A History of Mathematics - Midair MacCormaic

5000 BC: Scales. Standard weights, Egypt.

4000 BC: Sundials - can count days, now count parts of a day
(12

hours). Egypt.

1800 BC: Sumerians and Babylonians. Base 60 number system.
Still

followed today with minutes and seconds. Why? Divided easily
by 2, 3,

4, 5, 6, 10, 12, 15, 30. No need for hard fractions. Also, 360

degrees in circle, similar to 365 days for sun to move around
the

Earth. Also, seven days a week for the seven planets (including
sun

and moon). NOTE B.

1500 BC: Alphabet, from sounds.

520 BC: Irrational Numbers. Pythagorus thought whole numbers
and

fractions were basis of universe. These are rational numbers

(expressed as ratios). Now consider the rigth triangele, each
side a

length of one unit. What is the length of the hypotenuse? From

Pythagorean therom it equals square root of 2. (7/5)*(7/5)= 2.04.

(707/500)*(707/500)=1.999, but there is *no* fraction that equals
the

square root of two. It is irrational. NOTE C

500 BC: Abacus (at least this old). First important computing

device. Counters or pepples in grove. Aztec "quipus"
has wires with

wood pieces. Japan is similar, but 5 and 2. PICTURE: Ball p124

350 BC: Logic (greek for "word"). Aristotle. Book
"Organon" developed

logic in great detail, describing the art of reasoning from premise
to

necessary conclusion, demonstrating how to establish the validity
of a

line of thought.

300 BC: Geometry. Practical study may have begun in Egypt
with

pyramid building and boundaries measurements needed from the Nile

flooding. Greeks made it theoretical. Dealt with ideal points,
curves,

planes, and solids. Proof by reason, not by mesurement. Reason
was

for philosophers, measurement was for the artisan. Greeks were
snobs.

Euclid compiled all geometrical findings of earlier mathematicians
in

a textbook called "Elements". Added little himself,
but what he did

add was very important. He started with axiums, statements so
self

evident they required no proof. Then he proceeded systematically
to

prove theorem after theorem. Each proof depending on ly on the
axums

and previous proofs, giving geometry a firm foundation. The textbook

still used in modified form today. NOTE D.

150 BC: Trigonometry. Need to work with angles to do work
in

astronomy (measure angle between two bodies). In a right triangle
and

fix the angles, then the sides have fix ratios, called sine, cosine,

and tangent. These are trigonometric functions. The Greek astronomer

Hipparchus made up carefuol tables relating angles to side ratios.

Used this (and paralax)to calculate distance from Earth to the
Moon.

sin(A+-B) and cos(A+-B). Also first to indicate positions by
latitude

and longitude.

250 AD: Algebra Greek Diophantus presented problems that had
to be

solved by what we would call algebra. First text. Mainly delt
with

whole numbers (Diophantine equations). Showed fractions could
be

treated as numbers. NOTE E.

810: Zero. Marks form numbers. Think of abacus. Around 500
AD Indian

mathematician suggested using zero as the placeholder. Arabs
got it

from hindus around 700. Muhammad ibg Arkhwarizmi first used it
as

position notaion in 810. He also coined the word 'algebra'. "Algorism"

First to recognize two roots for quadratic equations (ax2+bx=c),
but

only positive real roots. sqr(a)sqr(b)=sqr(ab). Geometric proofs.

Other Hindu had only indirect effect (ball, p 150).

1202 Arabic Numerals. Italian mathematician Fibonacci wrote
"Book of

the Abacus" ("Liber Abaci", introduced arabic numerals
to

Europe. Break, NOTE F.

1436 Perspective. Lines the come together as they do in real
life,

but on paintings. Italian Leon Alberti wrote book, described
method

for achieving perspective in mathematical manner. Forerunner
of

projective geometry.

1535: Cubic Equations. Equations of third degree. Italian

mathematician Niccolo Tartaglia found general method. Kept his

discovery secret, but another mathematician Geronimo Cardano published

it and usually gets credit. Set trend for scientific method, stating

that the first person who publishes gets credit. Really discovered
by

Scipoione del Farro in Bologna (check?).

1545: Also Cardano, in solution to cubic equations, saw some
roots to

be square root of negative numbers. Called these "sophistic"
and these

results were "as subtile as it was useles." Not worked
on until two

centuries later by Euler and Bernoulli. Gauss introduced complex

numbers and 'i'.

1545 Can't have less than nothing, but they knew about debts,
which

means having less than no money. Cardano showed that debts and
the

like can be treated as negative numbers which foollow the rules
of

math similar to ordinary numbers. All types (int, fractions,

irrational)can be negative.

1545: Cardano, general solution to quartic equations, stolen
from

Ferrari.

1551: Rhaticus (german), trig tables, based not on circle
arc legnth

but on the rations of the length of the sides of triangles to
each

other. Also found sin2th and sin3th in terms of sinth + costh.

