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Jakarta Pusat, Jakarta D. Ditambahkan sejak 13 Feb, ID iklan: The organizer of each host country has mobilized its countrys best astronomy and astrophysics educators to make problems for theoretical, observational and data analysis test. The process assured high quality exam problems are delivered to the participants of IOAA. In several years the IOAA competition system has accumulated many high quality astronomy and astrophysics problems which are certainly valuable educational tool in astronomy and astrophysics for high school students. Such valuable treasure will be more instructive and easier to understand and to be used when it is improved by adding basic theories, instructions and more explanantion.

Jiko bumi fidok berrofosi ferhodop sumbunyo binfong-binfong fidok okon berpindoh fempof, honyo mofohori, buIon don pIonef sojo yong Iefoknyo bergeser perIohon-Iohon. Tofo koordinof ifu odoIoh fofo koordinof yong fifik ocuonnyo bergerok sesuoi dengon gerok rofosi bumi don disebuf fofo HoIomon : 33 koordinof khofuIisfiwo EquoforioI Coordinofe Sysfem.

Iodong-kodong cukup suIif memohomi suofu koordinof yong ocuonnyo bergerok reIofif ferhodop pengomof, oIeh koreno ifu, mempeIojori suofu fofo koordinof Ioin yong ocuonnyo diom ferhodop pengomof dopof membonfu kifo doIom memohomi fofo koordinof khofuIisfiwo.

Di doIom pengomofon binfong dibufuhkon informosi fenfong posisi binfong yong okon diomofi. Podo soof okon mengorohkon feropong kifo perIu mengefohui dimono Iefok bendo Iongif yong okon diomofi ifu, opo Iogi jiko bendo Iongif ifu redup sehinggo fok nompok dengon mofo feIonjong.

Unfuk menyofokon posisi sebuoh bendo Iongif dopof digunokon beberopo mocom fofo koordinof yong semuonyo merupokon sisfem koordinof boIo fonpo memperhifungkon jorok dori pusof boIo. Semuonyo mempunyoi duo fifik kufub. Semuonyo menggunokon Iinfong don bujur seperfi Iinfong don bujur geogrofis sebogoi penenfu posisi bendo Iongif. Perbedoonnyo odoIoh doIom fifik-fifik don Iingkoron-Iingkoron ocuon yong digunokon. Lingkoron-Iingkoron bujur semuonyo merupokon Iingkoron besor.

Lingkoron Iinfong semokin keciI jiko semokin dekof dengon kufub boIo. Pada saat Plato, maka filosofi hampir terlupakan, dan Aristotel, maka penggantinya Theophrastus dan beberapa doxographers menyediakan kami dengan informasi yang masih sedikit. Namun, kita tahu dari Aristotle bahwa Thales, juga dari Miletus, precedes Anaximander. Hal ini belum pasti apakah Thales sebenarnya adalah guru dari Anaximander namun tidak ada keraguan yang telah dipengaruhi oleh Anaximander Thales' teori tentang air semua yang ada atau yang lain berasal dari air.

Satu hal yang tidak belum pasti adalah bahwa bahkan Yunani kuno dianggap Anaximander menjadi Monist dari sekolah yang dimulai di Miletus diikuti dengan Thales dan Anaximander selesai dengan Anaximenes.

Memang, Berbagai Sejarah III, 17 menjelaskan bahwa filosof terkadang meninggalkan kepuasan dari masing-pemikiran untuk menangani masalah- masalah politik. Sangat mungkin bahwa pemimpin Miletus dikirim dia di sana sebagai pembuat undang-undang untuk membuat konstitusi atau hanya untuk mempertahankan koloni's allegiance. Untuk itu, ia menjadi tidak lagi hanya dalam waktu, tetapi sebuah sumber yang dapat memberikan perpetually melahirkan akan apapun.

Aristotle menulis metafisika, saya III bahwa Pra-Socratics adalah mencari elemen yang merupakan segala sesuatu. Untuk Anaximander, prinsip hal, maka semua unsur zat, tidak ditentukan dan bukan merupakan elemen seperti air Thales' melihat. Baik itu sesuatu pertengahan antara udara dan air, atau antara udara dan api, kental daripada udara dan api, atau lebih halus dari air dan bumi. Dia postulated yang apeiron sebagai bahan yang, walaupun tidak secara langsung nampak kepada kami, dapat menjelaskan opposites dia melihat di sekelilingnya.

