I think we agree that this discussion is difficult because expanding the toolbox from geometry to calculus raises ambiguities as to what to regard as "known" when attempting to prove so basic a theorem as PT.

I also agree that PT "works" only in "Euclidean" geometry. (I have contributed several pages to CTK on this subject.) Indeed, once the novelty of seeing many proofs of PT wears off, one aspect of these discussions which continues to be of interest (at least to me) is to ask where in the proof is the "Euclidean" nature of the context introduced or applied. In many cases, it is through the use of similar triangles (which, as Alex has observed, is, at least in part, the case here).

Of course, a straightforward proof of PT by means of similar triangles can be had without recourse to additional tools (such as the calculus), raising the stylistic question of what the additional machinery adds to our understanding.

Scott.

It also leads to another point of view on the PT: the essence of the latter is that the circle has an equation x² + y² = R².

Consider the figure:

The right triangle with legs x and y, and hypotenuse c. If the hypotenuse is rotated by a small angle, say clockwise, it determines a new right triangle, in this case with legs x + dx and y - dy. "To first order" (we are doing calculus here!), the small right triangle with with legs dx and dy is similar to the original triangle, as the hypotenuse of this little triangle is nearly perpendicular to the original hypotenuse, and we may read off the proportion,

x/y = -dy/dx,

and, as previously, xdx + ydy = 0, or x^2 + y^2 = Constant.

To evaluate the constant, perhaps we can appeal (invoking continuity as necessary) to the special case of the "degenerate" triangle where x or y is 0.

Happy New Year to all!

This discussion is making me think about the status of linear and areal measure in plane geometry.

In Euclid (please correct me if I'm wrong), area is not defined in terms of length at all.  It is taken as a primitive concept subject to certain assumptions, such as invariance under translation, rotation, mirroring and dissection/composition.  It is difficult for the modern mind (trained in analytic geometry) to refrain from seeing the squares in the Pythagorean Theorem as numerical entities, arising from the multiplication of the associated side length with itself.  (Of course there is nothing wrong with this point of view; I'm merely trying to assess the degree to which it colors our thinking about the theorem.)

In analytic geometry, in contrast, coordinate pairs are the fundamental quantities, and area and length are derived algebraically from them.  Consider the plane RxR.  For points of the form (a,c) and (b,c), distance can easily be defined as |a-b|; similarly for (c,a) and (c,b).  Distance is not automatically defined for point pairs of the form (a,b), (c,d), and deriving this distance is equivalent to proving the Pythagorean Theorem.

Just for variety, let's try to do it using integral calculus.  Define a line L in the plane to be the set of all points (a,b) satisfying pairwise A*(a1-a2) = B*(b1-b2) for some constants A, B.  And two lines L1, L2 are parallel when A1*B2 = A2*B1 or perpendicular when A1*A2 = -B1*B2.  Define a rectangle PQRS as a figure whose adjacent sides are perpendicular and opposite sides are parallel.  One special class of rectangles are those with sides parallel to the coordinate axes, and for them, we define the area to be the product of the lengths of one pair of adjacent sides.

Now consider an oblique rectangle, and define its area in two ways:  First, by extending the definition above, so that its area is the product of the length and width.  Second, by using the property of dissection/composition to write the area of the oblique rectangle as the sum of many small non-oblique rectangles.  This is essentially the method of exhaustion of Archimedes, or in modern dress, it is integration (and the dissection/ composition properties of area are properties of the product measure on the plane).  Requiring these definitions to agree will produce the Pythagorean Theorem.

Let PQRS have vertex coordinates (px,py), (qx,qy), (rx,ry), (sx,sy), satisfying A(px-qx) = B(py-qy), A(rx-sx) = B(ry-sy), B(qx-rx) = -A(qy-ry), B(sx-px) = -A(sy-py).  Assume for concreteness that px<sx<qx<rx and sx<px<rx<qx (other orientations of the rectangle will give similar calculations).  Of course neither A nor B is zero, since this is an oblique rectangle.  Then we may make a vertical cut through S and apply the cut-off triangle to the opposite end of the rectangle, to form a parallelogram, and then integrate.  (This depends on knowing that PS can be superimposed on QR -- but the general definition of oblique rectangle given above lets one calculate easily that rx-qx = sx-px and qy-ry = py-sy, so the dissection is seen to work.)  Integrating in thin vertical strips, we obtain an area (rx-sx)*((qy-py)*(sx-px)/(qx-px) + (py-sy)), where the second factor is the length of the vertical slice.  Since rx-sx = qx-px, this simplifies to (qy-py)*(sx-px) + (qx-px)*(py-sy).

