*Picture from MacTutor Archives.*

After the Pythagorean crisis with the square root of two, Greek mathematicians tried to avoid working with numbers. Instead, the Greeks used geometry to demonstrate mathematical concepts. A line can be drawn any length, so straight lines became a sort of non-algebraic variable.

You can see an example of this in The Pythagorean Proof, where Alexandria Jones represented the sides of her triangle by the letters *a* and *b*. These sides may be any length. The sizes of the squares will change with the triangle sides, but the relationship is always true for every right triangle.

## Euclidean Algebra: The Basics

When Euclid wrote, “Let AB be the given finite straight line,” he was saying to let the line segment AB = *x*. Then he might draw a square with *x* as a side, making the area of the square .

Or he might draw a second line (*y*) and make a rectangle (area *xy*).

These shapes can be combined to show relationships, as the triangles and squares were combined to demonstrate the Pythagorean Theorem.

## The Distributive Property

For example, we can draw a diagram that will make the Distributive Property obvious:

.

- First, draw any straight line and divide it into three segments.

This is . - Draw a rectangle using this line for the base.

The height of the rectangle is*a*.

When the drawing is done, you can easily see that the area of the three rectangles is the same as the area of the large rectangle . This is always true for any *a*, *b*, *c*, and *d*.

## Now for a Tough One

How might Euclid draw a diagram for ?

- First draw a square with sides
*a*, which is . - Draw lines to cut the sides of the square into two lengths:
*b*and the remainder, . This defines the square .

Notice the shaded areas in the diagram. The gray and hatched rectangles are both *ab*. They overlap each other, and the area of the overlap is .

Therefore, to make the square , start with . Cut off and throw away the rectangle *ab*. Now, you want to throw away another rectangle *ab*. But to make the second *ab*, you will first have to add in an extra .

This leaves only our goal, the square , which shows geometrically that:

.

## It’s Your Turn

Can you draw a diagram to demonstrate these identities?

## If You Enjoyed These Puzzles…

Take a look at the more advanced geometric algebra animations here:

## To Be Continued…

Read all the posts from the May/June 1999 issue of my ** Mathematical Adventures of Alexandria Jones** newsletter.

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Thank you for reminding me about that online edition of Euclid’s Elements — I had a lot of fun reading through it and playing with all the interactive demonstrations when I first found it a few years ago. I particularly concentrated on the first book, which I think is one of the most beautiful pieces of mathematics ever.

I keep meaning to read and follow through the later books, but have never quite got around to it :). Particularly with the books about three dimensional objects, the reasoning can get a little convoluted — see

http://aleph0.clarku.edu/~djoyce/java/elements/bookXII/propXII10.html

for example, which shows that the volume of a cone is a third the volume of a cylinder with equal base and height.

Nice article Denise.