Hilbert's construction of multiplication of two numbers

Question

I am now reading Hilbert's "Foundations of Geometry", section 15, where he describes there a geometric way to construct, given two segments of length $a$ and $b$, a segment of length $ab$ (in short: let $L$ be the line connecting the points $(1,0)$ and $(0,a)$. Let $T$ be the line parallel to $L$ and going through $(b,0)$. Then $T$ intersects the $y$-axis at the point $(0,ab)$).

was this construction known to the greeks? or to other cultures that dealt with geometry before the greeks, e.g. the babylonians?

thank you very much,

David

Answer 1

The Greeks had no coordinate systems, and they thought in terms of ratios and/or areas rather than in terms of products of numbers. With that proviso, then yes, this construction was known to the Greeks, and I'm almost sure it is in Euclid but I do not know Euclid well enough to give a citation.

I think the statement in Euclid would be close to something like this. Given three line segments $A,B$, and given another line segment $C$, one can construct a line segment $D$ such that the ratio of (the lengths of) $A$ to $C$ equals the ratio of $B$ to $D$.

Answer 2

The result can be thought of as done in Proposition 12, Book VI, of Elements.

For the proposition, I suggest going to this link. There you will find a brief discussion of Descartes' interpretation. For more detail on the proposition, please see Heath's heavily annotated Elements.