Quadrivium · the arts of number · art 7 of 7
Astronomy
Astronomia
Mathematical order and celestial motion
- Classical definition
- The science of magnitude in motion (Nicomachus's sphairike, 'sphaeric'). (Nicomachus, Introduction to Arithmetic I.3; Boethius, De institutione arithmetica I.1.)
- Proper object
- Continuous quantity in motion: the measurable motions of the heavens.
- Place in the order
- Quadrivium: magnitude in motion · see the whole order
Astronomy is the art of magnitude in motion, in Nicomachus's division of the quadrivium: it studies the sizes, distances and movements of the heavenly bodies. It is where arithmetic and geometry meet observation, and Eratosthenes' shadow measurement, Kepler's laws and Newton's moon test each turn angles and times into claims that can be checked. It also shows how a model is tested, kept or replaced, which is a habit every other art relies on.
The guiding questionWhat moves in the sky, how do we know, and which model accounts for it best?
Round 1 opens soon
The spiral
Three strands, three rounds
Each strand comes back in every round, a level deeper. Finish Round 1 across the arts before Round 2: breadth first, then depth.
Round I Foundations
- The skyDescribe the daily and yearly motions of the skyI can describe how the stars, the Sun and the Moon move over a day, a month and a year, and explain the seasons and the Moon's phases with a labelled diagram.
- Measuring the heavensReproduce Eratosthenes' measurement of the EarthI can reproduce Eratosthenes' calculation of the Earth's circumference from a shadow angle and a stated distance, and say which assumptions it needs.
- Models of the heavensCompare the three arrangements of the heavensI can state what Ptolemy's, Copernicus's and Tycho's arrangements each claim, and explain the retrograde motion of Mars in each.
Round II Practice
- The skyLocate and predict with sky coordinatesI can place an object by altitude and azimuth, predict the Sun's noon height through the year at a stated latitude, and predict the next two phases of the Moon from one observation.
- Measuring the heavensDistances to the Moon and the SunI can find the Moon's distance in Earth radii from its parallax, and explain why Aristarchus' method for the Sun's distance is sound in principle but gave a poor result.
- Models of the heavensKepler's laws and the tests the telescope made possibleI can state Kepler's three laws with their sources, check the third law against planetary data, and say which arrangements the phases of Venus and the moons of Jupiter rule out.
Round III Mastery
- The skyLong cycles: precession, eclipses and the calendarI can explain the precession of the equinoxes, the Saros eclipse cycle and the difference between the sidereal and synodic month, and use one of them to check a recorded prediction.
- Measuring the heavensStellar parallax and the scale of the universeI can use a stellar parallax in arcseconds to find a distance in parsecs, and explain why the absence of an observable parallax was a serious objection to Copernicus and why no direct measurement answered it until Bessel's in 1838.
- Models of the heavensFrom Kepler's laws to Newton's moon testI can derive Kepler's third law for a circular orbit from an inverse-square force, and repeat Newton's moon test to within a few per cent.
Core reading in the Library
- LibraryThe Seven Liberal Arts, Hands-On: Logic, Proof and the QuadriviumA practical guide to the trivium and quadrivium, with a practice and a made thing for each art, plus a path through Socratic questioning, Aristotle's logic and Euclid's method.
- LibraryGeometry, Euclid and TrigonometryThe bridge from classical geometry to trigonometry and precalculus, including a track for studying Euclid and Apollonius by reconstruction.
- LibraryHypotheses, Predictions and TestsHow to generate several competing explanations, derive checkable consequences, and commit to predictions before looking at the result.
- LibraryQuantitative Reasoning with Stated AssumptionsWorked examples in rates, packet sizes, recovery time and cost ledgers that show arithmetic is only as good as the assumptions you state.
- LibraryCalculus and Beyond: Change, Structure and UncertaintyWhat calculus, linear algebra, multivariable calculus, differential equations and statistics each accomplish, with gates a learner can use to check understanding.
Where it is used
- LibraryNutrient Coverage on a Plant-Based Diet · Nutrient by nutrientThe vitamin D entry names latitude and season among the factors in sunshine; astronomy explains both through the Sun's height in the sky.2 min glance · 5 min
- LibrarySystems Thinking: Feedback, Queues, Information and Dynamics · Time series and signalsThe yearly cycle is the model of seasonality: separate it from trend before you call a change an anomaly.2 min glance · 7 min
Seminars
Read the sources
One primary text for each round, with a question to think through. All 21 seminars →
Astronomy · Round 1
- Aristotle, On the Heavens (De Caelo), II.14 (Bekker 296a24-298a20; the arguments for a spherical Earth run from 297a8)J. L. Stocks's Oxford translation on the MIT Internet Classics Archive. The page carries Book II in full without Bekker numbers; chapter 14 is 'Part 14', the last on the page. Aristotle argues for a spherical Earth from the curved shadow in lunar eclipses and from the stars that change as one travels north or south, and cites the mathematicians' circumference of 400,000 stades.