1586: Stevinus: decimal notation (note A)

1589: Cryptanalysis. Spaid had 600 character cipher, changed

periodically, thought impossible to break. Henry IV gave problem
to

Francisus Vieta, who decoded it and French used it for two

years. Philip II complained to pope Sixtus V that French were
using

sorcery.

1591: Algebraic Symbols: used consonants b,c,d for known
quantities

and a,e,i for unknown. Use of a,b,c for known and x,y,z for variables

was introduced by Decartes in 1637. Vieta used A for x, A quadritics

for X2, and A cubus for x3, or their abriviations: Aq, Ac, Aqq,
etc.

3BAA-Da+AAA=Z is

B 3 in A quad - D plano in A + A cubo aequatus Z solido.

1596: PI: Dutch math Ludolf van Ceulen got pi to 20 decimals,
still

called Ludolf's number at times in Germany.

1614: Logarithms: Napier (scottish) published log table.
Nothing

better for computation for centuries, defined log of n as 10^n

loge. Tried to find base 10 logs, but died. Briggs did it in decimal

from 1 to 1000. Why good? multiplying is addition in logs.

1622: Slide rules, by William Oughtred, made log calculations

mechanical, not replaced until computers.

1637: Analytic Geometry. Descartes combined algebra and

geometry. Braw two perpendicular lines, mark intersection as 0
and

mark off units on each line, positive up and right, negative left
and

down. Every point in plane represented by two numbers. Can add
third

axis for every point in the universe. Can now define equations
by

f(x,y)=0 for any curve. This allows geometric problems to be solved

algebraically, and algebraic problems to be illustrated geometrically.

Also introduced the Method of Indivisibles (summing up small

rectangles).

1637: Fermats last theorem: x^n + y^n != z^n for n>2
and x,y,z

integers.

1642: Adding Machine. Pascal created one that could add and

subract. Patent in 1649, commercial failure.

1654: Probablility. Paxcal's triangle for coefficients of
(a+b)^n

for combinations of m choose n. Deals with large numbers of events.

1656: Willis, showed law of indicies. x^0 = 0, x^-1=1/x, etc...

1669: Calculus.

Newton: Fluxional or differential calculus. Whenever a quantity

changes according to some continuous law (as most natural things
do)

the dif calc allows us to measure its rate of increase or

decrease. Integral calc enabbles us to find the original quantitiy.

NOTE G (fluxons and fluents).

Liebnitz: dx and dy for smallest possible differences (differentials)

and int of y dx where int is an enlarged S for sums. Gaves rules
for

d(xy) and d(x/y). Proof: NOTE G.

Newton: Polar coordinates.

Liebnizs: Infinite series aren't just approximations, but real
*are*

the number or function (pi/4) = 1/1 - 1/3 + 1/5 - 1/7.

1693: Liebniz, calculating machine, multiply and divide.

1700: Liebniz, calculating machine, could multiply and divide.

1736: Euler. first to truly use algebra and calculus to discuss

Mechanics (even Newton used geometry). cos th + i sin th = e^(i
th).

Used 'e' and 'pi'. pi: Bernoilli used c, Euler used p, then
c, then

pi in 1742.

1742: Goldenback Conjecture: any even number equals of sum
of two

primes.

1744: Transcendental numbers, ones that aren't a soultion
for any

algebraic (power of x) equations (Ball p395).

1788: Lagrange: Analytical Mechanics, used *only* algebra
and

calculus to solve mechanics, no geometry at all.

1790: Metric System, during French Revolution, Laplace, Lagrange,
and

Lavasier meter = 1/(10,000,000) of diameter from N Pole to Equator.

1796: Heptadecagon. German mathematician Carl Freidrich Gause
worked

out a method for contructing this polygon built up of 17 sides
of

equal lengths using only a compass and stright edge. This is
the

first notable addition to geometry since ancient times. Also
showed

that only polygons of certain numbers of sides could be constructed
in

this manner - the first case of a geometric construct proved

impossible.

1799: Perturbation Theory. Laplace published first volume
of

Celestial Mechanics, showed that the small variations in the movement

of the planets (from other sourcers) are pareiodic and vary around

what would exist if the sun were the only gravity source. The
Solar

System is stable.

1822: Computers. Charles Babbage designed machines that would
work by

means of punch cards that would store answers, saving them later
for

additional operations, and that could print results. Could not
be done

using purely mechanical means.

1822: Projective Geometry. Frenchman Jen-Victor Pencelet published

book on the study of shadows cast by geometric figures - foundation
of

modern geometry.

1824: Quintic Equations. Since 1535 and 1545, mathemticians
wanted to

find solutions to fifth order equations. Norwegian mathematician

Niels Abel showed that a general algebraic solution of the quintic

equation was impossible. First impossibility in algebra.

