Cláudio Ptolomeu

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Nascimento: cerca de 85 no Egipto
Falecimento: cerca de 165 em Alexandria, Egipto

 

Cláudio Ptolomeu   ¾    Nome oriundo da forma latinizada Claudius Ptolemaeus, que por sua vez, para alguns historiadores o seu nome é uma mistura do egípcio grego ( Ptolemaios ) e do romano ( Claudius ) e, em consequência, conclui-se que ele era descendente de famílias grega  e romana. Matemático, astrónomo, geógrafo grego e que ainda era astrólogo por estar intimamente ligado, na época,  à astronomia, nasceu na cidade de Ptolomaida de Tebaida  no ano de 90 d.C. Não há conhecimento de  registro biográfico deste grande sábio, senão que passou um período entre 120 e 160 em Alexandria, Egipto, e foi o mais célebre astrónomo da Antiguidade. Este período foi determinado com base em observações astronómicas descritas nas suas obras e que tenham sido efectuadas entre 26 de Março de 127 e 2 de Fevereiro de 141.

Foram mencionadas pelos grandes historiadores da época várias obras, inclusive as de Matemática, que desapareceram, dentre elas podemos citar  " Peru ropon "  ( Sobre o equilíbrio ) e  " Peri diastáseos " ( Sobre a dimensão ) através da qual tenta provar que todo espaço é tridimensional. No campo da Geografia, Ptolomeu publicou uma obra, que tornou-se clássica até o século XVI,  intitulada  " Geographike Hyphegesis " ( Introdução à geografia ) que trouxe para a época, grande repercussão, tanto que foi editada por mais de duzentos anos. Ela foi dividida  em oito livros contendo 27 mapas sobre os países mediterrâneos e expõe uma concepção de que a extensão da Ásia era mais para o leste e existia um continente ao sul do oceano Índico. Apesar da obra ter apresentado vários erros e omissões de informações, Ptolomeu foi considerado um sábio que influenciou várias gerações. No campo da Física, destacamos duas de suas obras que foram traduzidas do árabe, no século III, para o latim, cuja título foi  " Optike  "  ( Óptica ) na qual aborda  cor, reflexão, refração e espelhos de várias formas, como também, estuda os fenômenos em relação aos astros. Na outra obra intitulada  " Harmonike " ( Harmonias ) ele trata de assuntos relacionados à acústica no qual emprega a teoria matemática dos sons na música grega. Na Astronomia, Ptolomeu sintetiza no ano de 140 d.C. através de sua principal obra  " He megále sýntaxis  "  ( A grande sintaxe ), também, intitulada  " Ho megas astronomos "  ( O grande astrônomo ) ou, ainda,  denominada pelos árabes de  " Almagesto " os conhecimentos astronômicos de seus antecessores. Dentre eles podemos citar: Pitágoras,  afirmava que por a Terra ser esférica, a esfera mais perfeita de todas as figuras geométricas, onde todas se moviam em torno de um fogo central, todos os corpos celestes devem ser esféricos. Ademais, ele fazia uma analogia entre o Sol e a Lua e afirmava que: O universo era infinito e eterno a partir da convicção na repetição cíclica dos acontecimentos em um período de dez mil anos que foi denominado de  " Grande Ano ". Filolau  concebeu um sistema cosmológico, composto de dez corpos celestes, no qual a Terra se deslocava no espaço como os outros astros; Platão, fixou as órbitas dos planetas e afirmou que os planetas movem-se em círculos perfeitos assim como todos os objetos celestes; Eudoxo, aceitou o modelo cosmológico de Platão introduzindo um sistema composto de esferas homocêntricas, centrada em uma Terra imóvel com eixos de rotação diferentes e agindo uns sobre os outros; Aristóteles **, aperfeiçoou o modelo de Eudoxo e propõe um modelo celestial composto por 55 esferas concêntricas às quais os corpos celestes estavam ligados e que rodavam a diferentes velocidades com a Terra ao centro. Além disso, o movimento  dos corpos celestes  era circular e uniforme; Heráclides do Ponto ou Pôntico foi o primeiro astrónomo que admitiu a rotação da Terra em torno do seu eixo, explicando assim o movimento observado nas estrelas e imaginou que Mercúrio e Vénus girassem em torno do Sol; Aristarco de Samo defendeu a tese de que  a Terra está em rotação em torno de si mesma e, ao mesmo tempo, em torno do Sol. Este é o primeiro modelo heliocêntrico do Universo; Apolônio, ao invés de utilizar a teoria da revolução das esferas para mostrar o movimento dos planetas, inventou a teoria dos epicíclos, segundo a qual cada estrela gira ao redor de um ponto.