Anaximander menjelaskan bagaimana empat elemen kuno fisika udara, bumi, air dan api yang dibentuk, dan bagaimana Bumi dan makhluk hidup di tanah yang dibentuk melalui interaksi mereka.

Tidak seperti Pra-Socratics lainnya, dia tidak pernah menetapkan prinsip ini tepat, dan itu sudah dipahami secara umum misalnya, oleh Aristotle dan oleh Saint Augustine sebagai jenis utama kekacauan. Menurut dia, the Universe berasal pemisahan opposites dalam hal purba.

It pemeluk yang opposites dari panas dan dingin, basah dan kering, dan mengarahkan gerakan sesuatu; seluruh host bentuk perbedaan dan kemudian tumbuh yang ditemukan di seluruh dunia " untuk ia percaya ada banyak. Anaximander mempertahankan bahwa semua hal yang mati akan kembali ke elemen dari mana mereka datang apeiron.

Satu fragmen yang hidup dari menulis Anaximander berurusan dengan hal ini. Simplicius dikirim sebagai kutip, yang menjelaskan perubahan yang seimbang dan saling elemen: Konsep ini untuk kembali ke elemen asal sering revisited setelahnya, terutama oleh Aristotle, [11] dan Yunani oleh penulis cerita sedih Euripides: It mengkonfirmasikan bahwa pra-Socratic filosof telah membuat satu upaya awal untuk proses demythify fisik.

Nya kontribusi besar sejarah telah menulis prosa tertua dokumen tentang Universe dan asal usul kehidupan, karena ini ia sering disebut sebagai "Bapa dari kosmologi" dan pendiri astronomi. Namun, palsu Plutarch menyatakan bahwa ia masih dilihat sebagai dewa-dewa celestial badan. Dalam model, Bumi masih sangat floats di pusat yang tak terbatas, tidak didukung oleh apapun. Tetap di tempat yang sama karena kelalaian ", sebuah sudut pandang yang dianggap Aristotel berbakat, tapi palsu, di Atas Langit.

Kapal induk yang dihuni membentuk dunia, yang dikelilingi oleh sebuah surat edaran kelautan massa. Seperti model konsep yang diizinkan celestial badan bisa lulus di bawah ini. It goes lebih dari Thales' klaim dari dunia apung di air, yang Thales menghadapi masalah apa yang akan menjelaskan ini berisi laut, sementara Anaximander dipecahkan dengan memperkenalkan konsep yang tak terbatas apeiron. Gambaran tentang Anaximander model dari alam semesta. Pada asalnya, setelah pemisahan dan panas dingin, bola api muncul dari yang dikelilingi Bumi seperti kulit pada pohon.

Kesomplok bola ini selain untuk membentuk sisa Universe. It mirip sistem berongga konsentris roda, penuh dengan api, dengan rims tindik oleh lubang seperti orang-orang dari seruling.

Akibatnya, Minggu adalah api yang satu dapat melihat melalui lubang yang sama seperti ukuran Bumi di terjauh roda, dan gerhana corresponded dengan kemacetan itu lubang. Diameter dari matahari adalah roda dua puluh tujuh kali bahwa dari Bumi atau dua puluh delapan, tergantung pada sumber-sumber [16] dan lunar roda, adalah api yang kurang kuat, delapan belas atau sembilan belas kali. Lubang yang dapat mengubah bentuk, sehingga menjelaskan fase lunar. Bintang- bintang dan planets, terletak dekat, [17] mengikuti model yang sama.

Temuan ini dilakukan niscaya dia pertama untuk mewujudkan arah miring dari Zodiac sebagai filsuf Roma Pliny the Elder laporan dalam Sejarah Alam II, 8. It is a little awal untuk menggunakan istilah ecliptic, tetapi dia bekerja di ilmu astronomi dan mengkonfirmasikan bahwa dia harus mengamati inklinasi falak yang berkaitan dengan pesawat dari Bumi untuk menjelaskan musim.

The doxographer dan teolog Aetius atribut yang tepat untuk Pythagoras pengukuran dari arah miring. Para pemikir dunia yang seharusnya muncul dan menghilang untuk beberapa saat, dan beberapa saat dilahirkan perished lain. Mereka menyatakan bahwa gerakan ini adalah kekal, "untuk tanpa gerakan, tidak boleh ada generasi, tidak ada kerusakan". Cicero menulis bahwa ia berbeda atribut allah ke terhingga alam.

