Editing Ptolemy (section)

==Other writings==
===''Geography''===
{{Main|Geography (Ptolemy)}}
{{further|Ptolemy's world map}}
[[File:Claudius Ptolemy- The World.jpg|thumb|upright=1.1|A printed map from the 15th century depicting Ptolemy's description of the ''[[Ecumene]]'' by Johannes Schnitzer (1482).]]
Ptolemy's second most well-known work is his ''Geographike Hyphegesis'' ({{Langx|el|Γεωγραφικὴ Ὑφήγησις}}; {{Lit|Guide to Drawing the Earth}}), known as the ''[[Geography (Ptolemy)|Geography]]'', a handbook on how to draw maps using [[geographical coordinates]] for parts of the [[Roman Empire|Roman world]] known at the time.<ref name="Graßhf-Mittnhbr-Rinner-2017">
{{cite journal
 |last1=Graßhoff |first1=G.
 |last2=Mittenhuber |first2=F.
 |last3=Rinner |first3=E.
 |year=2017
 |title=Of paths and places: The origin of Ptolemy's ''Geography''
 |journal=[[Archive for History of Exact Sciences]]
 |volume=71 |issue=6 |pages=483–508
 |doi=10.1007/s00407-017-0194-7
 |jstor=45211928 |s2cid=133641503
 |issn=0003-9519
}}
</ref><ref name=Isaksen-2011>
{{cite journal
 |last=Isaksen |first=L.
 |year=2011 |title=Lines, damned lines and statistics: Unearthing structure in Ptolemy's ''Geographia''
 |journal=E-Perimetron
 |volume=6 |issue=4 |pages=254–260
 |url=http://www.e-perimetron.org/Vol_6_4/Isaksen.pdf
}}
</ref>
He relied on previous work by an earlier geographer, [[Marinos of Tyre|Marinus of Tyre]], as well as on [[gazetteer]]s of the Roman and ancient [[History of Iran|Persian Empire]].<ref name=Isaksen-2011/><ref name="Graßhf-Mittnhbr-Rinner-2017"/> He also acknowledged ancient astronomer [[Hipparchus]] for having provided the elevation of the [[celestial pole#Finding the north celestial pole|north celestial pole]]<ref>The north celestial pole is the point in the sky lying at the common centre of the circles which the stars appear to people in the northern hemisphere to trace out during the course of a [[sidereal time|sidereal day]].</ref> for a few cities. Although [[maps]] based on scientific principles had been made since the time of [[Eratosthenes]] ({{Circa|276|195 BC}}), Ptolemy improved on [[map projection]]s.

The first part of the ''Geography'' is a discussion of the data and of the methods he used. Ptolemy notes the supremacy of astronomical data over land measurements or travelers' reports, though he possessed these data for only a handful of places. Ptolemy's real innovation, however, occurs in the second part of the book, where he provides a catalogue of 8,000 localities he collected from Marinus and others, the biggest such database from antiquity.<ref name=Mittnhbr-2010>
{{cite book
 |last=Mittenhuber |first=F.
 |year=2010
 |section=The tradition of texts and maps in Ptolemy's ''Geography''
 |title=Ptolemy in Perspective: Use and criticism of his work from antiquity to the nineteenth century
 |series=Archimedes
 |volume=23 |pages=95–119
 |place=Dordrecht, NL
 |publisher=Springer Netherlands
 |doi=10.1007/978-90-481-2788-7_4
 |isbn=978-90-481-2788-7
}}
</ref>
About {{gaps|6|300}} of these places and geographic features have assigned [[coordinate]]s so that they can be placed in a [[Grid (spatial index)|grid]] that spanned the globe.<ref name=Jones-2020/> [[Latitude]] was measured from the [[equator]], as it is today, but Ptolemy preferred to express it as ''[[clime|climata]]'', the length of the longest day rather than [[degree (angle)|degrees of arc]]: The length of the [[midsummer]] day increases from 12h to 24h as one goes from the equator to the [[polar circle]].<ref>
{{cite report
 |last=Shcheglov |first=D.A.
 |date=2002–2007
 |url=https://nw.academia.edu/DmitryShcheglov/Papers/142876/Hipparchus_Table_of_Climata_and_Ptolemys_Geography
 |title=Hipparchus' table of climata and Ptolemy's ''Geography''
 |series=Orbis Terrarum |volume=9 (2003–2007)
 |pages=177–180
}}
</ref>
One of the places Ptolemy noted specific coordinates for was the now-lost [[Stone Tower (Ptolemy)|stone tower]] which marked the midpoint on the ancient [[Silk Road]], and which scholars have been trying to locate ever since.<ref>
{{cite book
 |last=Dean |first=Riaz
 |year=2022
 |title=The Stone Tower: Ptolemy, the silk road, and a 2,000&nbsp;year-old riddle
 |publisher=Penguin Viking
 |isbn=978-0670093625
 |location=Delhi, IN
 |pages={{mvar|xi}}, 135, 148, 160
}}
</ref>

