at F, and the angles subtended by BG and CD at F are 
equal. The rest of the treatise is of the same character, and 
it need not be further described. 
in the same volume is 
not by Euclid, but is a compilation made at a much later date, 
possibly by Theon of Alexandria, from ancient works on the 
subject and mainly no doubt from those of Archimedes and 
Heron. Theon 1 himself quotes a Catoptrica by Archimedes, 
and Olympiodorus 2 quotes Archimedes as having proved the 
fact which appears as an axiom in the Catoptrica now in 
question, namely that, if an object be placed just out of sight 
at the bottom of a vessel, it will become visible over the edg( 
when water is poured in. It is not even certain that Euclid 
wrote Catoptrica at all, since, if the treatise was Theon’s, 
Proclus may have assigned it to Euclid through inadvertence. 
(y) Music. 
Proclus attributes to Euclid a work on the Elements of 
Music (ai ката ¡jLova-LKrjv (TTOL\euiocreis) 3 ] so does Marinus. 4 
As a matter of fact, two musical treatises attributed to Euclid 
are still extant, the Bectio Ganonis (Катато/лг] Kavovos) and the 
Introductio harmonica (Elaaycoyrj dppovLKrj). The latter, 
however, is certainly not by Euclid, but by Cleonides, a pupil 
of Aristoxenus. The question remains, in what relation does 
the Bectio Ganonis stand to the ‘Elements’ mentioned by 
Proclus and Marinus 1 The Bectio gives the Pythagorean 
theory of music, but is altogether too partial and slight to 
deserve the title ‘ Elements of Music ’. Jan, the editor of the 
Musici Graeci, thought that the Bectio was a sort of summary 
account extracted from the ‘ Elements ’ by Euclid himself, 
which hardly seems likely; he maintained that it is the 
genuine work of Euclid on the grounds (1) that the style and 
diction and the form of the propositions agree well with what 
we find in Euclid’s Elements, and (2) that Porphyry in his 
commentary on Ptolemy’s Harmonica thrice quotes Euclid as 
the author of a Bectio Ganonis. 5 The latest editor, Menge,