The study of past societies through their written records is:

The study of past societies through their written records is best described as historiography. Historiography encompasses the methods, principles, and practice of writing and interpreting history, fundamentally relying on the analysis of written documents - chronicles, letters, official records, inscriptions, treaties, and other textual sources - to reconstruct and understand past societies.

The other options each describe different, more specialised fields:

• Archaeology studies past societies through their material remains - artefacts, ruins, bones, and other physical evidence - rather than through written records. It is particularly important for prehistoric periods before writing existed.
• Palaeography is the study of historical handwriting and scripts. It focuses specifically on deciphering and dating old manuscripts and inscriptions based on the forms and styles of the writing itself, rather than on interpreting the broader content for historical understanding.
• Philology is the study of language in written historical sources, focusing on the literary, linguistic, and textual analysis of texts. While it engages with written records, its concern is with the language and its development rather than with reconstructing past societies.

Historiography is the broadest and most accurate answer because it describes the discipline that uses written records as its primary evidence base to study and interpret past societies as a whole.