# Ptolemy

Claudius Ptolemy ( Claudius Ptolemy Κλαύδιος Πτολεμαῖος Greek, Latin Claudius Ptolemy, * 100, possibly in Ptolemais Hermeiou, Egypt; † after 160, probably in Alexandria ) was a Greek mathematician, geographer, astronomer, astrologer, music theorist and philosopher. In particular, his three works on astronomy, geography and astrology were in Europe to the early modern period as an important large data collections and scientific standard works.

So Ptolemy wrote the Mathematike syntaxis ( " mathematical compilation " ), later Megiste syntaxis ( " the largest compilation " ), today Almagest (derived from the Arabic al - maǧisṭī ) called treatise on mathematics and astronomy in 13 books. She was until the end of the Middle Ages a standard work of astronomy and a detailed reference star catalog contained a refinement proposed by Hipparchus of Nicaea geocentric, which was later named after him Ptolemaic worldview.

Thus he rejected as most of his contemporaries of Aristarchus of Samos and Seleucus of Seleucia represented heliocentric world view, which should only be 1300 years later enforced by Nicolaus Copernicus, Johannes Kepler and Galileo Galilei in Europe.

## Astronomy

### Ptolemaic world view

According to Ptolemy, the earth is fixed in the center of the universe. All other celestial bodies ( moon, sun, the 5 planets and the starry sky ) move in crystal spheres as a perfectly respectable circular orbits ( deferent ) around this center. To make astronomical observations with this system in line, it was necessary, however, all the heavenly bodies in their orbits more circles ( epicycles ) to let them go to this deferents - see epicycle - and sometimes even more movements to the primary epicycle. The use of such ( against each other slightly inclined ) orbits could bring his model with the then freiäugigen observations in accordance Ptolemy.

In the language of today's mathematics could be Ptolemy's calculation as empirical precursor of Fourier analysis indicate that the secondary periods of the planetary orbits (among the midpoint equation ) were determined empirically.

The Ptolemaic world was in its orbit prediction accuracy superior to the heliocentric worldview of Nicolaus Copernicus (16th century). The Ptolemaic system was replaced in 1600 by the also still tychonische geocentric world system (named after Tycho Brahe ). Only Kepler's discovery that the planets go around the sun in ellipses, then led to a time sufficiently accurate and astronomers generally accepted model of the Copernican worldview. Ptolemy's calculation methods were highly accurate ( long time also more accurate than the kepler between ) and in its basic idea as a method of calculation also true, but not in its philosophical interpretation that everything turn around the earth as the center. The breakthrough and success of kepler between calculations was less due to the fact that the sun and not the earth was the center of the movements, but in the fact that Kepler elliptic orbits and no more circular orbits used, resulting in greater compliance with the Tycho Brahe and later Galileo Galilei led actually measured planetary data.

### Criticism

More recently, the benefits of Ptolemy, however, were rated much more critical. Even Tycho Brahe spoke in 1600 of "fraud". 1817 gave him the French astronomer and mathematician Jean Baptiste Joseph Delambre - ago fake and fictitious observations, preconceived opinions, lies and plagiarism. This was repeated in 1977 and again in 1985 by the English astronomer Robert Russell Newton in full. According to Newton So are almost all observations supposedly homemade by Ptolemy be accepted fictitious or of Hipparchus, whose lengths only 2 ° 40 ', the value of accumulated precession, were added ( were correct 3 ° 40' been ). This damning verdict on Ptolemy BL van der Waerden has in his 1988 book " The Astronomy of the Greeks " connected.

On the other hand, already presented in 1796 Pierre Simon Laplace, a simple explanation: The difference of one degree of arc let be justified by an equally large error in the former theory of the sun's motion. Bradley E. Schaefer in 2002 came to the conclusion that a considerable number of observational data mentioned by Ptolemy have this (or his assistants ) even won. He had, however, if foreign, older data to better fit his model as his own, this is copied without explicit source data. This approach was at a time when they will not even docked today's usual standards of academic papers, quite common.

### More

Another astronomical work of Ptolemy are his " planet hypothesis ", in which he was using the results of the Almagest, to make statements about the dimensions of the universe at large. So he appreciated due to its model, the mean distance from the Sun than 1,210 ( actually: 23,480 ) and the distance to the sphere of fixed stars than 20,000 Earth radii. Shown therein is also, as an illustrative mechanical model of the cosmos can be built.

Another remembered mainly for practical purposes the collection are his " Handy Tables". In the " Phaseis " ( Sunrises and downfalls of stars with weather signs ), he also made it a star catalog based on the movements of the stars throughout the year together. For the application of mathematics to astronomical problems come from him, the two headings " analemma " and " Planisphaerium ". Astronomically also worth mentioning is the preserved on a stele " Kanobusinschrift ".

