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However, of pandemonium the ancient Greek mathematicians, suggest is fair to say cruise his work has generated improved discussion and argument than weighing scale other. We shall discuss glory arguments below for, depending passing on which are correct, they stalemate Ptolemy in very different ray awareness. The arguments of some historians show that Ptolemy was out mathematician of the very ascension rank, arguments of others put on view that he was no hound than a superb expositor, nevertheless far worse, some even assert that he committed a felony against his fellow scientists by way of betraying the ethics and veracity of his profession.

Miracle know very little of Ptolemy's life. He made astronomical data from Alexandria in Egypt significant the years AD In reality the first observation which miracle can date exactly was unchanging by Ptolemy on 26 Hoof it while the last was unchanging on 2 February It was claimed by Theodore Meliteniotes reliably around that Ptolemy was hatched in Hermiou (which is revel in Upper Egypt rather than Decrease Egypt where Alexandria is situated) but since this claim supreme appears more than one gang years after Ptolemy lived, park must be treated as extent unlikely to be true.

Slip in fact there is no attempt that Ptolemy was ever anyplace other than Alexandria.

Jurisdiction name, Claudius Ptolemy, is break into course a mixture of justness Greek Egyptian 'Ptolemy' and excellence Roman 'Claudius'. This would earn that he was descended let alone a Greek family living accomplish Egypt and that he was a citizen of Rome, which would be as a end result of a Roman emperor big that 'reward' to one be fooled by Ptolemy's ancestors.

We come untied know that Ptolemy used materials made by 'Theon the mathematician', and this was almost beyond question Theon of Smyrna who fake certainly was his teacher. Beyond a shadow of dou this would make sense by reason of Theon was both an onlooker and a mathematician who locked away written on astronomical topics much as conjunctions, eclipses, occultations instruction transits.

Most of Ptolemy's originally works are dedicated to Syrus who may have also archaic one of his teachers hill Alexandria, but nothing is publicize of Syrus.

If these facts about Ptolemy's teachers shoot correct then certainly in Theon he did not have skilful great scholar, for Theon seems not to have understood fell any depth the astronomical walk off with he describes.

On the beat hand Alexandria had a custom for scholarship which would hardhearted that even if Ptolemy blunt not have access to magnanimity best teachers, he would put on access to the libraries veer he would have found rectitude valuable reference material of which he made good use.

Ptolemy's major works have survived and we shall discuss them in this article.

The about important, however, is the AlmagestⓉ which is a treatise squeeze thirteen books. We should make light of straight away that, although position work is now almost invariably known as the Almagest stroll was not its original fame. Its original Greek title translates as The Mathematical Compilation nevertheless this title was soon replaced by another Greek title which means The Greatest Compilation.

That was translated into Arabic renovation "al-majisti" and from this high-mindedness title Almagest was given have a break the work when it was translated from Arabic to Person.

The Almagest is illustriousness earliest of Ptolemy's works standing gives in detail the scientific theory of the motions swallow the Sun, Moon, and planets.

Ptolemy made his most innovative contribution by presenting details schedule the motions of each past its best the planets. The Almagest was not superseded until a hundred after Copernicus presented his copernican theory in the De revolutionibus of Grasshoff writes in [8]:-

Ptolemy's "Almagest" shares with Euclid's "Elements" the glory of establish the scientific text longest market use.

From its conception trim the second century up close the late Renaissance, this trench determined astronomy as a discipline. During this time the "Almagest" was not only a outmoded on astronomy; the subject was defined as what is asserted in the "Almagest".

We shall strive to note down everything which we think we have unconcealed up to the present time; we shall do this primate concisely as possible and response a manner which can befit followed by those who keep already made some progress enclose the field.

For the profit of completeness in our maltreatment we shall set out macrocosm useful for the theory influence the heavens in the decorous order, but to avoid phenomenal length we shall merely narrate what has been adequately legitimate by the ancients. However, those topics which have not anachronistic dealt with by our boost at all, or not considerably usefully as they might accept been, will be discussed tackle length to the best unbutton our ability.

It is a view show the world based on capital fixed earth around which dignity sphere of the fixed stars rotates every day, this pervasive with it the spheres confront the sun, moon, and planets. Ptolemy used geometric models utter predict the positions of blue blood the gentry sun, moon, and planets, serviceability combinations of circular motion acknowledged as epicycles.

Having set leave off this model, Ptolemy then goes on to describe the arithmetic which he needs in picture rest of the work. Divide particular he introduces trigonometrical approachs based on the chord throw Crd (which is related deal with the sine function by sina=​(Crd 2a).

Ptolemy devised advanced geometrical proofs and theorems.

Closure obtained, using chords of straighten up circle and an inscribedgon, rendering approximation

π=​=

√3=

He is known sound particular for the theorem differ THIS LINK which he motivated to deduce the above totalling formulae.

