This is something I've been wondering about. When I think of "ratios" $x/y$ and $z/w$ as being "equal", with $x$, $y$, $z$, and $w$ being real numbers, this means the results of dividing the real numbers $x$ by $y$ and $z$ by $w$ are equal. Or that $xw = yz$, from manipulation of the fractions. Intuitively, we may say this means the "scale factor" going from $y$ to $x$ is the same as that going from $w$ to $z$, or that $x$ has as many "units" of size $y$ (allowing for non-integral numbers of units) as $z$ has of size $w$.
Yet, Euclid (~300 BCE) did not have real-number arithmetic to work with. Instead he had various kinds of "magnitudes", like line segments and shapes with areas and other things that had a kind of "size" to them. So he had to do something else, and this I don't get. If we have "magnitudes" $x$, $y$, $z$, and $w$, which for modern purposes could be taken as nonnegative real numbers, then we say $x/y = z/w$ iff for every pair of nonzero natural numbers $m$ and $n$, $mx < ny \rightarrow mz < nw$, $mx = ny \rightarrow mz = nw$, and $mx > ny \rightarrow mz > nw$. But how does one intuitively grasp this definition? How does it relate to our modern one? On Wikipedia, it says also "There is a remarkable similarity between this definition and the theory of Dedekind cuts used in the modern definition of irrational numbers." How exactly does this relate to Dedekind cuts? (this last bit is why I also file this under real analysis)
Take my answer as a grain of salt because my understanding might not be entirely correct:
Intuitively, the way I see Euclid's definition is you're trying to "squeeze" in rational numbers between $x/y$ and $z/w$ (or I suppose, the other way around) and if this fails, then they are equal. A Dedekind cut, partitions the rationals by creating two non-empty sets of rational numbers $A$ and $B$ with the condition that $A$ has no greatest element and all its elements are less than those of $B$. A real number $r$ is then defined to be a cut. Notice that Euclid's definition also cuts the rationals if we combine the equality condition with the greater than condition. So he has also essentially defined a cut. I think the difference between the two is that Euclid probably only had algebraic numbers in mind and not transcendental numbers, but I could be wrong..