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Seminars · 21 primary texts

Read the sources

In a seminar the text is the teacher. Read the passage, then talk or write your way through the opening question. Good seminars end with a better question than they began with. Every link goes to a free edition.

This week's question is on the curriculum page. Notes you save here stay in this browser.

Round I

Foundations

Grammar · Round 1

  • Augustine of Hippo, Confessions, I.8.13 (Book I, chapter 8, section 13)J. G. Pilkington's translation (Nicene and Post-Nicene Fathers, First Series, vol. 1, 1887), public domain, as revised for New Advent. Read the whole of section 13, where Augustine describes how he learned to speak.
  • Dionysius Thrax (attributed), The Art of Grammar, sections 12 to 25 in Davidson's numbering (12 the word, 13 the sentence and the list of eight parts, 14 noun, 15 verb, 19 participle, 20 article, 21 pronoun, 23 preposition, 24 adverb, 25 conjunction)Thomas Davidson's 1874 translation, 'The Grammar of Dionysios Thrax' (Journal of Speculative Philosophy 8, pp. 326-339), public domain. Davidson thought the work genuine, but its authorship is now disputed: Vincenzo Di Benedetto argued in 1958 that only the opening sections go back to Dionysius (2nd century BC) and that the rest, including the parts of speech, was compiled in the 3rd or 4th century AD. Many specialists agree, though some still defend the attribution. Read it as the short school grammar that shaped later Greek and Latin teaching, not as the certain work of one author. Other editions number the sections differently.

Augustine says he learned to speak by watching which things his elders named and moved towards, and by reading their faces, glances and tones. Dionysius sorts the words of a sentence into eight kinds. What does each account assume a word is, and which of the two explains better how a child first learns what a word means?

  • Augustine says he gradually gathered what things words were the signs of. Are all eight of Dionysius's parts signs of things, or do some, such as the article and the conjunction, do a different job?
  • Dionysius defines a noun as a declinable part of speech signifying something concrete or abstract (section 14). Is 'nobody' a noun by that test, and what does the test miss?
  • Augustine's account suggests that a child learns 'this' by pointing. What is left to learn after the pointing, and who must already know what?

Logic or dialectic · Round 1

  • Plato, Euthyphro, 2a-16a (the whole dialogue)Read the whole dialogue in one sitting. Give most attention to 5d-6e (Socrates asks for the one form that makes every pious act pious), 10a-11b (is the pious loved by the gods because it is pious?) and 15c-16a (the ending). The link is H. N. Fowler's translation on Perseus, which renders the key word (hosion) as 'holy'; many other translations say 'pious'. Benjamin Jowett's translation is also free at Project Gutenberg (ebook 1642).

Euthyphro is certain he knows what piety is, yet each answer he gives fails under Socrates' questions. What does the questioning accomplish, and would you call the ending a failure?

  • At 6d-e Socrates asks for the one form that makes every pious act pious, not a list of pious acts. Why is a list of examples not a definition?
  • At 10a Socrates asks whether the gods love the pious because it is pious, or whether it is pious because they love it. Why does the order of the explanation matter for the definition?
  • At 11b-d Euthyphro complains that whatever they propose moves about and will not stay where they put it, and each speaker suggests the other is the Daedalus who sets it moving. Whose fault is it: the definitions', the speaker's or the questioner's?

Rhetoric · Round 1

  • Gorgias, Encomium of Helen, sections 6 to 10, with the summary at section 15 (Diels-Kranz 82 B11)LaRue Van Hook's 1913 translation (Classical Weekly 6, no. 16), public domain in the United States, but partial: it gives sections 1 to 10, 15 and 19 to 21, so sections 11 to 14 are not on the page. Section 6 lists the four possible causes of Helen's flight; section 8 begins the account of the power of speech.
  • Plato, Gorgias, 447a-466aW. R. M. Lamb's Loeb translation with the Greek, on Perseus. Socrates asks Gorgias what rhetoric is from 447c; belief and knowledge are separated at 454c-455a; rhetoric is called a knack and a part of flattery at 462b-466a.

Gorgias says that speech is a powerful lord that can end fear and stir pity, and he uses that power to excuse Helen. Plato's Socrates says the same craft is a knack for flattering people who do not know. Is the power of speech a reason to trust the speaker or to distrust him?

