To get started here, draw two line segments on the canvas at the right—first a horizontal segment and then a vertical segment. These will form two adjacent sides of a rectangle.
Next, click to choose any 2 points on the horizontal line segment between its endpoints. The animation should take it from there.
This is pretty close to what the beginning of Euclid's Proposition 2.1 (we're in Book 2!) looks like, though you a drew a vertical line and horizontal line instead of two horizontal lines. What appears as \(\small\mathtt{A}\) in the original, I have replaced with [\(\small\mathtt{BG}\)]:
There are four theorems from Book 1 that Euclid uses to construct this proof, which is interesting given how simple it is. They are 1.11, which demonstrates how to construct a line at a right angle to another line at a given point (necessary for making side \(\small\mathtt{BG}\) of the rectangle), 1.3, which allows us to snip off \(\small\mathtt{BG}\) to just the right length (doesn't quite make sense in terms of how we built the model), 1.31, which allows us to draw the bottom of the rectangle parallel to the top, and 1.34, which Euclid uses to show that the middle rectangle has sides equal to \(\small\mathtt{BG}\).
The Distributive Property
Of course, the most interesting thing about this theorem—and about several that start out Book 2—is that it's essentially the geometric equivalent of the Distributive Property of Multiplication over Addition for positive numbers. That is, \(\small\mathtt{A}\)(\(\small\mathtt{B}\) + \(\small\mathtt{C}\) + \(\small\mathtt{D}\)) = \(\small\mathtt{A}\)(\(\small\mathtt{B}\)) + \(\small\mathtt{A}\)(\(\small\mathtt {C}\)) + \(\small\mathtt{A}\)(\(\small\mathtt{D}\)).
What makes it equivalent to this property are the variables that Euclid uses in his proof—the same ones you used in your drawing above.
You decided, within limits, what the length of \(\small\mathtt{B}\) + \(\small\mathtt{C}\) + \(\small\mathtt{D}\) would be when you drew the horizontal line, and you also decided what the length of \(\overline{\small\mathtt{BG}}\), or \(\small\mathtt{A}\), would be when you drew the vertical line. You also determined that \(\small\mathtt{B}\), \(\small\mathtt {C}\), and \(\small\mathtt{D}\) all had individual values when you "randomly" selected 2 points on the horizontal line.
What Variables Mean
Often, in school mathematics, we come away from lessons with the sense that variables are "unknowns" that we need to figure out. But, if I could be so bold, that's not what variables most often mean. What they generally represent are values that we don't care about.
A systematic lack of concern is a soft skill in high demand for mathematical work. It takes years and years of practice to not care, and it is certainly something for which some have a strong innate ability. : )
P.S.: Check out the bonus toy here.
Image mask credit: Joe Goldberg
