Thus it is evident that the question of empire was uppermost in Athens during the year 417-6, even though it is impossible to say whether the renewed activity of the Chalcidians and their allies,31 or a reviving imperialism, was in the first instance responsible for the new interest in imperial questions. The extent to which Athens was thinking in terms of empire is apparent from the Melian debate where Thucydides discusses pro and con the probable results on the subjects of Athens of continued Melian neutrality; 32 and the advance in Athenian policy from mere restoration of empire to aggressive imperialism gives color to our conclusion that in matters of taxation the moderation of the assessment of 421 was then abandoned for exploitation such as had characterized the aggressive imperialism under Cleon when the tribute was first doubled. Nor can there be any doubt that the new policy was the result of a reaction in public opinion favoring the chauvinist Alcibiades to the detriment of Nicias under whom the tribute had been lowered in 421. The failure of Nicias in the north made it easier for Alcibiades to assume control of affairs, to persuade Athens to enslave Melos,33 and to double the tribute, thereby giving a tone to Athenian imperial policy that it had lacked since the death of Cleon. Even without the evidence of ancient authors, the responsibility of Alcibiades for the Melian affair can be taken for granted, since he was general during this year and since the action against Melos was what one might expect from this successor of Cleon, and we are probably warranted in concluding too that Thucydides in the Melian debate gives a summary of the arguments used by Alcibiades in his advocacy of the new aggressive policy.34

Whether the doubling of the tribute came as a precursor to the attack upon Melos, or whether it resulted from the

3 Thục. v, 80, 82.

32 Id. v, 91-99.

33 (Andoc.) 4, 22; Plut. Alcib. 16.

3 Thuc. v, 91–99.

refusal of Melos to submit, when a long and expensive siege was in prospect, it is now impossible to say, nor is the matter of the exact date within the year particularly important since the doubled tribute is such a natural correlative of the blossoming chauvinism displayed in the Melian affair and of the fullblown variety of the next year which under the evil influence of Alcibiades led Athens on to Sicily.

Thus every indication points to 417-6 as the year when the tribute was raised. The situation in Athens at that time, even though it may not have required a doubling of tribute, makes it necessary for us to assume that it was then doubled, and the several accounts of the affair agree both with themselves and with the facts as we have presented them. The tribute was doubled; it now amounted to between 1200 and 1300 talents, nearly three times as much as the tribute of Aristides; the assessment was sponsored by the demagogue Alcibiades during the period of peace that followed the Treaty of Nicias; and it was not the first increase since the death of Pericles.

Such is the literary tradition, and it is supported by the evidence of quota and assessment lists so far as they can be precisely dated, for we know that a tribute of more than 1200 talents in 417 would have required doubling of the rate in force after 421. But beyond that point the problem becomes more difficult, for the assessment lists are very fragmentary and the later quota lists almost all undated, and undatable too except by internal evidence.

In dealing with these stones most epigraphists and historians have committed a fundamental error in method. They have taken the assessment of 425 as the ultimate fact, and they have tried to relate to it everything, whether of a literary or of an epigraphical nature. We have seen what they did with Plutarch and the orators. They have done equal violence to the stones. At first after the discovery of an assessment list which could be dated, that of 425, there was a tendency among

scholars to assign to it all fragments of assessment lists. Thus two assessment lists were jumbled together in one document, and even though the fact of two assessments is now recognized, the allocation of fragments has been made carelessly.35 But there were four assessments within about ten years, and consequently many of the uncertain fragments may be, and one of them certainly is, from a third list. For example, I. G. 11, 543,36 an important fragment which has caused much trouble, does not necessarily belong to 425 because it can not be made to go with the list of 421.

Nor can the quota lists be used either to discredit or to corroborate the literary evidence as to the assessment of 417, for the later ones are so fragmentary that they offer almost no means of comparison. Only one of them can possibly fall within the assessment period of 417-3, and the date of that is uncertain.37

Consequently it would be rash to assert that our conclusion is false merely because no evidence for it is found in inscriptions. Correct historical method requires us to use the evidence of our literary sources as a means of dating our inscriptions, where other means are lacking, not to use the negative

35 The fragments are now divided between I. G. 12, 63, the list of 425, and I. G. 12, 64, the list of 421. I. G. 11, 543 can not go with the latter because of overlapping. See West and Meritt, A. J. A. xxix, 59 n. 1, Wilhelm, Anz. Wien. Akad. 1909, 48 f., 52 f. Moreover, fragments t-w' are not a part of I. G. 12, 64. See West and Meritt, A. J. A. xxx (1926), 149. If I. G. 11, 272 g is a part of I. G. 12, 63, fragments t-w' are from some nearly contemporary list or vice versa, for names are duplicated.

36 See West, A. J. A. xxix, 148-150. But a word of caution is necessary about this fragment. The restoration [μελλεσποντίο φόρο] κεφ[άλαιον] is not absolutely certain, for the order of districts did not remain the same from one assessment period to another. Consequently the district total partially preserved in this fragment may not be Hellespontine. But the allocation of the various fragments assigned to assessment lists must be reserved for further study.

37 I. G. 12, 219. For the dates of the late fragmentary quota lists, see West and Meritt, "A Revision of Athenian Tribute Lists, Part II," Harv. Stud. XXXVIII, (1927).

evidence of the inscriptions to force the literary evidence into a mould for which it was never intended.

In conclusion, Plutarch, Andocides, and a fourth century rhetorician, in their references to the increased tribute levied by Athens, agree with themselves, with the facts so far as they are known, and with historical probability. The final increase of which they write is the work of Alcibiades and the assessment of 417.

V.-The Harmonics of Ptolemy and the Lacuna in II, 14

JAMES FREDERICK MOUNTFORD

CORNELL UNIVERSITY

I. The value of Ptolemy's Harmonics; II. The manuscripts of the Harmonics and the revision of Nicephorus Gregoras; III. The lacuna in II, 14, and the material available for filling it; IV. The supplement of Qt, its value, and source.1

I. Claudius Ptolemaeus, mathematician, astronomer, and geographer, of the second century A.D., whose name is especially associated with the geocentric theory of the universe, has left us a work in three books entitled Tà 'Apμoviká. The science of harmonics he himself defined in its broadest terms as δύναμις καταληπτικὴ τῶν ἐν τοῖς ψόφοις περὶ τὸ ὀξὺ καὶ τὸ βαρὺ διαφορῶν; in effect, his work, like the 'Αρμονικά Στοιχεία of Aristoxenus, is a discussion of musical intervals, with special reference to their exact dimensions, their classification into concords and discords, and their various combinations in musical systems and scales.

From the point of view of the student of acoustics, a musical interval may be accurately represented by two numbers which are determined by the rates at which the instruments producing the two constituent notes vibrate. It is, therefore, in perfect accordance with the laws of physics to express an octave by the ratio 2:1, a Perfect Fifth by the ratio 3: 2, and a Perfect Fourth by the ratio 4:3. Since the rates at which two strings vibrate are in inverse proportion to the lengths of the strings (if other factors remain constant), it is equally legitimate to express a musical interval, as the Greeks commonly did, by a comparison of the lengths of two strings,

1 The material on which this paper is based could not have been collected without aid from the Moray Endowment in the University of Edinburgh, and from the Heckscher Research Foundation at Cornell University. To these sources I am indebted for the opportunity of inspecting many Ptolemy MSS. in Europe, and for the purchase of rotographs.