In order for this formula to be compatible with the first definition of area, it must be possible to write it as a product of factors which represent the lengths of the two adjacent sides of the oblique rectangle.  One side depends only on P,Q, and the other only on P,S, hence it must be true independent of the precise values of the coordinates that F(qx-px,qy-py)*F(px-sx,py-sy) = (qy-py)*(sx-px) + (qx-px)*(py-sy), where F is the function which assigns lengths to oblique line segments (F depends only on the coordinate differences, in order to secure translation invariance).  To clean up the notation a bit, let's write it as F(a,b)*F(c,d) = ad - bc.

By the perpendicularity of the sides of PQRS, A(px-qx) = B(py-qy) and  B(sx-px) = -A(sy-py), and multiplying these equations yields  (px-qx)(sx-px) = -(py-qy)(sy-py), or briefly ac = -bd.  Hence, squaring the equation above gives (F(a,b)*F(c,d))^2 = (ad - bc)^2 = (ad)^2 + (bc)^2 - 2adbc = (ad)^2 + (bc)^2 + (ac)^2 + (bd)^2 = (a^2 + b^2)*(c^2 + d^2).  Hence F(a,b) = sqrt(a^2+b^2), and similarly for F(c,d).  By itself, this does not show that the solution obtained is unique, only that it works.  Uniqueness follows when we consider that if there were two different solutions, we could find a situation in which one solution gave a larger numerical magnitude than the other for the sides of the same rectangle, hence producing disagreement with the area formula.

To summarize, in this viewpoint, the Pythagorean Theorem is viewed as a compatibility condition between length and area, which must be true in order to secure translation and rotation invariance for area.  Integral calculus allows one to associate areas of oblique rectangles with those of upright rectangles, while considerations of functional form drive the derivation from there.  Perpendicularity plays the role of ensuring that the area formula does indeed factor as required.

Well, that's enough for now, I think.

--Stuart Anderson

You know, I do not believe that there is a definition of area one way or another. Most definitely Euclid never used the term but, instead, identified a figure with a certain quantity that changed quadratically with the linear dimensions.

>It is taken as a
>primitive concept subject to certain assumptions, such as
>invariance under translation, rotation, mirroring and
>dissection/composition.

Yes. But even that was only implicit in his treatment.

>It is difficult for the modern mind
>(trained in analytic geometry) to refrain from seeing the
>squares in the Pythagorean Theorem as numerical entities,
>arising from the multiplication of the associated side
>length with itself.  (Of course there is nothing wrong with
>this point of view; I'm merely trying to assess the degree
>to which it colors our thinking about the theorem.)
>
>In analytic geometry, in contrast, coordinate pairs are the
>fundamental quantities, and area and length are derived
>algebraically from them.  Consider the plane RxR.  For
>points of the form (a,c) and (b,c), distance can easily be
>defined as |a-b|; similarly for (c,a) and (c,b).  Distance
>is not automatically defined for point pairs of the form
>(a,b), (c,d), and deriving this distance is equivalent to
>proving the Pythagorean Theorem.

In what you did below, the appearance of the expression ad - bc is revealing, being the twice the area of the triangle with vertices (a,b), (c,d) and (0,0).

One question I may have relates to the following:

>In order for this formula to be compatible with the first
>definition of area, it must be possible to write it as a
>product of factors which represent the lengths of the two
>adjacent sides of the oblique rectangle.  One side depends
>only on P,Q, and the other only on P,S, hence it must be
>true independent of the precise values of the coordinates
>that F(qx-px,qy-py)*F(px-sx,py-sy) = (qy-py)*(sx-px) +
>(qx-px)*(py-sy), where F is the function which assigns
>lengths to oblique line segments (F depends only on the
>coordinate differences, in order to secure translation
>invariance). 

A function that assigns lengths to oblique line segments is conceivably in the form F(P, Q). May not assuming that it depends on coordinate differences be in itself equivalent to adopting the Pythagorean theorem? Or, is it exactly what you say?

>You know, I do not believe that there is a definition of
>area one way or another. Most definitely Euclid never used
>the term but, instead, identified a figure with a certain
>quantity that changed quadratically with the linear
>dimensions.

That is also how I see it. The concept of area is defined only implicitly, via the properties Euclid assumed without question to hold.

>In what you did below, the appearance of the expression ad -
>bc is revealing, being the twice the area of the triangle
>with vertices (a,b), (c,d) and (0,0).

