Why does the hash come before the signature? The first figure in the original paper of Satoshi is a diagram of transactions.  This diagram seems to indicate that the transaction data is hashed before the sender's signature is appended.  Why not just hash the entire transaction, including the signature, to produce all-encompassing transaction hash?

Update:  I want to highlight the comment by @David Schwartz on the accepted answer.
Because it would not have a function. The hash listed in the picture of the original whitepaper is just to make the signature shorter (and possibly the process of creating it faster) without losing security, its only function is the authentication of a transaction, it is not used for anything else (or even stored).

To clarify, this is the diagram mentioned in the question: