## Unit Two: Euclid's Second Theorem

upon a given point to place a straight line equal to a given straight line

Let there be the given point A, and then the given straight line BC. It is necessary, then, upon the point A to place a straight line equal to the straight line BC.

Then let there be joined from the point A to the point B the straight line AB, and let there be constructed upon it the equilateral triangle DAB, and at the straight lines DA and DB, let the straight lines AE and BF be extended; and with the center B, and then with the distance BC, let there be drawn the circle CGH, and again with the center D and the distance DG, let a circle be drawn, GKL.

Since, therefore, the point B is the center of [the circle] CGH, BC is equal to BG. Again, since the point D is the center of the circle GKL, the line DL is equal to the line DG, of which the line DA is equal to the line DB. The remainder, then, the line AL is equal to the line BG. But it has also been shown that the line BC is equal to the line BG. Each of the lines AL and BC, therefore, is equal to the line BG. Things which are equal to the same thing are also equal to one another. The line AL, therefore, is equal to the line BC.

To the given point A, therefore, lies the line AL equal to the line BC: the very thing which was necessary to make.

A note from Sir Thomas Heath:

We should insist here upon the restrictions imposed by the first three postulates, which do not allow a circle to be drawn with a compass-carried distance; suppose the compasses to close of themselves the moment they cease to touch the paper. These two propositions (1.2, 1.3) extend the power of construction to what it would have been if all the usual power of the compasses had been assumed; they are mysterious to all who do not see that postulate iii does not ask for every use of the compasses. (The Thirteen Books of the Elements, 246, quoting De Morgan, 1849)

Think about this process as mentally discovering lines instead of mechanically drawing them. The lines and figures are already there, they just need to be described. Without theorems 2 and 3, we could not say, "draw line A equal to line B," nor "cut off segment B from A."

This theorem uses post. 1, of course, to draw lines between points; theorem 1 to make the equilateral triangle; post. 2 to produce the lines from the segments; post. 3 to make the circles; c.n. 3 to establish the equality of remainders; c.n. 1 transitive equality.

Assignment:

Read the history of The Elements and prepare to discuss the various "theorems" and how to "prove" them in your assigned groups during the next class. You should use at least two parts of the history as "axioms" or "prior theorems" to be able to prove each theorem. Remember that a historical proof has a different feel to it than a mathematical proof: perhaps less definite and incontrovertible but perhaps more intuitively true. The broader topic is how and why The Elements survived.

Study the historical articles and prepare to answer the essay questions.