How did latin philologists come to the conclusion that Cs in Latin are pronounced as Ks?

Answer 1

On the basis of the pronunciation of the voiceless velar 'k' in Old High German. The reconstructed pronunciation of ancient Latin and Greek relies on comparative evidence from within the Indo-European linguistic area, including mostly Sansckrit, Avestan (= old Persian), Greek, Latin, and Old High German. Grimm's law, for example, says that hard 'k' (= voiceless velar) in German corresponds to a palatal sound in Germanic languages like English (Kaiser ~ Caesar). The details of the reconstruction are given in W. Sidney Allen, Vox Latina (in English), Cambridge, 1965; see also http://www.plantilus.com/nomenclature/pronunciation.pdf.