  • Socrates separates persuasion that produces belief from persuasion that produces knowledge (454c-455a). Which kind does the Helen produce in its hearer?
  • Gorgias sums up that if Helen was won over by persuasion she deserves pity, not blame (section 15). Is a listener ever responsible for being persuaded?
  • At 462b-466a Socrates compares rhetoric with cookery. What would rhetoric need to have in order to count as a craft in his sense, and does the Helen have it?

Arithmetic · Round 1

  • Nicomachus of Gerasa, Introduction to Arithmetic, translated by Martin Luther D'Ooge (Macmillan, 1926), Book I, chapters 7 to 13Chapter 7 divides number into even and odd. Chapters 8 to 10 divide the even. Chapters 11 to 13 divide the odd into the prime and incomposite, the secondary and composite, and the kind that is composite in itself but prime and incomposite relative to another. Chapter 13 also describes the sieve of Eratosthenes and a test by repeated subtraction for whether two odd numbers have a common measure. D'Ooge's translation was published in 1926, so it entered the public domain in the United States on 1 January 2022. This Zenodo record holds a full PDF scan of the 1926 volume. Cite by book and chapter, not by page.
  • Euclid, Elements, Book VII (David Joyce's online edition, Clark University), Definitions 1 to 16; Propositions 1 and 2Definitions 11 to 14 define prime, relatively prime, composite and relatively composite numbers by what measures them. Propositions 1 and 2 are linked from the same page. Proposition 2 finds the greatest common measure of two numbers by repeated subtraction; Elements X.2 applies the same idea to magnitudes.

Euclid calls a prime a number 'measured by a unit alone' (Book VII, Definition 11), while Nicomachus calls it 'prime and incomposite' and files it among the odd numbers (I.11). What does each way of defining a prime make you notice, and what does each hide?

  • On Euclid's Definition 11, does the number 2 count as prime? Where would Nicomachus put it, given that he sorts primes among the odd numbers?
  • Nicomachus describes a sieve (I.13) that separates primes from composites by a procedure. Does knowing how to find primes tell you what a prime is?
  • In VII.1 and VII.2 Euclid subtracts the smaller number from the larger again and again. What does he assume about what it means for one number to 'measure' another?

Geometry · Round 1

  • Euclid, Elements, Book I, Definitions 1–23 (especially 15 and 20), Postulates 1–5, Common Notions 1–5, and Proposition 1D. E. Joyce's online edition (Clark University). Proposition 1 is at https://mathcs.clarku.edu/~djoyce/elements/bookI/propI1.html, and its guide points out that the proof never justifies the existence of the point C where the circles meet. The Academy quotes Heath's wording, which says 'distance' where Joyce says 'radius'.
  • Plato, Meno, 82b–85b (Socrates and the slave boy), with 85c–86b for Socrates' conclusionW. R. M. Lamb's English translation on Perseus, one Stephanus section per page. The Greek text (Burnet) is the same address with 0177 in place of 0178. Read the whole passage once, then reread it and mark every question the boy answers with a plain yes.

In both texts a figure is drawn and a conclusion follows. What does the figure contribute to the knowing, and what must already be in the mind of the one who looks at it?

  • Euclid asks us to grant five things instead of asserting them. Why does he ask, and what would it mean to refuse the fifth?
  • Socrates says he is only asking questions (82e). Which of his questions between 82b and 85b could the boy answer with a bare yes, and does that change what has been shown?
  • In I.1 the point C is named before it is shown to exist. Is a proof still a proof if the figure supplies a fact that the words do not?

