Quadrivium · the arts of number · art 5 of 7
Geometry
Geometria
Magnitude, spatial relationships, and demonstration
- Classical definition
- The science of magnitude, or continuous quantity, at rest. (Nicomachus, Introduction to Arithmetic I.3; Boethius, De institutione arithmetica I.1.)
- Proper object
- Continuous quantity that stays still: lines, surfaces and solids.
- Place in the order
- Quadrivium: magnitude at rest · see the whole order
Geometry is the art of magnitude at rest: lengths, angles and areas that stay put while we reason about them. Euclid's Elements (about 300 BC) builds a large body of geometry by proof from a few stated definitions, postulates and common notions, and its habits (define your terms, name what you assume, justify every step) are still how careful mathematics and technical writing are done. It also shows the limits of the method: Euclid's first proof has a gap that stayed open for more than two thousand years, and the parallel postulate turned out not to follow from the others, so that denying it gives a geometry as consistent as Euclid's.
The guiding questionWhat can be known for certain from a few agreed starting points, and what must simply be granted?
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
- PostulatesWhat each step rests onI can name the definition, postulate or common notion behind each step of a short Euclidean proof, and say which results depend on the parallel postulate.
- ConstructionConstruct and justifyI can construct an equilateral triangle on a given segment with compass and straightedge, justify every step, and say which step Euclid assumes without proof.
- MeasureSquares, diagonals and the incommensurableI can show that the square on a diagonal is double the square, prove the Pythagorean theorem by similar triangles, and explain why the diagonal and side of a square have no common measure.
Round II Practice
- PostulatesDependencies in Book II can trace a proposition in Book I back to the starting points it rests on, and show that the propositions before I.29 do not use Postulate 5 while I.29 does.
- ConstructionEuclid's basic constructionsI can carry out and justify Euclid's basic Book I constructions: placing a given length at a point, bisecting an angle and a segment, erecting and dropping a perpendicular, copying an angle, and drawing a parallel.
- MeasureArea and the Pythagorean theorem by areasI can compare the areas of parallelograms and triangles on equal bases and between the same parallels (I.35 to I.41), and follow Euclid's proof of I.47 and its converse I.48 using areas alone.
Round III Mastery
- PostulatesGaps, axiom systems and rival geometriesI can explain how a continuity principle repairs I.1, why a model of hyperbolic geometry shows that the parallel postulate cannot be proved from the rest of the axioms, and how triangle angle sums differ in Euclidean, spherical and hyperbolic geometry.
- ConstructionWhat compass and straightedge cannot doI can explain why doubling the cube, trisecting a general angle and squaring the circle cannot be done with compass and straightedge, using the fact that every constructible length comes from the given lengths by arithmetic and repeated square roots.
- MeasureMagnitudes without numbersI can state Euclid's definition of equal ratios (Elements V, Definition 5) and use it to compare an incommensurable ratio with fractions, explaining why it does not assume that every ratio is a fraction.
Labs
Core reading in the Library
- LibraryGeometry, Euclid and TrigonometryThe bridge from classical geometry to trigonometry and precalculus, including a track for studying Euclid and Apollonius by reconstruction.
- LibraryProof and Precise Reasoning: From Arguments to TheoremsThe grammar of valid inference: validity and soundness, truth tables, quantifiers, the main proof techniques and counterexamples, framed by the trivium habit of defining, testing and explaining.
- 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.
- LibraryHow to Read a Great Book: Levels, Modes and the Completion StandardA practical guide to reading difficult primary texts, and a per-book output standard that converts reading into durable understanding.
- LibraryA Map of Mathematics: Twenty-Five Areas and What Depends on WhatA guided tour of the branches of mathematics in learning order, with prerequisites and the philosophical and engineering parallels of each.
Where it is used
- LibraryEvidence-Based Troubleshooting: Separating Explanations · Roots in older reasoningBy analogy, a good incident write-up follows Euclid's order: assumptions stated first, then each conclusion tied to evidence.2 min glance · 6 min
- LibraryTechnical Writing as Reasoning · Where the proof structure comes inA recommendation has the shape of an argument: premises, an inference and a conclusion.2 min glance · 5 min
- LabString physicsFrets divide a string in a fixed ratio: each fret multiplies the sounding length by the same factor, about 0.944, so the frets get closer together up the neck.
- LabMonochordA divided string you can hear: simple ratios of length give consonant intervals.
Seminars
Read the sources
One primary text for each round, with a question to think through. All 21 seminars →
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?
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?
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?