[ 723 J 
Phrygian; rife a fifth to c# , for the Lydian; and 
fall a fourth to G#, for the Hypolydian. 
By this method, the pofitions of the feven modes 
come out exadtly, as I fixed them from Bacchius, in 
explaining the harmonic dodtrine ; and we fee, that, 
for fettling them, Ptolemey has really recourfe to no 
other theory of the modes, than that admitted by thole 
he contends with, though he makes the fpecies of the 
confonances, and thofe of the diapafon more particu- 
larly, the governing rule for fixing their pofitions, as 
the only means, by which the two dodtrines could be 
made to coincide. But it remained, after thus fettling 
the feven modes, to fhew more fully the confequence 
of following the method of the Ariftoxenians, and 
others, who divided the tonic fpaces found by his 
method, and placed the modes in a femitonic fuccef- 
fion, by which their number had been raifed to thir- 
teen, even within the compafs of the diapafon ; and, 
in doing this, we fhall find he ventures to affign the 
true reafon for his reduction, which was grounded 
on the mufical dodtrine. This argument, which is 
contained in the eleventh chapter of his fecond book, 
being very remarkable, and feeming ftrongly to fup- 
port the combination of the two dodtrines in the dia- 
gram I have given of the feven modes, I fhall give a 
tranflation of the whole chapter, left I fhould be 
thought to ftrain his arguments in favour of the mu- 
fical dodtrine, which has been thought by many to 
have little or no relation to the modes, and which, 
if we except what this author has delivered, feems 
indeed, upon a flight examination, and comparifon 
of the evidence, to have the weaker fupport. 
Vol.LI. 5 A Now, 