In the third part of the ''Geography'', Ptolemy gives instructions on how to create maps both of the whole inhabited world (''[[oikoumenē]]'') and of the Roman provinces, including the necessary [[Topographic map|topographic]] lists, and captions for the maps. His ''oikoumenē'' spanned 180&nbsp;degrees of longitude from the Blessed Islands in the [[Atlantic Ocean]] to the middle of [[China]], and about 80 degrees of latitude from [[Shetland]] to anti-Meroe (east coast of [[Africa]]); Ptolemy was well aware that he knew about only a quarter of the globe, and an erroneous extension of China southward suggests his sources did not reach all the way to the Pacific Ocean.<ref name="Graßhf-Mittnhbr-Rinner-2017"/><ref name=Isaksen-2011/>

It seems likely that the topographical tables in the second part of the work (Books&nbsp;2–7) are cumulative texts, which were altered as new knowledge became available in the centuries after Ptolemy.{{sfn|Bagrow|1945}} This means that information contained in different parts of the ''Geography'' is likely to be of different dates, in addition to containing many scribal errors. However, although the regional and [[Ptolemy's world map|world maps]] in surviving manuscripts date from {{Circa|1300&nbsp;AD}} (after the text was rediscovered by [[Maximus Planudes]]), there are some scholars who think that such maps go back to Ptolemy himself.<ref name=Mittnhbr-2010/>

===''Tetrabiblos''===
{{Main|Tetrabiblos}}
[[File:Ptolemaeus - Quadripartitum, 1622 - 4658973.tif|thumb|upright=.8|A copy of the ''Quadripartitum'' (1622)]]
Ptolemy wrote an astrological treatise, in four parts, known by the Greek term ''[[Tetrabiblos]]'' ({{lit|Four Books}}) or by its Latin equivalent ''Quadripartitum''.{{refn|
{{cite book
 |first=H. Darrel |last=Rutkin
 |year=2010
 |section=The use and abuse of Ptolemy's ''Tetrabiblos'' in Renaissance and early modern Europe
 |page=135
 |title=Jones (2010)
}}<ref name=Jones-2010/>{{rp|style=ama|p= 135}}
}}
Its original title is unknown, but may have been a term found in some Greek manuscripts, ''Apotelesmatiká'' (''biblía''), roughly meaning "(books) on the Effects" or "Outcomes", or "Prognostics".<ref name=Robbins-1940-intro/>{{rp|style=ama|p= {{mvar|x}} }}
As a source of reference, the ''Tetrabiblos'' is said to have "enjoyed almost the authority of a Bible among the astrological writers of a thousand years or more".{{refn|name=Robbins-1940-intro|
{{cite book
 |last=Robbins |first=Frank E.
 |section=Introduction
 |year=1940
 |editor=Robbins, F.E.
 |title=Ptolemy Tetrabiblos
}}<ref name=Robbins-1940/>
}}{{rp|style=ama|p= {{mvar|xii}} }}
It was first translated from Arabic into Latin by [[Plato of Tivoli]] (Tiburtinus) in 1138, while he was in Spain.<ref name=Robbins-1940/>

Much of the content of the ''Tetrabiblos'' was collected from earlier sources; Ptolemy's achievement was to order his material in a systematic way, showing how the subject could, in his view, be rationalized. It is, indeed, presented as the second part of the study of astronomy of which the ''Almagest'' was the first, concerned with the influences of the celestial bodies in the [[sublunary sphere]].<ref name=Jones-2010/><ref name=Heilen-2010/> Thus explanations of a sort are provided for the astrological effects of the [[planets]], based upon their combined effects of heating, cooling, moistening, and drying.<ref>
{{cite journal
 |last=Riley |first=M.
 |date=1988
 |title=Science and tradition in the ''Tetrabiblos''
 |journal=[[Proceedings of the American Philosophical Society]]
 |volume=132 |issue=1 |pages=67–84
 |jstor=3143825 |issn=0003-049X
}}
</ref> Ptolemy dismisses other astrological practices, such as considering the [[Numerology|numerological]] significance of names, that he believed to be without sound basis, and leaves out popular topics, such as [[electional astrology]] (interpreting astrological charts to determine courses of action) and [[medical astrology]], for similar reasons.<ref name=Riley1987>
{{cite journal
 |last=Riley |first=M.
 |date=1987
 |title=Theoretical and practical astrology: Ptolemy and his colleagues
 |journal=[[Transactions of the American Philological Association]]
 |volume=117 |pages=235–256
 |doi=10.2307/283969 |jstor=283969
}}
</ref>