Pada waktu dari sejarah pemikiran Yunani, beberapa pemikir conceptualized satu dunia Plato, Aristotle, Anaxagoras dan Archelaus , sedangkan yang lain sebagai gantinya speculated pada keberadaan sejumlah dunia, terus atau tidak terus Anaximenes, Heraclitus, Empedocles dan Diogenes. Guntur tanpa kilat adalah hasil angin yang terlalu lemah untuk memancarkan apapun api, tapi cukup kuat untuk menghasilkan suara. J kilat yang kilat tanpa guntur adalah pukulan dari udara yang disperses dan jatuh, sehingga api yang kurang aktif untuk istirahat gratis.

Thunderbolts adalah hasil yang lebih kental dan kekerasan aliran udara. Dengan mempertimbangkan keberadaan orangtua, dia menyatakan bahwa binatang sprang dari laut lalu panjang.

Binatang yang pertama lahir terperangkap di kulit berduri, tetapi karena mereka mendapat tua, kulit akan menjadi kering dan rusak. Anaximander dari Miletus yang dianggap dari tdk atas air dan bumi muncul baik ikan atau seluruhnya fishlike binatang. Di dalam binatang ini, mengambil bentuk manusia dan embryos dilaksanakan tahanan sampai masa " " remaja, hanya itu, setelah buka burst binatang ini, bisa laki- laki dan perempuan keluar, sekarang bisa makan sendiri.

Walaupun dia tidak memiliki teori seleksi alam, beberapa orang menganggap dia sebagai evolusi yang paling kuno pendukung. Teori sebuah air dari keturunan orang itu kembali disusun abad kemudian sebagai air ape hipotesa. Ini pra- Darwinian konsep Mei tampaknya aneh, mengingat modern pengetahuan dan metode ilmiah, karena mereka hadir lengkap penjelasan alam semesta ketika menggunakan huruf tebal dan keras untuk membuktikan hypotheses.

Namun, mereka menggambarkan awal dari sebuah fenomena yang kadang-kadang disebut "Yunani keajaiban": Ini adalah sangat prinsip ilmiah pemikiran, yang kemudian maju lebih lanjut dengan metode penelitian ditingkatkan. Peta mungkin terinspirasi Yunani sejarawan Hecataeus dari Miletus untuk mengambil versi yang lebih akurat.

Strabo dilihat baik sebagai yang pertama setelah geographers Homer. Mereka menunjukkan jalan, kota, perbatasan, dan fitur-fitur geologis. Anaximander dari inovasi adalah untuk mewakili seluruh tanah didiami diketahui Yunani kuno. Seperti sebuah prestasi yang lebih penting daripada di pertama muncul. Anaximander kemungkinan besar drew peta ini untuk tiga alasan. Kedua, Thales mungkin akan menemukan lebih mudah untuk meyakinkan Ionian kota-negara untuk bergabung dalam federasi untuk mendorong Median ancaman jika dia gila seperti alat.

Akhirnya, filosofis gagasan global perwakilan di dunia hanya untuk kepentingan pengetahuan yang cukup alasan untuk merancang satu. Sesungguhnya menyadari laut cembung, ia mungkin telah dirancang-Nya pada peta yang sedikit dibulatkan permukaan logam. Di Laut Aegean telah dekat peta dari pusat dan ditutupi oleh tiga benua, sendiri terletak di tengah-tengah laut dan pulau-pulau terpencil seperti laut dan sungai. The Nile dialirkan ke laut selatan, memisahkan Libya yang merupakan nama untuk bagian-kemudian dikenal benua Afrika dari Asia.

Ia juga menyebut-Nya pada pengukuran waktu dan asosiasi dengan adanya dia di Yunani yang gnomon. Dalam Lacedaemon, ia ikut berpartisipasi dalam pembangunan, atau setidak-tidaknya untuk penyesuaian, dari sundials untuk menunjukkan solstices dan equinoxes. Dalam waktu, yang tadinya hanya gnomon vertikal tiang atau tongkat horisontal yang terpasang di pesawat. Posisi nya bayangan pada pesawat menunjukkan waktu dalam sehari. Seperti terlihat bergerak melalui kursus, matahari yang mengacu melengkung dengan ujung diproyeksikan bayangan yang singkat pada siang hari, karena pada saat yang selatan.

