Ch. 4 Medieval Latin Astronomy

Ch. 4 Medieval Latin Astronomy

Chronology of Medieval Latin Astronomy 400 AD

Burning of library at Alexandria Sack of Rome Martianus Capella, Nuptials Philology & Mercury

Macrobius, Commentary on Dream of Scipio

Boëthius, Consolation of Philosophy 600 Venerable Bede, On the Divisions of Time; computus 800

1000 Fall of Toledo to Christians

Bologna first university

Gerard of Cremona: Latin trans. Almagest, Toledo Tables 1200

Sacrobosco, Treatise on Sphere Anon., Theory of the Planets

Profatius Judaeus: astrolabe quadrant

University of Paris Thomas Aquinas, Summa Theologica

Buridan, Oresme: impetus

1400

Gutenberg Bible (movable type) Peurbach, Regiomontanus: New Theory of the Planets, Epitome Bernhard Walther: observatory Fall of Constantinople to Turks Copernicus, On the Revolutions Reinhold, Prutenic Tables

Early Latin Astronomy

Early Latin Astronomy

Martianus Capella, Nuptials of Philology and Mercury • Elaborate allegory from early fifth century AD about the courtship of Philology (personifying learning and having been made immortal by the gods) by Mercury (personifying intelligent and profitable pursuit) • First two books accompanied by seven more dedicated to the liberal arts, handmaidens to the bride: 1) 2) 3) 4) 5) 6) 7)

Grammar Dialectics (metaphysics and logic) Rhetoric Geometry Arithmetic Astronomy Music (including poetry) Early Latin Astronomy

Martianus Capella, Nuptials of Philology and Mercury • Referred to by one modern commentator as “… an immense mass of learning, but the materials are ill-selected, ill-arranged, and ill-digested, though from amidst much that is dull and frivolous, we can occasionally extract curious and valuable information ...” • Was much admired during the Middle Ages and frequently copied (by hand, of course!) • Consequently accumulated a vast number of copying errors, with sometimes interesting results (next slide)

• Originally included idea attributed to Heraclides that Mercury and Venus go around Sun, not Earth; quoted by Copernicus Early Latin Astronomy

Martianus Capella, Nuptials of Philology and Mercury

Mercury and Venus’s orbits intersect!

Early Latin Astronomy

Macrobius, Commentary on Cicero’s ‘Dream of Scipio’ • Included handbook on astronomy • Cosmology presented based on Plato and Pythagoreans – number as basis • Spherical Earth with size according to Eratosthenes, seven planetary spheres, celestial sphere

• Placement of Sun ambiguous

Early Latin Astronomy

Macrobius, Commentary on Cicero’s ‘Dream of Scipio’

Universe with Earth at center, seven planetary spheres, and celestial sphere divided into twelve houses of zodiac Early Latin Astronomy

Macrobius, Commentary on Cicero’s ‘Dream of Scipio’

Diagram of solar eclipse

Diagram of lunar eclipse

Early Latin Astronomy

Boëthius, Consolation of Philosophy • Written while under house arrest just prior to his execution for treason by order of Theodoric the Great, King of the Ostrogoths • Is dialogue between author and Philosophy, personified as wise and compassionate lady • Has themes of transitory nature of fame and wealth, wheel of Fortune affecting everyone; consolation found in things of the mind, knowledge that God is the source of all good Early Latin Astronomy

Boëthius, Consolation of Philosophy • Has been described as “the single most important and influential work in the West on Medieval and early Renaissance Christianity, and is also the last great Western work that can be called Classical” (Wikipedia) • Was widely copied, edited, translated, and commented upon throughout the Middle Ages

Early Latin Astronomy

Boëthius • Apart from Consolation, aimed at preserving classical knowledge and especially Aristotle and Plato by translating Greek into Latin with comments • Plan largely unfulfilled; covered mostly mathematics and logic  influenced Scholastics in universities

• Organization of mathematical studies into what later became the quadrivium – arithmetic, harmony, geometry, and astronomy (not much about latter) Early Latin Astronomy

Date of Easter • Historically tied to Jewish observance of Passover – beginning 15th day of month Nisan • Early church: Pascha (Easter) on Nisan 14 • Independent of local Jewish observances, same for entire Church – First Council of Nicaea (325 AD):

