Longinus was Greek, not Roman. He is dated sometime between 1st and 3rd century AD, and probably lived in the eastern part of the Roman empire. There is an onging scholarly controversy about whether he can be identified with Cassius Longius, the philologist and teacher of Porphyry. See Malcolm Heath's work for a good analysis of the case for identity.
There is also a good case to be made that Longinus was involved in a very active 3rd century debate in the Platonic schools of the Greek East concerning the proper style for philosophical discourse -- Porphyry's Life of Plotinus and Eunapius are good evidence for the debate.
Longinus was an otherwise unknown first-century Roman literary critic, otherwise unknown because no other works of his are extant and because his correct name is not known. On the Sublime is a treatise that explores the emotive element of poetry and the human condition, particularly, the condition of sublimity of the human soul along with its product, poetry.
A lost essay by Caecilius omitted to discus the element of the sublime in poetry. Apparently, one of Longinus' students named Postumius was taken by Caecilius' essay and was challenging Longinus based on it. In response to Caecilius' essay and by way of confrontation with Postumius, Longinus wrote On the Sublime.
The sublime is defined as the elevation of the soul, and elevation is defined as loftiness, grandeur, dignity or nobleness (Random House Dictionary on Dictionary.com). The sublime--elevation of the soul by loftiness, nobleness, dignity and grandeur--is achieved through the five sources of the sublime. These are:
- grandeur of thought,
- inspired passion,
- effective use of stylistic figures,
- noble diction and phrasing, and
- elevated composition.
So, in part, Longinus' purpose in writing On the Sublime was to correct a wrong view of poetry that was attracting at least one of his students. Nonetheless, his greater purpose was to treat in a cogent manner the importance of grand thoughts and feelings and language and structure in poetry.
In addition, his foundational purpose was to state from where the sublime in poetry comes. Longinus asserts that great poetic writing stems from well-honed rhetorical skill combined with the cultivation of the poet's own sublimity (elevation) of soul. Further, the great poet is possessed of an inherent capacity for inspired passion and grandeur and for elevation of thought.