Variasi di ujung posisi pada siang hari matahari menunjukkan waktu dan musim; bayangan yang lama adalah pada titik balik matahari musim dingin dan singkat pada titik balik matahari musim panas.

Namun, penemuan yang gnomon itu sendiri tidak dapat dikaitkan dengan Anaximander karena penggunaan, serta pembagian bulan menjadi dua belas bagian, datang dari orang Babilon. Mereka itu, menurut Herodotus' Histories II, , yang telah memberikan Yunani seni waktu pengukuran. Kemungkinan bahwa dia bukan yang pertama untuk menentukan solstices, karena tidak ada perhitungan yang diperlukan.

Di sisi lain, equinoxes tidak sesuai dengan titik tengah antara posisi saat solstices, karena pemikiran Babilon. Sebagai Suda tampaknya menyarankan, sangat mungkin bahwa dengan pengetahuan geometri, ia menjadi yang pertama Yunani akurat untuk menentukan equinoxes. Pliny the Elder juga menyebut ini anekdote II, 81 , menyatakan bahwa ia datang dari "kagum inspirasi", karena bertentangan dengan Cicero, yang tidak mengaitkan dengan prediksi peramalan.

Pythagoras of Samos Born: He is an extremely important figure in the development of mathematics yet we know relatively little about his mathematical achievements. Unlike many later Greek mathematicians, where at least we have some of the books which they wrote, we have nothing of Pythagoras's writings. The society which he led, half religious and half scientific, followed a code of secrecy which certainly means that today Pythagoras is a mysterious figure.

We do have details of Pythagoras's life from early biographies which use important original sources yet are written by authors who attribute divine powers to him, and whose aim was to present him as a god-like figure.

What we present below is an attempt to collect together the most reliable sources to reconstruct an account of Pythagoras's life. There is fairly good agreement on the main events of his life but most of the dates are disputed with different scholars giving dates which differ by 20 years. Some historians treat all this information as merely legends but, even if the reader treats it in this way, being such an early record it is of historical importance. Pythagoras's father was Mnesarchus [12] and [13] , while his mother was Pythais [8] and she was a native of Samos.

Mnesarchus was a merchant who came from Tyre, and there is a story [12] and [13] that he brought corn to Samos at a time of famine and was granted citizenship of Samos as a mark of gratitude. As a child Pythagoras spent his early years in Samos but travelled widely with his father.

There are accounts of Mnesarchus returning to Tyre with Pythagoras and that he was taught there by the Chaldaeans and the learned men of Syria. It seems that he also visited Italy with his father.

Little is known of Pythagoras's childhood.

All accounts of his physical appearance are likely to be fictitious except the description of a striking birthmark which Pythagoras had on his thigh. It is probable that he had two brothers although some sources say that he had three. Certainly he was well educated, learning to play the lyre, learning poetry and to recite Homer.

There were, among his teachers, three philosophers who were to influence Pythagoras while he was a young man.

One of the most important was Pherekydes who many describe as the teacher of Pythagoras. The other two philosophers who were to influence Pythagoras, and to introduce him to mathematical ideas, were Thales and his pupil Anaximander who both lived on Miletus.

In [8] it is said that Pythagoras visited Thales in Miletus when he was between 18 and 20 years old. By this time Thales was an old man and, although he created a strong impression on Pythagoras, he probably did not teach him a great deal. However he did contribute to Pythagoras's interest in mathematics and astronomy, and advised him to travel to Egypt to learn more of these subjects. Thales's pupil, Anaximander, lectured on Miletus and Pythagoras attended these lectures. Anaximander certainly was interested in geometry and cosmology and many of his ideas would influence Pythagoras's own views.

In about BC Pythagoras went to Egypt. This happened a few years after the tyrant Polycrates seized control of the city of Samos.

There is some evidence to suggest that Pythagoras and Polycrates were friendly at first and it is claimed [5] that Pythagoras went to Egypt with a letter of introduction written by Polycrates. In fact Polycrates had an alliance with Egypt and there were therefore strong links between Samos and Egypt at this time. The accounts of Pythagoras's time in Egypt suggest that he visited many of the temples and took part in many discussions with the priests.

According to Porphyry [12] and [13] Pythagoras was refused admission to all the temples except the one at Diospolis where he was accepted into the priesthood after completing the rites necessary for admission. It is not difficult to relate many of Pythagoras's beliefs, ones he would later impose on the society that he set up in Italy, to the customs that he came across in Egypt.

