In the margin of my freshman year copy of Euclid’s Elements, I’ve written a question and answer about Proposition 12. “Why infinite? Why now? …to be able to choose any distance for circle.” Proposition 12 states “To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line.” This differs from Proposition 11 in that the starting line is infinite, rather than finite, and the point from which the perpendicular drops isn’t on the starting line.
So, still thinking about about plumb lines and right angles, I’ve included some squares and the protractor again in the collage layer. But, since the infinity of the starting point was of most interest to me back in 1978, I’ve drawn the proposition using a variety of sizes of circles.