Yes, I noticed that, too. I happened to assign the variables a,b,c,d to the coordinate differences, and the formula ad-bc popped out. Of course this is instantly recognizable as the determinant of a 2 by 2 matrix, which has the geometric interpretation of the area of the parallelogram spanned by the two column vectors. This was a useful checkpoint for me, since it'showed that I had (probably) not made any algebraic blunders in the derivation up to that point.

>
>One question I may have relates to the following:
>
>>In order for this formula to be compatible with the first
>>definition of area, it must be possible to write it as a
>>product of factors which represent the lengths of the two
>>adjacent sides of the oblique rectangle. One side depends
>>only on P,Q, and the other only on P,S, hence it must be
>>true independent of the precise values of the coordinates
>>that F(qx-px,qy-py)*F(px-sx,py-sy) = (qy-py)*(sx-px) +
>>(qx-px)*(py-sy), where F is the function which assigns
>>lengths to oblique line segments (F depends only on the
>>coordinate differences, in order to secure translation
>>invariance).
>
>A function that assigns lengths to oblique line segments is
>conceivably in the form F(P, Q). May not assuming that it
>depends on coordinate differences be in itself equivalent to
>adopting the Pythagorean theorem? Or, is it exactly what you
>say?

In response to your comment above, I have thought further about this point. It'seems that the translation invariance assumption is necessary, as well as an angle invariance assumption. In general, the only requirement for compatibility between the definition of oblique length and rectangular area is that F(P,Q)*F(P,R) = sqrt(a^2+b^2)*sqrt(c^2+d^2). It follows that F(P,Q) cannot be zero when a^2 + b^2 is nonzero; therefore this formula can be factored out, leaving F(P,Q) = f(P,Q)*sqrt(a^2+b^2).

Next, additivity of area implies that if we choose Q' so that q'x-px = ka and q'y-py = kb, then the rectangle constructed on PQ' and PR will have k times the area of PQRS. Hence, since F(P,R) is unchanged, it must be that F(P,Q') = kF(P,Q). Since the factor of k is already accounted for in sqrt((ka)^2 + (kb)^2), it follows that f(P,Q') is independent of the choice of Q', so long as PQ' is parallel to PQ, i.e. if P, Q, Q' are collinear. Therefore, f(P,Q) can depend at most on the location of P and line PQ.

One of the challenges of working at this primitive level is that many useful concepts are not yet defined. So far, all I have to work with is an algebraic definition of parallel and perpendicular, (which allows the definition of the line PQ as a set of points which includes distinct points P and Q and all points Q' for which PQ || PQ') and the assumption that the real line is mapped in some fashion onto each line in the Cartesian plane, giving a definition of oblique distance. In particular, angle has not been defined yet, but it is possible to associate a label theta with each line L, so that theta = theta' iff L || L' (theta is merely an equivalence class of lines, with no numerical value assigned to it).

With these ideas in mind, we can say that f(P,Q) = f(P,theta). This allows the assignment of length to line segments to depend on the location and orientation of the segment. This function must have several properties in order to be compatible with the basic idea that the area of a rectangle should be the product of the lengths of its sides: f(P,theta) must be nonzero, f(P,theta) = 1/f(P,theta') whenever theta and theta' refer to perpendicular lines, and f(P,theta) cannot depend on which direction we traverse the line to reach Q (because one can construct two equal rectangles sharing side PR, and we must obtain the same area for either). Also, of course, f=1 when PQ is parallel to either of the coordinate axes.

This leaves a great deal of freedom in defining f. Without further assumptions, f cannot be specified further. To see why not, consider what we have so far: a set of compatibility conditions applying to perpendicular pairs of line sets (i.e. equivalence classes theta and theta', where theta lines are perpendicular to theta' lines). These conditions are all satisfied by the original Cartesian coordinate axes, so what we have here is really a description of the collection of all possible alternative orthogonal coordinate systems on the Cartesian plane, which assign the same area to rectangles as the original coordinate system. In brief, f describes a choice, for each point P, of an orthogonal, area-preserving, affine coordinate transformation. Hence no contradiction can arise from any choice of function f (subject to the above conditions), and therefore no new conditions on f can be derived without introducing fresh assumptions.

Even assuming that f depends only on the coordinate differences between P and Q is not enough to pin down f(P,Q) exactly. Doing so means that F(P,Q) = f(theta)*sqrt(a^2 + b^2), but the scale could be defined independently for each theta. It seems that what is needed is something that ties together the scales of lines which are not orthogonal to one another. Of course, one could simply decree that f=1, but it would be more satisfying if there were some specific geometric idea which drove the decision beyond the mere aesthetics of simplicity -- that is, something more in the spirit of the original idea of extending the rectangle area formula to oblique rectangles.

