Prime journal, not the first treatise on animals (8)
I believe the answer is:
bestiary
'treatise on animals' is the definition.
'prime journal not the first' is the wordplay.
'prime' becomes 'best'.
'journal' becomes 'diary' (diary is a kind of journal).
'not the first' means to remove the first letter.
'diary' with its initial letter removed is 'iary'.
'best'+'iary'='BESTIARY'
(Other definitions for bestiary that I've seen before include "Medieval book of animals for sybarite" , "Medieval collection of descriptions of animals" , "medieval treatise" , "Medieval collection of animal descriptions for sybarite" , "Allegorical work" .)