I. The Four Divisions.
a. Fragment of Archytus, Pythagorean, friend and contemporary of Plato
b. Pythagorean Quadrivivium: arithmetic, geometry, sphaeric, and music
c. Plato's ordering based on dimension, included Stereometry
d. Theon of Smyrna and Proclus, offer different order-- AMGS
1.A concerned with number, M with number in relation to something else
2. G with magnitude at rest, S with magnitude in motion.
Aristotle on the Exactness of the different divisions
II. Aristotelian Reclassification
1. adds optics and mechanics
2. All other divisions are derived from arithmetic and geometry
3. classifies into two categories: physical (applied) and pure (theoretical)
4. The physical (applied) branches require the pure for their proof
(Aris. Physics, ii.2, 194 a 8)
III. Arithmetic
A. Pure -- Theory of Numbers
1. treated only in connection w/ the geometrical form of proof,
a. dots marking squares, triangles, gnomons, etc. (Pythagoreans) b. straight lines (Euclid).
B. Physical applications
1. Plato
a. arithmetic as a function of logic
b. numbers as expressions of worth
2. logistic
a. addition, subtraction, division, multiplication
b. applying to real life situations
c. simple equations, one unknown, or 2 simultaneous w/ 2 unknown
d. Plato: art of calculation/thought
3. harmonics (Aristotle)
IV. Geometry
A. Pure -- Theory
B. Physical applications
1. Geodesy -- practical measurement of surfaces and volumes (mensurnation)
2. Optics -- Geminus divides into 3 categories
a. Why placement and distance affect appearance of size & shape
b.Catoprica -- theory of mirrors
c. scene painting, i.e. applied perspective.
3. Mechanics, stereo also (Aris), Geminus-- 3 categories
a. theoretical -- centers of gravity, equilibrium, etc.
b. Phys apps: making weapons and "wonderful machines" (theater)
c. Phys app: imitation of movements of heavenly bodies
a. Archimedes said to have made such a sphere
4. Astronomy, Geminus divided into 2 categories
a. Plato
1. theoretical
2. phaenomena
b. Geminus
1. art of the gnomon, measuring time by sundials
2. measuring heights of stars crossing the meridian
3. determining positions of sun, moon, and stars