Elements, Euclid. Proposition 2

Question

What is wrong with the following argument?

By Postulate 3, we describe a circle centered at $A$ with a distance $BC$. Pick any point on the boundary of the circle. Call this point $F$. By Postulate 1, we connect $A$ and $F$ in a straight line. By the def. of circles, $AF$ = $BC$.

Related: https://math.stackexchange.com/questions/2682519/euclidean-proposition-8-of-book-i/2682526#2682526 — Ethan Bolker, Nov 24 '18 at 00:54

score 1 · Answer 1 · answered Nov 24 '18 at 00:48

1

"Pick any point" isn't a thing that any of the axioms allow you to do.

answered Nov 24 '18 at 00:48

user3482749

6,660

Book I, Proposition 11
"Take an arbitrary point D on AC..."
– ski Nov 24 '18 at 01:01
1

Nobody's claiming that Euclid actually stuck to his axiom list. – user3482749 Nov 24 '18 at 01:25

score 1 · Accepted Answer · answered Nov 24 '18 at 00:50

1

Postulate 3 allows you to construct a circle centered on one end of the given length (to be used as the radius). It does not justify constructing a circle centered on a point that is not an endpoint of the given segment.

Consequently, in your first line, one of $B$ or $C$ must be $A$.

answered Nov 24 '18 at 00:50

Eric Towers

67,037

"3. To describe a circle with any center and distance."
Maybe the notion got lost in the translation, but do you agree that it sounds like I can fix a center, and pick any arbitrary distance (to be used as radius), like BC in this case?
– ski Nov 24 '18 at 00:54
@ski : No. The proposition 2 is how you show you can transport a specified distance over to a given point. Postulate 3 allows you to produce a circle with a given center passing through a given point (so that the radius is the distance between the two given points). – Eric Towers Nov 24 '18 at 00:59

Elements, Euclid. Proposition 2

2 Answers2