The first paragraph of this chapter says, "the concept of dimension became identified with the exponents" used to describe a figure. The chapter starts out by giving fairly simple examples of this notion. For instance, when an object's overall size is doubled, each dimension is multiplied by a power of two. The exponent on the two indicates the dimension in which the measurement is taken. For example, the volume is eight (2 to the 3rd power) times what it was.
Next, the book explores the idea of dimensionality showing up in the denominator of a formula. For example, the pattern of n-dimensional volume of n-dimensional pyramid (starting with a 2D triangle or a 3D pyramid) is m^n/n , where m is the length of one side of the base.