A partir da  Idade Média o que prevalecia era o sistema cosmológico de Aristóteles e sua filosofia a qual era incorporada na teologia medieval, pois os teólogos rejeitavam qualquer teoria que não conferisse à Terra o lugar privilegiado de centro do universo. O Movimento Inicial de Aristóteles tornou-se o Deus da teologia Cristã, a esfera mais exterior do Movimento Inicial começou-se a identificar com o Céu Cristão, e a posição da Terra no centro de tudo era entendida em termos da preocupação de que o Deus Cristão tinha para os problemas da humanidade. Assim sendo, as idéias originadas, em grande parte, por filósofos gregos pagãos foram batizadas pela igreja Católica e assumiram o poder de dogmas religiosos: desafiar esta visão do Universo deixava de ser apenas um assunto científico, passava a ser também teológico, e sujeitava os incrédulos ao considerável e não sempre benevolente poder da Igreja.

Todas as teorias defendidas pelos sábios que confrontavam com as de Aristóteles  foram rejeitadas e, então começou uma nova fase com Hiparco, que para alguns historiadores foi o inventor do astrolábio, e o seu modelo geocêntrico clássico que preocupado em explicar com exactidão o movimento retrógrado dos planetas, estabeleceu um modelo em que a Terra permanecia fixa no centro de um círculo giratório ( deferente ).

A s t r o l á b i o     p l a n i s f é r i c o

Astrolábio [Do latim Astrolabium; Do grego Astrolábion: aster, estrela + lambanein, alcançar., pelo latim medieval astrolabiu.] Ele era usado para determinar a altura dos astros acima do horizonte, sobretudo a do Sol. Atualmente, foi aperfeiçoado, e é um dos instrumentos  fundamentais da astrometria. Feito em ferro, a rodela graduada era suspensa por um anel e apontado para o objeto celeste de modo a poder-se tomar a altura deste. Este instrumento astronômico, talvez, o mais antigo, cuja invenção é atribuída a Hiparco de Nicéia,  o pai da astronomia e trigonometria, foi astrônomo e matemático grego (séc. II a. C. entre 200 a.C. a 150 a.C.), remonta ao século III a.C., na Grécia. Ptolomeu designa por astrolábio a esfera armilar, ¾  Instrumento astronômico, constituído de numerosos anéis metálicos, que representam os principais círculos da esfera celeste   ¾   que os árabes combinaram com o globo celeste e aperfeiçoaram criando assim o astrolábio esférico.

 

Após a exposição dos sistemas cosmológicos dos precursores de Ptolomeu   ¾    que para outros historiadores, ele foi o inventor do Astrolábio  ¾    vamos apresentar a sua teoria que surgiu quando da sua análise dos sistemas anteriores.

Epiciclo  ¾    é um pequeno círculo imaginário da esfera celeste, cujo centro se encontra na circunferência de um outro círculo maior, ou seja é um círculo que um astro supostamente descrevia em torno de um ponto, o qual, por sua vez, descrevia um outro círculo em torno da Terra, chamado deferente, ou, às vezes, em torno do centro de outro epiciclo. Deferente  ¾   é um círculo imaginário da antiguidade para tentar explicar o movimento aparente dos planetas.

Ele desenvolveu seu sistema geocêntrico conhecido, também, por teoria ptolomaica ou geocêntrica  baseado no sistema de Hiparco e introduziu o equante na teoria do epiciclo, ou seja, ele supôs que a Terra esteja imóvel no centro do Universo, e que o Sol, a Lua e os planetas giram em torno dela, descrevendo órbitas complexas. Cada planeta realiza dois movimentos circulares simultâneos, excepto o Sol e a Lua por não possuírem epiciclos. Definiremos equante como sendo um ponto em que  o centro do epiciclo movia-se em volta desse ponto com movimento circular uniforme e que está simetricamente colocado do lado oposto do centro da terra.

