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Proposition 3 of 48 • Construction

To cut off from the greater of two given unequal straight lines a straight line equal to the less.

Given a long line and a short line, mark a segment on the long line that equals the short line's length.

Before You Read

You have a long stick and a short stick. Without any measuring tools, how would you mark a spot on the long stick so the piece from one end to the mark is exactly the length of the short stick? Think about what tools you've already built in Props 1 and 2.

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A B E D F C

What Euclid Is Doing

Setup: We have two lines: AB (the longer one) and C (the shorter one). We want to mark off a piece of AB that exactly equals C.

Approach: The postulates don't directly let us 'transfer' the length of C onto line AB. But now that we have Proposition 2, we can first copy length C to point A, then use a circle to mark that length on AB.

Conclusion: Here's why AE = C: First, we construct AD equal to C (using Prop 2). Then we draw a circle centered at A with radius AD. Point E is where this circle intersects AB. Since both AD and AE are radii of the same circle centered at A, we have AE = AD (Definition 15). And AD = C (by construction). Therefore AE = C. ✓

Key Moves

  1. Use Proposition 2 to place a line AD equal to C, starting at point A. Now AD = C.
  2. Draw a circle centered at A with radius AD (Postulate 3)
  3. Mark point E where this circle intersects AB
  4. AE = AD (both radii of circle centered at A, by Definition 15)
  5. Since AD = C (by construction), we have AE = C ✓

Try It Yourself

Draw a long line and a short line on your paper. Now use Props 1 and 2 to copy the short length to one end of the long line, then use a circle to mark it off. Notice how quick this is compared to Prop 2—the hard work was already done.

Proof Challenge

Available Justifications

1.

Place a line AD at point A equal to the shorter line C

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2.

Draw a circle centered at A with radius AD; let E be where it meets AB

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3.

AE = AD (both radii of the circle centered at A)

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4.

Since AD = C (by construction), we have AE = C

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0 of 4 steps completed

Curriculum Materials

Get the Teaching Materials

Structured lesson plan, student worksheet, and answer key for Proposition 3.

  • ✓Lesson Plan (prop-03-lesson-plan.pdf)
  • ✓Student Worksheet (prop-03-worksheet.pdf)
  • ✓Answer Key (prop-03-answer-key.pdf)
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Why It Matters

This completes our basic length-manipulation toolkit. We can now create equal lengths (Prop 1), copy lengths anywhere (Prop 2), and cut off specific lengths (Prop 3). These three constructions are the foundation for everything else.

Modern connection: This shows the power of building up from simple operations. Once Proposition 2 proves we can copy lengths, this proposition becomes straightforward. Each new capability unlocks further possibilities.

Historical note: Propositions 1, 2, and 3 form a unit: they establish that we can manipulate lengths freely using only the basic postulates. Every length operation we might want is now proven possible.

Discussion Questions

  • Why do we need Proposition 2 first before we can do this?
  • What if the shorter line were longer than AB? Would this construction fail?
  • How is this different from just using a ruler to measure?
Euclid's Original Proof
Let AB, C be the two given unequal straight lines, and let AB be the greater of them.

Thus it is required to cut off from AB the greater a straight line equal to C the less.

At the point A let AD be placed equal to the straight line C; [I. 2]
and with centre A and distance AD let the circle DEF be described. [Post. 3]

Now, since the point A is the centre of the circle DEF, AE is equal to AD. [Def. 15]

But C is also equal to AD.

Therefore each of the straight lines AE, C is equal to AD; so that AE is also equal to C. [C.N. 1]

Therefore, given the two straight lines AB, C, from AB the greater AE has been cut off equal to C the less.

(Being) what it was required to do.

What's Next

We now have a complete length-manipulation toolkit: create equal lengths (Prop 1), copy lengths (Prop 2), and cut to size (Prop 3). With these in hand, Euclid is ready for the first major theorem: proving when two triangles are identical. That's Proposition 4—Side-Angle-Side congruence.