Here is one possible idea: Consider a right triangle with its hypotenuse AB lying on the x axis. From its third corner C drop a perpendicular CD to side AB. All of these concepts can be defined algebraically within the context of the original Cartesian plane RxR. It is then an algebraic fact that the length of CD is the geometric mean of the lengths of AD and DB, as the segments in question are parallel to the original coordinate axes, hence their lengths are not in question. Now assume that this relation holds for right triangles with oblique legs. We now have a relation between the lengths of a perpendicular pair of oblique line segments, and this relation makes their sizes scale together for different choices of f(theta). The relation of length to area discussed earlier makes their sizes scale inversely with each other for different choices of f(theta). Together, these show f(theta) = constant, which must therefore be exactly 1 in order to make the are of a rectangle come out right.

So it appears that these two assumptions (geometric mean and translation invariance) suffice to force the Pythagorean theorem from an algebraic perspective, and in the absence of a previously defined metric.

Well, that was more subtle than I thought it would be. Do you see any major holes in the argument?

--Stuart Anderson

it'seems to me that you have overcomplicated the situation. The first definition of the area assumes translational and rotational invariance. For the two definitions to agree, it is then necessary to have f(theta) = const, as you actually show in the last paragraph.

The geometric mean requirement is a direct consequence of the Euclidean axioms via the similarity of right triangles formed by the altitude from the right angle. Rewritten differently, this condition is a proportion that guarantees the relation between perpendicular slopes. This proportion implies the PT in conventional (synthetic) ways but also via the differential calculus (as John showed) and now via the integral calculus. For the latter derivation, I believe you can consign yourself to considering squares which would related it more directly in a sense to the PT. But once you do, the areas could be evaluated synthetically, showing that implying the integral calculus is just another (like John's) round-about way of arriving at the same conclusion. This is not to say that there is anything wrong with doing that.

Curiously, John came up (see a nearby thread) with a proof based on Gauss' Shoelace formula which, when applied to a oblique square, provides a one-line proof of the PT. Roger Nelsen showed a synthetic way that entirely parallels an application of the shoelace formula. But, that formula is one of your intermediate flagpoles as well.

>it'seems to me that you have overcomplicated the situation.
>The first definition of the area assumes translational and
>rotational invariance. For the two definitions to agree, it
>is then necessary to have f(theta) = const, as you actually
>show in the last paragraph.

Well, yes the derivation is rather complicated. I think that this arose because my goal changed sometime between the first post and the second. Originally, I wanted to see if I could produce an integration + Euclidean hybrid proof, similar to John's differentiation + Euclidean hybrid proof. However, after your comment about the possibility that the length formula might depend on P and Q rather than the interval, I began to think that it would be more interesting to try to produce a "pure" analytic proof. In other words, I wanted to see how one might get to the Pythagorean theorem assuming ONLY the properties of the real numbers, the definition of the plane as RxR, and as few extra definitions and assumptions as possible.

The complication arises from the very restricted set of tools I chose to allow. I defined length originally only for segments along each axis and area only for algebraic rectangles (i.e. rectangles of the form IxJ where I and J are intervals in R). It was possible to give purely algebraic definitions of "line," "parallel," and "perpendicular." From there on,the rule I followed was that I would use only properties that could be defined within R or for intervals within R, and extend as few of these as possible to oblique figures.

Accordingly, I first made a postulate that the area formula should extend to oblique rectangles, which placed a restriction on how the concept of length could be extended to oblique line segments. Since the length formula |a-b| is translation invariant within R (an algebraic fact, hence admissible under my rules) I then assumed that this property extended to the oblique length formula as well.

However, rotational invariance cannot be used, or even formulated, under the restrictions I was trying to obey. In fact, how does one even define rotation without assuming the Pythagorean theorem? If I try to apply a rotation matrix to the coordinates of points in a figure, what tells me that my matrix is a rotation matrix? Only that it satisfies ad-bc=1 with a=d and b=-c, hence a^2+b^2 = 1. If I make this the definition of a rotation matrix, I seem to be begging the question, since I'm essentially building the Pythagorean theorem into my system pretty explicitly. On the other hand, if I try to define it by some geometric property, such as preserving lengths and angles, I'm begging the question about length (since I haven't finished defining it yet) and introducing the concept of angle. The latter will be difficult, since there is no adequate analog of angle within one dimensional R, from which I can construct the angle concept in RxR by extension.