The great respect in which later astrologers held the ''Tetrabiblos'' derived from its nature as an exposition of theory, rather than as a manual.<ref name=Riley1987 />

A collection of one hundred [[aphorism]]s about astrology called the ''[[Centiloquium]]'', ascribed to Ptolemy, was widely reproduced and commented on by Arabic, Latin, and Hebrew scholars, and often bound together in medieval manuscripts after the ''Tetrabiblos'' as a kind of summation.<ref name=Jones-2020/> It is now believed to be a much later [[pseudepigraph]]ical composition. The identity and date of the actual author of the work, referred to now as [[Pseudo-Ptolemy]], remains the subject of conjecture.<ref>
{{cite book
 |last=Boudet |first=J.-P.
 |year=2014
 |section=Astrology between rational science and divine inspiration: The pseudo-Ptolemy's centiloquium
 |editor1-last=Rapisarda |editor1-first=S.
 |editor2-last=Niblaeus  |editor2-first=E.
 |title=Dialogues among Books in Medieval Western Magic and Divination
 |series=Micrologus' library
 |volume=65  |pages=47–73
 |publisher=Sismel edizioni del Galluzzo
 |isbn=9788884505811
 |url=https://halshs.archives-ouvertes.fr/halshs-01628233
 |access-date=19 August 2021
}}
</ref>

===''Harmonics''===
[[File:Epogdoon.jpg|thumb|upright=.8|A diagram showing [[Pythagorean tuning]].]]
{{see also|Ptolemy's intense diatonic scale}}
Ptolemy's ''Harmonics'' ({{Langx|el|Ἁρμονικόν}}) is a work in three books on [[music theory]] and the mathematics behind musical scales<ref>
{{cite book
 |last=Wardhaugh |first=Benjamin
 |date=5 July 2017
 |title=Music, Experiment, and Mathematics in England, 1653–1705
 |publisher=Routledge
 |isbn=978-1-351-55708-5
 |location=London, UK / New York, NY
 |page=7
 |url=https://books.google.com/books?id=BzcrDwAAQBAJ&pg=PA7
}}
</ref>

''Harmonics'' begins with a definition of harmonic theory, with a long exposition on the relationship between reason and sense perception in corroborating theoretical assumptions. After criticizing the approaches of his predecessors, Ptolemy argues for basing musical intervals on mathematical ratios (as opposed to the ideas advocated by followers of [[Aristoxenus]]), backed up by empirical observation (in contrast to the excessively theoretical approach of the [[Pythagoreans]]).<ref>
{{cite journal
 |last=Barker |first=A.
 |year=1994
 |title=Ptolemy's Pythagoreans, Archytas, and Plato's conception of mathematics
 |journal=[[Phronesis]]
 |volume=39 |issue=2 |pages=113–135
 |doi=10.1163/156852894321052135
 |jstor=4182463  |issn=0031-8868
}}
</ref><ref>
{{cite journal
 |last=Crickmore |first=L.
 |year=2003
 |title=A re-valuation of the ancient science of ''Harmonics''
 |journal=[[Psychology of Music]]
 |volume=31 |issue=4 |pages=391–403
 |doi=10.1177/03057356030314004
 |s2cid=123117827
}}
</ref>