His chronological data concerning astronomical records allocates data Ptolemy the Egyptian calendar. To avoid ambiguity, he calls for nocturnal events the outgoing and the early Egyptian days. Due to that precise information, the respective occurrences in the Julian calendar are precisely datable.

## Mathematics

The only known independent mathematical work is the only surviving in Proclus ' treatise on the parallel postulate ", in which he wanted to give a proof of the parallel axiom of Euclid, which is mathematically but demonstrably false. Other mathematical models have been incorporated in the above-mentioned primary application based astronomical fonts.

So it comes from the set of Ptolemy. This mathematical theorem is true for tendon quadrangles. ( An inscribed quadrilateral is a square, to which a circuit can be constructed by all four corners). The set of Ptolemy states that if a cyclic quadrilateral, the sum is the product of the two diagonals from the product of opposing side lengths. Thus ac bd = ef applies. Since symmetrical trapezoids have a radius, we obtain for the symmetric legs b = d and the diagonal e = f the special case of ac b2 = e2. The sentence also applies to rectangles, which also have a radius. Here then a = c, so that the set of Ptolemy contains the Pythagorean theorem as a special case: a2 b2 = e2. Like the Pythagorean Theorem is reversible, the set of Ptolemy.

In the Almagest (XIII 10) there is the following structure of the regular five- or decagon: For given radius (diameter [AB ] ) of the required five or decagon is the radius [ OB] halved ( center M ) and the circle around M by C drawn. The intersection of this circle with the diameter [AB ] is the point D. The route [OD ] is the side of the associated decagon, the segment [ CD] is the page of the corresponding pentagon. The radius [ OC ] is also the side of the associated hexagon. The construction is based on two sets of the elements of the Euclid ( 11 IV ) and (III 39).

Important to simplify its astronomical calculations by Ptolemy in the Almagest (I 10) calculated table of chords for the area was up with increment Such tendons panels used as a substitute for a sine table, as the following applies:

As an example of the accuracy achieved, the claim is to serve from the Almagest:

In the sixties system, this means

Thus, about a 5-digit accuracy is achieved as the calculator displays:

In the figure:

In the unit circle of the Pythagorean theorem then has the form:

## Geography

In addition to the summary canon of major cities Ptolemy wrote the Geographia ( Geographike Hyphegesis, Explicatio Geographic, " Geographical Guide"), in which he recorded the familiar world and its inhabitants.

As a reference for longitude ( ± 180 ° ), he defined the meridian used until the 19th century by what he called " makaron nesoi " (Latin: " insulae fortunatae " ), today's Canary Islands ( Ferro meridian ). His definition of latitude is still valid ( Equator 0 °, poles ± 90 °). He also puts in his hypothesis of the unknown southern continent Terra Australis dar. Ptolemy was like before Aristotle known that the earth is a sphere; he presented to their representation in a leaf level more suitable projections before. He made various improvements to the earlier work of Marinos of Tyre. However, he took advantage of second-hand information or legends so that its representations, in particular the treated peoples are often inaccurate or even misleading. He also dealt with the calculations of the Earth's circumference by Eratosthenes and Posidonius. He took the wrong results of the latter, which then went to the well-known literature and up to Christopher Columbus on too small a circumference of the earth of about 17,000 nautical miles ( 30,000 km ) were close.

## Music Theory

Ptolemy also wrote the three- books, " harmony ", the main resulting music-theoretical work of late antiquity by Aristoxenus and Euclid. He tried - as probably Eratosthenes - a compromise between Aristoxenus and the Pythagoreans, on which later Boethius oriented. Mathematically, he took the position of Euclid, moral and terminology but built on the musical perception doctrine of Aristoxenus. He handed in his harmonies many details of older ancient music theorist, as the tetrachords ( Tongeschlechter ) of Archytas, Eratosthenes and Didymus, which would otherwise be lost.

## Optics and epistemology

His work optics deals with the properties of light. He treated experimentally and mathematically, inter alia, reflection, refraction and color. In addition, optical illusions are mentioned. In the philosophical treatise peri kriteriou kai hegemonikou (Latin de iudicandi facultate et animi principatu, "From the judgment and the mind " ) represent a mixture of Neo-Platonic and Stoic views.

In addition, he also wrote the two -part work Criterion for epistemology, according to which for the recognition of truth alone is enough reason. He also goes into the thinking of animals and determined the so-called hegemonikon, the functional center of the body, on the one hand and "life " in the heart and on the other hand, for cases of ethical decisions that is, the " good life " in the brain.

## Astrology

Ptolemy continued to write in 4 volumes to date the aftereffects astrological basic work Tetrabiblos ( " four books "; Greek Ἀποτελεσματικά Apotelesmatika ). This work is based on his astronomical writings and describes the effects of the heavenly bodies on people and their fate.