fluctuated, they have consistently been read and studied through the ages. Through Neoplatonism , he also greatly influenced both Christian and Islamic philosophy . In modern times, Alfred North Whitehead famously said: "the safest general characterization of

14798-768: The idea of a changeless, eternal universe and the view that change is an illusion. Plato's most self-critical dialogue is the Parmenides , which features Parmenides and his student Zeno , which criticizes Plato's own metaphysical theories. Plato's Sophist dialogue includes an Eleatic stranger. These ideas about change and permanence, or becoming and Being, influenced Plato in formulating his theory of Forms. In Plato's dialogues, Socrates and his company of disputants had something to say on many subjects, including several aspects of metaphysics . These include religion and science, human nature, love, and sexuality. More than one dialogue contrasts perception and reality , nature and custom, and body and soul. Francis Cornford identified

14949-547: The influence of Pythagoras , or in a broader sense, the Pythagoreans, such as Archytas also appears to have been significant. Aristotle and Cicero both claimed that the philosophy of Plato closely followed the teachings of the Pythagoreans . According to R. M. Hare , this influence consists of three points: Pythagoras held that all things are number, and the cosmos comes from numerical principles. He introduced

15100-447: The invention or improvement of several astronomical instruments, which were used for a long time for naked-eye observations. According to Synesius of Ptolemais (4th century) he made the first astrolabion : this may have been an armillary sphere (which Ptolemy however says he constructed, in Almagest V.1); or the predecessor of the planar instrument called astrolabe (also mentioned by Theon of Alexandria ). With an astrolabe Hipparchus

15251-449: The large total lunar eclipse of 26 November 139 BC, when over a clean sea horizon as seen from Rhodes, the Moon was eclipsed in the northwest just after the Sun rose in the southeast. This would be the second eclipse of the 345-year interval that Hipparchus used to verify the traditional Babylonian periods: this puts a late date to the development of Hipparchus's lunar theory. We do not know what "exact reason" Hipparchus found for seeing

15402-642: The lunar parallax . If he did not use spherical trigonometry, Hipparchus may have used a globe for these tasks, reading values off coordinate grids drawn on it, or he may have made approximations from planar geometry, or perhaps used arithmetical approximations developed by the Chaldeans. Hipparchus also studied the motion of the Moon and confirmed the accurate values for two periods of its motion that Chaldean astronomers are widely presumed to have possessed before him. The traditional value (from Babylonian System B) for

15553-556: The material cause; for the Forms are the cause of the essence in everything else, and the One is the cause of it in the Forms. He also tells us what the material substrate is of which the Forms are predicated in the case of sensible things, and the One in that of the Forms ;– that it is this the duality (the Dyad, ἡ δυάς ), the Great and Small ( τὸ μέγα καὶ τὸ μικρόν ). Further, he assigned to these two elements respectively

15704-479: The mean synodic month is 29 days; 31,50,8,20 (sexagesimal) = 29.5305941... days. Expressed as 29 days + 12 hours + ⁠ 793 / 1080 ⁠  hours this value has been used later in the Hebrew calendar . The Chaldeans also knew that 251 synodic months ≈ 269 anomalistic months . Hipparchus used the multiple of this period by a factor of 17, because that interval is also an eclipse period, and

15855-515: The monthly motion of the Moon in latitude"), a work mentioned in the Suda . Pliny also remarks that "he also discovered for what exact reason, although the shadow causing the eclipse must from sunrise onward be below the earth, it happened once in the past that the Moon was eclipsed in the west while both luminaries were visible above the earth" (translation H. Rackham (1938), Loeb Classical Library 330 p. 207). Toomer argued that this must refer to

16006-405: The muses". In other words, such people are willingly ignorant, living without divine inspiration and access to higher insights about reality. Many have interpreted Plato as stating – even having been the first to write – that knowledge is justified true belief , an influential view that informed future developments in epistemology. Plato also identified problems with

16157-539: The observation made on Alexandria 's large public equatorial ring that same day (at 1 hour before noon). Ptolemy claims his solar observations were on a transit instrument set in the meridian. At the end of his career, Hipparchus wrote a book entitled Peri eniausíou megéthous ("On the Length of the Year") regarding his results. The established value for the tropical year , introduced by Callippus in or before 330 BC

16308-419: The observations, rather than a single value for the distance. His results were the best so far: the actual mean distance of the Moon is 60.3 Earth radii, within his limits from Hipparchus's second book. Theon of Smyrna wrote that according to Hipparchus, the Sun is 1,880 times the size of the Earth, and the Earth twenty-seven times the size of the Moon; apparently this refers to volumes , not diameters . From

