C 751 ] 
chord hyperbolseon, would have fo little fpace left 
between the magas and the extremity of the firing, 
that it would be difficult for them to yield a diftindt 
found. And fecondly, The canon mud be crowded 
with thefe additional divifions, which would alfo 
have its inconvenience. To remedy this, he propofes 
feveral methods ; the firft of which is, to fit the in- 
ftrument with fifteen firings, of which the eight, that 
were to receive the divifions from mefe to nete hyper- 
bolason, fhould be in unifon, at the pitch of mefe, 
and the other feven in unifon, at the pitch of prof- 
lambanomenos. Now, as the fucceffion from the 
firing proflambanomenos to the firing mefe, though 
it varied with the modes, was yet, in the fame mode, 
always the fame with that from mefe to nete hyper- 
bolseon, the difference of a diapafon in pitch excepted, 
it is plain, that, under this adjuftment, a canon di- 
vided for the one diapafon would ferve equally for 
the other, the difference in pitch being eftablifhed 
before in the open firings. By this method, there- 
fore, the divifions for the acuter diapafon would be 
as large as thofe for the graver, and the canon needed 
not to be crowded with a greater number of divifions, 
than had been ufed for the fingle diapafon, by reafon 
of its double application. The other methods, which 
he propofed, I need not go through the explanation of, 
farther than to remark, that, for thofe, as well as for 
this, a freffi fet of numbers was neceffary, thofe con- 
trived for the fucceffion of the diapafon, from hypate 
mefon to nete diezeugmenon, being no-ways appli- 
cable to a fucceffion, which began from proflamba- 
nomenos or mefe. And this, it feems, was his reafon 
for giving the fecond fet of tables in his fifteenth 
chapter, 
