Search Results for Ptolemy


Biographies

  1. Ptolemy biography
    • Claudius Ptolemy .
    • One of the most influential Greek astronomers and geographers of his time, Ptolemy propounded the geocentric theory in a form that prevailed for 1400 years.
    • We shall discuss the arguments below for, depending on which are correct, they portray Ptolemy in very different lights.
    • The arguments of some historians show that Ptolemy was a mathematician of the very top rank, arguments of others show that he was no more than a superb expositor, but far worse, some even claim that he committed a crime against his fellow scientists by betraying the ethics and integrity of his profession.
    • We know very little of Ptolemy's life.
    • In fact the first observation which we can date exactly was made by Ptolemy on 26 March 127 while the last was made on 2 February 141.
    • It was claimed by Theodore Meliteniotes in around 1360 that Ptolemy was born in Hermiou (which is in Upper Egypt rather than Lower Egypt where Alexandria is situated) but since this claim first appears more than one thousand years after Ptolemy lived, it must be treated as relatively unlikely to be true.
    • In fact there is no evidence that Ptolemy was ever anywhere other than Alexandria.
    • His name, Claudius Ptolemy, is of course a mixture of the Greek Egyptian 'Ptolemy' and the Roman 'Claudius'.
    • This would indicate that he was descended from a Greek family living in Egypt and that he was a citizen of Rome, which would be as a result of a Roman emperor giving that 'reward' to one of Ptolemy's ancestors.
    • We do know that Ptolemy used observations made by 'Theon the mathematician', and this was almost certainly Theon of Smyrna who almost certainly was his teacher.
    • Most of Ptolemy's early works are dedicated to Syrus who may have also been one of his teachers in Alexandria, but nothing is known of Syrus.
    • If these facts about Ptolemy's teachers are correct then certainly in Theon he did not have a great scholar, for Theon seems not to have understood in any depth the astronomical work he describes.
    • On the other hand Alexandria had a tradition for scholarship which would mean that even if Ptolemy did not have access to the best teachers, he would have access to the libraries where he would have found the valuable reference material of which he made good use.
    • Ptolemy's major works have survived and we shall discuss them in this article.
    • The Almagest is the earliest of Ptolemy's works and gives in detail the mathematical theory of the motions of the Sun, Moon, and planets.
    • Ptolemy made his most original contribution by presenting details for the motions of each of the planets.
    • Grasshoff writes in [',' G Grasshoff, The history of Ptolemy’s star catalogue (New York, 1990).','8]:- .
    • Ptolemy's "Almagest" shares with Euclid's "Elements" the glory of being the scientific text longest in use.
    • Ptolemy describes himself very clearly what he is attempting to do in writing the work (see for example [',' G J Toomer (trs.), Ptolemy’s Almagest (London, 1984).','15]):- .
    • Ptolemy first of all justifies his description of the universe based on the earth-centred system described by Aristotle.
    • Ptolemy used geometric models to predict the positions of the sun, moon, and planets, using combinations of circular motion known as epicycles.
    • Having set up this model, Ptolemy then goes on to describe the mathematics which he needs in the rest of the work.
    • Ptolemy devised new geometrical proofs and theorems.
    • This occupies the first two of the 13 books of the Almagest and then, quoting again from the introduction, we give Ptolemy's own description of how he intended to develop the rest of the mathematical astronomy in the work (see for example [',' G J Toomer (trs.), Ptolemy’s Almagest (London, 1984).','15]):- .
    • In examining the theory of the sun, Ptolemy compares his own observations of equinoxes with those of Hipparchus and the earlier observations Meton in 432 BC.
    • Since, as Ptolemy himself knew, the accuracy of the rest of his data depended heavily on this value, the fact that the true value is 1/128 of a day less than 3651/4 days did produce errors in the rest of the work.
    • We shall discuss below in more detail the accusations which have been made against Ptolemy, but this illustrates clearly the grounds for these accusations since Ptolemy had to have an error of 28 hours in his observation of the equinox to produce this error, and even given the accuracy that could be expected with ancient instruments and methods, it is essentially unbelievable that he could have made an error of this magnitude.
    • A good discussion of this strange error is contained in the excellent article [',' J P Britton, Models and precision : the quality of Ptolemy’s observations and parameters, in Sources and Studies in the History and Philosophy of Classical Science 1 (New York, 1992).','19].
    • Based on his observations of solstices and equinoxes, Ptolemy found the lengths of the seasons and, based on these, he proposed a simple model for the sun which was a circular motion of uniform angular velocity, but the earth was not at the centre of the circle but at a distance called the eccentricity from this centre.
    • In Books 4 and 5 Ptolemy gives his theory of the moon.
    • Ptolemy also discusses, as Hipparchus had done, the synodic month, that is the time between successive oppositions of the sun and moon.
    • In Book 4 Ptolemy gives Hipparchus's epicycle model for the motion of the moon but he notes, as in fact Hipparchus had done himself, that there are small discrepancies between the model and the observed parameters.
    • Although noting the discrepancies, Hipparchus seems not to have worked out a better model, but Ptolemy does this in Book 5 where the model he gives improves markedly on the one proposed by Hipparchus.
    • An interesting discussion of Ptolemy's theory of the moon is given in [',' P Del Santo and G Strano, Observational evidence and the evolution of Ptolemy’s lunar model, Nuncius Ann.
    • Having given a theory for the motion of the sun and of the moon, Ptolemy was in a position to apply these to obtain a theory of eclipses which he does in Book 6.
    • The next two books deal with the fixed stars and in Book 7 Ptolemy uses his own observations together with those of Hipparchus to justify his belief that the fixed stars always maintain the same positions relative to each other.
    • He wrote (see for example [',' G J Toomer (trs.), Ptolemy’s Almagest (London, 1984).','15]):- .
    • In these two book Ptolemy also discusses precession, the discovery of which he attributes to Hipparchus, but his figure is somewhat in error mainly because of the error in the length of the tropical year which he used.
    • Much of Books 7 and 8 are taken up with Ptolemy's star catalogue containing over one thousand stars.
    • This must be Ptolemy's greatest achievement in terms of an original contribution, since there does not appear to have been any satisfactory theoretical model to explain the rather complicated motions of the five planets before the Almagest.
    • Ptolemy combined the epicycle and eccentric methods to give his model for the motions of the planets.
    • Ptolemy's really clever innovation here was to make the motion of C uniform not about the centre of the circle around which it moves, but around a point called the equant which is symmetrically placed on the opposite side of the centre from the earth.
    • The planetary theory which Ptolemy developed here is a masterpiece.
    • He created a sophisticated mathematical model to fit observational data which before Ptolemy's time was scarce, and the model he produced, although complicated, represents the motions of the planets fairly well.
    • We will return to discuss some of the accusations made against Ptolemy after commenting briefly on his other works.
    • These were not merely lifted from the Almagest however but Ptolemy made numerous improvements in their presentation, ease of use and he even made improvements in the basic parameters to give greater accuracy.
    • We only know details of the Handy Tables through the commentary by Theon of Alexandria but in [',' B van Dalen, On Ptolemy’s table for the equation of time, Centaurus 37 (2) (1994), 97-153.','76] the author shows that care is required since Theon was not fully aware of Ptolemy's procedures.
    • Ptolemy also did what many writers of deep scientific works have done, and still do, in writing a popular account of his results under the title Planetary Hypothesis.
    • Ptolemy does this rather cleverly by replacing the abstract geometrical theories by mechanical ones.
    • Ptolemy also wrote a work on astrology.
    • However, Ptolemy sees it rather differently for he claims that the Almagest allows one to find the positions of the heavenly bodies, while his astrology book he sees as a companion work describing the effects of the heavenly bodies on people's lives.
    • This is discussed in [',' R P Lorch, Ptolemy and Maslama on the transformation of circles into circles in stereographic projection, Arch.
    • In the stereographic projection treated by Ptolemy in the "Planisphaerium" the celestial sphere is mapped onto the plane of the equator by projection from the south pole.
    • Ptolemy does not prove the important property that circles on the sphere become circles on the plane.
    • Ptolemy's major work Geography, in eight books, attempts to map the known world giving coordinates of the major places in terms of latitude and longitude.
    • It is not surprising that the maps given by Ptolemy were quite inaccurate in many places for he could not be expected to do more than use the available data and this was of very poor quality for anything outside the Roman Empire, and even parts of the Roman Empire are severely distorted.
    • In [',' J P Britton, Models and precision : the quality of Ptolemy’s observations and parameters, in Sources and Studies in the History and Philosophy of Classical Science 1 (New York, 1992).','19] Ptolemy is described as:- .
    • Another work on Optics is in five books and in it Ptolemy studies colour, reflection, refraction, and mirrors of various shapes.
    • The establishment of theory by experiment, frequently by constructing special apparatus, is the most striking feature of Ptolemy's "Optics".
    • An English translation, attempting to remove the inaccuracies introduced in the poor Arabic translation which is our only source of the Optics is given in [',' A M Smith, Ptolemy’s theory of visual perception, Transactions of the American Philosophical Society 86 Pt.
    • The first to make accusations against Ptolemy was Tycho Brahe.
    • He discovered that there was a systematic error of one degree in the longitudes of the stars in the star catalogue, and he claimed that, despite Ptolemy saying that it represented his own observations, it was merely a conversion of a catalogue due to Hipparchus corrected for precession to Ptolemy's date.
    • After comments by Laplace and Lalande, the next to attack Ptolemy vigorously was Delambre.
    • He suggested that perhaps the errors came from Hipparchus and that Ptolemy might have done nothing more serious than to have failed to correct Hipparchus's data for the time between the equinoxes and solstices.
    • However Delambre then goes on to say (see [',' G Grasshoff, The history of Ptolemy’s star catalogue (New York, 1990).','8]):- .
    • One could explain everything in a less favourable but all the simpler manner by denying Ptolemy the observation of the stars and equinoxes, and by claiming that he assimilated everything from Hipparchus, using the minimal value of the latter for the precession motion.
    • However, Ptolemy was not without his supporters by any means and further analysis led to a belief that the accusations made against Ptolemy by Delambre were false.
    • To all appearances, one will have to credit Ptolemy with giving an essentially richer picture of the Greek firmament after his eminent predecessors.
    • Vogt showed clearly in his important paper [',' H Vogt, Versuch einer Wiederherstellung von Hipparchs Fixsternverzeichnis, Astronomische Nachrichten 224 (1925), 17-54.','77] that by considering Hipparchus's Commentary on Aratus and Eudoxus and making the reasonable assumption that the data given there agreed with Hipparchus's star catalogue, then Ptolemy's star catalogue cannot have been produced from the positions of the stars as given by Hipparchus, except for a small number of stars where Ptolemy does appear to have taken the data from Hipparchus.
    • This allows us to consider the fixed star catalogue as of his own making, just as Ptolemy himself vigorously states.
    • The most recent accusations of forgery made against Ptolemy came from Newton in [',' R R Newton, The crime of Claudius Ptolemy (Baltimore, MD, 1977).','12].
    • Towards the end Newton, having claimed to prove every observation claimed by Ptolemy in the Almagest was fabricated, writes [',' R R Newton, The crime of Claudius Ptolemy (Baltimore, MD, 1977).','12]:- .
    • [Ptolemy] developed certain astronomical theories and discovered that they were not consistent with observation.
    • Although the evidence produced by Brahe, Delambre, Newton and others certainly do show that Ptolemy's errors are not random, this last quote from [',' R R Newton, The crime of Claudius Ptolemy (Baltimore, MD, 1977).','12] is, I [EFR] believe, a crime against Ptolemy (to use Newton's own words).
    • The book [',' G Grasshoff, The history of Ptolemy’s star catalogue (New York, 1990).','8] is written to study validity of these accusations and it is a work which I strongly believe gives the correct interpretation.
    • Ptolemy, whose intention was to develop a comprehensive theory of celestial phenomena, had no access to the methods of data evaluation using arithmetical means with which modern astronomers can derive from a set of varying measurement results, the one representative value needed to test a hypothesis.
    • For methodological reason, then, Ptolemy was forced to choose from a set of measurements the one value corresponding best to what he had to consider as the most reliable data.
    • Ptolemy had to consider those values as 'observed' which could be confirmed by theoretical predictions.
    • As a final comment we quote the epigram which is accepted by many scholars to have been written by Ptolemy himself, and it appears in Book 1 of the Almagest, following the list of contents (see for example [',' O Neugebauer, A History of Ancient Mathematical Astronomy (3 Vols.) (Berlin-Heidelberg-New York, 1975).','11]):- .
    • A Poster of Ptolemy .
    • Ptolemy's hypotheses of astronomy .
    • Honours awarded to Ptolemy .
    • http://www-history.mcs.st-andrews.ac.uk/Biographies/Ptolemy.html .

