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According toJoyce' commentary, proposition 2 is only used in proposition 3 of Euclid's Elements, book I.

As Euclid states himself (I-3, the length of the shorter line is measured (as the radius of a circle) directly on the longer line (by letting the center of the circle reside on an extremity of the longer line). The incremental (deductive) chain of definitions, common notions, constructions and propositions seems thus broken by introducing a ''useless'' statement, namely I-2. - Or am I looking for the meaning of proposition I-2 in the wrong place?

A related question would be: Does the proposition in case need a purpose?

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2 Answers

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This statement is equivalent to the axiom of segment construction of modern textbooks. It is very important because it is needed to define the sum and difference of two segments.

EDIT.

Here's for instance the axiom (11.3 a.), as given in the textbook I'm using for first year high-school.

EDIT 2.

Some years ago a very interesting article appeared on The Mathematical Intelligencer (Vol. 15 N. 3): "A new look at Euclid's Second Proposition", by G. Toussaint. I think you will find there the ultimate answer to your doubts.

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I think that this proposition is interesting by itself. It tells us that, using Euclidian tools only, we can draw equilateral triangles given one of its sides. The natural question now is: “Which regular polygons can I draw using compass and straightege only?” And you know where that question led us, don't you?

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Note: Thanks to a comment, I realized that what I wrote is actually about proposition I.1, not I.2. Proposition I.2 tells us that we can transfer distances from a region of the plane to another one. One might think that this could be done using the compass alone (simply by not closing it), but Euclid actually never mentions compass (or straightedge); he just mentions the possibility of drawing line segments and circles. So, he needs to explain how is it possible, using these tools alone, to transfer distances. $\endgroup$ 14