Ptolemy introduces the ''harmonic canon'' (Greek name) or ''[[monochord]]'' (Latin name), which is an experimental musical apparatus that he used to measure relative pitches, and used to describe to his readers how to demonstrate the relations discussed in the following chapters for themselves. After the early exposition on to build and use monochord to test proposed tuning systems, Ptolemy proceeds to discuss [[Pythagorean tuning]] (and how to demonstrate that their idealized musical scale fails in practice). The Pythagoreans believed that the mathematics of music should be based on only the one specific ratio of 3:2, the [[perfect fifth]], and believed that tunings mathematically exact to their system would prove to be melodious, if only the extremely large numbers involved could be calculated (by hand). To the contrary, Ptolemy believed that musical scales and tunings should in general involve multiple different ratios arranged to fit together evenly into smaller [[tetrachord]]s (combinations of four pitch ratios which together make a [[perfect fourth]]) and [[octave]]s.<ref>
{{cite journal
 |last=Barker |first=A.
 |year=1994
 |title=Greek musicologists in the Roman Empire
 |journal=[[Apeiron]]
 |volume=27  |issue=4  |pages=53–74
 |doi=10.1515/APEIRON.1994.27.4.53
 |s2cid=170415282
 |url=https://www.degruyter.com/document/doi/10.1515/APEIRON.1994.27.4.53/html
}}
</ref><ref>
{{cite book
 |last=West |first=Martin Litchfield |author-link=Martin Litchfield West
 |year=1992
 |title=Ancient Greek Music
 |place=Oxford, UK
 |publisher=[[Oxford University Press]]
 |isbn=0-19-814975-1
}}
</ref>
Ptolemy reviewed standard (and [[enharmonic scale|ancient, disused]]) musical tuning practice of his day, which he then compared to his own subdivisions of the [[tetrachord]] and the [[octave]], which he derived experimentally using a [[monochord]] / harmonic canon. The volume ends with a more speculative exposition of the relationships between harmony, the soul (''psyche''), and the planets ([[Musica universalis|harmony of the spheres]]).<ref name=Feke-2012>
{{cite journal
 |last=Feke  |first=J.
 |year=2012
 |title=Mathematizing the soul: The development of Ptolemy's psychological theory from ''On the Kritêrion'' and ''Hêgemonikon'' to the ''Harmonics''
 |journal=Studies in History and Philosophy of Science Part&nbsp;A
 |volume=43  |issue=4  |pages=585–594
 |doi=10.1016/j.shpsa.2012.06.006
 |bibcode=2012SHPSA..43..585F
 |url=https://www.sciencedirect.com/science/article/abs/pii/S0039368112000428
}}
</ref>

Although Ptolemy's ''Harmonics'' never had the influence of his ''Almagest'' or ''Geography'', it is nonetheless a well-structured treatise and contains more methodological reflections than any other of his writings. In particular, it is a nascent form of what in the following millennium developed into the scientific method, with specific descriptions of the experimental apparatus that he built and used to test musical conjectures, and the empirical musical relations he identified by testing pitches against each other: He was able to accurately measure relative pitches based on the ratios of vibrating lengths two separate sides of the same [[monochord|single string]], hence which were assured to be under equal tension, eliminating one source of error. He analyzed the empirically determined ratios of "pleasant" pairs of pitches, and then synthesised all of them into a coherent mathematical description, which persists to the present as [[just intonation]] – the standard for comparison of consonance in the many other, less-than exact but more facile [[meantone temperament|compromise tuning]] systems.<ref>
{{cite journal
 |last=Barker |first=A.
 |year=2010
 |title=Mathematical beauty made audible: Musical aesthetics in Ptolemy's ''Harmonics''
 |journal=[[Classical Philology]]
 |volume=105 |issue=4 |pages=403–420
 |doi=10.1086/657028 |s2cid=161714215
}}
</ref><ref>
{{cite journal
 |last=Tolsa |first=C.
 |year=2015
 |title=Philosophical presentation in Ptolemy's ''Harmonics'': The ''Timaeus'' as a model for organization
 |journal=Greek, Roman, and Byzantine Studies
 |volume=55 |issue=3 |pages=688–705
 |issn=2159-3159
 |url=https://grbs.library.duke.edu/article/view/15395
}}
</ref>

During the [[Renaissance]], Ptolemy's ideas inspired [[Kepler]] in his own musings on the harmony of the world (''[[Harmonices Mundi|Harmonice Mundi]]'', Appendix to Book V).<ref>
{{cite book
 |last=Hetherington |first=Norriss S.
 |date=8 April 2014
 |title=Encyclopedia of Cosmology
 |series=Routledge Revivals
 |volume=Historical, Philosophical, and Scientific Foundations of Modern Cosmology
 |publisher=Routledge
 |isbn=978-1-317-67766-6
 |page=527
 |url=https://books.google.com/books?id=EP9QAwAAQBAJ&pg=PA527
}}
</ref>

===Optics===
{{main|Optics (Ptolemy)}}
The ''Optica'' ([[Koine Greek]]: {{math|Ὀπτικά}}), known as the ''Optics'', is a work that survives only in a somewhat poor Latin version, which, in turn, was translated from a lost Arabic version by [[Eugenius of Palermo]] ({{circa|lk=no|1154}}). In it, Ptolemy writes about properties of sight (not light), including [[reflection (physics)|reflection]], [[refraction]], and [[colour]]. The work is a significant part of the early [[history of optics]] and influenced the more famous and superior 11th-century ''[[Book of Optics]]'' by [[Ibn al-Haytham]].<ref name=Smith-1996>
{{cite book
 |last=Smith |first=A. Mark
 |year=1996
 |title=Ptolemy's Theory of Visual Perception: An English translation of the ''Optics''
 |publisher=[[The American Philosophical Society]]
 |isbn=0-87169-862-5
 |url=https://books.google.com/books?id=mhLVHR5QAQkC&pg=PP1
 |access-date=27 June 2009
}}
</ref> Ptolemy offered explanations for many phenomena concerning illumination and colour, size, shape, movement, and binocular vision. He also divided illusions into those caused by physical or optical factors and those caused by judgmental factors. He offered an obscure explanation of the Sun or [[Moon illusion]] (the enlarged apparent size on the horizon) based on the difficulty of looking upwards.<ref>
{{cite journal
 |first1=H.E. |last1=Ross
 |first2=G.M. |last2=Ross
 |year=1976
 |title=Did Ptolemy understand the moon illusion?
 |journal=[[Perception (journal)|Perception]]
 |volume=5 |issue=4
 |pages=377–395
|doi=10.1068/p050377
 |pmid=794813
 |s2cid=23948158
 }}
</ref><ref>
{{cite book
 |first=A.I. |last=Sabra
 |year=1987
 |section=Psychology versus mathematics: Ptolemy and Alhazen on the moon illusion
 |editor1=Grant, E.
 |editor2=Murdoch, J.E.
 |title=Mathematics and its Application to Science and Natural Philosophy in the Middle Ages
 |place=Cambridge, UK
 |publisher=Cambridge University Press
 |pages=217–247
}}
</ref>

The work is divided into three major sections. The first section (Book II) deals with direct vision from first principles and ends with a discussion of binocular vision. The second section (Books III-IV) treats [[Reflection (physics)|reflection]] in plane, convex, concave, and compound mirrors.<ref>
{{cite journal
 |last=Smith |first=A. M.
 |year=1982
 |title=Ptolemy's search for a law of refraction: A case-study in the classical methodology of "saving the appearances" and its limitations
 |journal=[[Archive for History of Exact Sciences]]
 |volume=26  |issue=3  |pages=221–240
 |doi=10.1007/BF00348501 |jstor=41133649
 |s2cid=117259123 |issn=0003-9519
}}
</ref> The last section (Book V) deals with [[refraction]] and includes the earliest surviving table of refraction from air to water, for which the values (with the exception of the 60° angle of incidence) show signs of being obtained from an arithmetic progression.<ref>
{{cite book
 |first=C.B. |last=Boyer |author-link=Carl Benjamin Boyer
 |year=1959
 |title=The Rainbow: From myth to mathematics
}}
</ref>
However, according to Mark Smith, Ptolemy's table was based in part on real experiments.<ref>
{{cite book
 |last=Smith |first=Mark
 |year=2015
 |title=From Sight to Light: The passage from ancient to modern optics
 |publisher=The University of Chicago Press
 |pages=116–118
 |bibcode=2014fslp.book.....S
}}
</ref>

Ptolemy's theory of vision consisted of rays (or flux) coming from the eye forming a cone, the vertex being within the eye, and the base defining the visual field. The rays were sensitive, and conveyed information back to the observer's intellect about the distance and orientation of surfaces. Size and shape were determined by the visual angle subtended at the eye combined with perceived distance and orientation.<ref name=Smith-1996/><ref>
{{cite journal
 |last=Riley |first=M.
 |year=1995
 |title=Ptolemy's use of his predecessors' data
 |journal=[[Transactions of the American Philological Association]]
 |volume=125 |jstor=i212542 |language=en
}}
</ref>
This was one of the early statements of size-distance invariance as a cause of perceptual size and shape constancy, a view supported by the Stoics.<ref>
{{cite book
 |first1=H.W. |last1=Ross
 |first2=C. |last2=Plug
 |year=1998
 |section=The history of size constancy and size illusions
 |editor1-first=V. |editor1-last=Walsh
 |editor2-first=J. |editor2-last=Kulikowski
 |title=Perceptual Constancy: Why things look as they do
 |place=Cambridge, UK
 |publisher=Cambridge University Press
 |pages=499–528
}}
</ref>