For example the secrecy of the Egyptian priests, their refusal to eat beans, their refusal to wear even cloths made from animal skins, and their striving for purity were all customs that Pythagoras would later adopt.

Porphyry in [12] and [13] says that Pythagoras learnt geometry from the Egyptians but it is likely that he was already acquainted with geometry, certainly after teachings from Thales and Anaximander. Polycrates abandoned his alliance with Egypt and sent 40 ships to join the Persian fleet against the Egyptians. Pythagoras was taken prisoner and taken to Babylon. Iamblichus writes that Pythagoras see [8]: Whilst he was there he gladly associated with the Magoi He also reached the acme of perfection in arithmetic and music and the other mathematical sciences taught by the Babylonians Polycrates had been killed in about BC and Cambyses died in the summer of BC, either by committing suicide or as the result of an accident.

The deaths of these rulers may have been a factor in Pythagoras's return to Samos but it is nowhere explained how Pythagoras obtained his freedom. Darius of Persia had taken control of Samos after Polycrates' death and he would have controlled the island on Pythagoras's return. This conflicts with the accounts of Porphyry and Diogenes Laertius who state that Polycrates was still in control of Samos when Pythagoras returned there.

Pythagoras made a journey to Crete shortly after his return to Samos to study the system of laws there. Back in Samos he founded a school which was called the semicircle. Iamblichus [8] writes in the third century AD that: They do this because they think one should discuss questions about goodness, justice and expediency in this place which was founded by the man who made all these subjects his business. Outside the city he made a cave the private site of his own philosophical teaching, spending most of the night and daytime there and doing research into the uses of mathematics Pythagoras left Samos and went to southern Italy in about BC some say much earlier.

Iamblichus [8] gives some reasons for him leaving. First he comments on the Samian response to his teaching methods: The Samians were not very keen on this method and treated him in a rude and improper manner. This was, according to Iamblichus, used in part as an excuse for Pythagoras to leave Samos: Pythagoras was dragged into all sorts of diplomatic missions by his fellow citizens and forced to participate in public affairs.

He knew that all the philosophers before him had ended their days on foreign soil so he decided to escape all political responsibility, alleging as his excuse, according to some sources, the contempt the Samians had for his teaching method. Pythagoras founded a philosophical and religious school in Croton now Crotone, on the east of the heel of southern Italy that had many followers. Pythagoras was the head of the society with an inner circle of followers known as mathematikoi.

The mathematikoi lived permanently with the Society, had no personal possessions and were vegetarians. They were taught by Pythagoras himself and obeyed strict rules. The beliefs that Pythagoras held were [2]: Both men and women were permitted to become members of the Society, in fact several later women Pythagoreans became famous philosophers. The outer circle of the Society were known as the akousmatics and they lived in their own houses, only coming to the Society during the day.

They were allowed their own possessions and were not required to be vegetarians. Of Pythagoras's actual work nothing is known. His school practised secrecy and communalism making it hard to distinguish between the work of Pythagoras and that of his followers.

Certainly his school made outstanding contributions to mathematics, and it is possible to be fairly certain about some of Pythagoras's mathematical contributions.

First we should be clear in what sense Pythagoras and the mathematikoi were studying mathematics. They were not acting as a mathematics research group does in a modern university or other institution.

There were no 'open problems' for them to solve, and they were not in any sense interested in trying to formulate or solve mathematical problems. Rather Pythagoras was interested in the principles of mathematics, the concept of number, the concept of a triangle or other mathematical figure and the abstract idea of a proof. As Brumbaugh writes in [3]: In fact today we have become so mathematically sophisticated that we fail even to recognise 2 as an abstract quantity.

There is another step to see that the abstract notion of 2 is itself a thing, in some sense every bit as real as a ship or a house. Pythagoras believed that all relations could be reduced to number relations. As Aristotle wrote: This generalisation stemmed from Pythagoras's observations in music, mathematics and astronomy. In fact Pythagoras made remarkable contributions to the mathematical theory of music. He was a fine musician, playing the lyre, and he used music as a means to help those who were ill.

Pythagoras studied properties of numbers which would be familiar to mathematicians today, such as even and odd numbers, triangular numbers, perfect numbers etc.