  2. Hipparchus biography
    • This seems to firmly place Hipparchus in Nicaea and indeed Ptolemy does describe Hipparchus as observing in Bithynia, and one would naturally assume that in fact he was observing in Nicaea.
    • Most of the information which we have about the work of Hipparchus comes from Ptolemy's Almagest Ⓣ but, as Toomer writes in [',' G J Toomer, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    • although Ptolemy obviously had studied Hipparchus's writings thoroughly and had a deep respect for his work, his main concern was not to transmit it to posterity but to use it and, where possible, improve upon it in constructing his own astronomical system.
    • Where one might hope for more information about Hipparchus would be in the commentaries on Ptolemy's Almagest Ⓣ.
    • There are two in particular by the excellent commentators Theon of Alexandria and by Pappus, but unfortunately these follow Ptolemy's text fairly closely and fail to add the expected information about Hipparchus.
    • Since when Ptolemy refers to results of Hipparchus he does so often in an obscure way, at least he seems to assume that the reader will have access to the original writings by Hipparchus, and it is certainly surprising that neither Theon or Pappus fills in the details.
    • Hipparchus's value of 46" for the annual precession is good compared with the modern value of 50.26" and much better than the figure of 36" that Ptolemy was to obtain nearly 300 years later.
    • His star catalogue, probably completed in 129 BC, has been claimed to have been used by Ptolemy as the basis of his own star catalogue.
    • However, Vogt shows clearly in his important paper [',' H Vogt, Versuch einer Wiederstellung von Hipparchs Fixsternverzeichnis, Astronomische Nachrichten 224 (1925), 17-54.','26] that by considering the Commentary on Aratus and Eudoxus and making the reasonable assumption that the data given there agreed with his star catalogue, then Ptolemy's star catalogue cannot have been produced from the positions of the stars as given by Hipparchus.
    • This last point shows that in any detailed discussion of the achievements of Hipparchus we have to delve more deeply than just assuming that everything in the Ptolemy's Almagest Ⓣ which he does not claim as his own must be due to Hipparchus.
    • So major shifts have taken place in our understanding of Hipparchus, first it was assumed that his discoveries were all set out by Ptolemy, then once it was realised that this was not so there was a feeling that it would be impossible to ever have detailed knowledge of his achievements, but now we are in a third stage where it is realised that it is possible to gain a good knowledge of his work but only with much effort and research.
    • For example Maeyama in [',' Y Maeyama, Ancient stellar observations : Timocharis, Aristyllus, Hipparchus, Ptolemy - the dates and accuracies, Centaurus 27 (3-4) (1984), 280-310.','13] sees major differences between the accuracy of the data in Commentary on Aratus and Eudoxus (claimed to be written around 140 BC) and Hipparchus's star catalogue (claimed to be produced around 130 BC).
    • Maeyama writes [',' Y Maeyama, Ancient stellar observations : Timocharis, Aristyllus, Hipparchus, Ptolemy - the dates and accuracies, Centaurus 27 (3-4) (1984), 280-310.','13]:- .
    • Although Hipparchus's treatise On sizes and distances has not survived details given by Ptolemy, Pappus, and others allow us to reconstruct his methods and results.
    • He showed that his model did not agree totally with observations but it seems to be Ptolemy who was the first to correct the model to take these discrepancies into account.
    • The documentary evidence comes from Ptolemy and Theon of Alexandria who explicitly says that Hipparchus wrote a work on chords in 12 books.

  3. Al-Battani biography
    • He composed a work on astronomy, with tables, containing his own observations of the sun and moon and a more accurate description of their motions than that given in Ptolemy's "Almagest" Ⓣ.
    • Apart from this, he took great interest in astrology, which led him to write on this subject too: of his compositions in this field I mention his commentary on Ptolemy's Tetrabiblos.
    • There is also the commentary on Ptolemy's Tetrabiblos referred to above and two other titles: On ascensions of the signs of the zodiac and On the quantities of the astrological applications.
    • The motions of the sun, moon and five planets are discussed in chapters 27 to 31, where the theory given is that of Ptolemy but for al-Battani the theory appears less important than the practical aspects.
    • Rather than using geometrical methods, as Ptolemy had done, al-Battani used trigonometrical methods which were an important advance.
    • However, as Swerdlow points out in [',' N Swerdlow, Al-Battani ’s determination of the solar distance, Centaurus 17 (2) (1972), 97-105.','8], the influence of Ptolemy was remarkably strong on all medieval authors, and even a brilliant scientist like al-Battani probably did not dare to claim a different value of the distance from the Earth to the Sun from that given by Ptolemy.
    • This was despite the fact that al-Battani could deduce a value for the distance from his own observations that differed greatly from Ptolemy's.
    • In [',' W Hartner, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1] Hartner gives a somewhat different opinion of the way that al-Battani is influenced by Ptolemy.
    • a very sound scepticism in regard to Ptolemy's practical results.
    • Thus, relying on his own observations, he corrects - be it tacitly, be it in open words - Ptolemy's errors.
    • In [',' Y Maeyama, Determination of the Sun’s orbit (Hipparchus, Ptolemy, al-Battani, Copernicus, Tycho Brahe), Arch.

  4. Levi biography
    • He calculated his sine tables using Ptolemy's methods and his tables are very accurate.
    • This is the first edition of any of Levi's astronomical writings in Hebrew; as the editor points out, because of the author's critical attitude towards Ptolemy his work did not find many readers who were willing to acknowledge that they had read it (which was translated into Latin at the papal court of Avignon).
    • These first 20 sections contain a general philosophical introduction, Levi ben Gerson's trigonometry, a description of the construction and use of an instrument invented by him (the Jacob staff) for exact determination of angular differences, remarks about the exact construction of the astrolabe and improvements invented by him, followed by practical and theoretical methods for the determination of the meridian and the position of stars and, finally, a proof that Ptolemy's theory does not account for the facts - neither for the moon (which in an epicycle motion would have to be seen from both its sides) nor for the planets, and a preliminary exploration of alternative (excentric circle) theories.
    • It was a bold move by Levi to reject Ptolemy's system and certainly shows his strength of character that he was prepared to argue against such a deeply held standard model of the universe.
    • However, he realised that Ptolemy's model does not match the facts.
    • For example, according to Ptolemy's model the size of Mars should vary by a factor of 6 but Levi observed only a factor of 2 in the size of the planet.
    • Having explained that Levi rejected Ptolemy's model, we must make it clear that the model which Levi gives is of a similar type consisting of 48 spheres.
    • He gives an iterative calculation for the size of the fluid layer and is led to a calculation of the size of the universe which differs very significantly from Ptolemy's.
    • For example, for Ptolemy the distance to the sphere of fixed stars is 20,000 Earth radii, while Levi calculates it to be 160 .
    • His lunar model, however, is very interesting for he was able to eliminate epicycles and come up with a model which agreed with observation much better than that of Ptolemy, see [',' B R Goldstein, Levi ben Gerson’s preliminary lunar model, Centaurus 18 (1973-74), 275-288.','28] and [',' B R Goldstein, Levi ben Gerson’s lunar model, Centaurus 16 (4) (1971-72), 257-284.','29].

  5. Eratosthenes biography
    • The library at Alexandria was planned by Ptolemy I Soter and the project came to fruition under his son Ptolemy II Philadelphus.
    • Ptolemy II Philadelphus appointed one of Eratosthenes' teachers Callimachus as the second librarian.
    • When Ptolemy III Euergetes succeeded his father in 245 BC and he persuaded Eratosthenes to go to Alexandria as the tutor of his son Philopator.
    • In his commentary on Proposition 1 of Archimedes' Sphere and cylinder Book II, Eutocius reproduces a letter reputed to have been written by Eratosthenes to Ptolemy III Euergetes.
    • Happy art thou, Ptolemy, in that, as a father the equal of his son in youthful vigour, thou hast thyself given him all that is dear to muses and Kings, and may be in the future, O Zeus, god of heaven, also receive the sceptre at thy hands.
    • Ptolemy tells us that Eratosthenes measured the tilt of the Earth's axis with great accuracy obtaining the value of 11/83 of 180°, namely 23° 51' 15".
    • Ptolemy’s reference to Eratosthenes in Almagest I.12, Centaurus 27 (2) (1984), 165-167.','17] are written just to examine the source of this value.
    • Perhaps the most commonly held view is that the value 11/83 is due to Ptolemy and not to Eratosthenes.
    • Heath [',' T L Heath, A History of Greek Mathematics (2 vols.) (Oxford, 1921).','4] argues that Eratosthenes used 24° and that 11/83 of 180° was a refinement due to Ptolemy.
    • Ptolemy’s reference to Eratosthenes in Almagest I.12, Centaurus 27 (2) (1984), 165-167.','17] agrees with attributing 11/83 to Ptolemy although he believes that Eratosthenes used the value 2/15 of 180°.

  6. Pappus biography
    • ','379 AD - 395 AD], when Theon the Philosopher, who wrote the Canon of Ptolemy, also flourished.
    • Similar insertions give the dates for Ptolemy, Hipparchus and other mathematical astronomers.
    • This fixes clearly the date of 320 for Pappus's commentary on Ptolemy's Almagest Ⓣ.
    • Book VI deals with the books on astronomy which were collected into the Little Astronomy so-called in contrast to Ptolemy's Almagest Ⓣ or Greater Astronomy.
    • Of Pappus's commentary on Ptolemy's Almagest Ⓣ only the part on Books 5 and 6 has survived.
    • When Ptolemy in the chapter on the apparent diameter of the sun, moon and shadow simply remarks that the tangential cones in question contact the spheres within a negligible error in great circles, then Pappus refers to Euclid's "Optics" to show that the circle of contact has a smaller diameter than the sphere, only to add a lengthy argument to demonstrate that the error committed in Ptolemy's construction is nevertheless negligible.
    • Or, when Ptolemy says that some phenomenon cannot take place, neither for the same clima nor for different geographical latitudes, Pappus feels obliged to explain "same clima" by "either in clima 3, or in 4, or in any other clima", and to illustrate "different" by referring to "Rome or Alexandria".
    • In case it might be thought that the quality of the Mathematical Collection and the commentary on Ptolemy's Almagest Ⓣ as of such different quality that Pappus may not have written both, then this is ruled out by his references which he makes in the Mathematical Collection (see for example [',' I Bulmer-Thomas, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    • We shall begin therefore after the Geography of Pappus of Alexandria, who followed the circle or special map of Claudius Ptolemy.

  7. Al-Haytham biography
    • The previous major work on optics had been Ptolemy's Almagest Ⓣ and although ibn al-Haytham's work did not have an influence to equal that of Ptolemy's, nevertheless it must be regarded as the next major contribution to the field.
    • Abu al-Qasim ibn Madan was an astronomer who proposed questions to ibn al-Haytham, raising doubts about some of Ptolemy's explanations of physical phenomena.
    • What should we think of Ptolemy's account in "Almagest" Ⓣ I.3 concerning the visible enlargement of celestial magnitudes (the stars and their mutual distances) on the horizon? Is the explanation apparently implied by this account correct, and if so, under what physical conditions? How should we understand the analogy Ptolemy draws in the same place between this celestial phenomenon and the apparent magnification of objects seen in water? ..
    • There are strange contrasts in ibn al-Haytham's work relating to Ptolemy.
    • In Al-Shukuk ala Batlamyus (Doubts concerning Ptolemy), ibn al-Haytham is critical of Ptolemy's ideas yet in a popular work the Configuration, intended for the layman, ibn al-Haytham completely accepts Ptolemy's views without question.

  8. Commandino biography
    • Already when he lived in Rome he had begun the task of editing Ptolemy's Planisphere and from that point on he spent the rest of his life publishing translations (mostly Greek into Latin), with commentaries, of the classic texts of Archimedes, Ptolemy, Euclid, Aristarchus, Pappus, Apollonius, Eutocius, Heron and Serenus.
    • Also in 1558 Commandino published the work which he had begun in Rome, namely Commentarius in planisphaerium Ptolemaei in which he gave an account of Ptolemy's stereographic projection of the celestial sphere.
    • What is interesting here is that Commandino recognised that Ptolemy's stereographic projection is related to the perspective studies made by architects in designing stage scenery.
    • In the letter Commandino outlined his plans for further publications including Ptolemy's De analemmate and an edition of Apollonius.
    • In fact Commandino had only a manuscript of a Latin translation of an Arabic version of this book by Ptolemy to work from.
    • He published his edition of De analemmate in 1562 together with a commentary on the text and a work his own On the calibration of sundials of which he felt added a practical aspect to Ptolemy's theoretical discussions.
    • This rapid ascent was assisted by Apollonius, Archimedes, Aristarchus, Euclid, Eutocius, Heron, Pappus, Ptolemy and Serenus - as published by Commandino.

