What each step rests on
I 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.
- Round IWhat each step rests on (this page)
- Round IIDependencies in Book IComing
- Round IIIGaps, axiom systems and rival geometriesComing
1 Learn
You will be able to
- Sort statements from Euclid's Book I into definitions, postulates and common notions.
- Cite the starting point behind each step of the proof of I.1.
- Name the postulate that I.32 and Thales' theorem depend on, and say what happens to them on a sphere.
- Say what an unstated assumption is, using the circles in I.1 as the example.
- LabDefinitionsDefines axiom, postulate and proof, so you can tell the kinds of starting point apart.
- LabEuclid's postulatesGives Euclid's five postulates and five common notions in Heath's wording, which later proofs cite by number.
- LabEuclid I.1: the equilateral triangleA short proof with every step labelled by its justification, and with the one assumption it never justifies.
- LabThe angle sum of a triangleShows the first result that needs Postulate 5 and what happens to it on a curved surface.
- LabThales' theoremA second proof whose dependence on Postulate 5 is two steps removed.
- LibraryProof and Precise Reasoning: From Arguments to Theorems · Anatomy of an argumentTeaches you to list premises, conclusion and hidden assumptions before judging an argument.2 min glance · 6 min
2 Practise
Answer each card from memory, then grade yourself honestly. The cards join your review deck and come back just before you would forget them.
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recall
What rule does the Academy draw from Euclid's fifth postulate, the parallel postulate?
Show answer
That some premises are chosen rather than proven, and must be named as chosen so they can be refused. The fifth resisted proof for two thousand years because it is not a theorem; denying it yields non-Euclidean geometry, such as hyperbolic geometry.
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recall
How does an axiom differ from a postulate, as the Academy defines them?
Show answer
Both are assumed without proof. Axioms, which Euclid called common notions, are shared by all reasoning; postulates are local to one system and stated openly so they may be refused. The parallel postulate proved to be optional.
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recall
Which postulate, definition and common notion does Euclid cite in Elements I.1 to build an equilateral triangle on a line AB, and where does each one enter?
Show answer
Postulate 3 gives the two circles (centre A through B, centre B through A) and Postulate 1 gives the lines CA and CB. Definition 15 gives AC = AB and BC = BA, because the lines from a circle's centre to the circle are equal, and Common Notion 1 gives CA = CB, because both equal AB.
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explain
What does Euclid assume without proof in Elements I.1, and why does a plane of points with rational coordinates show that the assumption is needed?
Show answer
He assumes that the two circles cut one another at a point C. In a plane containing only rational points, with A = (0, 0) and B = (1, 0), the circles would meet only at (1/2, ±√3/2), and √3/2 is irrational, so the circles can be drawn but do not meet. The proof therefore needs a continuity principle that Euclid never states.
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apply
In the figure for Elements I.1 the circles about A and B also meet on the other side of AB, at a second point F. What can you say about the lengths AF, BF and AB, and which steps of the proof justify it?
Show answer
AF = BF = AB. F lies on the circle with centre A through B and on the circle with centre B through A, so Definition 15 gives AF = AB and BF = BA, and Common Notion 1 gives AF = BF. The same argument that works for C works for F, giving a second equilateral triangle ABF on the other side of AB.
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recall
What does Euclid's Elements I.32 say about the interior angles of any triangle?
Show answer
The three interior angles of any triangle together equal two right angles, that is, 180°.
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recall
Which of Euclid's postulates does his proof that a triangle's angles sum to two right angles (Elements I.32) depend on, and how does the proof use parallels?
Show answer
Postulate V, the parallel postulate. The proof draws a line through one vertex parallel to the opposite side. The equal alternate and corresponding angles that parallels make with a crossing line (I.29) then carry the other two angles to that vertex, where they and the third angle together make a straight line.
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apply
On a sphere, take the north pole and two points on the equator a quarter of the way around from each other, and join them with great-circle arcs. What do this triangle's angles add up to, and why does this not refute Euclid's Elements I.32?
Show answer
Each of the three angles is a right angle, so they total 270°. Euclid's proof rests on Postulate V and begins by drawing a parallel to one side. On a sphere any two great circles meet, so no such parallel exists and the 180° result does not apply.
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recall
Which postulate does the proof of Thales's theorem (the angle in a semicircle) ultimately depend on, and what happens to the angle in a semicircle on a sphere?
Show answer
Euclid's fifth postulate, because the proof uses the theorem that a triangle's angles sum to two right angles. On a sphere the angle in a semicircle is greater than a right angle.
3 Prove it: the mastery check
5 questions drawn from a pool of 11. The pass mark is 80%. There is no time limit, and you can retake it with new questions; your best result counts.
4 Prove it: the performance task
A dependency chain for Thales' theorem
Take Thales' theorem (Elements III.31: the angle in a semicircle is a right angle) and the proof given on the Academy page. Work backwards from the theorem to the starting points it rests on. Write each link as one line: the claim, the earlier result or starting point it uses, and what kind of thing that is (definition, postulate, common notion or earlier proposition). Keep going until every line reaches a definition, a postulate, a common notion, or an earlier proposition that you name and mark 'not traced'. Finish with one sentence saying what the theorem needs that the Euclidean plane supplies and a sphere does not.
What to hand in: A numbered list of 5 to 8 lines in the form 'claim | relies on | kind', ending with one closing sentence. Save it on this page or download it.
Saved only in this browser. To keep a copy, download it or export your progress.
5 Discuss: the seminar
Read the text, then think, write or talk through the question with someone. There is no answer key: the aim is a better question.
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?
Where geometry 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.
Self-administered checks and self-assessed tasks: no credential is awarded. The design borrows from Khan Academy (mastery levels), WGU (competencies proved by assessment) and St. John's College (seminars on primary texts); this site is not affiliated with any of them.