Lecture 2.1: Early Mathematics

Early Near Eastern Developments
 
Egyptian mathematics relatively unimportant
Egyptians developed decimal counting system: Rhind Mathematical Papyrus 1650BCE
    3000 BC like Roman tallies
    1800 BC symbolic ciphers
    arithmetic is additive and uses unit fractions (very practical)
        18 divided by 3

        1    3
        2    6
        4    12
        2+4    18
-geometry - areas and volumes of simple figures
    areas of triangles; volumes of pyramids (1/3 bases area x height)


Mesopotamian Mathematics
    old Babylonian period: Hammurabi 1800 BCE; Seleucid period 300 BCE
    by 2000 BC decimal and sexagesimal system
    used place system for powers
    fraction calculation easy with sexagesimal system
    tables of reciprocals: A/B = 1; tables of squares, square roots, cubes, cube roots
    Pythagorean theorem known in old Babylonian times; tables of Pythagorean numbers
    “algebaic problems” - given product and sum of two numbers, what are the numbers; often very complex quadratic equations; virtually algebraic, without the explicit abstraction of letters
    geometry is relatively absent and is treated as just another entity in equations

Greek Mathematics

herodianic system of Greek number notation
    a system of abbreviations (like Roman numerals)
alphabetic numeral, based on the Phoenician alphabet
    first evidence from around 450 BC
    developed probably in Ionia
Greeks used multiplication tables

fractions
    sub-multiples e.g. 1/4, 1/3, etc.

Pythagoras (~572-497 BCE): guru, no writings, legendary figure
    esoteric school (pledge of secrecy), political leader
    sources obscure: Aristotle’s ‘the Pythagoreans’ or ‘the so-called Pythagoreans’

relationship bt math and the universe
    numbering the constellation (the number of stars and the shape of the figures)
    all things are numbers
    harmonic interval: octave 1:2; fifth 2:3; fourth 3:4 – tetraktys

    one is not a number, it is the denominator, the measure

Classification of numbers: odd and even
    even times even; odd times odd, etc.
    such search for factors implies interest in prime numbers (rectilinear numbers)

Perfect and Friendly numbers
    perfect number: equal to the sum of its factors e.g. 6 = 3+2+1; 28; 496
    Euclid shows how to derive them:  2n (2 n-1) where 2n is prime
    friendly number have aliquot parts (factors plus 1) equaling one another: e.g. 284 and 220

gnomons and the discovery of the irrational
    leads to the discovery of the progression of squares
        2n+1
        x2 + y2 = z2
    gnomons are picked up by Euclid’s elements ‘algebra’ in Elem II
    investigation of squares leads to right angle triangles: 3,4,5


Arithmetic, geometric and harmonic means
    arithmetic:  a + c
            2

    geometric:   square root of ac i.e  a:b::b:c

    harmonic: a/c = a-b/b-c,  e.g. 12,8,6

doubling of the square: used by Plato: recollection and reincarnation

Irrationality of root 2

    2aSQUARED = bSQUARED
    let a and b be expressed in their lowest terms
    therefore a must be odd, b must be even

    but if b is even, b = 2g
    and bSQUARED = 4c2SQUARED
    and 2aSQUARED = 4cSQUARED
    so, aSQUARED = 2cSQUARED
    so a is even, but a was odd
    therefore a and b cannot be expressed as integers
    
Pythagorean geometry

2R theorem from parallel lines
interior angles of polygons

*Pythagorean theorem (GS 21-2; from Proclus’ report, Euclid’s proof is not Pythagorean)
    no trustworthy evidence that it was discovered by him
YOU ARE RESPONSIBLE FOR KNOWING THE ‘SQUARE WITHIN A SQUARE’ PROOF AND THE EUCLIDEAN PROOF FROM GS 21-2)
may have been proved on the basis of Elements II.4: (a+b)SQUARED = aSQUARED + 2ab + bSQUARED