  9. Al-Khwarizmi biography
    • as opposed to most later Islamic astronomical handbooks, which utilised the Greek planetary models laid out in Ptolemy's "Almagest" Ⓣ ..
    • Although his astronomical work is based on that of the Indians, and most of the values from which he constructed his tables came from Hindu astronomers, al-Khwarizmi must have been influenced by Ptolemy's work too [',' G J Toomer, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    • It is certain that Ptolemy's tables, in their revision by Theon of Alexandria, were already known to some Islamic astronomers; and it is highly likely that they influenced, directly or through intermediaries, the form in which Al-Khwarizmi's tables were cast.
    • The book, which is based on Ptolemy's Geography, lists with latitudes and longitudes, cities, mountains, seas, islands, geographical regions, and rivers.
    • The manuscript does include maps which on the whole are more accurate than those of Ptolemy.
    • In particular it is clear that where more local knowledge was available to al-Khwarizmi such as the regions of Islam, Africa and the Far East then his work is considerably more accurate than that of Ptolemy, but for Europe al-Khwarizmi seems to have used Ptolemy's data.

  10. Regiomontanus biography
    • In 1450 George of Trebizond had translated and commented on Ptolemy's Almagest Ⓣ.
    • Bessarion encouraged Peurbach to produce an abridgment of Ptolemy's Almagest Ⓣ.
    • He had two motives, one being a desire to have a more easily understandable version of Ptolemy's work available; the second being to give support to Theon of Alexandria against the attack from George of Trebizond.
    • Regiomontanus, the foremost European astronomer of the time, was commissioned by the Cardinal to prepare an Epitome of Ptolemy's "Almagest" Ⓣ, also dedicated to him in 1462.
    • [for] it added later observations, revised computations, and critical reflections - one of which revealed that Ptolemy's lunar theory required the apparent diameter of the moon to vary in length much more than it really does.
    • commentary on the "Syntaxis" he will show with the utmost clarity to be worthless and his translation of Ptolemy's work not to be free of faults.

  11. Peurbach biography
    • In Theoricae Novae Planetarum (New theories of the planets), which he completed by 30 August 1454, Peurbach presented Ptolemy's epicycle theory of the planets in an elementary but thorough way.
    • The theory is derived from Ptolemy but also from Islamic astronomers.
    • 27 (3) (1996), 187-237.','4] puts forward a strong case that Peurbach used techniques developed by the Islamic astronomers to modify Ptolemy's model.
    • Based on Ptolemy's theory, they represented the best data available in Peurbach's time.
    • He wrote on the computation of sines and chords in Tractatus super propositiones Ptolemaei de sinubus et chordis using and explanation based on that of Islamic astronomers as well as one based on Ptolemy.
    • a shift from reverence for Ptolemy and antiquity to respect coupled with confident innovation.

  12. Al-Khazin biography
    • Al-Khazin wrote a commentary on Ptolemy's Almagest Ⓣ which was criticised by al-Biruni for being too verbose.
    • The fragment which has survived contains a discussion by al-Khazin of Ptolemy's argument that the universe is spherical.
    • Ptolemy wrote [',' R Lorch, Abu Ja’far al-Khazin on isoperimetry and the Archimedean tradition, Z.
    • Al-Khazin gives 19 propositions relating to this statement by Ptolemy.
    • Finally we should mention that al-Khazin proposed a different solar model from that of Ptolemy.
    • Ptolemy had the sun moving in uniform circular motion about a centre which was not the earth.

  13. Brahe biography
    • Neither tables based on Copernicus nor on Ptolemy gave the correct date for the conjunction, Ptolemy's being out by nearly a month and even Copernicus's being out by days.
    • Maeyama notes in [',' Y Maeyama, Determination of the Sun’s orbit (Hipparchus, Ptolemy, al-Battani, Copernicus, Tycho Brahe), Arch.
    • Among his many discoveries Tycho found that the obliquity of the ecliptic had decreased since the time of Ptolemy but, as explained in [',' K P Moesgaard, Tycho Brahe’s discovery of changes in star latitudes, Centaurus 32 (4) (1989), 310-323.','24], he obtained an incorrect value due to errors by Ptolemy.
    • He always thought a lot of himself and perhaps by this stage his view of his own importance (he saw himself as the natural successor to Hipparchus and Ptolemy, a far more important person than a King) had rather turned his head.

  14. Al-Tusi Nasir biography
    • It is fair to say that al-Tusi made the most significant development of Ptolemy's model of the planetary system up to the development of the heliocentric model in the time of Copernicus.
    • devised a new model of lunar motion, essentially different from Ptolemy's.
    • The aim of al-Tusi with this result was to remove all parts of Ptolemy's system that were not based on the principle of uniform circular motion.
    • These included revised Arabic versions of works by Autolycus, Aristarchus, Euclid, Apollonius, Archimedes, Hypsicles, Theodosius, Menelaus and Ptolemy.
    • Ptolemy's Almagest Ⓣ was one of the works which Arabic scientists studied intently.

  15. Conon biography
    • Conon of Samos is said to have served as court astronomer to Ptolemy III (also known as Ptolemy Euergetes) in Alexandria, see for example [',' I Bulmer-Thomas, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    • The story of the constellation Coma Berenices is that Queen Berenice II, the wife of Ptolemy Euergetes, swore a vow that she would dedicate a lock of her hair to the temple if her husband returned victorious from the Third Syrian War.
    • The war was fought by Ptolemy Euergetes to avenge the murder of his sister in Syria.
    • Ptolemy attributes seventeen "signs of the seasons" to Conon which he may have given in this work.

  16. Cleomedes biography
    • As [Cleomedes] seems to know nothing of the works of Ptolemy, he can hardly..
    • Using Ptolemy's positions for the stars at the time the Almagest Ⓣ was written and Ptolemy's value of 1° per 100 years for precession, Neugebauer gets his date of 371 AD for Cleomedes writings, to which Neugebauer estimates a maximum error of 50 years on either side.
    • Heath's comment that Cleomedes knows nothing of the works of Ptolemy is also less certain than it might at first appear.
    • In fact there are other features of the text which would tend to support the fourth century AD as a date, despite the lack of references to Ptolemy.

  17. Copernicus biography
    • Rather they were mathematics courses which introduced Aristotle and Ptolemy's view of the universe so that students could understand the calendar, calculate the dates of holy days, and also have skills that would enable those who would follow a more practical profession to navigate at sea.
    • my teacher always had before his eyes the observations of all ages together with his own, assembled in order as in catalogues; then when some conclusion must be drawn or contribution made to the science and its principles, he proceeds from the earliest observations to his own, seeking the mutual relationship which harmonizes them all; the results thus obtained by correct inference under the guidance of Urania he then compares with the hypothesis of Ptolemy and the ancients; and having made a most careful examination of these hypotheses, he finds that astronomical proof requires their rejection; he assumes new hypotheses, not indeed without divine inspiration and the favour of the gods; by applying mathematics, he geometrically establishes the conclusions which can be drawn from them by correct inference; he then harmonizes the ancient observations and his own with the hypotheses which he has adopted; and after performing all these operations he finally writes down the laws of astronomy ..
    • The problem that Copernicus faced was that he assumed all motion was circular so, like Ptolemy, was forced into using epicycles (see for example [',' E Rosen, Copernicus’ spheres and epicycles, Arch.
    • Through observations made by himself [Copernicus] discovered certain gaps in Ptolemy, and he concluded that the hypotheses established by Ptolemy admit something unsuitable in violation of the axioms of mathematics.

  18. Theon biography
    • Theon is famed for his commentaries on many works such as Ptolemy's Almagest Ⓣ and the works of Euclid.
    • Theon also wrote extensive commentaries on the astronomical works of Ptolemy, both on the Almagest Ⓣ and the Handy tables.
    • Theon wrote two commentaries on Ptolemy's Handy Tables.
    • As to Theon's commentary on Ptolemy's Syntaxis Heath writes [',' T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).','2]:- .

  19. Heron biography
    • The second was based on an argument which purported to show that he lived later that Ptolemy, and, since Pappus refers to Heron, before Pappus.
    • The fact that Ptolemy does not appear to have known of this method led historians to mistakenly believe Heron lived after Ptolemy; .
    • Catoptrica deals with mirrors and is attributed by some historians to Ptolemy although most now seem to believe that this is a genuine work of Heron.

  20. Jabir ibn Aflah biography
    • He also gave his name to a theorem in spherical trigonometry, and his criticisms of Ptolemy's Almagest Ⓣ are well known.
    • One sees that ibn Aflah even puts his argument regarding errors made by Ptolemy into the title of the work.
    • In [',' R P Lorch, The astronomy of Jabir ibn Aflah, Centaurus 19 (2) (1975), 85-107.','4] Lorch explains Jabir ibn Aflah's most famous criticism, namely Ptolemy's placement of Venus and Mercury below the Sun.
    • Ptolemy claimed that these planets could never be on a line between an observer on Earth and the sun., but ibn Aflah states that this is an error, and that Venus and Mercury are above the Sun.

  21. Campanus biography
    • The instrument was made of disks whose rotation recreated the motion of a planet using Ptolemy's deferent and epicycle description.
    • The work also contains detailed descriptions of Ptolemy's view of the universe giving the longitude of the planets as well as a geometrical description of the motion of the model.
    • Campanus, using Ptolemy's data, calculated the dimensions of the universe including a calculation of the area of the sphere of the fixed stars.
    • his works helped to establish the foundation on which later generations of astronomers would build their critical re-examination of Ptolemy.

  22. Theon of Smyrna biography
    • He was called 'the old Theon' by Theon of Alexandria and 'Theon the mathematician' by Ptolemy.
    • We know that he was making astronomical observations of Mercury and Venus between 127 and 132 since Ptolemy lists four observations which Theon made in 127, 129, 130 and 132.
    • In particular he wrote an important work on Ptolemy and another on Plato's Republic which he refers to himself in work which survives.

  23. Savile biography
    • On 10 October 1570 he began to lecture at Oxford on Ptolemy's Almagest Ⓣ and we are fortunate in that his lecture notes for this course have survived.
    • The lectures are far more than Ptolemy's text with added explanation.
    • The professor of astronomy had to meet similar requirements, but in this case the text was to be Ptolemy's Almagest Ⓣ but full details of the newer theories had also to be presented such as those of Jabir ibn Aflah in his Correction of the Almagest and Copernicus's heliocentric point of view.

  24. Mastlin biography
    • Despite his commitment to the views of Copernicus (which we state below in his own words) this teaching textbook was written purely as a description of astronomy based on the geocentric model of Ptolemy.
    • Although clearly believing in the system as proposed by Copernicus, he taught astronomy using his own textbook which was based on Ptolemy's system.
    • Therefore, I think that unless the common hypotheses are reformed (a task that I am not up to because of my inadequate abilities), I will accept the hypotheses and opinion of Copernicus - after Ptolemy, the prince of all Astronomers.

  25. Hypatia biography
    • However she assisted her father Theon of Alexandria in writing his eleven part commentary on Ptolemy's Almagest Ⓣ.
    • In addition to the joint work with her father, we are informed by Suidas that Hypatia wrote commentaries on Diophantus's Arithmetica, on Apollonius's Conics and on Ptolemy's astronomical works.
    • The passage in Suidas is far from clear and most historians doubt that Hypatia wrote any commentaries on Ptolemy other than the works which she composed jointly with her father.

  26. Euclid biography
    • This man lived in the time of the first Ptolemy; for Archimedes, who followed closely upon the first Ptolemy makes mention of Euclid, and further they say that Ptolemy once asked him if there were a shorted way to study geometry than the Elements, to which he replied that there was no royal road to geometry.