No entanto, ao observar os planetas, Ptolomeu notou que eles forneciam elementos incompatíveis com o esquema elaborado e introduziu um sistema em que foram introduzidos oitenta epiciclos através do qual os planetas se movimentariam.

Ptolomeu foi o primeiro a justificar a descrição do seu universo baseado no sistema relatado por Aristóteles. A sua justificativa foi elaborada usando modelos geométricos para prognosticar as posições do sol, da lua e dos planetas usando combinações de movimento circular conhecidas por epiciclos. Após ter montado o modelo ele descreveu matematicamente, introduzindo métodos trigonométricos baseados em uma função chamada de corda representada simbolicamente por  ' Crd ' que hoje equivale a  função  modernamente chamada de seno. O sena=(Crd 2a)/120

Ptolomeu criou novas provas geométricas e novos teoremas e obteve excelentes resultados. Posteriormente, ele descreveu a disposição geométrica dos corpos do sistema solar com o planeta Terra fixo e no centro do universo.

Os teólogos medievais adoptaram sua concepção por se tratar de um sistema em que a terra continuaria sendo o centro do Universo.

Este sistema dominou a astronomia durante catorze séculos quando surgiu outro pensador, o astrónomo polonês Nicolau Copérnico que, após análise de toda a teoria dos seus antecessores, construiu o modelo heliocêntrico onde os planetas deveriam girar em torno do Sol e introduziu  alguns epiciclos em seu sistema com a finalidade de explicar alguns movimentos planetários. Este sistema foi endossado por Galileo Galilei, rejeitado por Tycho Brahe e, posteriormente, Kepler, revendo o modelo de Copérnico e utilizando as mensurações do seu mestre Brahe com respeito as posições planetárias, concluiu que os círculos  poderiam ser substituídos pelas elipses, eliminando, desta forma, toda a ambiguidade da teoria heliocêntrica eliminando, assim, os epiciclos e as deferentes. Ademais, propôs uma órbita elípticas para o planeta Marte, após analisar os dados de Brahe sobre o referido planeta.

Nesta grande obra, que foi dividida em treze livros, além de apresentar uma vasta compilação dos conhecimentos astronómicos da antiguidade e o seu modelo geocêntrico, ela apresenta nos dois primeiros livros as bases científicas, da altura, para o resto da obra.  

No terceiro livro, Ptolomeu aborda temas relacionados com a solstício  e o equinócio  em que baseado em  suas observações encontrou a duração das estações que serviu para propor a teoria do Sol.

Nos livros quarto e quinto, ele sugere a teoria da Lua e que define as suas dimensões e a distância entre ela e o Sol. Aborda também, os eclipses do Sol e da Lua, etc. 

No livro sexto, Ptolomeu constrói a teoria dos eclipses, tendo por base as teorias do Sol e da Lua.

Nos dois livros seguintes, ele trata de assunto que diz respeito as estrelas fixas e no livro sete justifica as suas convicções de que as estrelas fixas sempre mantêm as mesmas posições em relação às outras. Nestes livros ele discute, também, a  precessão dos equinócios que foi atribuída por ele, a sua descoberta à Hiparco. Ptolomeu ainda inseriu nestes livros o catálogo de estrelas que contendo 1.022 estrelas fixas, sendo que 172 das quais, foram descobertas por ele e as demais por Hiparco, pois este elaborou o primeiro catálogo estelar.

Os cinco livros finais foram dedicados exclusivamente  à teoria planetária constituindo, assim, sua contribuição mais original à astronomia. 

Em um desses livros, Ptolomeu explicar a construção do Astrolábio e expõe questões de trigonometria retilínea e esférica. Estas questões foram posteriormente aprofundadas em " Haplosis epipháneias sphaíras " ( Sobre a projeção das esferas ).

Ptolomeu, ainda escreveu uma obra intitulada " Planetary Hypothesis " ( Hipóteses Planetárias ) o qual foi dividido em dois livros abordando assuntos relacionados aos planetas e a teoria geométrica. Os temas abordados foram de maior leveza e simplicidade. 

Em um outro livro intitulado " Analemma " ele discutiu  detalhes da projeção ortogonal  dos pontos da esfera celeste sobre três planos, pois, precisa construir um relógio de sol que envolvia a projeção de pontos na esfera celestial. Em  " Planisphaerium " ele se preocupa com projeção de estereográfica da esfera celestial sobre um objeto no espaço. Além disso, ele propõe nova demonstração para o postulado das paralelas de Euclides.

