In Penrose's The Road to Reality, after introducing the barest basics of classical Euclidean geometry, Penrose goes on to prove the Pythagorean theorem. As part of his first proof he promulgates:
Now, it is a general property of similar plane figures that their areas are in proportion to the squares of their corresponding linear dimensions.
My paraphrase of Penrose's restatement of Euclid's postulates is:
1) Connecting any two points, there is a unique line segment.
2) Every line segment lies on a (continuous) line (of infinite extent).
3) Associated with every point-radius pair, is a circle.
4) All right angles are congruent.
5) If the sum of consecutive interior angles is less than $\pi$, the two lines will intersect somewhere on that side of the transversal.
Nothing in those speaks directly about area.
In his Advanced Calculus of Several Variables, Edwards develops the general definition of area, beginning with the definition of the area of an interval in $\mathbb{R}^{2}$, as the product of the lengths of the component intervals, each in $\mathbb{R}^{1}$. That is, $\Delta{x}\times{\Delta{y}}$, where $\Delta{x},\Delta{y}\in\mathbb{R}^{1}$.
Edwards then expands that definition using limits of partitions. In short, Edwards doesn't treat the general definition of area as a trivial extension of the fundamental definition of the area of a rectangle. In the case of Edwards, there is no direct appeal to physical measurement. His definitions are strictly founded on abstract real numbers.
Is Penrose obligated to provide an argument similar to that of Edwards, in order to sustain his above quoted assertion?
Penrose's (proposed) proof is delightfully simple, if we accept his proposition.
One of the simplest ways to see that the Pythagorean assertion is indeed true in Euclidean geometry is to consider the confguration consisting of the given right-angled triangle subdivided into two smaller triangles by dropping a perpendicular from the right angle to the hypotenuse [figure reproduced below]. There are now three triangles depicted: the original one and the two into which it has now been subdivided. Clearly the area of the original triangle is the sum of the areas of the two smaller ones.
Now, it is a simple matter to see that these three triangles are all similar to one another. This means that they are all the same shape (though of different sizes), i.e. obtained from one another by a uniform expansion or contraction, together with a rigid motion. This follows because each of the three triangles possesses exactly the same angles, in some order. Each of the two smaller triangles has an angle in common with the largest one and one of the angles of each triangle is a right angle. The third angle must also agree because the sum of the angles in any triangle is always the same. Now, it is a general property of similar plane fgures that their areas are in proportion to the squares of their corresponding linear dimensions. For each triangle, we can take this linear dimension to be its longest side, i.e. its hypotenuse. We note that the hypotenuse of each of the smaller triangles is the same as one of the (non-hypotenuse) sides of the original triangle. Thus, it follows at once (from the fact that the area of the original triangle is the sum of the areas of the other two) that the square on the hypotenuse on the original triangle is indeed the sum of the squares on the other two sides: the Pythagorean theorem!
Of course, Penrose may sidestep my challenge by noting that a right triangle is half of a rectangle.
NB: I am sending a message to Dr. Penrose informing him that I have quoted from his book. If he objects, which I seriously doubt, I will remove the extended quote.
The definition of area in geometry can be done in several ways. See for instance Hilbert's "The foundations of geometry", Chapter IV. That areas of similar figures are in duplicate ratio than their sides is obvious, as the result holds for triangles and any polygon can be decomposed into triangles. Areas of non-polygonal figures can be defined by the method of exhaustion. See for instance Euclid's proof for the circle.
Notice that Penrose's proof of Pythagoras' theorem is not new: it also appears in Euclid's Elements, Prop. 31 in Book VI.