Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Almagest
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
== Contents == === The ''Syntaxis Mathematica'' books === [[File:Ptolemy-4.jpg|alt=Table of contents to a 1528 copy of Almagest, translated to Latin from Greek by George of Trebizond|thumb|Table of contents to a 1528 copy of ''Almagest'', translated to Latin from Greek by [[George of Trebizond]]]] The ''Syntaxis Mathematica'' consists of thirteen sections, called books. As with many medieval manuscripts that were handcopied or, particularly, printed in the early years of printing, there were considerable differences between various editions of the same text, as the process of transcription was highly personal. An example illustrating how the ''Syntaxis'' was organized is given below; it is a Latin edition printed in 1515 at Venice by Petrus Lichtenstein.{{sfn|Lichtenstein|1515|p={{Pages needed|date=April 2023}}}} * '''Book I''' contains an outline of [[Aristotle]]'s cosmology: on the spherical form of the heavens, with the spherical Earth lying motionless as the center, with the [[fixed stars]] and the various [[planets]] revolving around the Earth. Then follows an explanation of [[chord (geometry)|chords]] with [[Ptolemy's table of chords|table of chords]]; observations of the obliquity of the [[ecliptic]] (the apparent path of the Sun through the stars); and an introduction to [[spherical trigonometry]]. * '''Book II''' covers problems associated with the daily motion attributed to the heavens, namely risings and settings of celestial objects, the length of daylight, the determination of [[latitude]], the points at which the [[Sun]] is vertical, the shadows of the [[gnomon]] at the [[equinox]]es and [[solstice]]s, and other observations that change with the observer's position. There is also a study of the angles made by the ecliptic with the vertical, with tables. * '''Book III''' covers the length of the year, and the motion of the [[Sun]]. Ptolemy explains [[Hipparchus]]' discovery of the [[Precession (astronomy)|precession]] of the equinoxes and begins explaining the theory of [[epicycles]]. * '''Books IV''' and '''V''' cover the motion of the [[Moon]], lunar [[parallax]], the motion of the lunar [[apogee]], and the sizes and distances of the Sun and Moon relative to the Earth. * '''Book VI''' covers solar and lunar [[eclipse]]s. * '''Books VII''' and '''VIII''' cover the motions of the fixed stars, including precession of the equinoxes. They also contain a [[star catalogue]] of 1022 stars, described by their positions in the [[constellation]]s, together with [[ecliptic coordinates|ecliptic longitude and latitude]].{{refn|1= The catalogue actually contained 1,028 entries, but three of these were deliberate duplicates, because Ptolemy regarded certain stars as being shared between adjacent constellations. Three other entries were non-stellar, i.e. the Double Cluster in Perseus, M44 (Praesepe) in Cancer, and the globular cluster Omega Centauri.{{sfn|Ridpath|2023|loc=[http://www.ianridpath.com/startales/almagest.html "Ptolemy’s Almagest"]}} Ptolemy states that the longitudes (which increase due to [[precession]]) are for the beginning of the reign of [[Antoninus Pius]] (138 AD), whereas the latitudes do not change with time (but see [[#The star catalogue|the star catalog]]). The constellations north of the [[zodiac]] and the northern zodiac constellations (Aries through Virgo) are in the table at the end of Book VII, while the rest are in the table at the beginning of Book VIII. The brightest stars were marked first [[apparent magnitude|magnitude]] ({{mvar|m}} = 1), while the faintest visible to the naked eye were sixth magnitude ({{mvar|m}} = 6). Each numerical magnitude was considered twice the brightness of the following one, which is a [[logarithmic scale]]. (The ratio was subjective as no [[photodetector]]s existed.) This system is believed to have originated with Hipparchus. The stellar positions too are of Hipparchan origin, despite Ptolemy's claim to the contrary. Ptolemy identified 48 constellations: The 12 of the [[zodiac]], 21 to the north of the zodiac, and 15 to the south.