Brahe analisando os trabalhos de Ptolomeu,  foi o primeiro a descobrir que havia erro nas longitudes das estrelas constante nos livros em que foi inserido o catálogo de estrelas e pode demonstrar as imperfeições das Tábuas Alfonsinas, baseadas, no sistema de Ptolomeu. Esses erros eram de  um mês ao predizer o evento , e as tabelas de Copérnico erraram por vários dias. As análises não pararam por aí, pois, em seguida, apareceu os comentários de Laplace e Delambre e mais recentemente as de Newton que chega a declarar o seguinte.

Esta é a história de um crime científico. ... Eu quero dizer um crime cometido por um cientista contra os cientistas da mesma categoria e estudiosos, uma traição das éticas e integridade da profissão dele que privou gênero humano de informação fundamental sempre sobre uma área importante de astronomia e história.

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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. However, of all the ancient Greek mathematicians, it is fair to say that his work has generated more discussion and argument than any other. 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. He made astronomical observations from Alexandria in Egypt during the years AD 127-41. 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. Certainly this would make sense since Theon of Smyrna was both an observer and a mathematician who had written on astronomical topics such as conjunctions, eclipses, occultations and transits. 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 of Smyrna he did not have a great scholar, for Theon of Smyrna 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 most important, however, is the Almagest which is a treatise in thirteen books. We should say straight away that, although the work is now almost always known as the Almagest that was not its original name. Its original Greek title translates as The Mathematical Compilation but this title was soon replaced by another Greek title which means The Greatest Compilation. This was translated into Arabic as "al-majisti" and from this the title Almagest was given to the work when it was translated from Arabic to Latin.

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. The Almagest was not superseded until a century after Copernicus presented his heliocentric theory in the De revolutionibus of 1543. Grasshoff writes in [8]:-

Ptolemy's "Almagest" shares with Euclid's "Elements" the glory of being the scientific text longest in use. From its conception in the second century up to the late Renaissance, this work determined astronomy as a science. During this time the "Almagest" was not only a work on astronomy; the subject was defined as what is described in the "Almagest".

Ptolemy describes himself very clearly what he is attempting to do in writing the work (see for example [15]):-

We shall try to note down everything which we think we have discovered up to the present time; we shall do this as concisely as possible and in a manner which can be followed by those who have already made some progress in the field. For the sake of completeness in our treatment we shall set out everything useful for the theory of the heavens in the proper order, but to avoid undue length we shall merely recount what has been adequately established by the ancients. However, those topics which have not been dealt with by our predecessors at all, or not as usefully as they might have been, will be discussed at length to the best of our ability.

Ptolemy first of all justifies his description of the universe based on the earth-centred system described by Aristotle. It is a view of the world based on a fixed earth around which the sphere of the fixed stars rotates every day, this carrying with it the spheres of the sun, moon, and planets. 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. In particular he introduces trigonometrical methods based on the chord function Crd (which is related to the sine function by sin a = (Crd 2a)/120).

Ptolemy devised new geometrical proofs and theorems. He obtained, using chords of a circle and an inscribed 360-gon, the approximation

p = 3 ¹⁷/₁₂₀ = 3.14166

and, using 3 = chord 60,

3 = 1.73205.

He used formulas for the Crd function which are analogous to our formulas for sin(a + b), sin(a - b) and sin a/2 to create a table of the Crd function at intervals of 1/2 a degree.

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 [15]):-

[After introducing the mathematical concepts] we have to go through the motions of the sun and of the moon, and the phenomena accompanying these motions; for it would be impossible to examine the theory of the stars thoroughly without first having a grasp of these matters. Our final task in this way of approach is the theory of the stars. Here too it would be appropriate to deal first with the sphere of the so-called 'fixed stars', and follow that by treating the five 'planets', as they are called.

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. He confirmed the length of the tropical year as 1/300 of a day less than 365 ¹/₄ days, the precise value obtained by Hipparchus. 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 365 ¹/₄ 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 [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. This theory of the sun forms the subject of Book 3 of the Almagest.