  27. Al-Biruni biography
    • It is worth noting that he had a better feel for errors than did Ptolemy.
    • 2 (2) (1992), 299-306.','66] the author comments that Ptolemy's attitude was to select the observations which he thought most reliable (often that meant fitting in with his theory), and not to tell the reader about observations that he was discarding.
    • The author of [',' P G Bulgakov, al-Biruni and al-Khwarizmi (Russian), in Mathematics and astronomy in the works of scientists of the medieval East (Tashkent, 1977), 117-122; 140.','27] remarks that al-Biruni seemed to realise that for places given by both al-Khwarizmi and Ptolemy, the value obtained by al-Khwarizmi is the more accurate.

  28. Mercator Gerardus biography
    • Mercator published corrected and updated versions of Ptolemy's maps in 1578 as the first part of his 'atlas'.
    • Mercator's break from the methods of Ptolemy was as important for geography as was Copernicus for astronomy.

  29. Yunus biography
    • The underlying lunar theory is that of Ptolemy, but these tables are so devised that the user is spared the calculations which are associated with Ptolemy's lunar tables.

  30. Ulugh Beg biography
    • This excellent book records the main achievements which include the following: methods for giving accurate approximate solutions of cubic equations; work with the binomial theorem; Ulugh Beg's accurate tables of sines and tangents correct to eight decimal places; formulae of spherical trigonometry; and of particular importance, Ulugh Beg's Catalogue of the stars, the first comprehensive stellar catalogue since that of Ptolemy.
    • Observations made at the Observatory brought to light a number of errors in the computations of Ptolemy which had been accepted without question up to that time.

  31. Mansur biography
    • Abu Nasr Mansur's main achievements are his commentry on the Spherics of Menelaus, his role in the development of trigonometry from Ptolemy's calculation with chords towards the trigonometric functions used today, and his development of a set of tables which give easy numerical solutions to typical problems of spherical astronomy.
    • Menelaus's work formed the basis for Ptolemy's numerical solutions of spherical astronomy problems in the Almagest Ⓣ.

  32. Proclus biography
    • Proclus also wrote Hypotyposis, an introduction to the astronomical theories of Hipparchus and Ptolemy in which he described the mathematical theory of the planets based on epicycles and on eccentrics.
    • Nothing here is original and Proclus is proving results first given by Hipparchus and Ptolemy.

  33. Menelaus biography
    • Although we know little of Menelaus of Alexandria's life Ptolemy records astronomical observations made by Menelaus in Rome on the 14th January in the year 98.
    • He lived before Ptolemy, since the latter makes mention of him.

  34. Abul-Wafa biography
    • His trigonometric tables are accurate to 8 decimal places (converted to decimal notation) while Ptolemy's were only accurate to 3 places.
    • His other works include Kitab al-Kamil Ⓣ, a simplified version of Ptolemy's Almagest Ⓣ.

  35. Recorde biography
    • The Castle of Knowledge was first published in 1556 and gives an elementary introduction to Ptolemy's version of astronomy.
    • It is therefore essentially a mathematical work on the sphere and Recorde certainly read works by Ptolemy, Proclus, Sacrobosco and Oronce Fine before writing his text.

  36. Fine biography
    • Among the texts which he edited were Peurbach's Theoricae Novae Planetarum, which presented Ptolemy's epicycle theory of the planets, and Sacrobosco's Tractatus de Sphaera, a book on astronomy in four chapters.
    • In making maps, Fine had two distinct sources of information, one being the recently rediscovered maps of Ptolemy and others from that period, the other being the newly reported discoveries from his own time.

  37. Marinus biography
    • In the second commentary by Marinus described in [',' A Tihon, Notes sur l’astronomie grecque au Ve siecle de notre ere (Marinus de Naplouse - un commentaire au Petit commentaire de Theon, Janus 63 (1-3) (1976), 167-184.','7], he corrects the rules for the direction of parallax in longitude given by Theon of Alexandria in his small commentary on Ptolemy's Handy Tables.

  38. Werner biography
    • This book contains a translation of the Ptolemy's Geography with a commentary by Werner himself.

  39. Al-Nayrizi biography
    • Al-Nayrizi's works on astronomy include a commentary of Ptolemy's Almagest Ⓣ and Tetrabiblos.

  40. Wallis biography
    • Wallis made other contributions to the history of mathematics by restoring some ancient Greek texts such as Ptolemy's Harmonics, Aristarchus's On the magnitudes and distances of the sun and moon and Archimedes' Sand-reckoner.

  41. Ruan Yuan biography
    • Despite obvious limitations and misrepresentations, Western astronomy and mathematics made accessible to Chinese readers, was none the less, for the most part, based on significant excerpts from texts of primary importance such as Ptolemy's Almagest Ⓣ, Copernicus's De revolutionibus or Euclid's Elements and never on mythical accounts.

  42. Galileo biography
    • I confirm this view not only by refuting Ptolemy's and Aristotle's arguments, but also by producing many for the other side, especially some pertaining to physical effects whose causes perhaps cannot be determined in any other way, and other astronomical discoveries; these discoveries clearly confute the Ptolemaic system, and they agree admirably with this other position and confirm it.

  43. Jagannatha biography
    • Ptolemy's Almagest Ⓣ had been one of the works which Arabic scientists had studied intently and, in 1247, al-Tusi wrote Tahrir al-Majisti (Commentary on the Almagest) in which he introduced various trigonometrical techniques to calculate tables of sines.

  44. Apollonius biography
    • Ptolemy in his book Syntaxis says Apollonius introduced systems of eccentric and epicyclic motion to explain the apparent motion of the planets across the sky.

  45. Lansberge biography
    • Before we describe the contents of Lansberge's treatise, we note that his interest in astronomy, probably stimulated during his short stay in Leiden, led him to be dissatisfied with both Ptolemy and Copernicus.

  46. Posidonius biography
    • Later Ptolemy informs us via the writings of Cleomedes, Posidonius used the more accurate 3750 stadia for the Rhodes to Alexandria distance but kept his very inaccurate 7° 30' thus obtaining the figure of 180000 stadia for the circumference which is far too small.

  47. Barocius biography
    • He published Cosmographia in quatuor libros distributa summo ordine, miraque facilitate, ac brevitate ad magnam Ptolemaei mathematicam constructionem, ad universamque astrologiam institutens in 1585 which studied maps of the world following the style of Ptolemy.

  48. Cusa biography
    • Nicholas published improvements to the Alfonsine Tables which gave a practical method to find the position of the Sun, Moon and planets using Ptolemy's model.

  49. Horrocks biography
    • A very minor part of the course would cover Euclid's Elements and Ptolemy's astronomy.

  50. Aaboe biography
    • there is evidence of a planetary model of epicyclic type in which the planet travels on the epicycle in the opposite direction to that correctly adopted by Ptolemy.

  51. Ibrahim biography
    • It also provides a critical analysis of the observations underlying Ptolemy's solar theory, and Ibrahim ibn Sinan provides his own theory of the sun.

  52. Castelli biography
    • Castelli continued to study mathematics with a view to becoming a teacher, making a deep study of the works of Euclid, Ptolemy, Theodosius and Archimedes.

  53. Zenodorus biography
    • Despite the loss of Zenodorus's treatise On isometric figures, we do know something of the results which it contained since Theon of Alexandria quotes a number of propositions from Zenodorus's work when he is giving his commentary on Ptolemy's Syntaxis.

  54. Nunes biography
    • This book contained Nunes' edition of Sacrobosco's Tractatus de Sphaera, of Peurbach's Theoricae nouae planetarum, and of Book I of Ptolemy's Geographia.

  55. Gherard biography
    • For this reason Gherard went to Toledo in Spain where his intention was to learn Arabic so he could read Ptolemy's Almagest Ⓣ since no Latin translations existed at that time.

  56. Viete biography
    • 6 (3) (1975), 185-208.','22] where it is stated that Viete was interested purely in the geometry of the planetary theories of both Ptolemy and Copernicus, and did not consider the question of whether the theories represented the actual physical reality.

  57. Allen biography
    • It is likely that he produced teaching material for these lectures, almost certainly in the form of a commentary on an ancient text, but only one such survives, namely a commentary on the second and third books of Ptolemy's astrological text Tetrabiblos.

  58. Angeli biography
    • Also clearly influenced by Galileo is Angeli's writings on the two systems of Ptolemy and Copernicus which he writes in Galileo's dialogue style.

  59. Tibbon biography
    • He translated into Hebrew many Arabic versions of Greek mathematical and astronomical works, including Euclid's Elements, Euclid's Data, Euclid's Optics, Menelaus's Spherics, Autolycus of Pitane's On the Moving Sphere, Ptolemy's Almagest Ⓣ as well as certain Arabic works such as al-Haytham's Configuration of the World (intended for the layman), and works by al-Ghazali, al-Zarqali, and others.

  60. Wittich biography
    • Back in Wrocław he joined discussions with Henry Savile, who was visiting, about Ptolemy and Copernicus.

  61. Magini biography
    • We also note that in the Preface to De Planis Triangulis, Magini mentions writing an instruction book and an instrument: "if I had also made the instrument to use together with the book, I would have had it made in brass." Also in 1592 Magini published Tabula tetragonica, then in 1596 published a commentary on Ptolemy's Geographia.

  62. Callippus biography
    • Ptolemy gave us an accurate date for the beginning of this cycle in 330 BC in the Almagest Ⓣ saying that year 50 of the first cycle coincided with the 44th year following the death of Alexander.

  63. Roomen biography
    • This was his first work and it was essentially a work on astronomy, in particular on the number and nature of the heavenly spheres of Ptolemy.

  64. Borelli biography
    • He also showed convincingly that the comet was further from the earth than the moon - contradicting Ptolemy's beliefs which were still the Church's accepted system at the time.

  65. Wussing biography
    • These include: (i) Nicolaus Copernicus (1973); (ii) Carl Friedrich Gauss (1974); (iii) Isaac Newton (1977); (iv) Course on the history of mathematics (1979); (v) Adam Ries (1989); (vi) From counting pebbles to the computer (1997); (vii) From Gauss to Poincare (2009); (viii) From Leonardo da Vinci to Galileo Galilei (2010); and (ix) (with Menso Folkerts) From Pythagoras to Ptolemy (2012).

  66. Qadi Zada biography
    • The major work undertaken at the Observatory in Samarkand was the production of the Catalogue of the stars, the first comprehensive stellar catalogue since that of Ptolemy.

  67. Apianus biography
    • This was Cosmographia seu descriptio totius orbis and was a work based largely on Ptolemy.

  68. Al-Khujandi biography
    • Al-Khujandi says that the Indians found the greatest obliquity of the ecliptic, 24° ; Ptolemy 23° 51' ; himself 23° 32' 19".

  69. Ahmed biography
    • We know of a work by Ahmed on ratio and proportion, a book On similar arcs, a commentary on Ptolemy's Centiloquium and a book about the astrolabe.

  70. Delgado biography
    • The third chapter describes rising and setting of heavenly bodies from different geographical locations while the fourth chapter gives a brief introduction to Ptolemy's theory of the planets and of eclipses.

  71. Al-Kashi biography
    • There is little doubt that al-Kashi was the leading astronomer and mathematician at Samarkand and he was called the second Ptolemy by an historian writing later in the same century.

  72. Allman biography
    • Among the articles Allman contributed to the ninth edition of the Encyclopaedia Britannica were those on Thales, Pythagoras, Ptolemy and other Greek philosophers.

  73. Al-Tusi Sharaf biography
    • Certainly around 1165 Al-Tusi was in Damascus for there he taught Abu'l Fadl about the works of Euclid and Ptolemy.

  74. Sacrobosco biography
    • The third chapter describes rising and setting of heavenly bodies from different geographical locations while the fourth chapter gives a brief introduction to Ptolemy's theory of the planets and of eclipses.

  75. Halley biography
    • In 1710, using Ptolemy's catalogue, Halley deduced that the stars must have small motions of their own and he was able to detected this proper motion in three stars.

  76. Eutocius biography
    • Another chapter concerns isoperimetric problems, followed by a short section about the shape and size of the earth, based on Ptolemy's norm of 500 stades for the equatorial degree.

  77. Euler biography
    • as functions rather than as chords as Ptolemy had done.

  78. Al-Nasawi biography
    • On the one hand he says that it will act as an introduction to the Elements while on the other hand it will provide all the necessary background in geometry for anyone wanting to read Ptolemy's Almagest Ⓣ.

  79. Clavius biography
    • Of course the observations of Venus were the most disturbing since Ptolemy's version of the sun and planets would not account for the observed phases.

  80. Roberval biography
    • In this work he praises Aristarchus's heliocentric system but he did not totally reject Ptolemy's earth centered system with the sun and planets circling the earth, or Tycho Brahe's system which has the earth at the centre, but has the planets circling a sun which circles the earth.