{{sfn|Ridpath|2023|loc=[http://www.ianridpath.com/startales/almagest.html#48 Ptolemy’s 48 constellations]}}|group=note}} * '''Book IX''' addresses general issues associated with creating models for the five [[naked eye planet]]s, and the motion of [[Mercury (planet)|Mercury]]. * '''Book X''' covers the motions of [[Venus]] and [[Mars]]. * '''Book XI''' covers the motions of [[Jupiter]] and [[Saturn]]. * '''Book XII''' covers stations and [[apparent retrograde motion|retrograde motion]], which occurs when planets appear to pause, then briefly reverse their motion against the background of the zodiac. Ptolemy understood these terms to apply to Mercury and Venus as well as the outer planets. * '''Book XIII''' covers motion in latitude, that is, the deviation of planets from the ecliptic. The final topic of this chapter also covers how to determine when a planet first becomes visible after being hidden by the glare of the sun, as well as the last time it is seen before being hidden by the sun's glare. === Ptolemy's cosmos === The cosmology of the ''Syntaxis'' includes five main points, each of which is the subject of a chapter in Book I. What follows is a close paraphrase of Ptolemy's own words from Toomer's translation.{{sfn|Toomer|1984|p={{Pages needed|date=April 2023}}}} * The celestial realm is spherical, and moves as a sphere. * The Earth is a sphere. * The Earth is at the center of the cosmos. * The Earth, in relation to the distance of the fixed stars, has no appreciable size and must be treated as a mathematical point.{{sfn|Ptolemy|1952|loc= Book I, Chapter 5, p.9}} * The Earth does not move. === The star catalogue === [[File:Manuskript Ptolemy Labelled.png|thumb|Example of a Greek manuscript of the Almagest, showing a table layout for the stars of Ursa Minor. The functions of the columns, colours and rows are labelled in this depiction.]] The layout of the catalogue has always been tabular.{{sfn|Lichtenstein|1515|p=[https://daten.digitale-sammlungen.de/bsb00001192/images/index.html?id=00001192&groesser=&fip=xsxseayayztseayaeayafsdrxdsydeayaqrsfsdr&no=3&seite=161 79]}} Ptolemy writes explicitly that the coordinates are given as (ecliptical) "longitudes" and "latitudes", which are given in columns, so this has probably always been the case. It is significant that Ptolemy chooses the ecliptical coordinate system because of his knowledge of precession, which distinguishes him from all his predecessors. Hipparchus' celestial globe had an ecliptic drawn in, but the coordinates were equatorial.{{sfn|Hoffmann|2017|p={{Pages needed|date=April 2023}}}} Since Hipparchus' star catalogue has not survived in its original form, but was absorbed into the Almagest star catalogue (and heavily revised in the 265 years in between),{{sfn|Hoffmann|2018|p={{Pages needed|date=April 2023}}}}{{sfn|Hoffmann|2017|p={{Pages needed|date=April 2023}}}} the Almagest star catalogue is the oldest one in which complete tables of coordinates and magnitudes have come down to us. As mentioned, Ptolemy includes a star catalog containing 1022 stars. He says that he "observed as many stars as it was possible to perceive, even to the sixth magnitude". The [[ecliptic longitude]]s are given in terms of a zodiac sign and a number of degrees and fractions of a degree. The zodiac signs each represent exactly 30°, starting with [[Aries (constellation)|Aries]] representing longitude 0° to 30°. The degrees are added to the lower limit of the 30-degree range to obtain the longitude. Unlike the situation with the [[zodiac]] of modern-day [[astrology]], most of the stars of a given zodiac constellation in the catalog fall in the 30-degree range designated by the same name (the so-called 'zodiac sign'). The ecliptic longitudes are about 26° lower than those of AD 2000 (the [[J2000]] epoch).{{refn|1= For instance, the stars [[nu Orionis]] and [[62 Orionis]] which are now around ecliptic longitude 91° are attributed longitudes in the Almagest of 66° and 64°20' respectively.|group=note}} Ptolemy says that the ecliptic longitudes are for the beginning of the reign of [[Antoninus Pius]] (138 AD) and that he found that the longitudes had increased by 2° 40′ since the time of [[Hipparchus]] which was 265 years earlier (Alm. VII, 2). But calculations show that his ecliptic longitudes correspond more closely to around the middle of the first century CE (+48 to +58).