Date = first Sunday after first ecclesiastical full Moon following “vernal equinox” (actually Mar. 21; equinox often on Mar. 20) Early Latin Astronomy

Date of Easter • Ecclesiastical full Moon ≠ astronomical full Moon; based on day number for Moon’s age (14) with months of 29 or 30 days (synodic month) – see rightmost column in Farmer’s Almanac (+1) • Like Jewish calendar, based on Metonic 19-year cycle -- lunisolar

• Look for cycle that includes both lunisolar cycle and the day of the week (Easter must be Sunday)

Early Latin Astronomy

Date of Easter • Solution in Venerable Bede, On the Divisions of Time • Julian calendar: common years 365 days ÷ 7 = 52 remainder 1  day of week advances by one each year (Jan 1. on Mon. – next year on Tues.) • Leap years every fourth year: remainder 2 instead of 1  every four years add 5 days • In seven cycles of four years go through exactly 5 weeks, end up same day of week (28 years) Early Latin Astronomy

Date of Easter • Metonic cycle for tropical year and lunar phase, 28-year cycle for day of week  19 × 28 = 532 years • (Western Christianity later switched to Gregorian calendar; more complicated – three of four century years omitted)

• Latin name for calculation of date: computus

Early Latin Astronomy

Translations from Arabic to Latin • Gerbert of Aurillac (later Pope Sylvester I) studied astrology at Barcelona, wrote book on astrolabe (ca. 1000 AD) • Athelhard of Bath went to Middle East; translated zij of al-Khwarizmi (1126), introduced ideas of algebra to West • Translations often had awkward combinations of Latin and Arabic words Islamic Influence

Translations from Arabic to Latin • After Toledo captured by Christians (1085), numerous works translated into Latin, spread through Europe • Gerard of Cremona: Toledo Tables; later Almagest (1175; other translations elsewhere, some from Greek) • Gerard’s other translations included Aristotle’s On the Heavens, Euclid’s Elements, and alKhwarizmi’s Algebra – 87 works in all! Islamic Influence

Astrology in the Christian West • Astrology in precarious position, as in Islam – regarded by some as superstitious • At same time, established as part of medicine, based on Aristotle’s microcosm-macrocosm analogy (Chaucer quote in Hoskin) • Aristotle’s ideas themselves initially considered heathen, banned at University of Paris

Medieval Astronomy

Astrology in the Christian West • Later, astrology reconciled with Christianity to some degree: influence of stars not rigidly deterministic but only tendency • Man imbued with free will by the Creator, capable of resisting influence • Aristotle’s ideas themselves were completely reconciled with Church doctrine by Thomas Aquinas at University of Paris in Summa Theologica (1274) Medieval Astronomy

Universities • Some grown from cathedral schools; all tied to Church -- faculty clerical (some minor orders); faculty and students all under canon (church) law, not civil law • First chartered university Bologna (1088); second Oxford (1096) • Most distinguished medieval university Paris (1200) Medieval Astronomy

Universities Organization of faculty in descending order of status: • Theology • Medicine • Law • Arts (liberal arts) Function of university originally was teaching, not research. Students of Arts were very young, typically 13 or 14 and sometimes younger. Medieval Astronomy

Medieval Arts Curriculum Theology or Metaphysics HIGHER PHILOSOPHIES Moral Philosophy, Natural Philosophy

Quadrivium: Arithmetic, Harmony (Music), Geometry, Astronomy

Trivium: Grammar, Rhetoric, Logic

(Hoskin, Cambridge Illustrated History) Medieval Astronomy

Medieval Christian Cosmology • Illustration from Nuremberg Chronicles (1493) • Aristotelian with added Christian features

• Earth at center with four elements, planetary spheres, sphere of fixed stars = firmament (precession), crystalline sphere (trepidation), and Prime Mover (daily)

(Hoskin)

• Empyreum = Heaven with God and angels on outside of all

Medieval Christian Cosmology • Illustration from Peter Apian’s Cosmographia (1539) • Aristotelian with added Christian features

• Earth at center with four elements, planetary spheres, sphere of fixed stars = firmament (precession), crystalline sphere (trepidation), and Prime Mover (daily) • Empyreum = Heaven with God and angels on outside of all

Astronomy Textbooks • Need for simplified textbooks on astronomy for Arts students – Almagest far too hard! • Sacrobosco (John of Holywood) in 13th century:  Computus – introduction to time-reckoning  Algorismus – arithmetic used in astronomy  Tractatus de sphaera = Treatise on the Sphere – celestial sphere (stuff we did at beginning of course); comprised four parts: I. II.

