Given a rectangle of length ‘L’ and width of ‘W’, the area is calculated as L X W.
Is this result:
a) simply the definition of the area of a rectangle?
b) just an assumption?
c) provable (by a strictly geometric approach)?
I’ve tried to figure out the answer myself but it quickly became apparent to me that not only did I not know the answer, but that I really don’t have any sense how to procede. Part of my difficulty stems from the fact that this formula is so ingrained into me that I can’t even remember how to derive it, or if it is even derivable. But part of it also arises from an inevitable intrusion of algebraic thinking, as opposed to plain geometric reasoning, when I try to tackle the problem.
Well, if you don’t define area, finding a mathematical value for some specific area is gonna be a bit troublesome.
Having defined a surface bounded by one unit, per side, in the shape defined by a right angled rectange with equal one unit sides as being one square unit in area, the rest follows from analytic geometry.
You say you’re asking about Euclidean geometry, so why not go to the source?
An “area” is some amount of a plane bounded by a figure. There’s an extensive treatment of which areas are equal in magnitude to which other areas. Usually the attempt is to show a certain area is equal to the area of a recangular figure, particularly a square.
So within the context of Euclid, (a) is the correct answer, but your question is somewhat ill-posed since there’s no real sense of multiplication there. Within Euclidean geometry lengths are lengths and areas are areas. They’re two different sorts of quantities, and there’s no sense of number associated with either of them.
That’s a statement about the ratios of areas, not about the areas themselves. As I note elsewhere in the thread, lengths and areas are (to Euclid) two types of quantities and are not associated with the modern notion of real number.
According to my Plane Geometry text, admittedly very basic, a unit of surface is defined as a square one unit (any old unit will do) on a side. And the area is defined as the total number of these units of surface that can be fitted into the surface in question.
If the surface in question is irregular you have to keep making the unit of surface smaller and smaller so that you can fit them inside without leaving any space not covered by a unit of surface.
It looks like, in plane geometry area is a defined measure and the area of an irregular surface can only be approximated.
Plane geometry, which is what’s usually called Euclidean geometry these days, is not the geometry that Euclid invented. If you want to work in Euclid’s geometry, you have to stick to what’s in The Elements and not import the last 2500 years of progress. Good luck.
For what it’s worth, modern Euclidean geometry does use the methods of calculus, so proving that the area of a rectangle is the product of its length and width is very easy.
But ultrafilter is right that what you quoted also goes far beyond Euclidean geometry proper. Areas are just regions of the plane bounded by some collection of (generally curved) lines. They aren’t measured in any sort of unit.
I’m wondering if that might be a circular argument. From what I remember from my early Calc classes, integration of an area works by subdividing the area into a number of rectangular shapes with a given width x, then you take the limit of the sum of those areas as x goes to zero (conversely making the number of rectangles go towards infinity). This part of calculus is based upon the proposition that the area of a rectangle is L * W, so you can’t use it to prove the area of a rectangle is L * W.
From reading the links provided by yoyodyne and KarlGauss, Euclid’s geometry can prove that the ratios of areas are dependent on the ratios of the sides, and if you add the concept of a “unit” area being a square with sides one length unit, then maybe that’s enough to make the leap to the multiplication implementation of areas.
Of course, I went to college with an engineering focus, so I’m willing to accept “It just works, so use it” without the formalized proof.
Just for the record, I think it unduly restrictive to say that “Euclidean geometry” must have been done only by Euclid. Isn’t modern plane geometry a continuation of that work?
Modern calculus has been extended and put on a logical basis by people other than Leibnitz and Newton, but it starts from their original work.
Today’s Copenhagen interpretation of Quantum Mechanics wasn’t all done at the Copenhagen institute.