Music · Round 1

  • Boethius, De institutione musica, Book I, chapters 1-2 and 10-11 (Friedlein's chapter numbering)Latin only at this link (Friedlein's text); the standard English translation (Calvin Bower, Fundamentals of Music, Yale, 1989) is in print and not freely online. I.1: music is joined to us by nature and can ennoble or corrupt character. I.2: three kinds of music, of the world (mundana), of the human being (humana, the joining of soul and body) and of instruments. I.10: the smithy story, with hammers weighing 12, 9, 8 and 6. I.11: the tests Pythagoras is said to have made at home, with weights on strings, lengths of pipes and cups struck with a rod, ending with the length and thickness of strings and the 'regula', the measuring rule. Read I.10-11 as legend, not physics. Hammer pitch does not follow hammer weight in the ratios claimed. Where weights in these ratios are hung on equal strings, as in the parallel story in Nicomachus (Manual of Harmonics 6), frequency goes as the square root of the stretching weight, so a 2:1 weight gives a frequency ratio of about 1.41:1 (600 cents), not an octave. The ratios 2:1, 3:2 and 4:3 are right for string lengths, which the monochord shows.
  • Plato, Republic, Book VII, 530d-531cShorey's English translation at Perseus, with a link to the Greek. Jowett's translation is free at Project Gutenberg (ebook 1497), without Stephanus numbers. At 530d Socrates says that as the eyes are fitted for astronomy, the ears are fitted for harmonic motion, and that the two are sister sciences, as the Pythagoreans say (Shorey has 'kindred sciences'). At 530e-531a he says that students of harmony measure heard concords and sounds against one another and labour in vain. Glaucon mocks those who put their ears to the strings hunting for the smallest interval, and Socrates sets aside those who 'torture the strings' on the pegs (531b). His real target is the Pythagoreans, who seek numbers in heard concords but do not rise to problems, asking which numbers are concordant and which are not, and why (531c). Harmonics is paired with astronomy here, not rhythm.

At 531c Socrates faults the Pythagoreans: they look for the numbers in the concords they hear, but never rise to problems, asking which numbers are concordant and which are not, and why. What is the difference between those two activities, and which of them does a monochord let you do?

  • Boethius says music is joined to us by nature and can ennoble or corrupt character (I.1). What would count as evidence for or against that, and is it a claim about numbers at all?
  • The smithy story is false as physics, yet the ratios it reports are right for string lengths. What should a reader do with a source whose story fails but whose result holds?
  • Boethius's three kinds of music include the music of the world and the music of the human being, the joining of soul and body (I.2). Is that an analogy, a hypothesis or a claim, and how would you decide?

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?
Round II

Practice

Grammar · Round 2

  • Plato, Cratylus, 383a-391bH. N. Fowler's Loeb translation with the Greek, on Perseus. Hermogenes reports Cratylus's view and gives his own at 383a-384e; Socrates asks whether names can be true or false at 385b-d and calls a name an instrument of teaching and of separating reality at 388b-c; the user of the instrument is the dialectician at 390c-d. The etymologies that follow are outside this span.

Hermogenes says a name is whatever anyone decides to call a thing, and Cratylus says each thing has a name that belongs to it by nature. Socrates replies that a name is a tool for teaching and for sorting things out, like a shuttle in weaving. If a name is a tool, what makes it a good one, and who is entitled to judge?

  • At 385a Socrates asks what happens if a private person calls what we call a horse a 'man'. What would it cost Hermogenes to say that the name is then right for that speaker?
  • Socrates says the user of a tool, not its maker, is the judge of whether it is well made (390b-d). Who is the user of a word today: the speaker, the listener or the dictionary writer?
  • If names are conventions, can a convention still be better or worse? Use an example from a technical vocabulary you know.

Logic or dialectic · Round 2

  • Aristotle, Prior Analytics, Book I, I.1-7 (24a10-29b28); the definition of a syllogism is at 24b18-20, the perfect syllogism at 24b22-26, and the first figure begins at I.4, 25b26Translation by A. J. Jenkinson in The Works of Aristotle, vol. 1, ed. W. D. Ross (Oxford, 1928), on Wikisource, which prints Bekker numbers in the margin. The same translation is on the MIT Internet Classics Archive (classics.mit.edu/Aristotle/prior.1.i.html), divided into numbered parts that match the chapters but without Bekker numbers. Chapter 3 (conversion of modal premises) can be read lightly. The weight falls on chapters 1, 2 and 4-7: the definitions, conversion of premises, the three figures and the reduction of every syllogism to the universal syllogisms of the first figure.

Aristotle says that in a syllogism, certain things being stated, something other than what is stated 'follows of necessity' (24b18-20). What does it mean for a conclusion to follow of necessity, and how could you tell, in an argument you meet today, that it does?

