the long pause to regroup - york u

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11Math 1700 – Medieval Developments

The Long Pause to The Long Pause to RegroupRegroup

The transition from Antiquity to The transition from Antiquity to the Renaissance, with influences the Renaissance, with influences

from the Near Eastfrom the Near East


RomeRome

Italian peninsula occupied by 1000 BCE by Italian peninsula occupied by 1000 BCE by Latin speaking tribesLatin speaking tribes

800 BCE: 800 BCE: Greeks arrive in southGreeks arrive in southEtruscans in northEtruscans in north

Rome became a republic in 509 BCERome became a republic in 509 BCEPower invested in a SenatePower invested in a Senate


Roman TechnologyRoman Technology

Romans were great Romans were great engineersengineers

Roads, aqueducts, Roads, aqueducts, buildings, citiesbuildings, cities

The Rule of LawThe Rule of LawThe authority of The authority of Roman government Roman government was based upon was based upon written laws enacted written laws enacted by its senate. by its senate.

A surviving Roman aqueduct at Nîmes in France.

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Roman ScienceRoman Science

"Greek science written in Latin""Greek science written in Latin"Transmitted to Rome in the Hellenistic Transmitted to Rome in the Hellenistic period (300 BCE period (300 BCE -- 200200 CE)CE)

Mostly via the Museum in Alexandria and Mostly via the Museum in Alexandria and those trained there.those trained there.


Roman MathematicsRoman MathematicsVery difficult to find anything novel.Very difficult to find anything novel.Calculation in the Roman Empire was hampered Calculation in the Roman Empire was hampered by the system of Roman numerals:by the system of Roman numerals:

M=1000M=1000CM=900CM=900DCCC=800DCCC=800DCC=700DCC=700DC=600DC=600D=500D=500CD=400CD=400CCC=300CCC=300CC=200CC=200C=100C=100

C=100C=100XC=90XC=90LXXX=80LXXX=80LXX=70LXX=70LX=60LX=60L=50L=50XL=40XL=40XXX=30XXX=30XX=20XX=20X=10X=10

X=10X=10IX=9IX=9VIII=8VIII=8VII=7VII=7VI=6VI=6V=5V=5IV=4IV=4III=3III=3II=2II=2I=1I=1


Imagine the difficulty of arithmetic Imagine the difficulty of arithmetic operationsoperations

Compare 3629 x 2983 Compare 3629 x 2983 To MMMDCXXIX x MMCMLXXXIIITo MMMDCXXIX x MMCMLXXXIIINote: Note:

Like the Egyptians, the Romans used a decimalLike the Egyptians, the Romans used a decimal--based based system and had separate symbols for the units, tens, system and had separate symbols for the units, tens, hundreds, thousands, millions, etc.hundreds, thousands, millions, etc.Like the Babylonians, the Romans had place value in Like the Babylonians, the Romans had place value in that the larger numerals were always to the left of that the larger numerals were always to the left of lesser numerals, with the exception of the special lesser numerals, with the exception of the special symbols for the 4 and 9 position: IV and IX, XL and symbols for the 4 and 9 position: IV and IX, XL and XC, CD and CM, etc.XC, CD and CM, etc.

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Romans were practicalRomans were practical

The only interest that Romans took in The only interest that Romans took in mathematics was what was necessary for mathematics was what was necessary for commerce and engineering.commerce and engineering.They had no interest in, say, geometry for They had no interest in, say, geometry for its own sake. Theoretical knowledge was its own sake. Theoretical knowledge was little valued.little valued.


Roman engineeringRoman engineering

Roman engineering was amazing for its time. Roman engineering was amazing for its time. But it required relatively little mathematics. But it required relatively little mathematics. Roman engineers learned only what they Roman engineers learned only what they deemed necessary to know.deemed necessary to know.The famous engineerThe famous engineer--architect architect ArgippaArgippa, who , who undertook a survey of the entire Roman empire, undertook a survey of the entire Roman empire, had to call in specialists from Alexandria to carry had to call in specialists from Alexandria to carry out the measurements and calculations.out the measurements and calculations.


The Julian CalendarThe Julian Calendar

Introduced 45 BCE, by Julius Caesar Introduced 45 BCE, by Julius Caesar 365365--day year with leap years every fourth day year with leap years every fourth yearyearMuch better than all previous calendarsMuch better than all previous calendarsRecognition that the year is not evenly Recognition that the year is not evenly divided into days.divided into days.