In Books 4 and 5 Ptolemy gives his theory of the moon. Here he follows Hipparchus who had studied three different periods which one could associate with the motion of the moon. There is the time taken for the moon to return to the same longitude, the time taken for it to return to the same velocity (the anomaly) and the time taken for it to return to the same latitude. 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 [24].

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 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 [15]):-

If one were to match the above alignments against the diagrams forming the constellations on Hipparchus's celestial globe, he would find that the positions of the relevant stars on the globe resulting from the observations made at the time of Hipparchus, according to what he recorded, are very nearly the same as at present.

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.

The final five books of the Almagest discuss planetary theory. 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. The path of a planet P therefore consisted of circular motion on an epicycle, the centre C of the epicycle moving round a circle whose centre was offset from the earth. 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.

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

As a didactic work the "Almagest" is a masterpiece of clarity and method, superior to any ancient scientific textbook and with few peers from any period. But it is much more than that. Far from being a mere 'systemisation' of earlier Greek astronomy, as it is sometimes described, it is in many respects an original work.

We will return to discuss some of the accusations made against Ptolemy after commenting briefly on his other works. He published the tables which are scattered throughout the Almagest separately under the title Handy Tables. 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 [77] 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. This work, in two books, again follows the familiar route of reducing the mathematical skills needed by a reader. Ptolemy does this rather cleverly by replacing the abstract geometrical theories by mechanical ones. Ptolemy also wrote a work on astrology. It may seem strange to the modern reader that someone who wrote such excellent scientific books should write 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.

In a book entitled Analemma he discussed methods of finding the angles need to construct a sundial which involves the projection of points on the celestial sphere. In Planisphaerium he is concerned with stereographic projection of the celestial sphere onto a plane. This is discussed in [49] where it is stated:-

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 [19] Ptolemy is described as:-

... a man working [on map-construction] without the support of a developed theory but within a mathematical tradition and guided by his sense of what is appropriate to the problem.

Another work on Optics is in five books and in it Ptolemy studies colour, reflection, refraction, and mirrors of various shapes. Toomer comments in [1]:-

The establishment of theory by experiment, frequently by constructing special apparatus, is the most striking feature of Ptolemy's "Optics". Whether the subject matter is largely derived or original, "The Optics" is an impressive example of the development of a mathematical science with due regard to physical data, and is worthy of the author of the "Almagest".

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 [14].

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. There is of course definite problems comparing two star catalogues, one of which we have a copy of while the other is lost.

After comments by Laplace and Lalande, the 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 [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. Boll writing in 1894 says [4]:-

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 [78] 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. Vogt writes:-

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 [12]. He begins this book by stating clearly his views:-

This is the story of a scientific crime. ... I mean a crime committed by a scientist against fellow scientists and scholars, a betrayal of the ethics and integrity of his profession that has forever deprived mankind of fundamental information about an important area of astronomy and history.

Towards the end Newton, having claimed to prove every observation claimed by Ptolemy in the Almagest was fabricated, writes [12]:-

[Ptolemy] developed certain astronomical theories and discovered that they were not consistent with observation. Instead of abandoning the theories, he deliberately fabricated observations from the theories so that he could claim that the observations prove the validity of his theories. In every scientific or scholarly setting known, this practice is called fraud, and it is a crime against science and scholarship.

Although the evidence produced by Brahe, Delambre, Newton and others certainly do show that Ptolemy's errors are not random, this last quote from [12] is, I [EFR] believe, a crime against Ptolemy (to use Newton's own words). The book [8] is written to study validity of these accusations and it is a work which I strongly believe gives the correct interpretation. Grasshoff writes:-

... one has to assume that a substantial proportion of the Ptolemaic star catalogue is grounded on those Hipparchan observations which Hipparchus already used for the compilation of the second part of his "Commentary on Aratus". Although it cannot be ruled out that coordinates resulting from genuine Ptolemaic observations are included in the catalogue, they could not amount to more than half the catalogue.

... the assimilation of Hipparchan observations can no longer be discussed under the aspect of plagiarism. 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. When an intuitive selection among the data was no longer possible ... 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 [11]):-

Well do I know that I am mortal, a creature of one day.
But if my mind follows the winding paths of the stars
Then my feet no longer rest on earth, but standing by
Zeus himself I take my fill of ambrosia, the divine dish.