Currently I am planning on reading approximately 10 Propositions per week and finish book 1 in a little over a month before going on to 2nd book of the Elements. So today I wanted to begin discussing the propositions that I read this last week.
The first two propositions involve constructions. The first Construction is that of an Equilateral Triangle using a given segment as one of the sides. The second is to construct a segment congruent to a given segment using a point that is not on the segment. The interesting thing about the second proposition is that its starts by connecting an endpoint to the given point and constructing an equilateral triangle. Afterwards there are a couple circles involved.The second construction seems unnecessarily complex.
Naturally, this invokes the question as to why Euclid was so elaborate with this second construction. While in his third axiom he assumes that a circle can be drawn anywhere with any radius, he is very careful not to imply any sort of measurement. When we measure a Length we compare a standard measure with a length in question; technically, copying the length of a segment to another location is like using the former segment as a ruler. He appears to have wanted to avoid making any assumptions about measurement or refer to specific devices, thus adding logical rigor. That doesn't mean that he won't be more liberal in the future, in fact he is establishing his basis for doing so.
His reluctance to simply begin by transferring a length to some other location is sometimes referred to by commentators as having a "collapsible compass", meaning that once a compass or similar device is used to construct a circle of a given radius that it cannot be assumed to maintain the same radius upon completion, i.e. it collapses. Of course, Euclid doesn't say “use a compass” in the text at all, the construction is completely theoretical without any specific reference to device. In so doing, he adds to the timelessness of his work. As far as the further logical developments are concerned, afterwards he doesn't have to worry about sounding sloppy when he transfers lengths. This is generally true with all of his propositions, once proven rigorously, it is later assumed without discussion. For example his first proposition is to construct an equilateral triangle, in the second proposition his second step is to place an equilateral triangle on the page which he does with a simple "...on it let the equilateral triangle DAB be constructed." Likewise, after showing how a length could be transferred based only on very simple assumptions using his so-called "collapsible compass" he then feels free to transfer a length to a desired location in his third proposition describing how a longer segment can be cut to equal the length of a shorter segment.
