Easy Question - Simple Interest, Simple Discount

Looking over a text, it asks to prove that a simple discount rate $\displaystyle d$ is equivalent to a simple interest rate $\displaystyle r$ where $\displaystyle r = \frac{d}{1-dt}$. The solution in the book does this by setting total interest $\displaystyle I = Prt$ (where $\displaystyle P$ is the principal, and $\displaystyle t$ time) equal to the discount $\displaystyle D = Sdt$ (where $\displaystyle S$ is the amount of the loan, and $\displaystyle d$ the discount rate) and set the principal of the interest equal to the proceeds of the loan, both represented by $\displaystyle P$. They then solve for $\displaystyle r$.

With those assumptions in place, I totally get the math. But what I guess I don't understand about the problem is, why would we set these equal to each other? In what sense are we showing that the two are equivalent? Since the principal of the investment is what you start with before accruing any interest, I would think that we would equate this with $\displaystyle S$, the total loan amount before interest discounts.