[ 749 ] 
In the twelfth chapter of book fecond, he begins 
to treat of the fedions of the harmonic canon, for 
proving, by experiment, the truth of the ratios he 
had delivered for the genera, and propofes to exhibit 
thefe fedions for each genus, in all the feven tones. 
Then, after (hewing the defeats of the monochord 
canon in ufe, and (chapter thirteenth) the infufficiency 
of the improvements made thereon by Didymus, he 
propofes, and explains, his own method of applying 
the canon to eight firings tuned in unifon. But, be- 
fore he proceeds to give the fedions upon all the 
tones, he thinks it necelfary to fhew, by experiment, 
the advantage of his own numbers, above thofe of 
preceding harmonicians ; for which purpofe, the ex- 
pofition of them in one tone, that is, in the immu- 
table fyftem, was fufficient. Accordingly, in chapter 
fourteenth, (the greateft part of which has been loft, 
and is fupplied from conjecture by Wallis) and in the 
tables annexed, he gives the fedions of the Dorian 
fpecies of diapafon ; where the diazeudic tone, lying 
in the middle, left a complete tetrachord on each 
fide, for the expofition of the generic numbers ; and 
compares his own ratios with thofe of Archytas, 
Eratofthenes, Ariftoxenus, and Didymus. Then, in 
the fifteenth chapter, he proceeds, as he propofed, to 
give the fedions according to his own ratios, for the 
genera moft in ufe in all the feven tones : thefe fec- 
tions he has included in two fets of tables, each fet 
containing feven, viz. one for each mode; and each 
table being again fubdivided into five, for the generic 
differences. Now, of thefe two fets of tables, the 
firft juftifies the diagram; for here he has given his 
numbers upon the eight firings, from nete diezeug- 
5 D 2 menon 