16459-429: The one given by Archimedes of between 3 + 10 ⁄ 71 (≈ 3.1408) and 3 + 1 ⁄ 7 (≈ 3.1429). Perhaps he had the approximation later used by Ptolemy, sexagesimal 3;08,30 (≈ 3.1417) ( Almagest VI.7). Hipparchus could have constructed his chord table using the Pythagorean theorem and a theorem known to Archimedes. He also might have used the relationship between sides and diagonals of

16610-660: The other hand, if one derives one's account of something by way of the non-sensible Forms, because these Forms are unchanging, so too is the account derived from them. That apprehension of Forms is required for knowledge may be taken to cohere with Plato's theory in the Theaetetus and Meno . Indeed, the apprehension of Forms may be at the base of the account required for justification, in that it offers foundational knowledge which itself needs no account, thereby avoiding an infinite regression . Several dialogues discuss ethics including virtue and vice, pleasure and pain, crime and punishment, and justice and medicine. Socrates presents

16761-510: The other way around is debatable. Hipparchus also gave the value for the sidereal year to be 365 + ⁠ 1 / 4 ⁠ + ⁠ 1 / 144 ⁠ days (= 365.25694... days = 365 days 6 hours 10 min). Another value for the sidereal year that is attributed to Hipparchus (by the physician Galen in the second century AD) is 365 + ⁠ 1 / 4 ⁠ + ⁠ 1 / 288 ⁠ days (= 365.25347... days = 365 days 6 hours 5 min), but this may be

16912-411: The parallax of the Sun decreases (i.e., its distance increases), the minimum limit for the mean distance is 59 Earth radii—exactly the mean distance that Ptolemy later derived. Hipparchus thus had the problematic result that his minimum distance (from book 1) was greater than his maximum mean distance (from book 2). He was intellectually honest about this discrepancy, and probably realized that especially

17063-547: The people), and finally to tyranny (rule by one person, rule by a tyrant). Several dialogues tackle questions about art, including rhetoric and rhapsody. Socrates says that poetry is inspired by the muses , and is not rational. He speaks approvingly of this, and other forms of divine madness (drunkenness, eroticism, and dreaming) in the Phaedrus , and yet in the Republic wants to outlaw Homer's great poetry, and laughter as well. Scholars often view Plato's philosophy as at odds with rhetoric due to his criticisms of rhetoric in

17214-563: The period of 4,267 moons is approximately five minutes longer than the value for the eclipse period that Ptolemy attributes to Hipparchus. However, the timing methods of the Babylonians had an error of no fewer than eight minutes. Modern scholars agree that Hipparchus rounded the eclipse period to the nearest hour, and used it to confirm the validity of the traditional values, rather than to try to derive an improved value from his own observations. From modern ephemerides and taking account of

17365-413: The period of that precession caused by the slow gyration of the Earth's axis. Some time around the middle of the second century BC, the astronomer Hipparchus discovered that the fixed stars as a whole gradually shifted their position in relation to the annually determined locations of the Sun at the equinoxes and solstices... Otto Neugebauer argued that Hipparchus in fact believed that this [36,000 years]

17516-462: The philosopher could not have been named "Plato" because that name does not occur previously in his family line. Modern scholarship tends to reject the "Aristocles" story. Plato always called himself Platon . Platon was a fairly common name (31 instances are known from Athens alone), including people named before Plato was born. Robin Waterfield states that Plato was not a nickname, but

17667-441: The points where the angle intersects the circle. He may have computed this for a circle with a circumference of 21,600 units and a radius (rounded) of 3,438 units; this circle has a unit length for each arcminute along its perimeter. (This was “proven” by Toomer, but he later “cast doubt“ upon his earlier affirmation. Other authors have argued that a circle of radius 3,600 units may instead have been used by Hipparchus. ) He tabulated

17818-403: The positions of Sun and Moon when a solar or lunar eclipse is possible, are explained in Almagest VI.5. Hipparchus apparently made similar calculations. The result that two solar eclipses can occur one month apart is important, because this can not be based on observations: one is visible on the northern and the other on the southern hemisphere—as Pliny indicates—and the latter was inaccessible to

17969-417: The primary speaker is Socrates, who employs a method of questioning which proceeds by a dialogue form called dialectic. The role of dialectic in Plato's thought is contested but there are two main interpretations: a type of reasoning and a method of intuition. Simon Blackburn adopts the first, saying that Plato's dialectic is "the process of eliciting the truth by means of questions aimed at opening out what