  81. Kline biography
    • The conversion of mathematics by Greek philosophers into an abstract, deductive system of thought, the Greek and modern doctrine that nature is mathematically designed, the use of mathematics by Hipparchus and Ptolemy and later by Copernicus and Kepler to erect the most impressive astronomical theories, the development of a mathematical system of perspective by Renaissance painters who sought to achieve realism, the deduction by Galileo, Newton, and others of universal scientific laws which "united heaven and Earth", the reorganization of philosophy, religion, literature, and the social sciences in the Age of Reason, the rise of a statistical view of natural laws consequent upon the success of statistical procedures in the physical and social sciences, the effect of the creation of non-Euclidean geometry upon the belief in truth and on the common understanding of the nature of mathematics, and mathematics as an art are some of the illustrations of the cultural influences of mathematics.


History Topics

  1. Ptolemy Manuscript
    • Manuscript Tradition in Ptolemy's Geography .
    • The Geography of Ptolemy presents a great difficulty for scholars of ancient cartography, geography and philology.
    • The oldest of these manuscripts was created no earlier than the late thirteenth century[',' Ptolemy, Ptolemy’s geography : an annotated translation of the theoretical chapters / J.
    • Latin texts first appeared in the Renaissance, no doubt translated from the Greek manuscripts of the day.) [',' Diller, A., Review of Geography of Claudius Ptolemy by Edward Luther Stevenson.
    • The American Journal of Philology 62(2) (1941) 244-246.','4] The forebear of these two branches was not the Ptolemaic original, but a younger manuscript later than Ptolemy.
    • Such a manuscript does not survive today, but must be assumed [',' Ptolemy, Ptolemy’s geography : an annotated translation of the theoretical chapters / J.
    • It is possible that this missing link could have been written as far back as late antiquity.[',' Ptolemy, Ptolemy’s geography : an annotated translation of the theoretical chapters / J.
    • Ptolemy had his own original errors, as can be seen from variations between coordinates given in the Geography and Ptolemy's Handy Tables which are not likely to have been produced from scribal copying errors.
    • As time went on, however, distinguishing between original errors, copying errors, and improvements became more and more difficult.[',' Ptolemy, Ptolemy’s geography : an annotated translation of the theoretical chapters / J.
    • This was a major turning point in the history of the Geography, marking the beginning of a new proliferation of manuscripts.[',' Ptolemy, Ptolemy’s geography : an annotated translation of the theoretical chapters / J.
    • The probable explanation of this renewed interest in Ptolemy is the work of Byzantine scholar Maximos Planudes (c.1255-1305).
    • discovered through many toils the geographia of Ptolemy, which had disappeared for many years.(Kugeas in [',' Ptolemy, Ptolemy’s geography : an annotated translation of the theoretical chapters / J.
    • [',' Ptolemy, Ptolemy’s geography : an annotated translation of the theoretical chapters / J.
    • In places, the Greek has been corrected or reworded, corruptions from previous traditions have been fixed, and errors in the data possibly dating all the way back to Ptolemy have been adjusted.
    • These alterations seem to have been made due to discrepancies with the text and the maps drawn to the text's specifications.[',' Ptolemy, Ptolemy’s geography : an annotated translation of the theoretical chapters / J.
    • A tree of sources for Ptolemy's Geography .
    • While the rest of the Geography up to Book 8.2, excepting errors and corrections, can be attributed to Ptolemy with some certainty, Ptolemy's authorship of Book 8.29-30 and the world map is doubted, and, in fact, rejected by several scholars.
    • Classical Philology 35(3) (1940) 333-336.','5] Did Ptolemy have a map of the world in front of him, from which he wrote the text? Did Ptolemy create world and/or regional maps based on his text? Or did the maps come later, produced by a student or reader of Ptolemy's text? .
    • Very nearly all manuscripts containing any maps, have the world map drawn in the first projection that Ptolemy describes in Book 1 (the great exception being manuscript K, which uses Ptolemy's second projection).
    • Adopting the scheme of Berggren and Jones, I will denote two classes of manuscripts, A and B, based on the number of regional maps associated with the manuscripts.[',' Ptolemy, Ptolemy’s geography : an annotated translation of the theoretical chapters / J.
    • In both versions, the regional maps follow the projection described for them by Ptolemy, namely a cylindrical projection in which distances along the central latitude and longitude are in the proper ratio.[',' Ptolemy, Ptolemy’s geography : an annotated translation of the theoretical chapters / J.
    • Ptolemy does not give any data or coordinates for such a region.
    • The world maps of the A version, nonetheless, have the same, rough northern coastline for this unknown territory.[',' Ptolemy, Ptolemy’s geography : an annotated translation of the theoretical chapters / J.
    • To fit all of the maps into the manuscript, smaller maps were needed, and so the regional maps were divided into more manageable sizes.[',' Ptolemy, Ptolemy’s geography : an annotated translation of the theoretical chapters / J.
    • Did Ptolemy draw a map or maps to accompany his text? Berggren and Jones reason for the affirmative [',' Ptolemy, Ptolemy’s geography : an annotated translation of the theoretical chapters / J.
    • As Ptolemy insists in 1.17, the way to detect and eliminate inconsistencies such as those he detects in [his predecessor] Marinos' writing is to draw a map.
    • It does not necessarily follow that Ptolemy published any maps with the final version of the text.
    • Certainly the way Ptolemy speaks of the construction of maps and globes in Books 1 and 2, leads one to believe he is speaking through experience.
    • Furthermore, if the point of writing the text was to instruct how to draw a world map, surely he must have tried it himself.[',' Ptolemy, Ptolemy’s geography : an annotated translation of the theoretical chapters / J.
    • (some col.), maps (some col.).','2] Fischer takes this view a step further, concluding that Ptolemy in fact published a world map and several of the regional maps.
    • Fischer claims that the notion that Ptolemy only used drafts of maps to aid in the writing process rests on a false rendering of a passage in the text.[',' Hyde, W.W., Review of Text und Karten des Ptolemaus by Paul Schnabel.
    • Most maps of any size were often displayed publicly, painted or affixed to walls, not in manuscripts.[',' Ptolemy, Ptolemy’s geography : an annotated translation of the theoretical chapters / J.
    • Further evidence that the maps do not originate with Ptolemy has been put forward by Diller: [',' Diller, A., Studies in Greek manuscript tradition, Amsterdam :: Hakkert.','1] .
    • The coordinates on the maps attached to Ptolemy's "Geography" have one remarkable peculiarity.
    • It was inherited by Ptolemy from Eratosthenes, Hipparchus, and Marinus.
    • Whoever drew the maps used Ptolemy's data, but followed the tradition of Ptolemy's predecessors.
    • Why would Ptolemy have gone through the trouble of calculating the coordinates of all his localities in degrees only to use hours later? .
    • As far as the maps accompanying the extant medieval manuscripts are concerned, Fischer asserts that these maps cannot be traced back to Ptolemy as there are discrepancies between the maps and Ptolemy's text.[',' Hyde, W.W., Review of Text und Karten des Ptolemaus by Paul Schnabel.
    • The evidence of this lies mainly with a poem found in one of the manuscripts describing Planudes' discovery of Ptolemy.
    • The poem most likely means that Planudes [',' Ptolemy, Ptolemy’s geography : an annotated translation of the theoretical chapters / J.
    • Because of the lack of practically of fitting the maps into the relatively small manuscripts before 1300, [',' Ptolemy, Ptolemy’s geography : an annotated translation of the theoretical chapters / J.
    • Planudes and his assistants therefore probably had no pictorial models, and the success of their enterprise is proof that Ptolemy succeeded in his attempt to encode the map in words and numbers.
    • What is clear is that unless several lost manuscripts predating U and X come to light, we will never fully understand from where the extant manuscripts are derived, or, more importantly, how different today's texts are from Ptolemy's great original work.
    • http://www-history.mcs.st-andrews.ac.uk/HistTopics/Ptolemy_mss.html .

  2. Cartography
    • Ptolemy wrote his major work Guide to Geography, in eight books, which attempted to map the known world giving coordinates of the major places in terms of what are essentially latitude and longitude.
    • Right at the beginning Ptolemy identifies two distinct types of cartography, the first being [',' J L Berggren and A Jones, Ptolemy’s Geography : An annotated translation of the theoretical chapters (Princeton, 2000).','1]:- .
    • The second type is [',' J L Berggren and A Jones, Ptolemy’s Geography : An annotated translation of the theoretical chapters (Princeton, 2000).','1]:- .
    • Now the main part of Geography consisted of maps but Ptolemy knew that although a scribe could copy a text fairly accurately, there was little chance that maps could be successfully copied.
    • Thus the line of latitude through Rhodes and the Pillars of Hercules (present day Gibraltar) was 36° and this line divided the world as Ptolemy knew it fairly equally into two.
    • Ptolemy chose the Fortune Islands (which we believe are the Canary Islands) as longitude zero since it was the most western point known to him.
    • Ptolemy's map of the world.
    • Had Ptolemy taken the value of the circumference of the Earth worked out by Eratosthenes then his coordinates would have been very accurate.
    • Therefore instead of the Mediterranean covering 42° as it should, it covers 62° in Ptolemy's coordinates.
    • It is not surprising that the maps given by Ptolemy were quite inaccurate in many places for he could not be expected to do more than use the available data and, for anything outside the Roman Empire, this was of very poor quality with even some parts of the Roman Empire severely distorted.
    • Ptolemy used data from most of his predecessors, particularly Marinus of Tyre.
    • In order to estimate distances from such data, Ptolemy had to estimate the difficulty of the terrain, how straight the route the travellers taken had been, and many other unknowns.
    • Ptolemy's Geography was translated into Arabic in the 9th century and soon improvements were being made using data obtained from the explorations being carried out.
    • The book, which was based on Ptolemy's Geography, lists with latitudes and longitudes, cities, mountains, seas, islands, geographical regions, and rivers.
    • The manuscript does include maps which are more accurate than those of Ptolemy, in particular it is clear that where more local knowledge was available to al-Khwarizmi such as in the regions of Islam, Africa and the Far East then his work is considerably more accurate than that of Ptolemy, but for Europe al-Khwarizmi seems on the whole to have used Ptolemy's data.
    • Sezgin also argues that Ptolemy's Geography may not have included a world map, and that some later world maps are based, at least in part, on Islamic sources.
    • The western part of his map was partly based on portolan maps while the eastern part was based on Ptolemy's data.
    • The first steps involved the translation of Ptolemy's Geography into Latin which was begun by Emmanuel Chrysoloras and completed in 1410 by Jacobus Angelus.
    • Producing a map which did not follow Ptolemy clearly worried Mauro who wrote (see for example [',' G R Crone, Maps and their makers (London, 1953).','5]):- .
    • 67">I do not think it derogatory to Ptolemy if I do not follow his 'Cosmografia', because, to have observed his meridians or parallels or degrees, it would be necessary in respect to the setting out of the known parts of this circumference, to leave out many provinces not mentioned by Ptolemy.
    • Despite 1300 years passing since Ptolemy's time, Mauro is still not able to give a good approximation for the circumference of the Earth writing:- .
    • The first printed version of Ptolemy's Geography appeared in 1475 being the Latin translation referred to above.
    • 75">We have confined the Geography of Ptolemy to the first part of the work, in order that its antiquity may remain intact and separate.
    • He set up a new press in Nuremburg in 1472 and announced his intention to publish maps and books including Ptolemy's Geography.
    • His 1524 publication Cosmographia seu descriptio totis orbis was a work based largely on Ptolemy which provided an introduction to astronomy, geography, cartography, surveying, navigation, weather and climate, the shape of the earth, map projections, and mathematical instruments.

  3. references
    • References for: Manuscript Tradition in Ptolemy's Geography .
    • Ptolemy, Ptolemy's geography : an annotated translation of the theoretical chapters / J.
    • Diller, A., Review of Geography of Claudius Ptolemy by Edward Luther Stevenson.
    • http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Ptolemy_mss.html .

  4. references
    • References for: Manuscript Tradition in Ptolemy's Geography .
    • Ptolemy, Ptolemy's geography : an annotated translation of the theoretical chapters / J.
    • Diller, A., Review of Geography of Claudius Ptolemy by Edward Luther Stevenson.
    • [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Ptolemy_mss.html] .

  5. Ptolemy Source
    • The world map (employing Ptolemy's first projection in 1.24) follows the end of Book 7, while the twenty-six regional maps alternate with their respective captions in Book 8.

