PHYSICAL SUBJECTS AND THEIR BRANCHES 17 
mgular num- 
think, be no 
i to Logistic. 
Arithmetica 1 
the Diophan- 
ams, had pre- 
lumbers of 
>blems (V. 30) 
,sures of wine 
)ably saw that 
ould refer to 
n another, but 
)rm of finding 
!S, alone or in 
jim to be part 
numbers, 
jet ween arith- 
3 the time of 
With rare 
sieve, a device 
srs, the theory 
geometry, and 
>roof was used, 
ing out squares, 
igoreans), or of 
licomachus did 
from his work, 
ibers, of which 
roof is used. 
•ated out from 
\esy, or, as we 
measuring, but 
of surfaces and 
as well as from 
(y) Physical subjects, mechanics, optics, harmonics, 
astronomy, and their branches. 
In applied mathematics Aristotle recognizes optics and 
mechanics in addition to astronomy and harmonics. He calls 
optics, harmonics, and astronomy the more physical (branches) 
of mathematics, 1 and observes that these subjects and mechanics 
depend for the proofs of their propositions upon the pure 
mathematical subjects, optics on geometry, mechanics on 
geometry or stereometry, and harmonics on arithmetic; simi 
larly, he says, Phaenomena (that is, observational astronomy) 
depend on (theoretical) astronomy. 2 
The most elaborate classification of mathematics is that given 
by Geminus. 3 After arithmetic and geometry, which treat of 
non-sensibles, or objects of pure thought, come the branches 
which are concerned with sensible objects, and these are six 
in number, namely mechanics, astronomy, optics, geodesy, 
canonic {KavoviKT]), logistic. Anatolius distinguishes the same 
subjects but gives them in the order logistic, geodesy, optics, 
canonic, mechanics, astronomy. 4 Logistic has already been 
discussed. Geodesy too has been described as mensuration, 
the practical measurement of surfaces and volumes; as 
Geminus says, it is the function of geodesy to measure, not 
a cylinder or a cone (as such), but heaps as cones, and tanks 
or pits as cylinders. 5 Canonic is the theory of the musical 
intervals as expounded in works like Euclid’s KaraTo/xy 
kolvovos, Division of the canon. 
Optics is divided by Geminus into three branches. 0 (1) The 
first is Optics proper, the business of which is to explain why 
things appear to be of different sizes or different shapes 
according to the way in which they are placed and the 
distances at which they are seen. Euclid’s Optics consists 
mainly of propositions of this kind; a circle seen edge 
wise looks like a straight line (Prop. 22), a cylinder seen by 
one eye appears less than half a cylinder (Prop. 28); if the 
line joining the eye to the centre of a circle is perpendicular 
1 Arist. Phys. ii. 2, 194 a 8. 
2 Arist. Anal. Post. i. 9, 76 a 22-5 ; i. 18, 78 b 35-9. 
3 Proclus on Eucl. I, p. 88. 8-12. 
4 See Heron, ed. Hultsch, p. 278; ed. Heiberg, iv, p. 164. 
6 Proclus on Eucl. I, p. 39. 23-5. 6 lb., p. 40. 13-22. 
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