 To the Greeks, geometry was the all-encompassing mathematical science. This meant that Euclid's elements included many things we do not consider geometrical. For example, Euclid's book 2 is an example of what we call geometric algebra, the use of geometric language, to express algebraic relationships. To translate between algebra and geometry, we want to identify key geometric objects with key algebraic objects. In general, algebraic quantities, x, can be identified with the lengths of line segments. And that means the squares or products of these quantities, x squared or ab, can be identified with the areas of squares or rectangles. If we look at cubes or products of these quantities, we can identify those with the volumes of solid objects. And the fourth powers of these quantities are non-geometric. And so there is an inherent limitation to geometric algebra. Or is there? We'll take a look at that in a bit. For example, book 2 opens up with the following proposition. If there are two straight lines, and one of them is cut into any number of segments whatsoever, then the rectangle contained by the two straight lines equals the sum of the rectangles contained by the uncut straight line and each of the segments. So let's see what this means. Now it's important to keep in mind that when Euclid or other Greek geometers talked about a line, they meant a finite object. A line had a beginning and an end. So this statement that we have two straight lines, nowadays we might say something like we have two straight line segments. Well let a and bc be two straight lines, and let bc be cut at points d, e, and so on. Now what we'll do is we'll form a rectangle with base bc and height a, and we'll also form the rectangles with base bd, de, ec, and height a. And the claim is that the whole rectangle is equal to the sum of the smaller rectangles. And if we think about this when Euclid is talking about the rectangle, what he really means is what we would think about as the area of the rectangle. So if we let the length of a, b, a, and the length of bd, de, e, c be equal to b, c, and d, then what this proposition is claiming is that the area of the whole rectangle, that's a times the length b plus c plus d, is equal to the areas of all these other rectangles put together, ab, ac, and ad. And this is the distributive property in algebra, but phrased using geometric language. A couple propositions later Euclid proves an important one. If a straight line is cut at random, the square on the whole is the squares on the segments together with twice the rectangle on the segments. So let ab be our line, and let it be cut at c someplace at random. So we have the square on the whole, and if we partition this square, we get a couple of other figures. First, we have the square on the part ac, the square on the part cb, and let's slide that square up to get it out of the way. You should take a moment to convince yourself that this is what it would actually look like, the rectangle with with ac and height equal to cb, and the rectangle with with cb and height equal to ac. Now if we let the length of ab be a plus b, then this says that the square on the whole, that's the square of a plus b, is equal to the squares on the parts, a squared plus b squared, plus twice the rectangle on the segments. That's two ab. Now the geometric algebra offers a visual representation of algebraic relations. So from the geometry, it's obvious that a plus b squared is not the same as a squared plus b squared. Unfortunately, geometric algebra doesn't handle negative quantity as well. For example, an algebraic quantity like a minus b would be the geometric quantity. How about the remainder when a line is removed from a line? For example, we might try to express the identity in geometric algebra. A minus b squared is a squared plus b squared minus two ab. And so we need to interpret the quantities as geometric objects. If a and b are the lengths of line segments, then a minus b, well, we can view that as the remainder when a line is removed from a line. A squared, well, that's not too bad. That's the square on the original line. And likewise, b squared is the square on the line removed. And ab is a rectangle with sides a and b. And so we might state this as the following. Let two lines, a and b, be given. The square on the line remaining when one line is removed from the other is equal to the squares on the two lines reduced by twice the rectangle on the two lines. Now that really is just a translation of this algebraic statement into geometry. The real problem occurs when you try to draw a picture. If we try to draw that picture, we have a line. We have another line, but we want to remove one line from the other. So here's our line remaining, and we want to take a look at the square on the line remaining. It's supposed to be equal to the squares on the two lines. And now for the complicated part, we're going to reduce these two squares by twice the rectangle on the two lines. So we need to remove these two rectangles. And while it's possible to see how that can be done, maybe there's a better way. The problem with subtracting one quantity from another is that order matters. The subtracted quantity must be less than the quantity being subtracted from. Unfortunately, this can't be guaranteed in a general statement. So we can at least modify our statement a little bit so we can guarantee it. So to avoid this, we might say that A minus B is the difference between two lines with A the larger. As a second option, we can eliminate subtracted quantities entirely because remember, subtracted quantities can change sides. For example, let's take a look at that expression again. And now I can move this subtracted quantity minus 2AB to the other side. And our wording becomes a lot easier. So what do we have? We have the square on the difference and twice the rectangle. So we might say something like the following. The square on the difference between two line segments plus twice the rectangle formed by the two line segments is equal to the squares on the segments. And here it's a little bit easier to draw a picture. So we're going to take the square of the difference and add two rectangles. And because we're adding, it's easier to see how we fit these pieces together to get the big square and the small square. There's a third way to handle negative quantities and that's to interpret them as an excess or a deficit. For example, the statement 8 equals 13 minus 5 could be read as 8 falls short of 13 by 5 or that 13 exceeds 8 by 5. And so that gives us another way to express a subtracted quantity in geometric algebra. So we could express it as a deficit. The square on the difference between two line segments falls short of the sum of the squares by twice the rectangle of the line segments. Or we could also express it as an excess. The squares on two line segments exceeds the square on their difference by twice the rectangle on the line segments.