  • In chapter 4 Aristotle states the first-figure arguments with letters (A, B, C) instead of examples. What do letters let him do that examples cannot?
  • In chapter 4 he shows that some pairs of premises yield no syllogism by giving two sets of terms that make the same premises true: one where the first term belongs to all of the last (animal, man, horse) and one where it belongs to none (animal, man, stone). How does this compare with the counterexample method on the Proof and Precise Reasoning page?
  • In chapter 1 (24b22-26) a perfect syllogism needs nothing beyond what has been stated to make the conclusion plain, and in chapter 7 every syllogism is reduced to the universal syllogisms of the first figure. Why might a less obvious argument need to be made perfect before you trust it?

Rhetoric · Round 2

  • Aristotle, Rhetoric, I.1-2 (Bekker 1354a1 to 1358a35)J. H. Freese's Loeb translation (1926) on Perseus; the Greek text on the same site can be browsed by Bekker page. Chapter 1 argues that rhetoric is the counterpart of dialectic and defends its use; chapter 2 defines it at 1355b25-26 and gives the three kinds of proof at 1356a1-20.

Aristotle calls rhetoric the counterpart of dialectic, and says it is useful because the true and the just are by nature stronger than their opposites (1355a21). If that is so, why are true and just causes ever lost, and what does skill in speaking add that the truth does not already supply?

  • He names three kinds of proof: the character of the speaker, the state of mind of the hearer, and the argument itself (1356a1-20). Which of these can be separated from the other two, and which cannot?
  • He answers the charge that rhetoric can do harm by saying that the same objection applies to every good thing except virtue (1355b2-7). Is that a good defence?
  • Aristotle says rhetoric has no subject of its own but observes the means of persuasion in any given case (1355b25-34). How does this differ from a craft such as medicine, and is it a strength or a weakness?

Arithmetic · Round 2

  • Euclid, Elements, Book IX (David Joyce's online edition, Clark University), Book IX, Proposition 20Read Euclid's own proof, not a summary. It takes the least number measured by the assigned primes (their product), adds a unit, and considers two cases: the result is prime, or it is measured by some prime that is not among those assigned (VII.31). The statement does not use the words 'infinitely many'.
  • Euclid, Elements, Book IX (David Joyce's online edition, Clark University), Book IX, Proposition 36, with Book VII, Definition 22Definition 22 of Book VII says a perfect number is equal to the sum of its own parts. The first four perfect numbers, 6, 28, 496 and 8128, all have the form Euclid builds. Joyce's page links each step of the proof to the earlier propositions and definitions it uses.

IX.20 shows that no assigned list of primes is complete. IX.36 shows how a prime of one particular form yields a perfect number. What does each proposition let us say about how many primes and how many perfect numbers there are, and where does each stop?

  • Euclid starts from a unit and doubles: 1, 2, 4, 8, 16, 32. Find the running sums, say where the sum is prime, and compute the perfect numbers that IX.36 then gives.
  • IX.20 gives a way to produce a prime that is missing from any list. Does IX.36 give an equally good way to produce a new perfect number? What must be true before it applies?
  • IX.36 says that numbers of this form are perfect. Does it say that every perfect number has this form?

Geometry · Round 2

  • Euclid, Elements, Book I, Propositions 47 and 48Joyce's edition and guide. I.47 is proved by areas, using the squares of I.46 and the area result I.41, and I.48 (propI48.html on the same site) is its converse. Compare Euclid's proof with the similar-triangles proof on the Academy page, which is not his.
  • Plato, Republic, Book VI, 510c–511aPaul Shorey's English translation (Loeb) on Perseus. The Greek (Burnet) is document 1999.01.0167 on the same site. Socrates says that geometers start from assumptions of which they give no account (510c), and that they reason for the sake of 'the square as such and the diagonal as such', not the ones they draw (510d–e). The Academy's Divided Line (Republic 509d–511e) sets the passage in its larger context.

When Euclid proves I.47, what is the proof about: the squares drawn on the page, or something that the drawing only stands for?

  • Plato says geometers give no account of their hypotheses (510c). Does Euclid give an account of his starting points, and does it matter for I.47?
  • I.48 is the converse of I.47 but is not the same argument read backwards. What does the proof of I.48 need that I.47 does not?
  • In the Meno the double square is found by drawing on the diagonal. What does I.47 keep of that figure, and what does it add?

