Squares, diagonals and the incommensurable
I 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 ISquares, diagonals and the incommensurable (this page)
- Round IIArea and the Pythagorean theorem by areasComing
- Round IIIMagnitudes without numbersComing
1 Learn
You will be able to
- Show with the Meno figure that the square on a square's diagonal has twice its area, and explain why doubling the side quadruples the area instead.
- Prove a² + b² = c² from similar triangles, and use the theorem and its converse on given lengths.
- Reproduce the proof that a square's diagonal and side are incommensurable.
- Say which part of the Meno passage is demonstrated and which is a philosophical claim.
- LabThe Meno: doubling the squareDoubles a square by its diagonal, shows why doubling the side fails, and separates the geometry from Socrates' inference.
- LabPythagoras' theorem, provedThe Pythagorean theorem proved by similar triangles, with where it stops being true.
- Lab√2 is irrationalThe proof that a square's diagonal and side have no common measure.
- LabThe AcademyStates what incommensurable means and what it does not mean.
- LabThe AcademyExplains why a ratio of magnitudes is a comparison and not a fraction.
- LibraryGeometry, Euclid and Trigonometry · Why geometry comes before trigonometryCovers the converse of the theorem (I.48), area in square units and proportion.2 min glance · 5 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
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.
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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).
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apply
A square has sides of 5 cm. By the theorem in this section, what is the area of the square on its diagonal, and why can the diagonal not be written as a whole-number ratio to the side?
Show answer
The area is twice 25, so 50 cm². The diagonal is 5√2 cm, about 7.07 cm, and its ratio to the side is √2, which Prop. II shows is not a ratio of whole numbers.
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recall
In the similar-triangles proof of the Pythagorean theorem, triangle ABC has its right angle at C, with a = BC, b = CA and c = AB. A perpendicular from C meets AB at D, with AD = q and DB = p. Which two equations do the similar triangles give, and how are they combined?
Show answer
Triangle ACD is similar to ABC, giving b² = cq; triangle BCD is similar to BAC, giving a² = cp. Adding them: a² + b² = c(p + q) = c · c = c², since p + q = c.
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recall
Babylonian scribes used the Pythagorean relation more than a thousand years before Euclid (c. 300 BC). If the fact was already known, what was the Greek contribution?
Show answer
A reason: a proof that the relation must hold, not just a rule that works. The novelty was the obligation to justify the claim, and once a reason exists, accepting it on authority becomes optional.
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explain
Why does a² + b² = c² not hold exactly for long right triangles on the Earth's surface, and what relation replaces it on a sphere?
Show answer
The theorem is true of flat (Euclidean) space, and the Earth's surface is curved. On a sphere a right triangle obeys cos c = cos a · cos b, with sides measured as angles at the sphere's centre, so long triangles measurably depart from the flat version.
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recall
The proof that a square's diagonal d has no common measure with its side s supposes d : s = p : q, with whole numbers p and q in lowest terms. Which theorem then gives p² = 2q², and how?
Show answer
The Pythagorean theorem. The diagonal is the hypotenuse of a right triangle whose legs are two sides, so d² = s² + s² = 2s². Since d : s = p : q, this becomes p² = 2q².
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explain
In the proof that a square's diagonal and side are incommensurable, how does p² = 2q², with p : q in lowest terms, lead to a contradiction?
Show answer
p² is even, and the square of an odd number is odd, so p is even: p = 2m. Then 4m² = 2q², so q² = 2m², and q is even by the same step. Both even contradicts lowest terms, so no common measure exists.
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recall
What does it mean for two magnitudes to be incommensurable?
Show answer
They have no common measure, however small: no unit divides both a whole number of times. The diagonal and side of a square are the classic example.
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apply
A carpenter marks 3 units from the corner along one wall and 4 units from the corner along the adjacent wall, then finds the diagonal between the marks is exactly 5 units. What can she conclude, and which result justifies it?
Show answer
Since 3^2 + 4^2 = 9 + 16 = 25 = 5^2, the sides satisfy a^2 + b^2 = c^2, so by the converse of the Pythagorean theorem (Elements I.48) the corner is a right angle. The converse is the direction that tests a shape rather than deriving a length.
3 Prove it: the mastery check
5 questions drawn from a pool of 12. 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
Double the square, then show the diagonal has no common measure
Part A (the figure). On squared paper draw a square ABCD with side 2. Add three more 2 × 2 squares to make a square of side 4, then draw the diagonal across each of the four small squares so that the four diagonals enclose one square. Shade the enclosed square. Part B (the argument). Write four to six sentences showing that the shaded square has twice the area of ABCD and that its side is a diagonal of ABCD. Count in triangles, not in decimals. Part C (the proof). From memory, write the proof that the diagonal and side of a square have no common measure. Mark where you use the Pythagorean theorem (or the figure of Part A) and where you use 'the square of an odd number is odd'.
What to hand in: A written solution in three parts: a description of your figure with its labelled points (a sketch is optional), the argument for Part B, and the proof for Part C. 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.