18120-586: The printing press  [ it ] at the Dominican convent of San Jacopo di Ripoli  [ it ] . The 1578 edition of Plato's complete works published by Henricus Stephanus ( Henri Estienne ) in Geneva also included parallel Latin translation and running commentary by Joannes Serranus ( Jean de Serres ). It was this edition which established standard Stephanus pagination , still in use today. The text of Plato as received today apparently represents

18271-438: The problem of denoting locations on the Earth's surface. Before him a grid system had been used by Dicaearchus of Messana , but Hipparchus was the first to apply mathematical rigor to the determination of the latitude and longitude of places on the Earth. Hipparchus wrote a critique in three books on the work of the geographer Eratosthenes of Cyrene (3rd century BC), called Pròs tèn Eratosthénous geographían ("Against

18422-420: The processes of collection and division . More explicitly, Plato himself argues in the Timaeus that knowledge is always proportionate to the realm from which it is gained. In other words, if one derives one's account of something experientially, because the world of sense is in flux, the views therein attained will be mere opinions. Meanwhile, opinions are characterized by a lack of necessity and stability. On

18573-613: The public in his lecture On the Good ( Περὶ τἀγαθοῦ ), in which the Good ( τὸ ἀγαθόν ) is identified with the One (the Unity, τὸ ἕν ), the fundamental ontological principle. The first witness who mentions its existence is Aristotle, who in his Physics writes: "It is true, indeed, that the account he gives there [i.e. in Timaeus ] of the participant is different from what he says in his so-called unwritten teachings ( Ancient Greek : ἄγραφα δόγματα , romanized :  agrapha dogmata )." In Metaphysics he writes: "Now since

18724-406: The ratio of the epicycle model ( 3122 + 1 ⁄ 2  : 247 + 1 ⁄ 2 ), which is too small (60 : 4;45 sexagesimal). Ptolemy established a ratio of 60 : 5 + 1 ⁄ 4 . (The maximum angular deviation producible by this geometry is the arcsin of 5 + 1 ⁄ 4 divided by 60, or approximately 5° 1', a figure that is sometimes therefore quoted as the equivalent of

18875-695: The reality of the material world, considering it only an image or copy of the real world. According to this theory of Forms, there are these two kinds of things: the apparent world of material objects grasped by the senses, which constantly changes, and an unchanging and unseen world of Forms, grasped by reason ( λογική ). Plato's Forms represent types of things, as well as properties , patterns, and relations , which are referred to as objects. Just as individual tables, chairs, and cars refer to objects in this world, 'tableness', 'chairness', and 'carness', as well as e.g. justice , truth , and beauty refer to objects in another world. One of Plato's most cited examples for

19026-551: The representative figure for astronomy. It is not certain that the figure is meant to represent him. Previously, Eudoxus of Cnidus in the fourth century BC had described the stars and constellations in two books called Phaenomena and Entropon . Aratus wrote a poem called Phaenomena or Arateia based on Eudoxus's work. Hipparchus wrote a commentary on the Arateia —his only preserved work—which contains many stellar positions and times for rising, culmination, and setting of

19177-687: The requirements. The eccentric model he fitted to these eclipses from his Babylonian eclipse list: 22/23 December 383 BC, 18/19 June 382 BC, and 12/13 December 382 BC. The epicycle model he fitted to lunar eclipse observations made in Alexandria at 22 September 201 BC, 19 March 200 BC, and 11 September 200 BC. These figures are due to the cumbersome unit he used in his chord table and may partly be due to some sloppy rounding and calculation errors by Hipparchus, for which Ptolemy criticised him while also making rounding errors. A simpler alternate reconstruction agrees with all four numbers. Hipparchus found inconsistent results; he later used

19328-509: The second and third centuries, coins were made in his honour in Bithynia that bear his name and show him with a globe . Relatively little of Hipparchus's direct work survives into modern times. Although he wrote at least fourteen books, only his commentary on the popular astronomical poem by Aratus was preserved by later copyists. Most of what is known about Hipparchus comes from Strabo 's Geography and Pliny 's Natural History in

19479-417: The shadow of the Earth is a cone rather than a cylinder as under the first assumption. Hipparchus observed (at lunar eclipses) that at the mean distance of the Moon, the diameter of the shadow cone is 2 + 1 ⁄ 2 lunar diameters. That apparent diameter is, as he had observed, 360 ⁄ 650 degrees. With these values and simple geometry, Hipparchus could determine the mean distance; because it

