28 b ab a² a 2 Euclid's approach to geometry 62 ab b Figure 2.8: The square of a sum of line segments • The square on the line is what we write as (a + b)². • The squares on the two segments a and b are a² and b², respectively. The rectangle "contained" by the segments a and b is ab. • The square (a+b)² equals (in area) the sum of a², b², and two copies of ab. It should be emphasized that, in Greek mathematics, the only inter- pretation of ab, the "product" of line segments a and b, is the rectangle with perpendicular sides a and b (or "contained in" a and b, as Euclid used to say). This rectangle could be shown "equal" to certain other regions, but only by cutting the regions into identical pieces by straight lines. The Greeks did not realize that this "equality of regions" was the same as equal- ty of numbers-the numbers we call the areas of the regions-partly be- ause they did not regard irrational lengths as numbers, and n ey did not think the product of lengtho al 2.4 Area of parallelograms and triangles Exercises In Figure 2.8, the large square is subdivided by two lines: one of them perpendic- ular to the bottom side of the square and the other perpendicular to the left side of the square. 29 2.3.1 Use the parallel axiom to explain why all other angles in the figure are necessarily right angles. Figure 2.8 presents the algebraic identity (a + b)2 = a² + 2ab + b² in geo- metric form. Other well-known algebraic identities can also be given a geometric presentation. 2.3.2 Give a diagram for the identity a(b+c) = ab + ac. 2.3.3 Give a diagram for the identity a²-b² = (a+b)(a - b). Euclid does not give a geometric theorem that explains the identity (a+b)³ = a³ +3a²b+3ab² + b3. But it is not hard to do so by interpreting (a+b)3 as a cube with edge length a + b, a³ as a cube with edge a, a²b as a box with perpendicular edges a, a, and b, and so on. 2.3.4 Draw a picture of a cube with edges a + b, and show it cut by planes (parallel to its faces) that divide each edge into a segment of length a and a segment of length b. 2.3.5 Explain why these planes cut the original cube into eight pieces: a cube with edges a, • a cube with edges b, • three boxes with edges a, a,b, • three boxes with edges a, b,b. 2.4 Area of parallelograms and triangles un "equal" to a rectar