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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1623