Just discovered this wonderful website with Euclid's Elements. Have wanted to read this for a while now, and just read a few pages of it tonight. Discovered an interesting idea called "Neusis", or fitting a line into a diagram.
The operation is thus : You take a line segment of a fixed length (say the radius of a circle) and mark out it's length on a straight edge. You then move the straight edge and use the specific length to make another line segment of that same length. This is used in a very intuitive proof of trisection of a given angle (see link below) - ie. Given an angle, to find another angle whose measure is exactly one-third the measure of the original angle.
Apparently, this process of Neusis cannot be justified using Euclid's postulates and is thus outside the scope of the Elements. Seems reasonable as it is does intimately depend on the measurement of a length, but what's neater is that a mathematician named Whatzel actually proved that neusis would be unjustified under Euclid's axioms. This rabid quest for correctness in all respects is what makes this stuff so exciting!
Euclid's Elements, Introduction
The operation is thus : You take a line segment of a fixed length (say the radius of a circle) and mark out it's length on a straight edge. You then move the straight edge and use the specific length to make another line segment of that same length. This is used in a very intuitive proof of trisection of a given angle (see link below) - ie. Given an angle, to find another angle whose measure is exactly one-third the measure of the original angle.
Apparently, this process of Neusis cannot be justified using Euclid's postulates and is thus outside the scope of the Elements. Seems reasonable as it is does intimately depend on the measurement of a length, but what's neater is that a mathematician named Whatzel actually proved that neusis would be unjustified under Euclid's axioms. This rabid quest for correctness in all respects is what makes this stuff so exciting!
Euclid's Elements, Introduction
I'm creating this version of Euclid's Elements for a couple of reasons. The main one is to rekindle an interest in the Elements, and the web is a great way to do that. Another reason is to show how Java applets can be used to illustrate geometry. That also helps to bring the Elements alive.
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