  6. Ptolemy Source

  7. Ptolemy Source

  8. Ptolemy Source

  9. Greek astronomy
    • Meton's calendar never seems to have been adopted in practice but his observations proved extremely useful to later Greek astronomers such as Hipparchus and Ptolemy.
    • Aristyllus was a pupil of Timocharis and in Maeyama [',' Y Maeyama, Ancient stellar observations : Timocharis, Aristyllus, Hipparchus, Ptolemy - the dates and accuracies, Centaurus 27 (3-4) (1984), 280-310.','23] analyses 18 of their observations and shows that Timocharis observed around 290 BC while Aristyllus observed a generation later around 260 BC.
    • Maeyama writes [',' Y Maeyama, Ancient stellar observations : Timocharis, Aristyllus, Hipparchus, Ptolemy - the dates and accuracies, Centaurus 27 (3-4) (1984), 280-310.','23]:- .
    • Apollonius used his geometric skills to mathematically develop the epicycle theory which would reach its full importance in the work of Ptolemy.
    • We will not describe the contributions of Hipparchus and Ptolemy in detail in this article since these are given fully in their biographies in our archive.
    • 91">Alexandria in the second century AD saw the publication of Ptolemy's remarkable works, the 'Almagest' and the 'Handy Tables', the 'Geography', the 'Tetrabiblos', the 'Optics', the 'Harmonics', treatises on logic, on sundials, on stereographic projection, all masterfully written, products of one of the greatest scientific minds of all times.
    • The eminence of these works, in particular the 'Almagest', had been evident already to Ptolemy's contemporaries.
    • 93">Ptolemy had no successor.

  10. Ptolemy Source

  11. Ptolemy Source

  12. Ptolemy Source

  13. Ptolemy Source

  14. Ptolemy Source

  15. Ptolemy Source

  16. Ptolemy Source

  17. Ptolemy Source

  18. Cartography references
    • J L Berggren and A Jones, Ptolemy's Geography : An annotated translation of the theoretical chapters (Princeton, 2000).
    • H Stevens, Ptolemy's Geography : A brief account of all the printed editions down to 1730 (London, 1908).
    • K Andersen, The central projection in one of Ptolemy's map constructions, Centaurus 30 (2) (1987), 106-113.
    • R P Lorch, Ptolemy and Maslama on the transformation of circles into circles in stereographic projection, Arch.
    • O Neugebauer, Ptolemy's Geography, book VII, chapters 6 and 7, Isis 50 (1959), 22-29.
    • M A Tolmacheva, Ptolemy's East Africa in early medieval Arab geography, J.

  19. Cartography references
    • J L Berggren and A Jones, Ptolemy's Geography : An annotated translation of the theoretical chapters (Princeton, 2000).
    • H Stevens, Ptolemy's Geography : A brief account of all the printed editions down to 1730 (London, 1908).
    • K Andersen, The central projection in one of Ptolemy's map constructions, Centaurus 30 (2) (1987), 106-113.
    • R P Lorch, Ptolemy and Maslama on the transformation of circles into circles in stereographic projection, Arch.
    • O Neugebauer, Ptolemy's Geography, book VII, chapters 6 and 7, Isis 50 (1959), 22-29.
    • M A Tolmacheva, Ptolemy's East Africa in early medieval Arab geography, J.

  20. Trigonometric functions
    • Ptolemy was the next author of a book of chords, showing the same Babylonian influence as Hipparchus, dividing the circle into 360° and the diameter into 120 parts.
      Go directly to this paragraph
    • Ptolemy, together with the earlier writers, used a form of the relation sin2 x + cos2 x = 1, although of course they did not actually use sines and cosines but chords.
      Go directly to this paragraph
    • Similarly, in terms of chords rather than sin and cos, Ptolemy knew the formulas .
    • Ptolemy calculated chords by first inscribing regular polygons of 3, 4, 5, 6 and 10 sides in a circle.
    • Using these methods Ptolemy found that sin 30' (30' = half of 1°) which is the chord of 1° was, as a number to base 60, 0 31' 25".
    • although it could have easily have been deduced from Ptolemy's formula sin(x + y) = sin x cos y + cos x sin y with x = y.

  21. Physical world
    • The most famous of the ancient models of the heavenly bodies put forward to "save the appearances" was that by Ptolemy.
    • Ptolemy is quite clear in stating that his model is not intended to represent physical reality, rather it is only a mathematical model that will represent what is observed.
    • Copernicus, however, maintained that his Sun centred system was superior for it provided an explanation of the retrograde motion of the planets as opposed to Ptolemy's model which was devised to produce the observed effect.
    • For Galileo the simplicity of the mathematics of the Copernican system over the complex mathematics of Ptolemy's system was a strong proof of the reality of the Copernican hypothesis.
    • Now Kepler could claim that the Copernican system was real since it provided an explanation for the planetary motions while that of Ptolemy did not.

  22. Arabic mathematics
    • Finally, the translation of Ptolemy's Almagest furnished important astronomical material.
    • Diocles' treatise on mirrors, Theodosius's Spherics, Pappus's work on mechanics, Ptolemy's Planisphaerium, and Hypsicles' treatises on regular polyhedra (the so-called Books XIV and XV of Euclid's Elements) ..
    • Al-Battani (born 850) made accurate observations which allowed him to improve on Ptolemy's data for the sun and the moon.
      Go directly to this paragraph
    • Nasir al-Din al-Tusi (born 1201), like many other Arabic mathematicians, based his theoretical astronomy on Ptolemy's work but al-Tusi made the most significant development of Ptolemy's model of the planetary system up to the development of the heliocentric model in the time of Copernicus.
      Go directly to this paragraph

  23. Doubling the cube
    • This purports to be a letter written by Eratosthenes to King Ptolemy and, although the letter is a forgery, the writer does quote some genuine writings of Eratosthenes [',' M R Cohen and I E Drabkin (trs.), A source book in Greek science (Harvard, 1948).','1]:- .
    • Eratosthenes to King Ptolemy, greetings.
    • He erected a column at Alexandria dedicated to King Ptolemy with an epigram inscribed on it relating to his own mechanical solution to the problem of doubling the cube [',' T L Heath, A history of Greek mathematics I (Oxford, 1931).','2]:- .
    • Happy art thou, Ptolemy, in that, as a father the equal of his son in youthful vigour, thou hast thyself given him all that is dear to muses and Kings, and may be in the future, O Zeus, god of heaven, also receive the sceptre at thy hands.

  24. Christianity and Mathematics
    • The developments of Aristotle's astronomy by Ptolemy became more than a scientific belief; for many Christians it became part of their religious belief.
    • It was Ptolemy's mathematical tricks of deferent and epicycle which Copernicus modified to his own mathematical model.
    • He realised the mathematical superiority of Copernicus over Ptolemy but he proposed a modification of Copernicus's system in which the planets revolved around the Sun while the Sun itself moved around the stationary Earth.
    • I confirm this view not only by refuting Ptolemy's and Aristotle's arguments, but also by producing many for the other side, especially some pertaining to physical effects whose causes perhaps cannot be determined in any other way, and other astronomical discoveries; these discoveries clearly confute the Ptolemaic system, and they agree admirably with this other position and confirm it.

  25. Size of the Universe
    • Ptolemy, in the 2nd century AD, calculated the distance to the Moon using a parallax method.
    • Using the same method as Hipparchus to determine the distance to the Sun led Ptolemy to the same serious underestimate in its distance.
    • In fact Copernicus, although proposing a very different model of the universe from Ptolemy, took essentially the same values for the distances to the Sun and Moon.
    • One could argue that this is still relatively far from the true value of 93 million miles, but it was a vast improvement over Ptolemy's value which was of the order of 4 million miles.

  26. Cosmology
    • This model was further developed in the following centuries, culminating in the second century AD with Ptolemy's great system.
      Go directly to this paragraph
    • Despite its complicated structure, Ptolemy produced a model so successful at reproducing the apparent motion of the planets that when, in the sixteenth century, Copernicus proposed a heliocentric system, he could not match the accuracy of Ptolemy's Earth-centred system.
      Go directly to this paragraph

  27. Longitude1

  28. function concept
    • If we move forward to Greek mathematics then we reach the work of Ptolemy.
    • Surely, one might think, if he was computing trigonometric functions then Ptolemy must have understood the concept of a function.
    • Ptolemy dealt with functions, but it is very unlikely that he had any understanding of the concept of a function.

  29. test2.html
    • The world map (employing Ptolemy's first projection in 1.24) follows the end of Book 7, while the twenty-six regional maps alternate with their respective captions in Book 8.
    • Ptolemy, Ptolemy's geography : an annotated translation of the theoretical chapters, ed.

  30. Zero
    • Ptolemy in the Almagest written around 130 AD uses the Babylonian sexagesimal system together with the empty place holder O.
    • By this time Ptolemy is using the symbol both between digits and at the end of a number and one might be tempted to believe that at least zero as an empty place holder had firmly arrived.
    • The idea of the zero place (certainly not thought of as a number by Ptolemy who still considered it as a sort of punctuation mark) makes its next appearance in Indian mathematics.

  31. Science in the 17th century
    • For centuries the Catholic Church had worked to 'Christianise' ancient philosophies, for instance to accept the teachings of Aristotle, Ptolemy and Galen, shaping them so that their writings were compatible with the Scriptures and could be read according to the doctrines of the Church.
    • At the same time, the role of the Catholic Church translating and adapting the works of the classical philosophers, mathematicians and astronomers, helped to spread the ideas of Aristotle and Ptolemy and other thinkers of the past.

  32. Sundials
    • Both Vitruvius and Ptolemy describe analemmas which for given solar positions serve to determine length and direction of the shadow cast by a gnomon on the face of a planar sundial.
    • Specifically, in his book, 'On the Analemma', Ptolemy gives methods for deriving, both by trigonometric and also by graphic means, three pairs of spherical coordinates for the sun relative to a given place on earth, given solar declination, terrestrial latitude, and hour of the day.

  33. Non-Euclidean geometry
    • Proclus (410-485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four, in particular he notes that Ptolemy had produced a false 'proof'.
      Go directly to this paragraph

  34. Pi chronology
    • 5Ptolemy15033.14166 .

  35. Greek astronomy references
    • Y Maeyama, Ancient stellar observations : Timocharis, Aristyllus, Hipparchus, Ptolemy - the dates and accuracies, Centaurus 27 (3-4) (1984), 280-310.

  36. Mathematics and Art
    • In this work he gave an account of Ptolemy's stereographic projection of the celestial sphere, but its importance for perspective is that he broadened the study of that topic which had up until then been concerned almost exclusively with painting.

  37. Mathematics and Art references
    • V Valerio, Projective knowledge and perspective of lines in the works of Ptolemy and in late Hellenistic culture (Italian), Nuncius Ann.

  38. Greek astronomy references
    • Y Maeyama, Ancient stellar observations : Timocharis, Aristyllus, Hipparchus, Ptolemy - the dates and accuracies, Centaurus 27 (3-4) (1984), 280-310.

  39. Classical light
    • Ptolemy, about 80 years after Heron, studied light in his astronomical work.

  40. Pi history

  41. Kepler's Laws
    • Apart from frequent application of the trigonometrical propositions of Ptolemy (fl.

  42. Classical time
    • Eclipses of the sun and moon and phases of the moon were also shown, and there was an astrolabe which was designed on Ptolemy's version of the universe.

  43. Indian mathematics
    • 52">The Hindu astronomers did not possess a general method for solving problems in spherical astronomy, unlike the Greeks who systematically followed the method of Ptolemy, based on the well-known theorem of Menelaus.

  44. Mathematics and Art references
    • V Valerio, Projective knowledge and perspective of lines in the works of Ptolemy and in late Hellenistic culture (Italian), Nuncius Ann.

