How is this even remotely related to associative law of multiplication? I'm going through the Euclid's proof of associative law of multiplication and I'm having difficulty seeing how this proof works. It basically says this:  

If A = kB and C = kD, then when you take tA and tC, they will be the
  same multiples of B and D respectively(that is tA = uB and tC = uD for some u).

I don't see the connection between above and associative law of multiplication. Any help?
 A: 
I think Euclid's point is that you can calculate $GL$ either by multiplying B by $n={A\over B}$ to get $A$ and then scaling it by $m={C\over A}$ or by first scaling to get $D$ and then multiplying it by $m$.  In other words $m(Bn)=(mB)n$ if you allow the different multiplications to be written out that way.
Yikes.  And I woke up this morning thinking that the proof of the associativitity of multiplication from the Peano axioms was thorny.
A: I believe OP is right, that Euclid Elements V, 3 is unrelated to the associative law of multiplication. It surely is not an attempt to prove it. Not until Elements VII, 16 does Euclid even prove the commutative law of multiplication, and I don't find the associative law anywhere explicitly shown in Euclid's arithmetical books (Elements VII-IX). Book V is a general treatment of the theory of proportion, applicable to magnitudes of any kind, even incommensurables, and it seems on its face unlikely that Euclid would deal with a purely arithmetical matter at this place and in this context.
In his notes on V, 3, T. L. Heath, to whom wiki refers, says "we are practically shown that the multiple of a arrived at, viz. $m\cdot na$, is the same multiple denoted by the product of the numbers m, n, i.e. the (mn)th multiple, or in other words that $m\cdot na=mn\cdot a$". Thus he may seem to say we are in effect shown the associative law of multiplication in this proof (although his "practically" is ambiguous). But when Euclid proves the commutative law of multiplication in VII, 16, Heath alerts us to it at the beginning of his note and quite clearly, rather than at the end and only obscurely, so I don't believe he is here saying we have a proof of the associative law.
The associative law of multiplication is neither the subject, nor an implied premise, nor a possible corollary, of Elements V, 3. Heath's notes, though often helpful, are here misleading.