- Cleomedes, On the Circular Motion of the Celestial Bodies (Caelestia), Book I, chapter 10 in Ziegler's numbering (chapter 7 in Todd's numbering)T. L. Heath's English translation of the chapter, reproduced on a freely readable page; it also gives Posidonius's method. Cleomedes reports the measurement of Eratosthenes (about 276 to 194 BC). The length of the stade is uncertain, so the result in kilometres is too. Bowen and Todd (2004) is the standard translation but is not free.
Aristotle argues that the Earth is a sphere from the curved shadow in an eclipse and from the stars that change as you travel, and Cleomedes reports that Eratosthenes found its size from one shadow angle and one distance. Which of these arguments depends most on what we already assume, and which could you check yourself this week?
- Eratosthenes needs Syene and Alexandria on one meridian and the two lines to the Sun to be parallel. Cleomedes states both. Does his reason for the parallel lines convince you, and what would go wrong if they were not parallel?
- Cleomedes also reports Posidonius, who used the star Canopus, on the horizon at Rhodes and one forty-eighth of a circle above it at Alexandria. Compare the two methods: which hypotheses do they share, and which does each add?
- Aristotle gives 400,000 stades as the figure of unnamed mathematicians. How much weight can a figure carry when neither its method nor its unit is given?
Astronomy · Round 2
- Ptolemy, Almagest, Book I, chapters 1 to 7 (chapter 8, on the two primary motions of the heavens, completes the group)The linked page gives I.1 to I.7 in R. Catesby Taliaferro's English translation (Great Books of the Western World, 1952) without crediting it; it is freely readable, but its copyright status is not stated. G. J. Toomer's translation (1984) is the standard one but is not free. Heiberg's Greek text (Teubner, 1898-1903) is public domain and digitised on archive.org. Chapters 3 to 7 argue that the heavens move as a sphere and that the Earth is spherical, central, point-like and at rest; chapter 7 discusses the rotation of the Earth.
- Nicolaus Copernicus, De revolutionibus orbium coelestium, Book I, chapters 8 and 9 (the reply to the ancient arguments, and the several movements of the Earth)The linked page carries an older English translation of Book I, chapters 1 to 9, which it does not credit, together with the unsigned preface 'To the reader' that Osiander added. Chapter 10, on the order of the spheres, follows chapter 9 but is not on that page; the Dobson and Brodetsky translation (Royal Astronomical Society, 1947) or Rosen's (1978) will be in a library.
Ptolemy grants that an Earth turning once a day might make the heavenly phenomena simpler to account for, but says it would be absurd given what happens around us in the air (Almagest I.7). Copernicus answers that a motion natural to the Earth would not scatter it, and asks why Ptolemy did not fear for the far larger heavens (De revolutionibus I.8). Who has the better argument on the evidence each could have, and what would you need to observe to decide?
- Ptolemy's objections are about falling bodies, clouds and birds, and Copernicus replies with a different account of natural motion. Is the disagreement about the sky at all, or about how things move on the Earth?
- Both writers treat the simplicity of an arrangement as a reason. Is simplicity evidence, and does each mean the same thing by it?
- Osiander's unsigned preface says the hypotheses of astronomers need not be true, only fit the observations. Do Ptolemy's and Copernicus's own words in these chapters claim more than that?
Astronomy · Round 3
- Galileo Galilei, Sidereus Nuncius (The Sidereal Messenger, Venice, 1610), the sections on the Moon, on the fixed stars and the Galaxy, and on the four satellites of Jupiter (observations from 7 January to 2 March 1610)E. S. Carlos's 1880 translation, public domain; Van Helden's 1989 translation is standard but not free. The book does not report the phases of Venus, which Galileo observed later in 1610; the Gutenberg volume also includes part of Kepler's preface to his Dioptrice (1611), which prints Galileo's letters of December 1610 and January 1611 announcing that Venus shows phases like the Moon.
- Isaac Newton, Principia, Book III, Proposition IV, Theorem IV (the moon test), read with the Rules of Reasoning that open Book IIIMotte's translation of 1729 in the 1846 American edition, public domain. Proposition IV is the fourth of Propositions I to IX on the linked page. The Rules of Reasoning are two pages earlier, before the Phaenomena, at BookIII-Rules under the same address.
Galileo's moons of Jupiter show a body circling a centre that is itself moving, and Newton's moon test shows that the force holding the Moon in its orbit is the force that makes a stone fall, weakened by the square of the distance. What does each show about the Earth's place and the nature of the heavens, and what does neither show?
- Galileo says the satellites of Jupiter remove the scruple of those who accept the planets circling the Sun but cannot accept one Moon circling a moving Earth. Does this show that the Earth moves? What does it show?
- Newton takes the Moon to be about 60 Earth radii away and notes that astronomers give anything from 56 and a half (Tycho) to 60 and two-fifths (Street). How much does the result depend on that figure?
- Rule IV says propositions gathered by induction from the phenomena are to be held as accurately or very nearly true until other phenomena make them more exact or liable to exceptions. How does that rule treat Kepler's laws, and what has happened to Newton's own law since?