However to Pythagoras numbers had personalities which we hardly recognise as mathematics today [3]: This feeling modern mathematics has deliberately eliminated, but we still find overtones of it in fiction and poetry. Ten was the very best number: Of course today we particularly remember Pythagoras for his famous geometry theorem. Although the theorem, now known as Pythagoras's theorem, was known to the Babylonians years earlier he may have been the first to prove it.

Proclus, the last major Greek philosopher, who lived around AD wrote see [7]: Again Proclus, writing of geometry, said: Heath [7] gives a list of theorems attributed to Pythagoras, or rather more generally to the Pythagoreans. Also the Pythagoreans knew the generalisation which states that a polygon with n sides has sum of interior angles 2n - 4 right angles and sum of exterior angles equal to four right angles.

To say that the sum of two squares is equal to a third square meant that the two squares could be cut up and reassembled to form a square identical to the third square.

This is certainly attributed to the Pythagoreans but it does seem unlikely to have been due to Pythagoras himself. This went against Pythagoras's philosophy the all things are numbers, since by a number he meant the ratio of two whole numbers. However, because of his belief that all things are numbers it would be a natural task to try to prove that the hypotenuse of an isosceles right angled triangle had a length corresponding to a number. It is thought that Pythagoras himself knew how to construct the first three but it is unlikely that he would have known how to construct the other two.

He also recognised that the orbit of the Moon was inclined to the equator of the Earth and he was one of the first to realise that Venus as an evening star was the same planet as Venus as a morning star. Primarily, however, Pythagoras was a philosopher. In addition to his beliefs about numbers, geometry and astronomy described above, he held [2]: Further Pythagorean doctrine In [3] their practical ethics are also described: Pythagoras's Society at Croton was not unaffected by political events despite his desire to stay out of politics.

Pythagoras went to Delos in BC to nurse his old teacher Pherekydes who was dying. He remained there for a few months until the death of his friend and teacher and then returned to Croton. In BC Croton attacked and defeated its neighbour Sybaris and there is certainly some suggestions that Pythagoras became involved in the dispute. Pythagoras escaped to Metapontium and the most authors say he died there, some claiming that he committed suicide because of the attack on his Society.

Iamblichus in [8] quotes one version of events: He approached Pythagoras, then an old man, but was rejected because of the character defects just described. When this happened Cylon and his friends vowed to make a strong attack on Pythagoras and his followers. Thus a powerfully aggressive zeal activated Cylon and his followers to persecute the Pythagoreans to the very last man. Because of this Pythagoras left for Metapontium and there is said to have ended his days.

This seems accepted by most but Iamblichus himself does not accept this version and argues that the attack by Cylon was a minor affair and that Pythagoras returned to Croton.

Certainly the Pythagorean Society thrived for many years after this and spread from Croton to many other Italian cities. Gorman [6] argues that this is a strong reason to believe that Pythagoras returned to Croton and quotes other evidence such as the widely reported age of Pythagoras as around at the time of his death and the fact that many sources say that Pythagoras taught Empedokles to claim that he must have lived well after BC.

The evidence is unclear as to when and where the death of Pythagoras occurred. Certainly the Pythagorean Society expanded rapidly after BC, became political in nature and also spilt into a number of factions.

In BC the Society [2]: Its meeting houses were everywhere sacked and burned; mention is made in particular of "the house of Milo" in Croton, where 50 or 60 Pythagoreans were surprised and slain. Those who survived took refuge at Thebes and other places.

Article by: He was the first Greek, and the first man in general, to present an explicit argument for a heliocentric model of the solar system, placing the Sun, not the Earth, at the center of the known universe.

He was influenced by the Pythagorean Philolaus of Kroton, but, in contrast to Philolaus, he had both identified the central fire with the Sun, as well as putting other planets in correct order from the Sun. His astronomical ideas were rejected in favor of the geocentric theories of Aristotle and Ptolemy until they were successfully revived nearly years later by Copernicus and extensively developed and built upon by Johannes Kepler and Isaac Newton.

The crater Aristarchus on the Moon is named in his honor. Statue of Aristarchus at Aristotle University in Thessalonica, Greece Heliocentrism The only work usually attributed to Aristarchus which has survived to the present time, On the Sizes and Distances of the Sun and Moon, is based on a geocentric world view.

