According to the theory which appears now to prevail among students of Greek music, these famous names had a double application. There was a Dorian mode as well as a Dorian key, a Phrygian mode and a Phrygian key, and so on. This is the view set forth by Boeckh in the treatise which may be said to have laid the foundations of our knowledge of Greek music (_De Metris Pindari_, lib. III. cc. vii-xii). It is expounded, along with much subsidiary speculation, in the successive volumes which we owe to the fertile pen of Westphal; and it has been adopted in the learned and excellent _Histoire et Théorie de la Musique de l'Antiquité_ of M. Gevaert. According to these high authorities the Greeks had a system of key ([Greek: tonoi]), and also a system of modes ([Greek: harmoniai]), the former being based solely upon difference of pitch, the latter upon the 'form' or species ([Greek: eidos]) of the octave scale, that is to say, upon the order of the intervals which compose it. § 3. _The Authorities._ The sources of our knowledge are the various systematic treatises upon music which have come down to us from Greek antiquity, together with incidental references in other authors, chiefly poets and philosophers. Of the systematic or 'technical' writers the earliest and most important is Aristoxenus, a pupil of Aristotle. His treatise on _Harmonics_ ([Greek: harmonikê]) has reached us in a fragmentary condition, but may be supplemented to some extent from later works of the same school. Among the incidental notices of music the most considerable are the passages in the _Republic_ and the _Politics_ already referred to. To these we have to add a few other references in Plato and Aristotle; a long fragment from the Platonic philosopher Heraclides Ponticus, containing some interesting quotations from earlier poets; a number of detached observations collected in the nineteenth section of the Aristotelian _Problems_; and one or two notices preserved in lexicographical works, such as the _Onomasticon_ of Pollux.