  45. Golden ratio
    • With Ptolemy trigonometric tables, at least in terms of chords of circles, begin to be computed.


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References

  1. References for Ptolemy
    • References for Ptolemy .
    • http://www.encyclopedia.com/topic/Ptolemy.aspx .
    • http://www.britannica.com/biography/Ptolemy .
    • V A Bronshten, Claudius Ptolemy : Second century A.D (Russian) 'Nauka' (Moscow, 1988).
    • O Gingerich, The Eye of Heaven: Ptolemy, Copernicus, Kepler (1993).
    • G Grasshoff, The history of Ptolemy's star catalogue (New York, 1990).
    • R R Newton, The crime of Claudius Ptolemy (Baltimore, MD, 1977).
    • A M Smith, Ptolemy's theory of visual perception, Transactions of the American Philosophical Society 86 Pt.
    • G J Toomer (trs.), Ptolemy's Almagest (London, 1984).
    • K Andersen, The central projection in one of Ptolemy's map constructions, Centaurus 30 (2) (1987), 106-113.
    • J L Berggren, Ptolemy's maps of earth and the heavens : a new interpretation, Arch.
    • J P Britton, Models and precision : the quality of Ptolemy's observations and parameters, in Sources and Studies in the History and Philosophy of Classical Science 1 (New York, 1992).
    • J P Britton, Ptolemy's determination of the obliquity of the ecliptic, Centaurus 14 (1969), 29-41.
    • J Chabas and A Tihon, Verification of parallax in Ptolemy's 'Handy tables', J.
    • B Chatterjee, Geometrical interpretation of the motion of the sun, moon and the five planets as found in the mathematical syntaxis of Ptolemy and in the Hindu astronomical works, J.
    • P Del Santo and G Strano, Observational evidence and the evolution of Ptolemy's lunar model, Nuncius Ann.
    • J Dobrzycki, Historians of science on the astronomical observations of Ptolemy (Polish), Wiadom.
    • S Drake, Ptolemy, Galileo, and scientific method, Stud.
    • Yu N Efremov and E D Pavlovskaya, Determination of the epoch of the star catalogue 'Almagest' by analyzing the proper motion of the stars (on a problem of the authorship of Ptolemy's star catalogue) (Russian), Istor.-Astronom.
    • J Evans, On the function and the probable origin of Ptolemy's equant, Amer.
    • A T Fomenko, V V Kalashnikov and G V Nosovsky, The dating of Ptolemy's 'Almagest' based on the coverings of the stars and on lunar eclipses, Acta Appl.
    • A T Fomenko, V V Kalashnikov and G V Nosovskii, Statistical analysis and dating of the observations on which Ptolemy's 'Almagest' star catalogue is based, in Probability theory and mathematical statistics I (Vilnius, 1990), 360-374.
    • A T Fomenko, V V Kalashnikov and G V Nosovsky, When was Ptolemy's star catalogue in 'Almagest' compiled in reality? Statistical analysis, Acta Appl.
    • O Gingerich, Was Ptolemy a fraud?, Quart.
    • O Gingerich, Ptolemy revisited : A reply to R R Newton, Quart.
    • O Gingerich and B L Welther, Some puzzles of Ptolemy's star catalogue, Sky and Telescope 67 (1984), 421-423.
    • B R Goldstein, Saving the phenomena : the background to Ptolemy's planetary theory, J.
    • S J Goldstein, Problems raised by Ptolemy's lunar tables, J.
    • A A Gurshtein, Ptolemy and Copernicus (Russian), Priroda (3) (1988), 85-92.
    • N T Hamilton, N M Swerdlow and G J Toomer, The 'Canobic inscription' : Ptolemy's earliest work, in From ancient omens to statistical mechanics, Acta Hist.
    • W Hartner, Ptolemy and Ibn Yunus on solar parallax, Arch.
    • W Hartner, Ptolemy, Azarquiel, Ibn al-Shatir, and Copernicus on Mercury : A study of parameters, Arch.
    • W Hartner, Ptolemy's and Copernicus' Mercury models : An accuracy test, Arch.
    • P Kunitzsch, Fragments of Ptolemy's 'Planisphaerium' in an early Latin translation, Centaurus 36 (2) (1993), 97-101.
    • P Kunitzsch, The second Arabic manuscript of Ptolemy's 'Planisphaerium', Z.
    • P Kunitzsch and R Lorch, Maslama's notes on Ptolemy's 'Planisphaerium' and related texts, Bayer.
    • R P Lorch, Ptolemy and Maslama on the transformation of circles into circles in stereographic projection, Arch.
    • D MacMinn, An analysis of Ptolemy's treatment of retrograde motion, J.
    • Y Maeyama, Determination of the Sun's orbit (Hipparchus, Ptolemy, al-Battani, Copernicus, Tycho Brahe), Arch.
    • Y Maeyama, Ancient stellar observations : Timocharis, Aristyllus, Hipparchus, Ptolemy - the dates and accuracies, Centaurus 27 (3-4) (1984), 280-310.
    • K P Moesgaard, In chase of an origin for the mean planetary motions in Ptolemy's 'Almagest', in From ancient omens to statistical mechanics, Acta Hist.
    • A Murschel, The structure and function of Ptolemy's physical hypotheses of planetary motion, J.
    • O Neugebauer, Ptolemy's Geography, book VII, chapters 6 and 7, Isis 50 (1959), 22-29.
    • J Nevalainen, The accuracy of the ecliptic longitude in Ptolemy's Mercury model, J.
    • A Pannekoek, Ptolemy's precession, Vistas in Astronomy 1 (1955), 60-66.
    • V M Petersen, The three lunar models of Ptolemy, Centaurus 14 (1969), 142-171.
    • V M Petersen and O Schmidt, The determination of the longitude of the apogee of the orbit of the sun according to Hipparchus and Ptolemy, Centaurus 12 (1967/1968), 73-96.
    • D Rawlins, Ancient heliocentrists, Ptolemy, and the equant, Amer.
    • A I Sabra, Psychology versus mathematics : Ptolemy and Alhazen on the moon illusion, in Mathematics and its applications to science and natural philosophy in the Middle Ages (Cambridge-New York, 1987), 217-247.
    • J Samso and F Castello, An hypothesis on the epoch of Ptolemy's star catalogue according to the authors of the Alfonsine tables, J.
    • O Schmidt, On the theorems of Ptolemy and Menelaus (Danish), Nordisk Mat.
    • M Shevchenko, An analysis of errors in the star catalogues of Ptolemy and Ulugh Beg, J.
    • M Yu Shevchenko, Claudius Ptolemy's star catalogue : the specific character of ancient astrometrical observations (Russian), Istor.-Astronom.
    • A M Smith, Ptolemy's search for a law of refraction : a case-study in the classical methodology of 'saving the appearances' and its limitations, Arch.
    • A M Smith, The psychology of visual perception in Ptolemy's 'Optics', Isis 79 (297) (1988), 189-207.
    • N M Swerdlow, The enigma of Ptolemy's catalogue of stars, J.
    • N M Swerdlow, Ptolemy's theory of the inferior planets, J.
    • N M Swerdlow, Ptolemy on trial, Amer.
    • C M Taisbak, Eleven eighty-thirds : Ptolemy's reference to Eratosthenes in 'Almagest' I.12, Centaurus 27 (2) (1984), 165-167.
    • M A Tolmacheva, Ptolemy's East Africa in early medieval Arab geography, J.
    • G J Toomer (ed.), Ptolemy's 'Almagest' (New York-Berlin, 1984).
    • G Van Brummelen, Lunar and planetary interpolation tables in Ptolemy's 'Almagest', J.
    • V Valerio, Projective knowledge and linear perspective in the works of Ptolemy and in late Hellenistic culture (Italian), Nuncius Ann.
    • B van Dalen, On Ptolemy's table for the equation of time, Centaurus 37 (2) (1994), 97-153.
    • D T Whiteside, A refined computation of the perigee angle in Ptolemy's Mercury model, J.
    • C Wilson, The sources of Ptolemy's parameters, J.
    • J Wlodarczyk, Notes on the compilation of Ptolemy's catalogue of stars, J.
    • http://www-history.mcs.st-andrews.ac.uk/References/Ptolemy.html .

  2. References for Hipparchus
    • G Grasshoff, The history of Ptolemy's star catalogue (New York, 1990).
    • Y Maeyama, Ancient stellar observations : Timocharis, Aristyllus, Hipparchus, Ptolemy - the dates and accuracies, Centaurus 27 (3-4) (1984), 280-310.
    • Y Maeyama, Determination of the Sun's orbit (Hipparchus, Ptolemy, al-Battani, Copernicus, Tycho Brahe), Arch.
    • V M Petersen and O Schmidt, The determination of the longitude of the apogee of the orbit of the sun according to Hipparchus and Ptolemy, Centaurus 12 (1967/1968), 73-96.

  3. References for Copernicus
    • A A Gurshtein, Ptolemy and Copernicus (Russian), Priroda (3) (1988), 85-92.
    • W Hartner, Ptolemy, Azarquiel, Ibn al-Shatir, and Copernicus on Mercury : A study of parameters, Arch.
    • W Hartner, Ptolemy's and Copernicus' Mercury models : An accuracy test, Arch.

  4. References for Francesca
    • A reading of the diagrams: Piero della Francesca, Plato, Ptolemy and others, in Between art and science: Piero della Francesca (Italian), Arezzo-Sansepolcro, 1992 (1996), 385-406.

  5. References for Thabit
    • K P Moesgaard, Thabit ibn Qurra between Ptolemy and Copernicus : an analysis of Thabit's solar theory, Arch.

  6. References for Yunus
    • W Hartner, Ptolemy and Ibn Yunus on solar parallax, Arch.

  7. References for Galileo
    • S Drake, Ptolemy, Galileo, and scientific method, Stud.

  8. References for Ulugh Beg
    • M Shevchenko, An analysis of errors in the star catalogues of Ptolemy and Ulugh Beg, J.

  9. References for Al-Battani
    • Y Maeyama, Determination of the Sun's orbit (Hipparchus, Ptolemy, al-Battani, Copernicus, Tycho Brahe), Arch.

  10. References for Theon
    • D Pingree, An illustrated Greek astronomical manuscript : Commentary of Theon of Alexandria on the 'Handy tables' and scholia and other writings of Ptolemy concerning them, J.

  11. References for Brahe
    • Y Maeyama, Determination of the Sun's orbit (Hipparchus, Ptolemy, al-Battani, Copernicus, Tycho Brahe), Arch.

  12. References for Regiomontanus
    • M H Shank, Regiomontanus on Ptolemy, physical orbs, and astronomical fictionalism : Goldsteinian themes in the 'Defense of Theon against George of Trebizond', in New foundations in the history of astronomy, Pittsburgh, PA, 1999, Perspect.

  13. References for Neuberg
    • C Jacob, On a mechanical meaning of Ptolemy, Mobius, Neuberg and Pompeiu theorems, Rev.

  14. References for Al-Tusi Nasir
    • G Saliba, The role of the 'Almagest' commentaries in medieval Arabic astronomy : a preliminary survey of Tusi's redaction of Ptolemy's 'Almagest', Arch.

  15. References for Menelaus
    • O Schmidt, On the theorems of Ptolemy and Menelaus (Danish), Nordisk Mat.

  16. References for Eratosthenes
    • Ptolemy's reference to Eratosthenes in Almagest I.12, Centaurus 27 (2) (1984), 165-167.

  17. References for Mobius
    • C Jacob, On a mechanical meaning of Ptolemy, Mobius, Neuberg and Pompeiu theorems, Rev.


Additional material

  1. Ptolemy's hypotheses of astronomy
    • Ptolemy's hypotheses of astronomy .
    • Near the beginning of Book I of the Almagest, Ptolemy sets out the hypotheses of Ptolemaic astronomy:- .
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Ptolemy_hypotheses.html .

  2. Rudio's talk
    • Already under the first Ptolemy, at around 300 BC, Euclid had written his famous "Elements", which you all know, here.
    • And it was here, in Alexandria, where Claudius Ptolemy wrote his immortal work μεγάλη Σνταξiς, the Great Treatise, about 300 years later.
    • The name of Ptolemy, the founder of the world-view named after him, is connected with the culture of an era spanning almost one and a half millennia, and particularly with the culture of the Renaissance so profoundly that I cannot but dwell on this magnificent figure for a while.
    • But this view alone, which, after all, eventually suggests itself to any naive observer, did not suffice to establish the everlasting fame of Ptolemy.
    • Hipparchus had already observed that the seasons are of different durations 300 years before Ptolemy.
    • Now, how should one go about explaining this complicated motion without infringing the fundamental view of Antiquity that only allowed the use of uniform motion on a circle in order to explain celestial motions, and to which people adhered partly due to metaphysical reasons? Ptolemy solved this problem by assuming that every planetary motion was composed of two circular motions.
    • According to Ptolemy, every planet initially moves uniformly along a circle, the so-called epicycle, whose centre, in turn, moves along a second circle, the so-called deferent, which is circumscribed eccentrically around the Earth.
    • This is a brief sketch of the basis of Ptolemy's famous theory of epicycloids, which testifies to commendable ingenuity.
    • The first Greek manuscripts that were translated to Arabic were Ptolemy's Σύνταξiς, Euclid's Elements, Apollonius's conic sections and Archimedes's treatises on measuring the circle and on the sphere and cylinder.
    • The name "Almagest", which we use for Ptolemy's work even today, stems from this period.
    • On the other hand we notice that Greek geometry, culminating in Ptolemy's theories, is booming, particularly in Germany and Italy; and we marvel at the global scientific process that comes to a close in the mid-16th century, when the Ptolemaic world-view gives way to our modern one.
    • When he left Italy in 1468, he was in possession of a whole collection of valuable Greek manuscripts; among them the manuscript of Ptolemy's σunτaξiζ in particular, a gift from Bessarion.