Celestial sphere with Earth at center, daily motion Celestial equator, ecliptic and zodiac, meridian, altitude of celestial pole III. Rising and setting of celestial bodies, length of day and night at different latitudes IV. Motions of Sun, Moon, planets; outline of eclipses Medieval Astronomy

Astronomy Textbooks • Part IV completely unsatisfactory • Most popular replacement: anonymous Theory of the Planets fairly soon (later 13th century)  Hipparchus’s model for Sun  Ptolemaic-like models for Moon, Mars, Jupiter, and Saturn  Models for Venus and Mercury  Treatment of eclipses not entirely satisfactory Medieval Astronomy

Universities Beginning of new thought – research? -- in universities, not just teaching ancients

University of Paris in 14th century – concept of impetus in connection with Aristotle’s theory of violent or forced motion

Medieval Astronomy

Aristotle • Sublunary (terrestrial) region:  everything consisting of varying proportions of four elements (taken from Empedocles) – earth and water (heavy) and air and fire (light)  mutability (change) is endemic  natural motions are vertical – down (heavy elements) and up (light elements)

• Superlunary (celestial) region:  only one element = “fifth” element or quintessence

 immutability = no change  natural motion is uniform circular motion

Motion other than natural is “violent/forced motion”

Universities Development of impetus concept -• Jean Buridan: effect of air resistance on motion

• Nicole Oresme: impetus as “incorporeal force” (like fuel) initially infused into body, then used up (also anticipated Descartes with analytic geometry) • Albert of Saxony: projectile’s trajectory in three stages: impetus dominates; impetus exhausted and gravity becomes important; gravity dominates Medieval Astronomy

Universities Oresme, Commentary on Aristotle’s On the Heavens • Impetus allows one to evade the argument of fall against the Earth’s rotation

• Arrow is imparted impetus in forward direction by Earth’s rotation before it is released • Arrow moves eastwards as it rises and falls • Conclusion: One cannot tell if Earth rotates or is at rest. Medieval Astronomy

Aristotle • Earth does not rotate around an axis: Argument of fall: If Earth rotates, an arrow shot vertically will fall to the west of the archer; instead it falls on the archer’s head unless he moves away.

(Actually, the arrow does fall to the west, but by an extremely small amount.)

Instruments of Late Medieval Old Quadrant foresight

curves for finding time of day

angular scale for altitude

backsight plumb line

(Hoskin, Cambridge Illustrated History)

Medieval Instruments

Instruments of Late Medieval New Quadrant = Astrolabe Quadrant ecliptic from rete of astrolabe, folded twice

angular scale for altitude

backsight

foresight Medieval Instruments

Instruments of Late Medieval New Quadrant = Astrolabe Quadrant • Invented by Profatius Judaeus (Jacob ben Mahir) • Both quadrants more efficient than astrolabe: for given size, baseline is twice as long for angular scale (radius vs. diameter) • Like astrolabe, new quadrant is both demonstration instrument and measuring instrument Medieval Instruments

Instruments of Late Medieval Cross-Staff

(Hoskin)

Medieval Instruments

Instruments of Late Medieval Cross-Staff • Invented by Levi ben Gerson = Gersonides in first half of 14th century • Observer sights across top of cross-piece at one object, then across bottom at the other, adjusting the position of the cross-piece to do so simultaneously • Long piece is marked so that distance of crosspiece from eye can be measured Medieval Instruments

Instruments of Late Medieval Cross-Staff • Cross-staff (also known as Jacob’s staff) had two uses: 1. Measuring angular separation between two objects, as with the star and the Moon in the illustration 2. Measuring the altitude of an object as angular distance above the horizon; if Polaris, estimates latitude for navigation

• (Could also be used to measure Sun’s altitude with eye protection, but was replaced by backstaff, which uses a shadow, and later sextant)

Medieval Instruments

Instruments of Late Medieval Nocturnal

Little Dipper sighting hole

(Polaris)

Big Dipper pivot arm

handle

• Sight through hole on Polaris • Line up pivot arm on pointer stars – Big or Little Dipper Medieval Instruments