1826: Non-Euclidean Geometry. Axiom - "Through a given
point, not on

a given line, one and only one line can be drawn parellel to the
given

line." Not every self evident, and mathematicians tried
to prove this

axium from the others, and failed. Italian Girolamo Saccheri
started

by supposing the axium was false, and tried to build a geometry
in the

hope of finding a contradiction, so he could conclude the axium
was

true. He couldn't find it, and wrote, in 1733 a book entitled
Euclid

Cleared of Every Flaw, where he claimed (falsely) to prove the
axium.

Later, Russian Nikolay Lobachevsky, just removed the axium all

together, and started with "Through a given point, not on
a given

line, any number of lines can be drawn parallel to a given line".
This

along with Euclids other axioms made a "non-Elclidean"
geometry -

still self consistent. Gauss figured this out in 1816, but never

published it.

1830: Group Theory. Frech Evariste Galois (died in a duel
on his 21st

birthday) generalized the work of Abel (quintic equations), showed

that no equation of any degree higher than the fourth could be
solved

algebraically by inventing a technique called group theory, which
is

usefull for working out quantum mechanics.

1837: Trisecting an Angle. Greeks established the principle
that

geometrical contructs must be carried through using only a straghtedge

and a compass. Three constructions couldn't be solved by them:

squaring the circle (equal areas), doubling the cube (volume),
and

trisecting the angle. French Pierre Wantsel proved that the last
two

were impossible using greek rules (the squaring circle was proved

impossible later).

1843: Quaternions. Irish William Hamilton showed that you
could

change the axioms in algebra, also, and still have a self consistent

system. Developed hypercomplex numbers that could be presented
as

points in three or more dimemtions by abandoning the AxB=BxA rule.

1843: Higher Analytic Geomtry. British Arthur Cayley worked
out an

analytic geomtry in three or more dimentions.

1847: Symbolic Logic. English George Bool applied a set of
symbols to

logical operatoins that resemble those of algebra and used

algebra-like manipulations, yielding logical results.

1854: Non-Euclidean Geometry. German Georg Reimann used an
axium

stating that it was impossible for any two lines to be parallel
and

all lines intersected, with lines of finite length. A triangle's

angles: Euclidean - 180, Lobachevskian - < 180, Riemannian
- > 180.

Reimann is a surface of a sphere. Also generalize geomtry to
consider

it in any number of dimensions.

1865: Topology. German August Mobius preseting a long flat
strip of

paper that is given a half twist and the two ends are connect
to give

a circular figure (Mobius strip). It has one edge and one side.
The

founder of topology, which deals with propoerties of figures that
are

not altered by deformations (aside from tearing and puncturing).

1873: Transcendental Numbers. An algebraic number is one
that can be

used as a solution to a polynomial equation made up of powers
of x

(intergers, fractions, and some irrational numbers serve this

purpose). The ones that can't are called transcendetal (latin
- to

blimb beyond). French Charles Hermite proved that e (2.71828)
is

transcendetal, the first to be identified.

1874: Transfinite Numbers. German Georg Canton used one-to-one

correspondence to show that all fractions are denumerable and
can be

counted by integers. Real numbers cannot be, therefore, the group
of

real numbers represents a higher infinity, a transfinite numbers,
and

are nondenumerable.

1880: Electromechanical Calculator. American Herman Hollerith
used

punch cards to create a cencus calculating device using not just

mechanical but electrical switches. His company became IBM.

1881: Venn Diagram. British John Venn extended Boole's work
by

representing logical states as intersecting circles, a gemetric
logic

in comparison to Boole's algebraic logic.

1882: PI as Transcendental. German Ferdinand von Lindemann
showed

that pi was transcendtal, which means that it is impossible to
square

the circle using a finite number of steps.

1899: Logic and Geometry. German David Hilbert proposed the
most

satisfactory set of self consistent axiums to date by describing
the

properties points, lines, and planes possessed, as well as

relationships such as between, parallel, and continuous.

1902: Logic and Math. German Gottlob Frege extented symbolic
logic,

working over 20 years to make a symbolic logic that would be the
basis

of all mathematics. But a friend (Betrand Russell) discovered
a

self-contradiction in Frege's system, so he felt his project was

worthless.

1910: Logic and Math. Russel and British Alfred Whitehead wrote

Principia Mathematica, another effort to establish mathematics
as a

branch of logic, building it out of basic processes and definitions.

1928: Game Theory. Hungarian-born American John von Neumann
developed

a new branch of mathematics which dealt with strategies to follow
when

playing fixed rule games.

1930: Computer. American engineer Vannevar Bush produced the
first

machine capable of solving diferential equations. Only partly

electronic.

1931: Godel's Proof. Austrian Kurt Godel put an end to schemes
of

placing all mathematics on a formal logical basis and fully rigorous.

He showed that if you began with any set of axioms, there would
always

be statements within the system governed by those axioms that
could be

neither proved nor disproved on the basis of those axioms. Modifying

the axioms would result in a different statement with the same

feature. Ended the search for certainty in mathematics, just
as

Heisenberg did in physics.

*Last Updated: **28 June 98
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