  3. Rose's Greek mathematical literature
    • The Optics is preserved in a later compilation (we may render this freely, Outlines of Astronomical Mathematics; the epithet 'little' implies a contrast with Ptolemy's Almagest), dating from the third century A.D.
    • But the most famous name in this connexion is that of Ptolemy (Claudius Ptolemaeus), a contemporary of Theon, whose works, through the fault chiefly of readers and not of himself, kept astronomy at a standstill for more than a millennium after his death.
    • Besides some minor writings, Ptolemy also composed an astrological treatise in four books, hence known as the Tetrabiblos; for astrology was flourishing in his day, and had done so since about the third century B.C.
    • There exist also fragments of a more famous poet of this kind, Dorotheos, who is supposed to have lived about the same time as Ptolemy.
    • Not unconnected with mathematics is Music, and it may be mentioned here that Ptolemy wrote three books of Harmonies (i.e., musical theory, not what our musicians call harmony, for that did not exist in Greek music), which still survive, and another writer, perhaps of the third century A.D., Aristeides Quintilianus, also produced three books, On Music.
    • The worst, stylistically, is perhaps Ptolemy, who is prone to heavy, over-long sentences and a certain pretentious pedantry.
    • A native of Knidos, he spent a great part of his life in Alexandria, where he was tutor to the young Ptolemy VIII.
    • More important than any of these is the geography, or rather gazetteer, of Ptolemy, in eight books, under the general title of 'Greek Title'.
    • The absence of literary merit, and therefore of interest to the non-specialist, the immense amount of work involved in sifting the manuscripts tradition - this is extensive and includes maps, whose exact relation to Ptolemy himself is anything but certain - and the recurrent problems of identifying doubtful names and deciding whether a given mistake is due to the author or a copyist have combined to prevent a full critical edition being published hitherto.

  4. Galileo: 'Dialogue
    • So that hither may be referred the argument taken from a shot fired directly upwards from a cannon, as also that other used by Aristotle and Ptolemy, of the heavy bodies that, falling from on high, are observed to descend by a direct and perpendicular line to the surface of the Earth.
    • Now, that I may begin to untie these knots, I demand this of Simplicius: in case one should deny to Ptolemy and Aristotle that weights in falling freely from on high descend by a right and perpendicular line, that is, directly to the centre, what means would he use to prove it? .
    • On this fact many based the opinion, and Ptolemy amongst the rest, that, if the Earth should turn round with such great velocity, the stones and creatures upon it should be tossed into the sky and that there could not be a mortar strong enough to fasten buildings to their foundations so that they should not suffer a like extrusion.

  5. Proclus and the history of geometry as far as Euclid
    • Euclid lived in the time of the first Ptolemy, for Archimedes, whose life overlapped the reign of this Ptolemy too, mentions Euclid.
    • Furthermore, there is a story that Ptolemy once asked Euclid whether there was any shorter way to a knowledge of geometry than by the study of the Elements.

  6. Mathematicians and Music 2.1
    • And last among the Greeks to whom we shall refer is the celebrated mathematician, astronomer and geographer, Claudius Ptolemy, who flourished in the second century of the Christian era.
    • Apart from the Almagest, works on optics and mechanics, a book on stereographic projection, a book in which he tried to show that the possible number of dimensions is limited to three, and other works, Ptolemy wrote a remarkable treatise on music.
    • Ptolemy derived many scales in which the relations were similar; for example.

  7. Mathematics at Aberdeen 1
    • The man chosen for the Chair had to be well versed in Euclid, Ptolemy, Copernicus, Archimedes and other Mathematics.
    • Various ancient beliefs were mentioned and the theories of Ptolemy, Copernicus and Tycho Brahe considered.

  8. Plato Mathematics
    • From their efforts stem many of the greatest achievements of Greek mathematical science down to Ptolemy.
    • Ptolemy's astronomy is the ultimate descendant of the astronomy done in the Academy with the backing of Plato's recommendations for the sciences.

  9. Proclus on the Parallel Postulate
    • Posidonius, Ptolemy, Proclus and others tried either to prove the postulate from the others or to replace it with another they deemed more satisfactory.
    • This [fifth postulate] ought even to be struck out of the Postulates altogether; for it is a theorem involving many difficulties which Ptolemy, in a certain book, set himself to solve, and it requires for the demonstration of it a number of definitions as well as theorems.

  10. Pappus on mechanics
    • These matters have been treated by Ptolemy in his Mathematica.

  11. Women mathematicians by Dubreil-Jacotin
    • It is also believed that a commentary on the tables of Ptolemy, which has come down to us under Theon's name, is due to her.

  12. De Coste on Mersenne
    • But in the immortal sciences we have today two men who know precisely all that was known by Eudoxus and Hipparchus, those two famous antagonists, who became both the rivals of Euclid and the legitimate successors of Ptolemy; I am speaking of the Reverend Father Mersenne, Religious of the Minim Order, and Pierre Gassendi, two intellects who, despite the ignorance of the age, represent us in some measure, these two live monuments who, in spite of the waters of the universal deluge, preserve for the world all the arts and all the sciences, in which they excel, the one vying with the other.

  13. Heath: Everyman's Library 'Euclid' Introduction
    • They purport by their titles to be either "from the edition of Theon" or "from the school (i.e., lectures) of Theon." Moreover, Theon himself says in a passage of his commentary on the Syntaxis of Ptolemy:- "But that sectors in equal circles are to one another as the angles on which they stand has been proved by me in my edition of the Elements at the end of the sixth book," from which it is plain that the second part of Euclid VI, 33 containing this proof was added by Theon to the original.

  14. Valdivia aspects of maths
    • Euclid was director of the famous Museum of Alexandria when Egypt was ruled by Ptolemy I and published his work, later called the Elements of Euclid, at the age of thirty, in the year 300 BC.

  15. Apostol Project
    • Animation relates the sine and cosine of an angle with chord lengths of a circle, as explained in Ptolemy's Almagest.

  16. Brinkley Copley Medal
    • The notions of Ptolemy of cycles and epicycles, and the moving spheres of the heavens, were then current.

  17. Biography of Mathematics
    • Subsequently to Euclid new results were obtained, especially with the contributions of Archimedes, Ptolemy and Apollonius, but it does not vary the fundamental conception of reasoning on concrete magnitudes.

  18. Mathematicians and Music 2.2
    • The first of these is a Greek and Latin edition of Ptolemy's Harmony, and Porphyry's third century commentary on the same, with an extensive appendix by Wallis on ancient and modern music.

  19. Craig books
    • The literature of projections is very large, and its history presents the names of many of the most eminent mathematicians that have lived between the time of Ptolemy and the present day.

  20. Santalo honorary doctorate
    • In both his and Claudius Ptolemy's (90, 168) creations in trigonometry and his world system, it is difficult to decide which parts correspond to pure mathematics and which to the applied one.

  21. Kepler's Planetary Laws
    • Apart from frequent application of the trigonometrical propositions of Ptolemy (fl.

  22. George William Hill's new theory of Jupiter and Saturn
    • The earliest tables of the motions of Jupiter and Saturn, as well as those of the other large planets which have come down to us, are those contained in the Syntaxis of Claudius Ptolemy.

  23. Carathéodory: 'Conformal representation
    • Disregarding as trivial the Euclidean magnification (Ahnlichkeitstransformation) of the plane, we may say that the oldest known transformation of this kind is the stereographic projection of the sphere, which was used by Ptolemy (flourished in the second quarter of the second century; died after A.D.

  24. James Jeans addresses the British Association in 1934, Part 2
    • The plays of Aeschylus and Sophocles still thrill us with their vital human interest, but the scientific writings of Aristarchus and Ptolemy are dead - mere historical curiosities which leave us cold.

  25. De Morgan 1859 Preface
    • In this language we find a system of logic and of metaphysics: an astronomy worthy of comparison with that of Greece in its best days; above comparison, if some books of Ptolemy's Syntaxis be removed.


Quotations

  1. A quotation by Ptolemy
    • A quotation by Ptolemy .
    • http://www-history.mcs.st-andrews.ac.uk/Quotations/Ptolemy.html .

  2. Quotations by Kepler
    • And from this such small difference of eight minutes [of arc] it is clear why Ptolemy , since he was working with bisection [of the linear eccentricity], accepted a fixed equant point.
    • For Ptolemy set out that he actually did not get below ten minutes [of arc], that is a sixth of a degree, in making observations.
    • To us, on whom Divine benevolence has bestowed the most diligent of observers, Tycho Brahe, from whose observations this eight-minute error of Ptolemy's in regard to Mars is deduced, it is fitting that we accept with grateful minds this gift from God, and both acknowledge and build upon it.
    • So the uncertainty of the observation or (as they say) its spread, is greater than this error in Ptolemy's calculation [of the longitude].
    • Since divine goodness has bestowed on us Tycho Brahe the most diligent observer, from whose observations this error of 8 minutes in Ptolemy's calculation for Mars has been deduced; it is right that with grateful minds we acknowledge and profit by God's good gift.


Famous Curves

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Chronology

  1. Mathematical Chronology
    • Ptolemy produces many important geometrical results with applications in astronomy.
    • Arabic numerals are introduced into Europe with Gherard of Cremona's translation of Ptolemy's Almagest.
    • He uses Ptolemy's epicycle theory of the planets but believes they are controlled by the sun.

  2. Chronology for 1AD to 500
    • Ptolemy produces many important geometrical results with applications in astronomy.

  3. Chronology for 1300 to 1500
    • He uses Ptolemy's epicycle theory of the planets but believes they are controlled by the sun.

  4. Chronology for 1100 to 1300
    • Arabic numerals are introduced into Europe with Gherard of Cremona's translation of Ptolemy's Almagest.


EMS Archive

  1. 1912-13 Nov meeting
    • Swaminarayan, J C: "A determinantal proof of Ptolemy's theorem", [Title] .

  2. EMS Proceedings papers
    • Ptolemy's theorem and certain trigonometrical formulae .

  3. Edinburgh Mathematical Society Lecturers 1883-2016
    • (Gujerat College, Ahmedabad, India) A determinantal proof of Ptolemy's theorem .


BMC Archive

  1. BMC 2011


Gazetteer of the British Isles

  1. Oxford individuals
    • He died and is buried at Eton, but there is a large monument, with portrait bust resting on a southern hemisphere and statuettes of Ptolemy and Euclid, on the west wall of Merton Antechapel, near the south door.

  2. London Museums
    • In the Maps & Views case is a c1300 copy of Planudes' version of Ptolemy's world map, which estimates the circumference of the world as 18,000 miles.

  3. Cork
    • There is a memorial window in the Aula Maxima of the College, including figures of Archimedes, Leonardo da Vinci(?), Copernicus, Hipparchus, Galileo, Bacon, Napier, Newton, Pascal, Leibniz, Descartes, Strabo, Ptolemy surrounding a central panel of Euclid and Aristotle behind Boole.


Astronomy section

  1. The Structure of the Solar System
    • Hipparchus and Ptolemy who introduced epicycles, eccentrics, and equants respectively to increase the accuracy of the prediction.
    • Over the coming centuries debates arose over the validity of the Ptolemaic model with astronomers such as Thabit ibn Qurra drawing out inconsistencies between Ptolemy's geometrical model of the solar system and his physical view of it.
    • However, to achieve anything like accuracy, Copernicus had to place the sun a bit away from the center of the universe, and use even more epicycles than Ptolemy, which complicated the model.

  2. The Reaches of the Milky Way
    • Since heliocentric theories were unable to predict celestial motion any better than the epicycles from Ptolemy's work [See: Structure of the Solar System], the inability to determine parallax appeared to provide scientific evidence against the Copernican system.

  3. List of astronomers

  4. List of astronomers
    • Ptolemy, Claudius .


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