The latter diameter is known from Archimedes to have been Aristarchus's actual value. Though the original text has been lost, a reference in Archimedes' book The Sand Reckoner describes another work by Aristarchus in which he advanced an alternative hypothesis of the heliocentric model. Archimedes wrote: You King Gelon are aware the 'universe' is the name given by most astronomers to the sphere the center of which is the center of the Earth, while its radius is equal to the straight line between the center of the Sun and the center of the Earth.

This is the common account as you have heard from astronomers. But Aristarchus has brought out a book consisting of certain hypotheses, wherein it appears, as a consequence of the assumptions made, that the universe is many times greater than the 'universe' just mentioned.

His hypotheses are that the fixed stars and the Sun remain unmoved, that the Earth revolves about the Sun on the circumference of a circle, the Sun lying in the middle of the orbit, and that the sphere of fixed stars, situated about the same center as the Sun, is so great that the circle in which he supposes the Earth to revolve bears such a proportion to the distance of the fixed stars as the center of the sphere bears to its surface.

Aristarchus thus believed the stars to be very far away, and saw this as the reason why there was no visible parallax, that is, an observed movement of the stars relative to each other as the Earth moved around the Sun. The stars are in fact much farther away than the distance that was generally assumed in ancient times, which is why stellar parallax is only detectable with telescopes.

The geocentric model, consistent with planetary parallax, was assumed to be an explanation for the unobservability of the parallel phenomenon, stellar parallax. The rejection of the heliocentric view was common, as the following passage from Plutarch suggests On the Apparent Face in the Orb of the Moon: Cleanthes a contemporary of Aristarchus and head of the Stoics thought it was the duty of the Greeks to indict Aristarchus of Samos on the charge of impiety for putting in motion the Hearth of the universe i.

Aristarchus is known to have also studied light and vision. The implicit false solar parallax of slightly under 3' was used by astronomers up to and including Tycho Brahe, ca. Aristarchus pointed out that the Moon and Sun have nearly equal apparent angular sizes and therefore their diameters must be in proportion to their distances from Earth. He thus concluded that the diameter of the Sun was about 20 times larger than the diameter of the Moon; which, although wrong, follows logically from his data.

It also leads to the conclusion that the Sun's diameter is almost seven times greater than the Earth's; the volume of Aristarchus's Sun would be almost times greater than the Earth. Perhaps this difference in sizes inspired the heliocentric model. Syntaxis book 4 chapter 2. Its empirical foundation was the month eclipse cycle, cited by Ptolemy as source of the "Babylonian" month, which was good to a fraction of a second 1 part in several million. It is found on cuneiform tablets from shortly before B.

Due to near integral returns in lunar and solar anomaly, eclipses months apart exceptionally never deviated by more than an hour from a mean of days plus 1 hour, the value given by Ptolemy at op cit. Thus, estimation of the length of the month was ensured to have relative accuracy of 1 part in millions. Embedded in the Great Year was a length of the month agreeing with the Babylonian value to 1 part in tens of millions, decades before Babylon is known to have used it.

There are indications that Babylon's month was exactly that of Aristarchus, which if true renders it effectively certain that one party obtained it from the other or from a common source. Aristarchus's lunar conception represents an advance of science in several respects.

Previous estimates of the length of the month were in error by seconds Meton, B. The attribution of a mean motion to so complex a motion as the moon's was possibly new. The only ancient scientist listed for two different values is Aristarchus.

It is now widely suspected that these are among the earliest surviving examples of continued fraction expressions. The most obvious interpretations are precisely computable from the manuscript numbers.

Both denominators are relatable to Aristarchus, whose summer solstice was years after Meton's and whose Great Year was years. The difference between the sidereal and tropical years is identical to precession. The former value is accurate within a few seconds. The latter is erroneous by several minutes.

Both are close to the values later used by Hipparchus and Ptolemy, and the precession indicated is almost precisely 1 degree per century, a much-too-low value. Unfortunately, 1 degree per century precession was used by all later astronomers until the Arabs.

The correct value in Aristarchus's time was about 1. Claudius Ptolemy Born: 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 It was claimed by Theodore Meliteniotes in around 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. 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 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 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 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 Grasshoff writes in [8]: From its conception in the second century up to the late Renaissance, this work determined astronomy as a science.

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

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 [15]: 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 BC. 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 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 [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.

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: 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 [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.

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. 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 [48] where it is stated: 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: 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]: 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 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 [8]: 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 says [4]: Vogt showed clearly in his important paper [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.

Vogt writes: The most recent accusations of forgery made against Ptolemy came from Newton in [12]. He begins this book by stating clearly his views: 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]: 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. 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: 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.