19630-456: The so-called "middle dialogues" provide more clearly stated positive teachings that are often ascribed to Plato such as the theory of Forms. The remaining dialogues are classified as "late" and are generally agreed to be difficult and challenging pieces of philosophy. It should, however, be kept in mind that many of the positions in the ordering are still highly disputed, and also that the very notion that Plato's dialogues can or should be "ordered"

19781-509: The solstice observation of Meton and his own, there were 297 years spanning 108,478 days; this implies a tropical year of 365.24579... days = 365 days;14,44,51 (sexagesimal; = 365 days + ⁠ 14 / 60 ⁠ + ⁠ 44 / 60 ⁠ + ⁠ 51 / 60 ⁠ ), a year length found on one of the few Babylonian clay tablets which explicitly specifies the System B month. Whether Babylonians knew of Hipparchus's work or

19932-406: The stars — were to be increasingly confused." Ptolemy has even been accused of committing scientific fraud by making up observations that would give the figure of 36,000 years even though the data available to him were good enough to get very near the true figure of 26,000. Josephus (first century AD) refers to a 'Great Year' ( Ancient Greek : μέγας ἐνιαυτός ) of 600 years. God afforded them

20083-513: The stereographic projection is found in Ptolemy 's Planisphere (2nd century AD). Besides geometry, Hipparchus also used arithmetic techniques developed by the Chaldeans . He was one of the first Greek mathematicians to do this and, in this way, expanded the techniques available to astronomers and geographers. There are several indications that Hipparchus knew spherical trigonometry, but

20234-473: The summer solstices in 146 and 135 BC both accurately to a few hours, but observations of the moment of equinox were simpler, and he made twenty during his lifetime. Ptolemy gives an extensive discussion of Hipparchus's work on the length of the year in the Almagest III.1, and quotes many observations that Hipparchus made or used, spanning 162–128 BC, including an equinox timing by Hipparchus (at 24 March 146 BC at dawn) that differs by 5 hours from

20385-468: The synodic and anomalistic periods that Ptolemy attributes to Hipparchus had already been used in Babylonian ephemerides , specifically the collection of texts nowadays called "System B" (sometimes attributed to Kidinnu ). Hipparchus's long draconitic lunar period (5,458 months = 5,923 lunar nodal periods) also appears a few times in Babylonian records . But the only such tablet explicitly dated,

20536-418: The term "Great Year" can be used for any concept of eternal return in the world's mythologies or philosophies . Historian Otto Neugebauer writes: The difficulty with the term "great year" lies in its ambiguity. Almost any period can be found sometime or somewhere honored with this name. The plane of the ecliptic is the plane described by the apparent motion of the Sun against the starry background. It

20687-492: The term "featherless biped", and later ζῷον πολιτικόν ( zōon politikon ), a "political" or "state-building" animal ( Aristotle 's term, based on Plato's Statesman ). Diogenes the Cynic took issue with the former definition, reportedly producing a recently plucked chicken with the exclamation of "Here is Plato’s man!" (variously translated as "Behold, a man!"; "Here is a human!" etc.). Plato never presents himself as

20838-466: The term "perfect year" to describe the return of the celestial bodies ( planets ) and the diurnal rotation of the fixed stars (circle of the Same) to their original positions; there is no evidence he had any knowledge of axial precession . The cycle which Plato describes is one of planetary and astral conjunction, which can be postulated without any awareness of axial precession. Hipparchus (c. 120 BC)

20989-556: Was 365 + 1 ⁄ 4 days. Speculating a Babylonian origin for the Callippic year is difficult to defend, since Babylon did not observe solstices thus the only extant System B year length was based on Greek solstices (see below). Hipparchus's equinox observations gave varying results, but he points out (quoted in Almagest III.1(H195)) that the observation errors by him and his predecessors may have been as large as 1 ⁄ 4 day. He used old solstice observations and determined

21140-406: Was Aristocles ( Ἀριστοκλῆς ), meaning 'best reputation'. "Platon" sounds like "Platus" or "Platos", meaning "broad", and according to Diogenes' sources, Plato gained his nickname either from his wrestling coach, Ariston of Argos, who dubbed him "broad" on account of his chest and shoulders, or he gained it from the breadth of his eloquence, or his wide forehead. Philodemus , in extracts from