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EncyclopaedistsEncyclopaedists

In the late Roman Empire.In the late Roman Empire.Attempted to write down Attempted to write down ““everythingeverything””they knew in they knew in ““Epitomes.Epitomes.””Often totally disorganized, but great Often totally disorganized, but great source for historians to figure out what source for historians to figure out what people knew and believed.people knew and believed.


MartianusMartianus CapellaCapella

An exception: An exception: MartianusMartianus CapellaCapella’’ssorganization of knowledge into the 7 organization of knowledge into the 7 Liberal ArtsLiberal Arts

TriviumTriviumGrammar, Dialectic, RhetoricGrammar, Dialectic, Rhetoric

QuadriviumQuadriviumGeometry, Arithmetic, Astronomy, MusicGeometry, Arithmetic, Astronomy, Music


The Roman Empire at its heightThe Roman Empire at its height

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ByzantiumByzantium

In 395 C. E., the Roman Empire was In 395 C. E., the Roman Empire was divided into an Eastern and a Western divided into an Eastern and a Western branch.branch.The Eastern Empire, The Eastern Empire, ““Byzantium,Byzantium,”” based based in Constantinople, thrived, lasting until in Constantinople, thrived, lasting until 1453.1453.

Largely Greek influenceLargely Greek influence


Emperor Justinian 527Emperor Justinian 527--565565

Emperor of the Eastern Roman EmpireEmperor of the Eastern Roman EmpireCalled later the "Byzantine Empire"Called later the "Byzantine Empire"

Justinian tried to reJustinian tried to re--unite the Eastern and unite the Eastern and Western empires, unsuccessfullyWestern empires, unsuccessfully

Tried to establish Latin across his empire.Tried to establish Latin across his empire.After his death, Greek became the official After his death, Greek became the official language.language.


The Corpus The Corpus JurisJuris

Justinian Justinian commissioned a summary of commissioned a summary of Roman LawRoman Law..

Part of his effort to rePart of his effort to re--unite the empire.unite the empire.Was completed by a team of scholars in only Was completed by a team of scholars in only eight years, but was a fraction of the body of eight years, but was a fraction of the body of law.law.

Centuries later, in the west, the Centuries later, in the west, the Corpus Corpus JurisJuris was taken to be the final and perfect was taken to be the final and perfect expression of Roman Law.expression of Roman Law.

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The Byzantine EmpireThe Byzantine Empire

Successfully defended itself from attack Successfully defended itself from attack from North, East, and West when other from North, East, and West when other cultures fell.cultures fell.

Superior knowledge of military technologies, Superior knowledge of military technologies, notnot involving guns.involving guns.

Finally fell to the Ottoman Turks in 1453Finally fell to the Ottoman Turks in 1453Famous victory for guns.Famous victory for guns.One of the traditional dates for the end of the One of the traditional dates for the end of the Middle Ages and the beginning of the Modern Middle Ages and the beginning of the Modern Era.Era.


Greek FireGreek Fire

A Byzantine terror weapon.A Byzantine terror weapon.A petroleum based liquid.A petroleum based liquid.

Caught fire when it hit the water. It was a mainstay of ByzantinCaught fire when it hit the water. It was a mainstay of Byzantine e defence from both Arabs and Slavs from as early as the 7th defence from both Arabs and Slavs from as early as the 7th century.century.


The Decline of the (Western) The Decline of the (Western) Roman EmpireRoman Empire

The Roman Empire had spread across Europe, North Africa, and theThe Roman Empire had spread across Europe, North Africa, and theNear East during the thousand years from 500 BCE to 500 CE.Near East during the thousand years from 500 BCE to 500 CE.The western empire then collapsed, leaving a governmental void The western empire then collapsed, leaving a governmental void across much of Western civilization. across much of Western civilization.

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The Fall of RomeThe Fall of RomeMany reasons Many reasons are given for the are given for the fall of the fall of the Roman Empire.Roman Empire.Possibly they Possibly they were no longer were no longer able to feed able to feed themselves by themselves by importing food importing food from their from their colonies.colonies.

Ruins of Leptis Magna (now in Libya). A great city in a region that supplied food to Rome. Overfarming made the land unproductive. It is now a desert.