Therefore, I decided to avoid using rotation invariance altogether, and instead looked for some other property which I could extend. The geometric mean property works, because it can be proven purely algebraically within RxR, and can be extended in an obvious way to oblique right triangles.

>
>The geometric mean requirement is a direct consequence of
>the Euclidean axioms via the similarity of right triangles
>formed by the altitude from the right angle. Rewritten
>differently, this condition is a proportion that guarantees
>the relation between perpendicular slopes. This proportion
>implies the PT in conventional (synthetic) ways but also via
>the differential calculus (as John showed) and now via the
>integral calculus. For the latter derivation, I believe you
>can consign yourself to considering squares which would
>related it more directly in a sense to the PT. But once you
>do, the areas could be evaluated synthetically, showing that
>implying the integral calculus is just another (like John's)
>round-about way of arriving at the same conclusion. This is
>not to say that there is anything wrong with doing that.

I would agree with the above if I felt free to use other Euclidean ideas, but I'm afraid I must quibble. Since I was trying to use algebraic ideas only as far as possible, I did not want to derive the geometric mean from the similar right triangles argument (I can't use that because not all the triangles are oriented the same way in that proof, and at that stage in the derivation, I still had not secured the fact that length was rotation invariant -- in fact, I was trying to use the geometric mean theorem to avoid just that.) Without the similar right triangles, and its underlying Euclidean assumptions, the PT would not follow.

Also, I could not restrict the construction to squares, because I cannot define the term "square" until after I know that the length formula is independent of the orientation of the line segment. That was why I cast the derivation in terms of rectangles in the first place -- simply because I could define them.

>
>Curiously, John came up (see a nearby thread) with a proof
>based on Gauss' Shoelace formula which, when applied to a
>oblique square, provides a one-line proof of the PT. Roger
>Nelsen showed a synthetic way that entirely parallels an
>application of the shoelace formula. But, that formula is
>one of your intermediate flagpoles as well.

Yes, I had noticed that -- the ad-bc formula is a special case of the shoestring formula. Of course, I couldn't use that either, as my goal was to generate the Euclidean metric, not just the Pythagorean theorem. The latter can be conceived entirely in terms of the intuitive properties of areas, in the way Euclid used them, "square on the side" instead of "square of the length of the side". The shoestring formula deals with area directly, and never addresses the question of the lengths of segments, which I was trying to get at.

Obviously, I've made quite a mess here. That's what happens when you try to work with very, very few tools, of course. It could be worse; for instance, Russel and Whitehead didn't finish proving 1+1=2 until several hundred pages into Principia Mathamatica. Compared to them, I'm being an absolute slob!

Thanks for a very stimulating discussion!

--Stuart Anderson

I believe then that I misunderstood your intention. My interpretation of

"Distance is not automatically defined for point pairs of the form (a,b), (c,d), and deriving this distance is equivalent to proving the Pythagorean Theorem.

Just for variety, let's try to do it using integral calculus."

was that you would try to follow in John's footsteps with a twist of employing integral instead of differential calculus. This would make the Elements available to an extent, for a cautious use, as well as parts of analytic geometry that are not based on the PT, or scalar product.

I asked that question because translation invariance of the area is a distinguished feature of Euclidean geometry and just on the basis of that you practically delivered this quadratic formula of the Pythagorean theorem (in F(P, Q)).

Now, the first definition of the area of a rectangle is clearly rotationally invariant because so are Euclid's segment lengths and angles as used in the compass and straightedge constructions. Had you picked that up you would have f = const right away. I wondered why you did not. Now I understand. I missed that turn. Sorry about that, and thank you for sharing.

Reverse engineering here, if one is to "follow in (my) footsteps" using first an integral and then a derivative, it'seems this would be massively difficult since the integral of y = sqrt(c^2 - x^2) is:

https://www.wolframalpha.com/input/?i=integrate+sqrt(c^2-x^2),x

I could be wrong, but it seems building that "area" function for the semicircle would be hard to do strictly from a geometric standpoint.

The only other option I can see in this is to use some kind of limit to either define the integral expression or its derivative (or both) from the geometric ideas Stuart presents. This seems to me even more massively difficult.

Of course, I may be missing Stuart's point like you did. I suppose I will know in time as Stuart works out the details.

By the way Stuart, I am cheering you on. I think anyone who goes to the primitive level you are choosing to use stands a good chance at bypassing a lot of assumptions, axioms, lemma, etc... (like I did with Euclid, etc...) Good luck!