Dictionary.com on subject:
9. Grammar (in English and many other languages) a syntactic unit that functions as one of the two main constituents of a simple sentence, the other being the predicate, and that consists of a noun, noun phrase, or noun substitute which often refers to the one performing the action or being in the state expressed by the predicate, as He in He gave notice.
8. Grammar (in many languages, as English) a syntactic unit that functions as one of the two main constituents of a simple sentence, the other being the subject, and that consists of a verb, which in English may agree with the subject in number, and of all the words governed by the verb or modifying it, the whole often expressing the action performed by or the state attributed to the subject, as is here in Larry is here.
The subject, then is defined as the subject noun (e.g. man) along with the words which modify the noun (e.g. the fat blind). The predicate is the verb (e.g. ran), direct object (none in this sentence), and any modifying adverbs or prepositional phrases (e.g. from the dog):
Subject = The fat blind man
Predicate = ran from the dog.