Europe in the Dark AgesEurope in the Dark AgesThe period between antiquity (ancient times), The period between antiquity (ancient times), marked by the fall of Rome around 500 C.E., marked by the fall of Rome around 500 C.E., and the beginning of the modern era (around and the beginning of the modern era (around the middle of the 15the middle of the 15thth century) is called the century) is called the Medieval period, or the Middle Ages.Medieval period, or the Middle Ages.

The first 500 years of that is called the Dark Ages The first 500 years of that is called the Dark Ages (about 500(about 500--1000 C.E.).1000 C.E.).

During this time Europe was overrun by nomadic During this time Europe was overrun by nomadic tribes from the north and west.tribes from the north and west.Literacy was lost, farming techniques forgotten, Literacy was lost, farming techniques forgotten, infrastructure deteriorated.infrastructure deteriorated.Villages became isolated from each other.Villages became isolated from each other.


CharlemagneCharlemagne

Europe began to recover Europe began to recover under Charles the Great under Charles the Great ((CharlemagneCharlemagne).).In 800 he was crowned In 800 he was crowned Holy Roman Emperor.Holy Roman Emperor.He saw illiteracy as one He saw illiteracy as one of the major problems of the major problems facing Europe.facing Europe.

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SchoolsSchools

Charlemagne founded schools in Charlemagne founded schools in cathedrals all over Europe.cathedrals all over Europe.

The language of instruction was Latin.The language of instruction was Latin.The curriculum was the seven liberal arts, The curriculum was the seven liberal arts, using using MartianusMartianus CapellaCapella’’ss work.work.

When Charlemagne died many of his When Charlemagne died many of his reforms were lost, but some of the schools reforms were lost, but some of the schools remained and became the bases of later remained and became the bases of later European universities.European universities.


The Recovery of EuropeThe Recovery of Europe

By the early 1400s, Europe had pulled By the early 1400s, Europe had pulled itself out of its disorganization after the itself out of its disorganization after the fall of Rome.fall of Rome.The feudal system provided stable The feudal system provided stable government and protection from raiders.government and protection from raiders.The economy was flourishing and the The economy was flourishing and the population rising and in good health.population rising and in good health.But there was little literacy and almost no But there was little literacy and almost no knowledge of ancient science.knowledge of ancient science.


The Rise of IslamThe Rise of Islam

Before MohammedBefore MohammedArab culture disorganizedArab culture disorganized

nomadic, polytheisticnomadic, polytheisticcenteredcentered in Meccain Mecca

MohammedMohammedborn 570 in Meccaborn 570 in Meccaat age of 40 had revelation, became prophet and fled at age of 40 had revelation, became prophet and fled Mecca in 622 (traditional date for start of Islam)Mecca in 622 (traditional date for start of Islam)

Mohammed's revelations = the Koran (Mohammed's revelations = the Koran (QuQu’’ranran))One God, one Prophet, one BookOne God, one Prophet, one Book

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The The Spread of IslamSpread of Islam

In 100 years (630In 100 years (630--730) Islam spread from Arabia to 730) Islam spread from Arabia to India in the East and to Spain in the West.India in the East and to Spain in the West.


The founding of BaghdadThe founding of Baghdad

In 762, the rulers of the Islamic empire, In 762, the rulers of the Islamic empire, the caliphs (i.e., the caliphs (i.e., ““successorssuccessors””), then in ), then in Damascus, decided to found a new capital Damascus, decided to found a new capital on the Tigris river at the site of a small on the Tigris river at the site of a small town known by the Persian name, town known by the Persian name, Baghdad.Baghdad.By the ninth century, Baghdad had a By the ninth century, Baghdad had a population of 800,000, making it larger population of 800,000, making it larger than Constantinople. than Constantinople.


The rise of ArabicThe rise of Arabic

Arabic became the language of learning.Arabic became the language of learning.All Moslems had to learn Arabic to read All Moslems had to learn Arabic to read the Koran.the Koran.Anything written in Arabic carried the Anything written in Arabic carried the prestige previously reserved for works in prestige previously reserved for works in Greek.Greek.