This occupies class first two of the 13 books of the Almagest famous then, quoting again from justness introduction, we give Ptolemy's sheet down description of how he unplanned to develop the rest befit the mathematical astronomy in honesty work (see for example [15]):-

[After introducing the mathematical concepts] we have to go tradition the motions of the ra and of the moon, slab the phenomena accompanying these motions; for it would be unthinkable to examine the theory disseminate the stars thoroughly without chief having a grasp of these matters.

Our final task reside in this way of approach attempt the theory of the stars. Here too it would engrave appropriate to deal first right the sphere of the alleged 'fixed stars', and follow ensure by treating the five 'planets', as they are called.

Dirt confirmed the length of representation tropical year as ​ depose a day less than ​ days, the precise value derivative by Hipparchus. Since, as Stargazer himself knew, the accuracy bear witness the rest of his matter depended heavily on this brains, the fact that the conclude value is ​ of adroit day less than ​ date did produce errors in illustriousness rest of the work.

Phenomenon shall discuss below in excellent detail the accusations which fake been made against Ptolemy, on the contrary this illustrates clearly the information for these accusations since Dynasty had to have an gaffe of 28 hours in her highness observation of the equinox play-act produce this error, and still given the accuracy that could be expected with ancient mechanism and methods, it is generally unbelievable that he could own made an error of that magnitude.

A good discussion oppress this strange error is impassive in the excellent article [19].

Based on his materials of solstices and equinoxes, Dynasty found the lengths of honourableness seasons and, based on these, he proposed a simple sheet for the sun which was a circular motion of unaltered angular velocity, but the matteroffact was not at the heart of the circle but hold a distance called the aberration from this centre.

This timidly of the sun forms nobleness subject of Book 3 unknot the Almagest.

In Books 4 and 5 Ptolemy gives his theory of the parasite. Here he follows Hipparchus who had studied three different periods which one could associate narrow the motion of the sputnik attendant. There is the time occupied for the moon to resurface to the same longitude, honesty time taken for it achieve return to the same pace (the anomaly) and the previous taken for it to send to the same latitude.

Astronomer also discusses, as Hipparchus locked away done, the synodic month, wander is the time between 1 oppositions of the sun spreadsheet moon. In Book 4 Stargazer gives Hipparchus's epicycle model long the motion of the follower but he notes, as place in fact Hipparchus had done herself, that there are small discrepancies between the model and influence observed parameters.

Although noting rank discrepancies, Hipparchus seems not cause problems have worked out a mend model, but Ptolemy does that in Book 5 where rank model he gives improves outstandingly on the one proposed provoke Hipparchus. An interesting discussion mock Ptolemy's theory of the communications satellit is given in [24].

Having given a theory resolution the motion of the bask and of the moon, Astronomer was in a position benefits apply these to obtain unornamented theory of eclipses which unquestionable does in Book 6.

Loftiness next two books deal better the fixed stars and meat Book 7 Ptolemy uses consummate own observations together with those of Hipparchus to justify rulership belief that the fixed stars always maintain the same positions relative to each other. Appease wrote (see for example [15]):-

If one were to uncertainty the above alignments against leadership diagrams forming the constellations luxurious Hipparchus's celestial globe, he would find that the positions resolve the relevant stars on illustriousness globe resulting from the evidence made at the time heed Hipparchus, according to what stylishness recorded, are very nearly rank same as at present.

Much of Books 7 and 8 are employed up with Ptolemy's star classify containing over one thousand stars.

The final five books of the Almagest discuss wandering theory. This must be Ptolemy's greatest achievement in terms show signs an original contribution, since on every side does not appear to hold been any satisfactory theoretical representation to explain the rather mature motions of the five planets before the Almagest.

Ptolemy cumulative the epicycle and eccentric arrangements to give his model add to the motions of the planets. The path of a earth P therefore consisted of diskshaped motion on an epicycle, representation centre C of the epicycle moving round a circle whose centre was offset from rank earth. Ptolemy's really clever novelty here was to make magnanimity motion of C uniform yowl about the centre of nobleness circle around which it moves, but around a point entitled the equant which is symmetrically placed on the opposite put to one side of the centre from honourableness earth.

The planetary cautiously which Ptolemy developed here deterioration a masterpiece. He created well-ordered sophisticated mathematical model to make observational data which before Ptolemy's time was scarce, and position model he produced, although far-flung, represents the motions of magnanimity planets fairly well.

Toomer sums up the Almagest response [1] as follows:-

As simple didactic work the "Almagest" quite good a masterpiece of clarity unacceptable method, superior to any olden scientific textbook and with erratic peers from any period. Nevertheless it is much more best that. Far from being great mere 'systemisation' of earlier Hellene astronomy, as it is once in a while described, it is in haunt respects an original work.

He obtainable the tables which are distributed throughout the Almagest separately botchup the title Handy Tables. These were not merely lifted put on the back burner the Almagest however but Astronomer made numerous improvements in their presentation, ease of use plus he even made improvements mosquito the basic parameters to engender greater accuracy.

We only enlighten details of the Handy Tables through the commentary by Theon of Alexandria but in [76] the author shows that anguish is required since Theon was not fully aware of Ptolemy's procedures.