Music · Round 2

  • Plato, Timaeus, 35b-36bLamb's English translation at Perseus, with a link to the Greek; Jowett's translation is at Project Gutenberg (ebook 1572), without Stephanus numbers. At 35b-c the maker marks off portions 1, 2, 3, 4, 9, 8 and 27, a series of doubles and a series of triples. At 35c-36a he fills the double and triple intervals with two kinds of mean, which yields intervals of 3:2, 4:3 and 9:8. At 36a-b he fills each 4:3 with 9:8 tones, leaving a remainder of 256:243, about 90.2 cents. The music is exact arithmetic using only the factors 2 and 3. The cosmic setting is cosmology offered as a 'likely story' (29d), not a measurement.

The maker of the world builds the soul from portions 1, 2, 3, 4, 9, 8 and 27, fills the gaps with means to get intervals of 3:2, 4:3 and 9:8, then fills each 4:3 with 9:8 tones until only a small remainder, 256:243, is left. Why might a maker who wants perfect order choose these numbers, and what does the remainder say about the order that can be achieved?

  • The numbers use only the factors 2 and 3. What would be gained or lost if the factor 5 were allowed?
  • Is the 256:243 remainder a flaw in the plan or part of it?
  • Timaeus calls his account a likely story (29d). Which parts of this passage could be tested, and which could not?

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?
Round III

Mastery

Grammar · Round 3

  • John Locke, An Essay Concerning Human Understanding, Book III, chapter 2 (Of the Signification of Words), sections 1 to 5, and chapter 9 (Of the Imperfection of Words), sections 4 to 6Project Gutenberg's edition of Books III and IV (volume 2), public domain, which it describes as based on the second edition. Locke goes on to propose remedies in chapter 11, which is optional here.

Locke says that words, in their primary signification, stand for nothing but the ideas in the mind of the person who uses them, yet we all suppose our words also stand for the ideas in other people's minds and for the things themselves (III.2.2, 4, 5). If that is so, what exactly is shared when two people understand one another, and what does Locke think can go wrong?

  • Locke says the doubtfulness of words has its cause more in the ideas they stand for than in the sounds (III.9.4), and that it is worst for the names of complex ideas such as moral words and for the names of substances (III.9.5-6). Test this on two words of your own choosing, one you can point to and one you cannot.
  • If every word had one exact definition agreed by all, would we have a perfect language? What would be lost?
  • Locke treats words as signs of ideas. How would a speaker who held that words stand directly for things answer him?

Logic or dialectic · Round 3

Hilbert states a conviction, which he admits no one has yet proved, that every definite mathematical problem can be settled, and that in mathematics there is no ignorabimus. Gödel's introduction sketches a statement that a formal system can neither prove nor refute, and then says it has been decided by other means. Does Gödel contradict Hilbert?

  • Gödel says the undecidable statement 'has hence been decided by meta-mathematical considerations' (p. 176). From what standpoint is it decided, and does that standpoint lie inside or outside the system?
  • Gödel notes a close kinship between his undecidable statement and the liar antinomy (p. 175). What separates an undecidable statement from a paradox?
  • Hilbert's Problem 2 asks for a proof that the arithmetical axioms are consistent, and Gödel's introduction points ahead to surprising results about consistency proofs in section 4 (p. 176). What would count as success for Problem 2 after 1931?

Rhetoric · Round 3

  • Plato, Phaedrus, 257b-279cH. N. Fowler's Loeb translation with the Greek, on Perseus. Key places: the speaker must know the truth (259e-262c); rhetoric as leading the soul by words (261a, 270b-272b); Theuth and Thamus on writing (274c-275b); writing cannot defend itself (275d-e); the farmer and the seeds (276b-277a); the speaker is better called a lover of wisdom (278d).

Socrates says that no one can be a true rhetorician without knowing the truth about his subject and the kinds of soul he addresses (259e-262c, 270b-272b), and that writing, which cannot answer questions or defend itself, is only a reminder (275d-278a). What should we make of a written argument meant to persuade, including a book?