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The House of WisdomThe House of Wisdom

Caliph alCaliph al--MaMa’’mmûûnn established a great established a great academy, the House of Wisdom, in academy, the House of Wisdom, in Baghdad, comparable to the Museum in Baghdad, comparable to the Museum in Alexandria.Alexandria.The House of Wisdom set out to acquire The House of Wisdom set out to acquire all the great Greek manuscripts and works all the great Greek manuscripts and works of other cultures.of other cultures.

An emissary was sent to Constantinople to An emissary was sent to Constantinople to obtain a copy of Euclidobtain a copy of Euclid’’s s Elements Elements from the from the Byzantine Emperor.Byzantine Emperor.


The House of Wisdom, 2The House of Wisdom, 2By the beginning of the tenth century, almost all By the beginning of the tenth century, almost all the available Greek scientific and philosophical the available Greek scientific and philosophical works had been translated into Arabic at the works had been translated into Arabic at the House of Wisdom.House of Wisdom.Copies of these and other works were then Copies of these and other works were then made and sent to Islamic libraries throughout made and sent to Islamic libraries throughout Islam.Islam.Islamic scholars then began to extend and refine Islamic scholars then began to extend and refine all these works, creating a huge body of all these works, creating a huge body of scholarship.scholarship.


Arab mathematicsArab mathematics

One of the greatest Arab mathematicians One of the greatest Arab mathematicians whose work was preserved in these whose work was preserved in these libraries was Mohammed libraries was Mohammed ibnibn MMûûsâsâ alal--KhowârizimiKhowârizimi, who lived approximately , who lived approximately from 780 to 850. His patron was Caliph al from 780 to 850. His patron was Caliph al MaMa’’mmûûnn..He was the court astronomer to the Caliph He was the court astronomer to the Caliph and one of the early scholars at the House and one of the early scholars at the House of Wisdom.of Wisdom.

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AlAl--KhowârizimiKhowârizimi’’ss worksworks

AlAl--KhowârizimiKhowârizimi wrote two works that wrote two works that became enormously influential in both the became enormously influential in both the Islamic world and later in Europe when Islamic world and later in Europe when they were translated into Latin.they were translated into Latin.One was on arithmetic, the other on One was on arithmetic, the other on algebra.algebra.


AlAl--KhowârizimiKhowârizimi’’ss ArithmeticArithmetic

AlAl--KhowârizimiKhowârizimi’’ss book on arithmetic had a book on arithmetic had a title that could be translated as title that could be translated as Book of Book of Addition and Subtraction According to the Addition and Subtraction According to the Hindu Calculation.Hindu Calculation.It explained the Hindu method of writing It explained the Hindu method of writing numerals and the decimal system.numerals and the decimal system.


0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 00, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0

AlAl--KhowârizimiKhowârizimi mentions mentions ““nine lettersnine letters””meaning the symbols for the numerals 1 meaning the symbols for the numerals 1 to 9.to 9.He also introduces the symbol and use of He also introduces the symbol and use of the place holder, zero:the place holder, zero:

““When nothing remains [in subtraction], put When nothing remains [in subtraction], put down a small circle so that the place be not down a small circle so that the place be not empty, but the circle must occupy it.empty, but the circle must occupy it.””

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The influence of The influence of alal--KhowârizimiKhowârizimi’’ssarithmetic textarithmetic text

All copies of All copies of alal--KhowârizimiKhowârizimi’’ss original original Arabic text have been lost.Arabic text have been lost.We have only a Latin translation We have only a Latin translation AlgoritmiAlgoritmide de numeronumero IndorumIndorum by John Seville from by John Seville from the early 12the early 12thth century.century.So great was its influence that the new So great was its influence that the new numerals were misnamed numerals were misnamed ““ArabicArabic””because of albecause of al--KhowârizimiKhowârizimi’’ss book, despite book, despite being referred to there as of Indian origin.being referred to there as of Indian origin.


AlAl--KhowârizimiKhowârizimi’’ss algebra textalgebra text

Though the Though the ArithmeticaArithmetica of of DiophantusDiophantus, a Greek, , a Greek, was written long before alwas written long before al--KhowârizimiKhowârizimi’’ss and, and, despite the name, was really about algebra, despite the name, was really about algebra, Western Europe first learned about algebra from Western Europe first learned about algebra from alal--KhowârizimiKhowârizimi’’ss book.book.

DiophantusDiophantus’’ss work was not translated into Latin until work was not translated into Latin until long after allong after al--KhowârizimiKhowârizimi’’ss..