Instruments of Late Medieval Nocturnal 1) Use reading to correct for displacement of Polaris from true North Celestial Pole 1) Use reading to tell sidereal time of night; correct to solar time by using scale for days of year

Medieval Instruments

Instruments of Late Medieval Mechanical Clock

(Hoskin, Cambridge Illustrated History) Medieval Instruments

Instruments of Late Medieval Mechanical Clock • Mechanical clocks for time of observation -not yet accurate enough for precision time measurement • Richard of Wallingford – astronomical clock replica medieval universe: Moon’s phases and eclipses, variable motion in longitude

• Giovanni de Dondi – astrarium at Padua; geared Ptolemaic mechanism (Remember Antikythera Mechanism?)

Medieval Instruments

Instruments of Late Medieval Astrarium of de Dondi (1364)

Moon dial

Modern reproduction from detailed plans

Venus dial

Medieval Instruments

Prague Astronomical Clock (1410) Medieval Instruments

Instruments of Late Medieval Prague Astronomical Clock, complete view

astrolabe face

calendar face

Medieval Instruments

Instruments of Late Medieval Astronomical Clock of Strasbourg Cathedral (1354)

astrolabe face

Medieval Instruments

Astronomy Textbooks • Printing using moveable type introduced to Europe by Gutenberg, 1450 – two major effects:  More rapid and wider dissemination of ideas  Elimination of copying errors (but more important to proofread!)

• Georg Peurbach (sometimes Purbach), professor at University of Vienna; gave private lectures on Ptolemaic system • Lecture notes written up, published posthumously 20 years later by protegé Johannes Müller, known as Regiomontanus – New Theory of the Planets (1474)

New Theory of the Planets (1474) • Peurbach self-taught in astronomy (professor of philology!)

• Studied Almagest and Alhazen’s On the Configuration of the World among others • Sufficiently knowledgeable to be appointed mathematicus (court astrologer) to King of Bohemia and Hungary, later to Holy Roman Emperor

Renaissance Textbooks

• Elementary but thorough treatment of Ptolemaic system • Based on planets embedded in crystalline spheres; system resembles aspects of al-Tusi’s work – connection? • Most widely used textbook of sixteenth century

New Theory of Planets

• Spheres rolling within spheres – 3-d Tusi couple? New Theory of Planets

Regiomontanus • Prodigy – almanac at 11 superior to Gutenberg’s!

• Cardinal Bessarion from Constantinople met Peurbach, Regiomontanus; invited to Italy to study Almagest in Greek, critique translation by George of Trebizond • Result was Epitome of the Almagest (1462), a clarification of Ptolemaic system read by Copernicus and Galileo • Finished after Peurbach’s death Regiomontanus

• Wrote textbooks on trigonometry and algebra

• Moved to Nuremberg 1471 – center of trade, learning, publishing – for two reasons: 1. Recruit highly qualified observers to obtain more and better observational data  improve planetary tables 2. Print books – key element in advancing astronomy and mathematics (dissemination, eliminate copying errors)

Regiomontanus

Regiomontanus

• Published Ephemerides (almanacs) for years 14751506 -- day-by-day tables  Left-hand page: positions Sun, Moon, planets, nodes for day  Right-hand page: positions of Sun w.r.t. Moon, times of new and full Moon, Moon w.r.t. planets, planets w.r.t. each other (conjunctions)  Feast days, Sunday dominical letters

Regiomontanus

• Ephemerides famously used by Christopher Columbus in 1504 to frighten natives of Jamaica into giving him provisions

Regiomontanus

• Called to Rome by Pope Sixtus IV – calendar reform • Died shortly thereafter, reform didn’t happen; rumor poisoned by George of Trebizond’s relatives – more likely plague • Calendar reform postponed until much later

Regiomontanus

Bernhard Walther • Wealthy merchant of Nuremberg; humanist, friend of Albrecht Dürer • Trained by Regiomontanus • Converted part of house into observatory • Acquired Regiomontanus’s instruments after death; observed over 30 years

Walther

Observatory located on roof

Albrecht Dürer House (museum)

Walther

• First long European series of accurate observations

• Recognized and allowed for atmospheric refraction • Most accurate observations until Tycho Brahe (1’ for Sun, 5’ for Moon and planets) • Used mechanical clock in astronomical observation • Three Mercury observations used by Copernicus Walther

Copernicus • Born Niklas Koppernigk in Thorn (Torun in modern Poland), 1473 • Name spelled differently in vernacular at various times, as was customary • In Latin Nicolaus Copernicus; only last used • Thorn in German-speaking region  both Germans and Poles claim Copernicus!