{{sfn|Toomer|1984|p={{Pages needed|date=April 2023}}}}{{sfn|Graßhoff|1990|p={{Pages needed|date=April 2023}}}}{{sfn|Hoffmann|2017|p={{Pages needed|date=April 2023}}}} Since Tycho Brahe found this offset, astronomers and historians investigated this problem and suggested several causes: * that all coordinates were calculated from Hipparchus' observations, whereby the precession constant, which was known too inaccurately at the time, led to a summation error (Delambre 1817);{{sfn|Graßhoff|1990|p={{Pages needed|date=April 2023}}}}{{sfn|Peters|Knobel|1915|p=[https://archive.org/details/cu31924012300491/page/n30/mode/1up 15]}} * that the data had in fact been observed a century earlier by Menelaus of Alexandria (Björnbo 1901);{{sfn|Graßhoff|1990|p={{Pages needed|date=April 2023}}}} * that the difference is a sum of individual errors of various kinds, including calibration with outdated solar data;{{sfn|Dreyer|1917|p=528}}{{sfn|Graßhoff|1990|p={{Pages needed|date=April 2023}}}} * that Ptolemy's instrument was wrongly calibrated and had a systematic offset.{{sfn|Vogt|1925|pp=17–54}} [[File:Errors in Ptolemy’s star catalogue.svg|thumb|upright=1.35|Errors in Ptolemy’s star catalogue: In blue are the stars in the Almagest’s list; in orange, the points of modern measurements transformed to the second century, taking proper motion into account; the orange and blue band is the ecliptic; and the blue curve is the equator. Figure inspired by Hoffmann 2017.{{sfn|Hoffmann|2017|p={{Pages needed|date=April 2023}}}}]] Subtracting the systematic error leaves other errors that cannot be explained by precession. Of these errors, about 18 to 20 are also found in Hipparchus' star catalogue (which can only be reconstructed incompletely).{{sfn|Vogt|1925|pp=17–54}}{{sfn|Hoffmann|2017|p={{Pages needed|date=April 2023}}}} From this it can be concluded that a subset of star coordinates in the Almagest can indeed be traced back to Hipparchus,{{sfn|Graßhoff|1990|p={{Pages needed|date=April 2023}}}} but not that the complete star catalogue was simply "copied". Rather, Hipparchus' major errors are no longer present in the Almagest{{sfn|Hoffmann|2017|p={{Pages needed|date=April 2023}}}} and, on the other hand, Hipparchus' star catalogue had some stars that are entirely absent from the Almagest.{{sfn|Hoffmann|2017|p={{Pages needed|date=April 2023}}}} It can be concluded that Hipparchus' star catalogue, while forming the basis, has been reobserved and revised.{{sfn|Hoffmann|2017|p={{Pages needed|date=April 2023}}}}{{sfn|Hoffmann|2018|p={{Pages needed|date=April 2023}}}} ==== Errors in the coordinates ==== The figure he used is based on Hipparchus' own estimate for precession, which was 1° in 100 years, instead of the correct 1° in 72 years. Dating attempts through [[proper motion]] of the stars also appear to date the actual observation to Hipparchus' time instead of Ptolemy.{{sfn|Dambis|Efremov|2000|pp=115–134}} Many of the longitudes and latitudes have been corrupted in the various manuscripts. Most of these errors can be explained by similarities in the symbols used for different numbers. For example, the Greek letters Α and Δ were used to mean 1 and 4 respectively, but because these look similar copyists sometimes wrote the wrong one. In Arabic manuscripts, there was confusion between for example 3 and 8 (ج and ح). (At least one translator also introduced errors. [[Gerard of Cremona]], who translated an Arabic manuscript into Latin around 1175, put 300° for the latitude of several stars. He had apparently learned from [[Moors]], who [[Abjad numerals|used the letter س (sin) for 300]] (like the Hebrew [[Shin (letter)#Hebrew shin/sin|ש (shin)]]), but the manuscript he was translating came from the East, where س was used for 60, [[Hebrew numerals|like the Hebrew ס (samekh)]].){{sfn|Peters|Knobel|1915|pp=[https://archive.org/details/cu31924012300491/page/n24/mode/1up 9–14]}} Even without the errors introduced by copyists, and even accounting for the fact that the longitudes are more appropriate for 58 AD than for 137 AD, the latitudes and longitudes are not fully accurate, with errors as great as large fractions of a degree. Some errors may be due to [[atmospheric refraction]] causing stars that are low in the sky to appear higher than where they really are.{{sfn|Peters|Knobel|1915|p=[https://archive.org/details/cu31924012300491/page/n29/mode/1up 14]}} A series of stars in [[Centaurus]] are off by a couple of degrees, including the star we call [[Alpha Centauri]]. These were probably measured by a different person or persons from the others, and in an inaccurate way.{{sfn|Peters|Knobel|1915|p=[https://archive.org/details/cu31924012300491/page/n133/mode/1up 112]}} ==== Constellations in the star catalogue ==== The star catalogue contains 48 constellations, which have different surface areas and numbers of stars. In Book VIII, Chapter 3, Ptolemy writes that the constellations should be outlined on a globe, but it is unclear exactly how he means this: should surrounding polygons be drawn or should the figures be sketched or even line figures be drawn? This is not stated. Although no line figures have survived from antiquity, the figures can be reconstructed on the basis of the descriptions in the star catalogue: The exact celestial coordinates of the figures' heads, feet, arms, wings and other body parts are recorded.