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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