21291-495: Was Heinrich Gomperz who described it in his speech during the 7th International Congress of Philosophy in 1930. All the sources related to the ἄγραφα δόγματα have been collected by Konrad Gaiser and published as Testimonia Platonica . Plato's thought is often compared with that of his most famous student, Aristotle , whose reputation during the Western Middle Ages so completely eclipsed that of Plato that

21442-506: Was a Greek astronomer , geographer , and mathematician . He is considered the founder of trigonometry , but is most famous for his incidental discovery of the precession of the equinoxes . Hipparchus was born in Nicaea , Bithynia , and probably died on the island of Rhodes , Greece. He is known to have been a working astronomer between 162 and 127 BC. Hipparchus is considered the greatest ancient astronomical observer and, by some,

21593-572: Was also observed in Alexandria, where the Sun was reported to be obscured 4/5ths by the Moon. Alexandria and Nicaea are on the same meridian. Alexandria is at about 31° North, and the region of the Hellespont about 40° North. (It has been contended that authors like Strabo and Ptolemy had fairly decent values for these geographical positions, so Hipparchus must have known them too. However, Strabo's Hipparchus dependent latitudes for this region are at least 1° too high, and Ptolemy appears to copy them, placing Byzantium 2° high in latitude.) Hipparchus could draw

21744-831: Was announced that a part of it was discovered in a medieval parchment manuscript, Codex Climaci Rescriptus , from Saint Catherine's Monastery in the Sinai Peninsula , Egypt as hidden text ( palimpsest ). Hipparchus also constructed a celestial globe depicting the constellations, based on his observations. His interest in the fixed stars may have been inspired by the observation of a supernova (according to Pliny), or by his discovery of precession, according to Ptolemy, who says that Hipparchus could not reconcile his data with earlier observations made by Timocharis and Aristillus . For more information see Discovery of precession . In Raphael 's painting The School of Athens , Hipparchus may be depicted holding his celestial globe, as

21895-542: Was apparently compiled by Hipparchus, who is consequently now known as "the father of trigonometry". Earlier Greek astronomers and mathematicians were influenced by Babylonian astronomy to some extent, for instance the period relations of the Metonic cycle and Saros cycle may have come from Babylonian sources (see " Babylonian astronomical diaries "). Hipparchus seems to have been the first to exploit Babylonian astronomical knowledge and techniques systematically. Eudoxus in

22046-514: Was buried in the garden of his academy in Athens, close to the sacred shrine of the Muses. In 2024, a scroll found at Herculaneum was deciphered, that confirmed some previous theories. The papyrus says that before death Plato "retained enough lucidity to critique the musician for her lack of rhythm", and that he was buried "in his designated garden in the Academy of Athens". Plato never speaks in his own voice in his dialogues ; every dialogue except

22197-417: Was calculated by Delambre based on clues in his work. Hipparchus must have lived some time after 127 BC because he analyzed and published his observations from that year. Hipparchus obtained information from Alexandria as well as Babylon , but it is not known when or if he visited these places. He is believed to have died on the island of Rhodes, where he seems to have spent most of his later life. In

22348-404: Was computed for a minimum distance of the Sun, it is the maximum mean distance possible for the Moon. With his value for the eccentricity of the orbit, he could compute the least and greatest distances of the Moon too. According to Pappus, he found a least distance of 62, a mean of 67 + 1 ⁄ 3 , and consequently a greatest distance of 72 + 2 ⁄ 3 Earth radii. With this method, as

22499-609: Was sold as a slave as early as in 404 BC, when the Spartans conquered Aegina, or, alternatively, in 399 BC, immediately after the death of Socrates. After Dionysius's death, according to Plato's Seventh Letter , Dion requested Plato return to Syracuse to tutor Dionysius II , who seemed to accept Plato's teachings, but eventually became suspicious of their motives, expelling Dion and holding Plato against his will. Eventually Plato left Syracuse and Dion would return to overthrow Dionysius and rule Syracuse, before being usurped by Callippus ,

22650-405: Was the first to be able to measure the geographical latitude and time by observing fixed stars. Previously this was done at daytime by measuring the shadow cast by a gnomon, by recording the length of the longest day of the year or with the portable instrument known as a scaphe . Ptolemy mentions ( Almagest V.14) that he used a similar instrument as Hipparchus, called dioptra , to measure

22801-492: Was the maximum figure and that he also computed the true rate of one complete precession cycle at just under 26,000 years... It is argued that a confusion between the two originated with the astronomer Ptolemy (c. 170 AD), who "adopted the larger, erroneous, figure, with the result that henceforth the two versions of the Great Year — the Platonic Great Year, defined by the planets, and the precessional, defined by

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