{{sfn|Hoffmann|2017|p={{Pages needed|date=April 2023}}}} It is therefore possible to draw the stick figures in the modern sense so that they fit the description in the Almagest.{{refn|1= These modern stick figures as a reconstruction of the historical constellations of the Almagest are available in the free planetarium software [[Stellarium (software)|Stellarium]] since 2019.{{citation needed|date=April 2023}}|group=note}} These constellations form the basis for the modern [[constellation]]s that were formally adopted by the International Astronomical Union in 1922, with official boundaries that were agreed in 1928. Of the stars in the catalogue, 108 (just over 10%) were classified by Ptolemy as 'unformed', by which he meant lying outside the recognized constellation figures. These were later absorbed into their surrounding constellations or in some cases used to form new constellations.{{sfn|Ridpath|2023|loc=[http://www.ianridpath.com/startales/ptolemy-spare.html "How Ptolemy’s spare stars became new constellations"]}} === Ptolemy's planetary model === [[File:Ptolemaicsystem-small.png|thumb|16th-century representation of Ptolemy's [[geocentric model]] in [[Peter Apian]]'s ''Cosmographia'', 1524]] In Almagest, Ptolemy assigned the following order to the [[planetary spheres]], beginning with the innermost:<ref>See Book IX, Chapter I, "On the order o f the spheres o f sun, moon and the 5 planets" in the Toomer's translation.</ref> {{Colbegin|colwidth=30em}} # Moon # Mercury or Venus (undecided)<ref>For example, in Albert Van Helden , Measuring the Universe Cosmic Dimensions from Aristarchus to Halley, Chicago University Press, 1985, page 20, it is stated that "He left the order of Venus and Mercury with respect to each other undecided in the Almagest."</ref> # Mercury or Venus (undecided) # Sun # Mars # Jupiter # Saturn # Sphere of fixed stars {{colend}} Later, in his "Planetary Hypothesis", he concludes that Mercury is the second closest planet. Other classical writers suggested different sequences. [[Plato]] ({{Circa|427|347 BC}}) placed the Sun second in order after the Moon. [[Martianus Capella]] (5th century AD) put Mercury and Venus in motion around the Sun. Ptolemy's authority was preferred by most [[Astronomy in medieval Islam|medieval Islamic]] and late medieval European astronomers. <!-- Comparison with Aristotle's model should be added --> Ptolemy inherited from his Greek predecessors a geometrical toolbox and a partial set of models for predicting where the planets would appear in the sky. [[Apollonius of Perga]] ({{Circa|262|190 BC}}) had introduced the [[deferent and epicycle]] and the eccentric deferent to astronomy. Hipparchus (2nd century BC) had crafted mathematical models of the motion of the Sun and Moon. Hipparchus had some knowledge of [[History of Astronomy#Mesopotamia|Mesopotamian astronomy]], and he felt that Greek models should match those of the Babylonians in accuracy. He was unable to create accurate models for the remaining five planets. [[File:HipparchusConstruction.svg|thumb|upright=1.35 |Geometric construction used by Hipparchus in his determination of the distances to the Sun and Moon, which was later incorporated into Ptolemy's work]] The ''Syntaxis'' adopted Hipparchus' solar model, which consisted of a simple eccentric deferent. For the Moon, Ptolemy began with Hipparchus' epicycle-on-deferent, then added a device that historians of astronomy refer to as a "crank mechanism":{{sfn|Hoskin|1999|p=44}} he succeeded in creating models for the other planets, where Hipparchus had failed, by introducing a third device called the [[equant]]. Ptolemy wrote the ''Syntaxis'' as a textbook of mathematical astronomy. It explained geometrical models of the planets based on combinations of circles, which could be used to predict the motions of celestial objects. In a later book, the ''Planetary Hypotheses'', Ptolemy explained how to transform his geometrical models into [[three-dimensional space|three-dimensional spheres]] or partial spheres. In contrast to the mathematical ''Syntaxis'', the ''Planetary Hypotheses'' is sometimes described as a book of [[cosmology]].
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)