• Born in Thorn (Torun in modern Poland)

(POLAND)

Copernicus

• Birthplace is house at left (not too shabby!) • Father was well-to do merchant; died when Copernicus 10; mother predeceased • Taken under care of maternal uncle; priest, later Bishop of Warmia – political as well as spiritual authority Copernicus

• Uncle planned for career in Church = canon • Entered University of Cracow at 18 (had been in several schools previously) • Studied astronomy with Brudzewski; Peurbach’s New Theory of the Planets and Aristotle’s On the Heavens among others

Copernicus

• Early opposition to uncle’s plan  Copernicus sent to Italy • Matriculation University of Bologna (uncle’s alma mater) as member of the German nation (one of four)

• Studies for degree in canon (church) law; humanities • Also learned more astronomy from Domenico Maria Novara; boarded, observed with him

Copernicus

• Ended up taking degree later on at Ferrara (cheaper than Bologna) • After brief trip home permitted return to Italy, study medicine – physician to uncle and chapter of cathedral

• Studied medicine two years at Padua • Returned to Poland to take up position as canon at Frauenberg (Frombork) cathedral Copernicus

Frauenberg (Frombork) Cathedral, 2010

Spent rest of life carrying out his duties in Frombork …

Copernicus

… while pursuing his understanding of planets ...

Copernicus

… and, of course, observing them.

Copernicus

• According to account much later, invited to lecture on astronomy at Jubilee in Rome (1500, before degree)

• Early statement of ideas – Commentariolus = Brief Commentary (1512); circulated in manuscript

Copernicus

Brief Commentary Seven theses: 1. Celestial bodies do not all revolve around the same point. 2. Earth’s center is only the center of the Moon’s orbit and the center of gravity (tendency of things to fall). 3. All planets revolve around the Sun, which is the center of the Universe. 4. The Earth-Sun distance is extremely small compared to the distances of the stars  no parallax 5. The stars are immobile; their apparent motion is caused by Earth’s rotation. Copernicus

Seven theses (cont’d): 6. Earth revolves around the Sun, causing Sun’s apparent motion around the ecliptic, in addition to other motions. 7. Earth’s revolution around Sun causes the retrograde motion of the other planets

Copernicus

• Reception among those who saw it not unfavorable • Interest on part of high Church officials, including Pope Clement VII and several cardinals • Observed especially from 1512 through 1529, accumulating data to prove his theses

Copernicus

Copernicus • Situation changed with 1539 visit by Rheticus, professor of astronomy and mathematics at Wittemberg University (Protestant).

• Rheticus spent two years with Copernicus, learned about results. • Rheticus wrote Brief Narration describing Copernicus’s work. • Lack of controversy persuaded Copernicus to publish complete heliocentric system.

Copernicus

On the Revolutions, Six Books • Above is original title given by Copernicus, translated from Latin De revolutionibus, libri VI • Dedicated to Pope Paul III

• Printed Nuremberg; originally Rheticus oversaw printing • Oversight turned over to Andreas Osiander, Lutheran minister and theologian, who saw it through to completion in 1543 Copernicus

On the Revolutions of the Heavenly Spheres, Six Books • Above is translation of title given by Osiander (next slide)

Nuremberg Copernicus

• Osiander made two unauthorized changes, probably with the best of intentions: 1) Title changed to On the Revolutions of the Heavenly Spheres, Six Books 2) Preface added without Osiander’s name attached

• First suggests Earth excluded, especially to reader of that time (Aristotle’s cosmology) – immobile? • Second contradicts Copernicus’s main point Copernicus

Osiander’s Preface

• No consultation with Copernicus or Rheticus • Admitted contents controversial • Denied models are real; confused reader about Copernicus’s position Copernicus

Why did Osiander make these changes? He was afraid of the reaction of the Protestants! He was right, too: Martin Luther himself referred to Copernicus as “that fool [or possibly ‘that fellow’] who would overturn all of astronomy.” Philipp Melanchthon, one of the foremost Lutheran intellectuals, denounced Copernicanism. Copernicus

Why were the Protestants opposed?