Did alDid al--KhowârizimiKhowârizimi learn algebra from learn algebra from DiophantusDiophantus’’ss Greek text?Greek text?

Unlikely, since their approaches are entirely different.Unlikely, since their approaches are entirely different.


AlAl--jabrjabr

The title of alThe title of al--KhowârizimiKhowârizimi’’ss algebra text, algebra text, in transliterated Arabic, is in transliterated Arabic, is HisâbHisâb alal--jabrjabrww’’alal muqâbalahmuqâbalah, which means something , which means something like like ““the science of reunion and reduction.the science of reunion and reduction.

Reunion meaning transferring negative terms Reunion meaning transferring negative terms to the other side of the equation.to the other side of the equation.Reduction meaning combining like terms on Reduction meaning combining like terms on the same side into a single term, or cancelling the same side into a single term, or cancelling like terms on both sides.like terms on both sides.

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AlgebraAlgebra

The title, The title, HisâbHisâb alal--jabrjabr ww’’alal muqâbalahmuqâbalah, was , was translated into Latin astranslated into Latin asLiberLiber AlbebraeAlbebrae et et AlmucabolaAlmucabola, , from which we from which we have taken have taken ““algebraalgebra”” as the mathematics of as the mathematics of solving equations for unknowns.solving equations for unknowns.Likewise our work Likewise our work ““algorithmalgorithm””, meaning a , meaning a specified procedure to follow to solve a problem specified procedure to follow to solve a problem is merely a Latinized version of alis merely a Latinized version of al--KhowârizimiKhowârizimi’’ssname.name.


AlAl--KhowârizimiKhowârizimi the transmitter, not the transmitter, not the inventorthe inventor

Just as EuclidJust as Euclid’’s work was the means by s work was the means by which much of Greek mathematics which much of Greek mathematics became known to a wider world, became known to a wider world, alal--KhowârizimiKhowârizimi’’ss books were the vehicles for books were the vehicles for the transmission of the old Persian the transmission of the old Persian mathematical knowledge to the rest of the mathematical knowledge to the rest of the world.world.


Algebra as divine revelationAlgebra as divine revelation

Algebraic formulae and algorithms were Algebraic formulae and algorithms were expressed as rhetorical pronouncements, expressed as rhetorical pronouncements, in verse, or as aphorisms. No symbols in verse, or as aphorisms. No symbols were used. No proofs were supplied.were used. No proofs were supplied.Any demonstrations given as justification Any demonstrations given as justification were usually in a geometric form, inspired were usually in a geometric form, inspired by Euclid.by Euclid.

Perhaps it was felt that an argument had to Perhaps it was felt that an argument had to be geometric to convince.be geometric to convince.

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A geometric solutionA geometric solutionConsider a problem that we would Consider a problem that we would write aswrite as

xx22 + 10x = 39+ 10x = 39The central square is xThe central square is x22

The four rectangles equal 10xThe four rectangles equal 10xThe four small squares at the The four small squares at the edges have area 4(10/4)edges have area 4(10/4)2 2

=100/4=25=100/4=25Therefore the entire large square Therefore the entire large square is equal to xis equal to x22 + 10x + 25.+ 10x + 25.Since xSince x22 + 10x = 39, then+ 10x = 39, thenxx22 + 10x + 25 = 39 + 25 = 64+ 10x + 25 = 39 + 25 = 64

Therefore the side of the large Therefore the side of the large square is square is √√64 = 8.64 = 8.Therefore x = 8 Therefore x = 8 –– 10/2 = 3.10/2 = 3.This is the method called This is the method called completing the square.completing the square.


The CrusadesThe Crusades

The Byzantine Empire, feeling threatened The Byzantine Empire, feeling threatened by the encroaching Islamic Caliphate by the encroaching Islamic Caliphate appealed to Christian Europe for military appealed to Christian Europe for military assistance.assistance.Thus began the Crusades to free the Holy Thus began the Crusades to free the Holy Land, starting in 1092, and lasting about Land, starting in 1092, and lasting about 300 years.300 years.


The Surprise DiscoveryThe Surprise Discovery

Much to the surprise of the invading Christians, Much to the surprise of the invading Christians, the Muslim world was much more literate and the Muslim world was much more literate and culturally advanced than Europe. culturally advanced than Europe. Europeans discovered a wealth of literature, Europeans discovered a wealth of literature, much of it originally written in Greek, that was much of it originally written in Greek, that was unknown in medieval Europe.unknown in medieval Europe.

