Construct and justify: Geometry, Round 1

1 Learn

You will be able to

Carry out Euclid's construction for I.1 in the right order.
Justify the equality of the three sides with Definition 15 and Common Notion 1.
Point out the step that Euclid assumes without proof, and say why it needs a continuity principle.
Show, by the figure of the Meno, how a square of double the area is constructed and justified.

LabEuclid I.1: the equilateral triangleThe construction and its proof, step by step, with the postulates it uses and the gap it leaves.
LabEuclid's postulatesPostulates 1 and 3 say what the straightedge and the compass may do.
LabThe Meno: doubling the squareA second construction with its justification: the square on the diagonal doubles a square.
LibraryThe Seven Liberal Arts, Hands-On: Logic, Proof and the Quadrivium · Three teachers of methodPlaces the gap in I.1 and the Meno figure in the history of method.2 min glance · 7 min
LibraryProof and Precise Reasoning: From Arguments to Theorems · What a proof is, and is notExplains why a diagram can suggest a truth and still hide an assumption.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.

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.
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.
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.
recall
In the Meno (82b–85b) the boy first proposes a side of 4 feet and then 3 feet for a square of area 8. What areas do those sides give, and on what line does the square of area 8 stand?

Show answer
A side of 4 feet gives 16 square feet and a side of 3 feet gives 9 square feet. The square of area 8 stands on the diagonal of the original 2-foot square: the four diagonals of four 2-foot squares enclose a square made of four half-squares, 4 × 2 = 8 square feet.
explain
Which part of the Meno passage on doubling the square is demonstrated, and which part is only Socrates' philosophical claim?

Show answer
The geometry is demonstrated: the square on the diagonal of a square is double the square. Socrates' inference that the boy is recollecting knowledge from before this life, and so that the soul is immortal, is a philosophical claim, not demonstrated. His questions also lead the boy, and Socrates himself says that he cannot confidently assert most of the argument (86b).
explain
In the proof of Thales's theorem, with AB a diameter, O the centre and C on the circle, why are triangles OAC and OBC isosceles, and how does that give angle ACB = 90°?

Show answer
OA, OB and OC are equal radii, so OAC has equal angles α at A and C, and OBC has equal angles β at B and C. Angle ACB = α + β, and triangle ABC's angles sum to 180°: 2(α + β) = 180°, so ACB = 90°.

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

Build I.1, justify it, then find its gap

Part A. On paper, or in free geometry software, draw a segment AB. Construct the equilateral triangle on it as Euclid does: one circle with centre A through B, one with centre B through A, then join a point C where they meet to A and to B. Write the construction as numbered steps, each ending with the postulate it uses. Part B. Write the proof that triangle ABC is equilateral as numbered steps, each ending with its reason (a definition or common notion). Part C. Put A = (0, 0) and B = (1, 0). Solve x² + y² = 1 and (x − 1)² + y² = 1 to find C. Then explain in two or three sentences why a plane containing only points with rational coordinates has no such point, and what that shows about Euclid's step 'the circles meet'.

What to hand in: A written solution in three parts (A, B and C), with the numbered steps and your coordinates for C. A sketch of the figure is optional. Save it on this page or download it.

Check your work against the rubric Part A builds both circles before the lines, and every construction step cites Postulate 3 (circles) or Postulate 1 (lines). Part B shows CA = AB and CB = BA by Definition 15, once for each circle, and then CA = CB by Common Notion 1. Every justification in Parts A and B is a citation of a postulate, definition or common notion, not a remark that the figure looks right. Part C finds x = 1/2 and y = ±√3/2, so C = (1/2, √3/2). Part C says that √3/2 is irrational, so C is missing from a plane of rational points, and names the assumption that the circles meet as the step Euclid does not justify.

Your work Or a link to it (optional)

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?

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Construct and justify