• They believed that each man could and should interpret the Bible for himself. • There are passages that seem to be contradicted by Copernicanism – Sun standing still in Joshua, other plain references to Sun going around Earth.

Copernicus

Why was the Roman Catholic Church not opposed? • The bishops and cardinals thought the work interesting and encouraged him. • Church officials saw no threat to Church doctrine – Church’s theologians interpreted Bible. • Copernicus was one of their own. Church’s opposition came later, with Galileo’s challenge. Copernicus

On the Revolutions, Six Books I.

II. III. IV. V. VI.

Outline of Copernican system and arguments for it, plus two chapters on plane and spherical triangles Spherical astronomy Earth’s motions: rotation, trepidation, precession Moon’s motions Planetary motions in (celestial) longitude Planetary motions in (celestial) latitude

Copernicus

Arguments for Copernican system – two kinds:

1. Citations from ancients – Philolaus, Heraclides, and others, but not Aristarchus (struck out in MS); also, idea of Sun as “hearth of the Universe” 2. Criticisms of Ptolemy – equant, celestial sphere rotating more rapidly than Earth; also, Sun’s role “Escape clause” from argument of parallax (as in Brief Commentary)

Copernicus

• In Book I, sketch of heliocentric model, no details • Orbital periods given roughly • Orbital periods calculated from separate formulae for inferior, superior planets Copernicus

Formulae for sidereal (orbital) period P using synodic period S, which is the observed quantity:

Copernicus

d

Inferior planet: assuming circular orbit, size of orbit in astronomical units (AU) = sine of greatest elongation Copernicus

d

Superior planet: knowing planet’s orbital period, calculate angle P’SE’ and find length SP’ = distance. Copernicus

Distances mostly pretty good except Saturn, most distant (Earth doesn’t count – 1 AU by definition) Copernicus

• Model for Earth’s orbit • T is Earth (Terra), S is Sun, E is “mean Sun”

(Note that Earth’s orbit isn’t centered on the real Sun!)

• Circle with small circle going around it is primary epicycle; much smaller circle is secondary epicycle, both moved near center Copernicus

• Model for superior planet orbit • T is Earth (Terra), E is mean Sun, P is planet

Copernicus

• Model for inferior planet orbit • T is Earth (Terra), E is mean Sun, Q is planet • Epicycle moved to center from outside

Copernicus

• Model for Moon’s orbit • T is Earth (Terra), E is mean Sun, M is Moon • Identical to al-Shatir’s lunar model

Copernicus

Copernicus Several misconceptions about Copernicus’s work: • Won adherents because Copernican system much simpler than Ptolemaic

Actually at least as complicated – eliminated separate epicycles for retrograde motion, but added secondary epicycles to eliminate equant  ended up with roughly as many as Ptolemy

Several misconceptions about Copernicus’s work: • Proved Sun at center and Earth rotating, revolving

Only presented arguments, no proof; proof of Earth’s revolution around Sun in 1728, rotation a century later

Copernicus

Several misconceptions about Copernicus’s work: • Hardly read by anyone – few could understand it

There were at least 500 copies, many of which were marked with comments, especially about the mathematical models if not the arguments for the Sun being at the center. Also, it went to a second edition (see next slide). Copernicus

On the Revolutions, Six Books • Title page from second edition (yes, there was one!). • Published in Basel (Switzerland) in 1566

Copernicus

Copernicus In some ways Copernicus’s work was of the past – epicycles and uniform circular motion. However, the one “big idea” in it – heliocentrism - was to have an enormous impact. It led to an understanding of the planets’ actual orbits and, in the process, to the establishment of the foundations of modern physics. That is really what the Copernican Revolution is all about. That idea was also a giant first step in understanding our true place in the Universe. Copernicus

• Approached by Vatican about calendar reform (remember Regiomontanus) • Opined that time wasn’t ripe – needed more and better data before undertaking • Reform followed his death by a few decades

Copernicus

Prutenic Tables (1550) • Erasmus Reinhold, professor of astronomy and mathematics at Wittenberg, author • Based on On the Revolutions, used models • Superseded Alfonsine Tables

• Somewhat better, partly because much newer • Replaced in turn by Kepler’s Rudolphine Tables about 80 years later (much better) Copernicus

End Ch. 4