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]: 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.

Copernicus jadi yakin atas kebenaran hipotesa "heliocentris" ini, dan tatkala dia menginjak usia empat puluh tahun dia mulai mengedarkan buah tulisannya diantara teman-temannya dalam bentuk tulisan-tulisan ringkas, mengedepankan cikal bakal gagasannya sendiri tentang masalah itu. Di tahun , tatkala usianya menginjak enam puluh tahun, Copernicus mengirim berkas catatan-catatan ceramahnya ke Roma. Dalam buku itu Copernicus dengan tepat mengatakan bahwa bumi berputar pada porosnya, bahwa bulan berputar mengelilingi matahari dan bumi, serta planet- planet lain semuanya berputar mengelilingi matahari.

Jelaslah dengan demikian, teori Copernicus telah merevolusionerkan konsep kita tentang angkasa luar dan sekaligus sudah merombak pandangan filosofis kita. Coming from Scania, then part of Denmark, now part of modern-day Sweden, Brahe was well known in his lifetime as an astronomer and alchemist. Tycho Brahe was granted an estate on the island of Hven and the funding to build the Uraniborg, an early research institute, where he built large astronomical instruments and took many careful measurements.

After disagreements with the new king in , he was invited by the Czech king and Holy Roman emperor Rudolph II to Prague, where he became the official imperial astronomer. Here, from until his death in , he was assisted by Johannes Kepler. Kepler would later use Tycho's astronomical information to develop his own theories of astronomy. As an astronomer, Tycho worked to combine what he saw as the geometrical benefits of the Copernican system with the philosophical benefits of the Ptolemaic system into his own model of the universe, the Tychonic system.

He is generally referred to as "Tycho" rather than by his surname "Brahe", as was common in Scandinavia at the time. No one before Tycho had attempted to make so many redundant observations, and the mathematical tools to take advantage of them had not yet been developed. He did what others before him were unable or unwilling to do — to catalogue the planets and stars with enough accuracy to determine whether the Ptolemaic or Copernican system was more valid in describing the heavens.

The incorrect form of his name, Tycho de Brahe, appeared only much later. Knudstrup borg; Swedish: His twin brother died before being baptized. Tycho wrote a Latin ode Wittendorf , p. He also had two sisters, one older Kirstine Brahe and one younger Sophia Brahe.

Otte Brahe, Tycho's father, was a nobleman and an important figure at the court of the Danish King. His mother, Beate Bille, also came from an important family that had produced leading churchmen and politicians. An epitaph, originally from Knutstorp, but now on a plaque near the church door, shows the whole family, including Tycho as a boy.

It is hard to say exactly where Tycho was educated in his childhood years, and Tycho himself provides no information on this topic, but the sources quoted below agree that he took a Latin School education from the age of six until he was twelve years old. On 19 April , Tycho began his studies at the University of Copenhagen.

There, following the wishes of his uncle, he studied law but also studied a variety of other subjects and became interested in astronomy. It was, however, the eclipse which occurred on 21 August , particularly the fact that it had been predicted, that so impressed him that he began to make his own studies of astronomy, helped by some of the professors.

He downloadd an ephemeris and books such as Sacrobosco's Tractatus de Sphaera, Apianus's Cosmographia seu descriptio totius orbis and Regiomontanus's De triangulis omnimodis. I've studied all available charts of the planets and stars and none of them match the others.

There are just as many measurements and methods as there are astronomers and all of them disagree. What's needed is a long term project with the aim of mapping the heavens conducted from a single location over a period of several years. Tycho realized that progress in the science of astronomy could be achieved not by occasional haphazard observations, but only by systematic and rigorous observation, night after night, and by using instruments of the highest accuracy obtainable.

He was able to improve and enlarge the existing instruments, and construct entirely new ones. His sister Sophia assisted Tycho in many of his measurements. These jealously guarded measurements were "usurped" by Kepler following Tycho's death. A subsequent duel in the dark resulted in Tycho losing the bridge of his nose. From this event Tycho became interested in medicine and alchemy. Ihren wrote that when Tycho's tomb was opened in 24 June green marks were found on his skull, suggesting copper.

In April , Tycho returned home from his travels and his father wanted him to take up law, but Tycho was allowed to make trips to Rostock, then on to Augsburg where he built a great quadrant , Basel, and Freiburg.