  • Theuth offers writing as a cure for forgetting and Thamus replies that it will produce forgetfulness (274e-275b). Name a technology that has its own Theuth and Thamus, and say what each would claim.
  • Socrates says the art of speaking must resemble medicine and know the nature of the whole it treats (270b-d). Is such a rhetoric possible for a speaker addressing a crowd he cannot know one by one?
  • At the end Socrates says 'lover of wisdom' suits such a speaker better than 'wise' (278d). What does that change about the demand at 259e that the speaker must know the truth?

Arithmetic · Round 3

Dedekind says that arithmetic shall be 'developed out of itself' and that irrational numbers must be defined 'by means of the rational numbers alone' (Section III). When he defines the square root of 2 by a cut in the rational numbers, has he found a number or made one?

  • Dedekind writes that he cannot prove the principle of continuity of the straight line and that it is 'nothing else than an axiom'. What does he say would follow if space were not continuous, and what does that suggest about the link between number and space?
  • In the Preface of 1888 he writes both that 'nothing capable of proof ought to be accepted without proof' and that numbers are 'free creations of the human mind'. Can both stand? Which parts of arithmetic are chosen, and which are proved?
  • Section IV gives its own indirect proof that no rational number squares to D. Compare it with the proof you reconstructed for the diagonal of a square. What do the two proofs share, and what differs?

Geometry · Round 3

  • Euclid, Elements, Book I, Postulate 5 and Propositions 27–29I.27 and I.28 (propI27.html and propI28.html on the same site) show that lines are parallel without using Postulate 5. Joyce's guide notes that the parts of I.29 are converses of I.27 and I.28, and that I.29 is the first proposition to depend on the parallel postulate.
  • Nikolai Ivanovich Lobachevsky, Geometrical Researches on the Theory of Parallels (Berlin, 1840), translated by George Bruce Halsted, Introduction (the paragraphs before section 1) and sections 16–23Halsted's English translation (Open Court, new edition 1914; the translator's preface is dated 1891), public domain, as scanned pages with OCR text on archive.org. The section numbers are Lobachevsky's own. Section 16 defines parallels as the boundary between lines that cut a given line and lines that do not. Section 20 shows that if the angles of one triangle sum to two right angles, so do the angles of every triangle. That his geometry is as consistent as Euclid's was shown only later, by Beltrami in 1868.

Lobachevsky does not claim that Euclid's fifth postulate is false. He asks what follows if it is not granted, and works out a geometry in which no contradiction appears. What would it take to decide which geometry describes the space we live in, and is that a question for geometry at all?

  • Euclid proves I.27 without Postulate 5 and needs it for I.29. Why might he have postulated it instead of proving it, and why did geometers for two thousand years want it to be a theorem?
  • Euclid's Definition 23 calls lines parallel if they never meet. Lobachevsky's section 16 defines parallels as a boundary between two classes of lines. What does each definition let you do that the other does not?
  • Section 20 shows that if one triangle has an angle sum of two right angles, every triangle does. What does that imply about what a measurement of a single triangle could and could not tell us?

Music · Round 3

  • Hermann von Helmholtz, On the Sensations of Tone as a Physiological Basis for the Theory of Music (Alexander J. Ellis, translator), Part II, Chapter VIII, On the Beats of Simple Tones (pp. 159-173), and Chapter X, Beats of the Upper Partial Tones (pp. 179-197)Ellis's English translation is public domain and scanned at the Internet Archive. This scan is the 1895 printing, called the third edition but reprinted from the 1885 second edition, and the page numbers above are taken from its table of contents. Helmholtz explains the smoothness or roughness of an interval by whether the partials of the two notes coincide, beat slowly or beat fast. It is a physical account of roughness in sustained tones. Whether roughness is the same thing as dissonance in music is a question for the seminar.

Helmholtz accounts for the smoothness of some intervals by the way the partials of two notes coincide or beat. Is that an explanation of consonance in music, or only of roughness in sustained tones?

  • Which of his steps would fail for a note whose partials are not whole-number multiples of the fundamental, as in a stiff string?
  • The tempered fifth beats slowly and the tempered major third beats faster. What does the tolerance of players and listeners for those beats tell us about what consonance is?
  • What would a counterexample to his account look like, and how would you look for one?

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?