Among these were works of Euclid, Ptolemy, and Among these were works of Euclid, Ptolemy, and much of Aristotle.much of Aristotle.

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Greek texts in ArabicGreek texts in Arabic

EuclidEuclid’’s s ElementsElements was available in Arabic was available in Arabic by the year 800.by the year 800.PtolemyPtolemy’’s s MegaleMegale SyntaxisSyntaxis ((““Great Great CompositionComposition””) appeared in Arabic in 827, ) appeared in Arabic in 827, where the title was transliterated as where the title was transliterated as ““Al Al MagestMagest””

The Arabic title stuck when it was translated The Arabic title stuck when it was translated into Latin later. We know it in English as into Latin later. We know it in English as The The Almagest.Almagest.


The Great Translation ProjectThe Great Translation ProjectA huge project was undertaken to translate A huge project was undertaken to translate much of this literature from Arabic into Latin to much of this literature from Arabic into Latin to be accessible to European scholars.be accessible to European scholars.

At Toledo, Spain, over 150 years.At Toledo, Spain, over 150 years.The typical process: The typical process:

Jewish rabbis, fluent in Arabic and Spanish, translated into Jewish rabbis, fluent in Arabic and Spanish, translated into Spanish.Spanish.Christian scholars, fluent in Spanish and Latin, translated Christian scholars, fluent in Spanish and Latin, translated from Spanish to Latin.from Spanish to Latin.

The translated works were sent to Italy and The translated works were sent to Italy and became the foundation for scholarship at the became the foundation for scholarship at the new European universities.new European universities.


Some notable translators: Some notable translators: Gerard of CremonaGerard of Cremona

Gerard of Cremona (1114Gerard of Cremona (1114--1187), a widely 1187), a widely educated Italian with a special interest in educated Italian with a special interest in astronomy. He moved to Toledo and learned astronomy. He moved to Toledo and learned Arabic there. Then he produced Latin Arabic there. Then he produced Latin translations of:translations of:

PtolemyPtolemy’’s s AlmagestAlmagestArchimedesArchimedes’’ Measurement of a CircleMeasurement of a CircleApolloniusApollonius’’ Conic SectionsConic Sectionsalal--KhowârizimiKhowârizimi’’ss HisâbHisâb alal--jabrjabrEuclidEuclid’’s s ElementsElementsAnd in total at least 90 complete Arabic texts on And in total at least 90 complete Arabic texts on mathematics and science.mathematics and science.

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Some notable translators:Some notable translators:AdelardAdelard of Bathof Bath

AdelardAdelard of Bath (1090of Bath (1090--1150), an English monk who 1150), an English monk who travelled widely to Spain, Italy, Sicily, Greece, Syria, and travelled widely to Spain, Italy, Sicily, Greece, Syria, and Palestine.Palestine.In Cordoba, Spain, around 1120 he disguised himself as In Cordoba, Spain, around 1120 he disguised himself as an Islamic scholar and managed to obtain an Arabic an Islamic scholar and managed to obtain an Arabic copy of Euclidcopy of Euclid’’s s Elements, Elements, which he translated into Latin. which he translated into Latin. This was the first translation of Euclid available in This was the first translation of Euclid available in Europe.Europe.

150 years later, Johannes 150 years later, Johannes CampanusCampanus brought out a new Latin brought out a new Latin translation which was clearer and more complete, being based translation which was clearer and more complete, being based upon the original Greek text. This translation became the basis upon the original Greek text. This translation became the basis for the first printed edition of Euclid in 1482. Euclidfor the first printed edition of Euclid in 1482. Euclid’’s s Elements Elements was the first mathematical work of any importance to be was the first mathematical work of any importance to be printed.printed.


ScholasticismScholasticism

The European universities tried to digest The European universities tried to digest the new learning in a Christian context.the new learning in a Christian context.AristotleAristotle’’s world view was taken as a base s world view was taken as a base for Christian dogma, and all scientific for Christian dogma, and all scientific understanding was fit to that context, understanding was fit to that context, including Ptolemyincluding Ptolemy’’s s AlmagestAlmagest, which , which therefore became theologically correct.therefore became theologically correct.

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