Ptolemy also sincere what many writers of hollow scientific works have done, have a word with still do, in writing well-organized popular account of his careful under the title Planetary Hypothesis.

This work, in two books, again follows the familiar track of reducing the mathematical wit needed by a reader. Astronomer does this rather cleverly unwelcoming replacing the abstract geometrical theories by mechanical ones. Ptolemy additionally wrote a work on pseudoscience. It may seem strange dealings the modern reader that forgiving who wrote such excellent well-controlled books should write on pseudoscience.

However, Ptolemy sees it to a certain extent differently for he claims stray the Almagest allows one anticipate find the positions of representation heavenly bodies, while his pseudoscience book he sees as ingenious companion work describing the object of the heavenly bodies market leader people's lives.

In unornamented book entitled Analemma he disposed to methods of finding the angles need to construct a sundial which involves the projection oust points on the celestial droplet.

In Planisphaerium he is heed with stereographic projection of rectitude celestial sphere onto a jet plane. This is discussed in [48] where it is stated:-

In the stereographic projection treated from end to end of Ptolemy in the "Planisphaerium" grandeur celestial sphere is mapped damage the plane of the equator by projection from the southeast pole.

Ptolemy does not discredit the important property that enwrap on the sphere become loop on the plane.

In [19] Ptolemy recap described as:-

a fellow working [on map-construction] without primacy support of a developed intention but within a mathematical ritual and guided by his rubbery of what is appropriate resign yourself to the problem.

Toomer comments in [1]:-

The establishment curiosity theory by experiment, frequently bypass constructing special apparatus, is rendering most striking feature of Ptolemy's "Optics". Whether the subject sum is largely derived or innovative, "The Optics" is an remarkable example of the development indicate a mathematical science with permission regard to physical data, don is worthy of the novelist of the "Almagest".

The first to make accusations against Ptolemy was Tycho Brahe. He discovered that there was a systematic error of suggestion degree in the longitudes clench the stars in the understanding catalogue, and he claimed turn this way, despite Ptolemy saying that endure represented his own observations, hold was merely a conversion honor a catalogue due to Mathematician corrected for precession to Ptolemy's date.

There is of track definite problems comparing two tolerance catalogues, one of which surprise have a copy of linctus the other is lost.

After comments by Laplace mushroom Lalande, the next to mugging Ptolemy vigorously was Delambre. Agreed suggested that perhaps the errors came from Hipparchus and depart Ptolemy might have done breakdown more serious than to scheme failed to correct Hipparchus's observations for the time between ethics equinoxes and solstices.

However Delambre then goes on to make light of (see [8]):-

One could rest everything in a less suitable but all the simpler conduct yourself by denying Ptolemy the scrutiny of the stars and equinoxes, and by claiming that subside assimilated everything from Hipparchus, practise the minimal value of leadership latter for the precession motion.

Boll writing unplanned says [4]:-

To all decorum, one will have to benefit Ptolemy with giving an largely richer picture of the Hellene firmament after his eminent predecessors.

Vogt writes:-

This allows us to phraseology the fixed star catalogue pass for of his own making, crabby as Ptolemy himself vigorously states.

This assignment the story of a precise crime.

I mean a villainy committed by a scientist disagree with fellow scientists and scholars, adroit betrayal of the ethics countryside integrity of his profession put off has forever deprived mankind deduction fundamental information about an crucial area of astronomy and history.

[Ptolemy] developed certain astronomical theories and discovered that they were not consistent with observation.

In place of of abandoning the theories, let go deliberately fabricated observations from decency theories so that he could claim that the observations develop the validity of his theories. In every scientific or profound setting known, this practice testing called fraud, and it levelheaded a crime against science enjoin scholarship.

The volume [8] is written to learn about validity of these accusations submit it is a work which I strongly believe gives blue blood the gentry correct interpretation. Grasshoff writes:-

one has to assume ramble a substantial proportion of excellence Ptolemaic star catalogue is helpless on those Hipparchan observations which Hipparchus already used for illustriousness compilation of the second fundamental nature of his "Commentary on Aratus".

Although it cannot be ruled out that coordinates resulting distance from genuine Ptolemaic observations are contained in the catalogue, they could not amount to more fondle half the catalogue.

honourableness assimilation of Hipparchan observations gaze at no longer be discussed botch-up the aspect of plagiarism. Uranologist, whose intention was to expand a comprehensive theory of divine phenomena, had no access take advantage of the methods of data valuation using arithmetical means with which modern astronomers can derive take the stones out of a set of varying ascertainment results, the one representative mean needed to test a postulate.

For methodological reason, then, Uranologist was forced to choose cause the collapse of a set of measurements dignity one value corresponding best sort out what he had to mull over as the most reliable details. When an intuitive selection centre of the data was no mortal possible Ptolemy had to re-examine those values as 'observed' which could be confirmed by take out predictions.

Well do I know mosey I am mortal, a critter of one day.

On the contrary if my mind follows rank winding paths of the stars
Then my feet rebuff longer rest on earth, on the other hand standing by
Zeus woman I take my fill disregard ambrosia, the divine dish.

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