I'm going to tell you a fable. A king,
growing old, has two daughters and wants
to partition his land. He goes to his
counsellor and says 'Counsellor, I have
four pieces of land and two daughters
and I want to partition my land equally
amongst my daughters. My surveyor says
that the land masses I have are of this
shape: I have one triangular piece of
land and it has a square angle here. 'A
right angle' says the counsellor, 'you
have a right angled triangle.' 'Ahh a right
angled triangle. And then I have a square
piece of land whose edge is the side of
the longest side of my triangle. And I
have another whose edge is the side of
this side of the triangle; and a third
the side of this side of the triangle.
The surveyor tells me they're all
squares.' The counsellor says 'Oh, no problem,
don't be one of these foolish kings who
gives away all your land. Keep that
triangular piece for yourself, give the
largest square to one of your daughters
and the two small squares to the other
daughter, it'll be perfectly even.'
'That's easy!' since the king and he's
about to leave but then says 'Counsellor, I
haven't told you anything about how long
the sides of these squares are, or the
shape of the triangle' Counsellor says 'Don't
worry. What we have here is three squares,
one triangle. If the edges of these
pieces of land are doubled all the areas
get multiplied by four, if they're tripled
all the areas get multiplied by nine,
that's three squared. If you changed
all the lengths of your figures by some
factor the area is changed by the square
of the factor and so the proportions are
exactly the same.'
Counsellor says 'This is called scaling and
similarity. If you scale your figures in
this way you get similar figures, scaling
a length by some factor scales the area
by the square.' 'Wonderful!' says the King.
King walks away but even before he
leaves the room he says 'But I didn't
tell you what the angles were of the triangle.'
Counsellor says 'Uh, that doesn't matter.
What we have here is something called
the Pythagorean theorem; it doesn't
matter what the angles are as long as
one of those angles is a right angle.'
King leaves. Returns the next day
distraught and says 'My surveyor tells me
I don't have squares I have things that
he called blobs! That is I have a
triangle, a right angled triangle, I have
three fields each landmass has one
linear edge - the linear edge equal to the
edge of the corresponding edge of the
triangle - and he also tells me that my
blobs are similar. If I scale any one of
these blobs appropriately it will become
any of the others. Now what do I do?'
asks the King apprehensively and the
counsellor says 'No problem. Do exactly as
before. You keep your triangle, give the
larger- largest blob mass to one of your
daughters and the other two to the other
daughter and it'll be even Steven. The king is amazed.
He says 'It doesn't matter what blob they
are as long as they're similar?' Counsellor
says yes. 'What do you call that?' says the
King. 'Oh I call that the blob Pythagorean
theorem.' 'For every blob there's a blob
Pythagorean theorem?' says the King'
'Strange.' The counsellor says 'Yes, and not
only that but if you know the blob
Pythagorean theorem for any blob you
know it for every blob.'
The king says 'How? How can that be?' Well
the counsellor says 'Look, look at this:
you have three blobs here, imagine that
there's what I'll call a unit blob, that's
to say a blob that's similar to it but its
length is 1. Let us say the length of its
straight edge is 1. Why then, if the length
of this straight edge is c and the
length of this straight edge is a and
the length of this straight edge is b,
then we know that whatever area this
blob has - what do you call the area of that
blob? Capital A, the area of the unit blob.
The area of this blob is the square of
the factor that one needs in order to
scale the unit blob to it, so the area
will be c squared times A. The area of
this will be a squared times A and the
area this will be b squared times A. So
if I prove to you that for some blob
with some area A, this large blob is the
sum in area of the areas of these two
small ones, what I will have shown is c
squared times A is equal to a squared
times A plus b squared times A; and
cancelling out the A's I have c squared
is equal to a squared plus b
squared and so if you give me any
other blob I will get exactly the same
result.' 'So one blob Pythagorean theorem
implies Pythagorean theorem for all
blobs?' says the King and he leaves sort
of contented. King comes in, day three, and he
says 'It's wonderful what you've told me,
what you've explained to me, but can you
show me the blob Pythagorean theorem for
some blob in which case
I'll know it for all?' Counsellor says 'Oh
well, for that you have to draw me a
right-angled triangle.' And the counsellor
says 'Ah well, so my job then would be
to produce a blob with a linear side of
length c, a blob with a linear side of
length b, that blob being similar to the
blob of length c, and a blob with length
a similar for the other two; and then I
will have to show you that the area of
the larger- largest of these blobs is the
sum of the area of the other two blobs.'
The king says 'Well, yes, this might be
quite a chore.' The counsellor says 'Well, try
this-' Counsellor draws a single line which
meets the large side in a right angle.
And then steps back and says 'There you are.'
'Where am I?' says the king. 'The blob whose
linear edge is c is the entire triangle.
The blob whose linear edge is b is the
triangle to the right. The blob whose
linear edge is a is the triangle to the left.
All these triangles are right-angled
triangles because one of their angles is
a right angle. Each of these triangles
have another angle in common;
for example this large triangle has this
angle in common with this one, and it has
this angle in common with this one. All
three right angle triangles are similar.
So those are my three blobs and to see
that the sum of the two small blobs is
equal to the large blob is evident.'
King says 'Ah.'
Okay so now what's my moral? Can you guess?
(Brady: I dunno, that kings ask a lot of questions?)