History of Mathematics

;-
"All men by nature,
history
tllroughtout
desire to know."l
the need to know has been a prime source of
I
governing
mens actions.
izations,
it has started wars, and it has led man to
his ultimate
the causes
control
This need has founded civil-
of his environment 1
and developments
of mathematics.
with early Egypt and Babylon,
and finally
the 17th century
to Babylon,
through modern
in the development
where ancient
methods
dynasty,
around 1700 B. C ..
nor Rome advanced
arithmetic
developed
existed
Greece,
times; I
of mathematics.
of mathematics
land numeration,
geometry
Starting
then on to classical
will trace the need and development
"Priority
I shall examine
belongs
algebra,
and
at least from the Hammurabi
beyond
although
,,2
the level of elementary
and mensuration."]
mostly practical
"Neither Egypt
These ancient civilizations
mathematics,
even upon modern mathematics,
practical
but thier effect,
is still enormous.
shall cover both the need and the methods
I
used in this
ancient mathematics.
"There is little doubt that mathematics
necessity.
example,
The annual flooding
arose from
of the Nile valley,
forced the ancient Egyptions
to develop
for
some
1 Aristotle
2 "Mathematics, History",
volume 11,page 642
J Ibid
Encyclopedia
Britannica,
system of reastablishing land boundries.,,4 Increased
barter increased the need for early practical arithmatic.5
The need for a calender, if a basic one, led to development in mathematics; "'theastronomy of the old Babylonian
period was just adequate for maintaining the calender,
on which the irragation system supporting the civilization
depended.,,6 Civilization and mathematics are inseperable
i. ~. "Mathematics beyond primitive counting originated
wi*n the evolution of advanced form ..:-t)f
society. 7
It
As
Aris-t.Q)tle
once pointed out; a civilization is necessary
"to; Sl'eperatea thinking class from the working class.
Early mathematios~consisted
trfual and error.
almost exclusively of
Early Egyptian mathematics was geometry.8
The Egyptians also developed formulas for the areas and
volumes of many shapes, but used trial and error rather
than proofs, so they were not entirely correct in their
formulas. 9
The Babyloneans were only more advanced than
the Egyptians.
"The Babylonians were interested in
number relations beyond the merely practical mathematics. ,,10
i. e. "An old Babylonian text (1700 B. C. ) investigates
4 "Mathematics", Encyclopedia Americana, volume 17,
page 392
5 Ibid
6 Michal Moffatt, The ages of Mathematics vol. 1,
Page 35
7 "Mathematics",Encyclopedia Americana, volume 17,
Page 388
8 The word geometry is from a ~reek word meaning
_."measure of the land".
9 Michal
Moffatt, The ages of Mathematics vol. 1,
page 43
10 Ibid
3
triples of Pythagoran
that is, numbers
satisfying
a2 + b2 :::c2 .,,11 "The facility of
the relationship
numerical
numbers;
computation,
as the result of the place value
notation, is everywhere visible in Babylonian mathematics."l~
The Babylonians had a sort of combination between base 10
and base 60 with a zero.13
The acomplishments of ancient Greece dwarfed those
of Egypt and Babylon,
and rival the achievements
17th and 18th centuries!
It was during this period
that such great men as Pythagoras,
lived.
"The Greeks insisted
must be established
deductive
Euclid, and Archimedes
that mathematical
facts
not by empirical procedures
reasoning.,,14
terms, axioms, theorems
of the
but by
This led to a system of undefined
et cetera.15
Although most
\~
Greek mathematic;
"Purely algebraic notation
had been
\
11 "Mathematics, History",
volume 11, page 644
Encyclopedia
Britannica,
12 Ibid
13 Bases are a part of the place value notation,
that is the position of each digit tells its
magnit~de.
For ~xample, t~e number ~94071 means
3 x 10 ". 9 x 10 + 4 x 10 + 0 x 10 +7 x 10 +
1 x 1. If we were to write a number in another
base besides base 10 ~t would be like:2 13702
(base 8) equals 1 x 8_+ 3 x 83 + 7 x 8 + 0 x 8 +
2 ~ 1, of B3AI06 (base 12) = B (11) x 125 + 3 x
124 + A (10) x 123 + 1 x 122+
x 12 + 6 x 1.
All bases need the number of different charactors
equal to the number of the base (zero is one of the
digits).
~ithout zero a place value system cannot
exist, the number 57,302, for example, would have
to be written 5732, which could mean 500,732 or
5,732,000Qr just 5732 or any number of other things.
°
14 "Mathematics",
Page 393
15 Michal Moffatt,
page 67
Encyclopedia
Americana,
The ages of Mathematics
volume
17,
vol. 1,
used
by Aristotle
in his investigation
and liThe concept
of area
of integration17
even
liThe first
Miletus.1I19
from
the priests.
they
established
After
rise
in ancient
great
Thales
gave
thinker
rigorous
history
to Egypt
surpassing
and error
all their
logic
1116
theory
times. illS
went
He was soon
he had learned
to a near
in Greek
as a youth
by trial
of formal
was Thales
to learn
thier
and held
knowledge,
of
the methods
methods,
which
in mystic
regard.
he wil1t)back
to Greece
and set up a s c h 0 0 1 .
In his sc h 0 0 1 he set u pas
ci sit ion s (a x i 0 m s) and
d e r iv edt h in g s wit h d e due t iv e me..
t hod s .~ ll' rth
Thales
lived
from
e.
567 B.
to 497 B.
the seven
Wise men of Greece,
one due to politics.22
The next major
Greek
oras was a student
Pythagoras
women,
formed
learn
a deductive
a school
thereV
system
c.
mat~ematician
of Thales,
in Croton
who
~.-I.J1r
~!'
/jJp4'f'C,
if
/$
.
d ;d n 't bee 0 m ~,
was Pythagoras.
he went
Pythagto Egypt.
and let everyone,
furthered
building
even
the work
theorems
on
upon
theorems.
Pythagoras
made
five
propositions
lIP IIMathematics,
Historyll,
VOLUME 11, page 647
17 Integration
19 Michal Moffatt,
page 73
20 Leon
21 Ibid
22 Michal
~3
Perry,
Encyclopedia
is a function
18 IIMathematics,
History",
volume 11, page 647
Moffatt,
-?
The ages
he 'proved;
from
Britannica,
of calculus.
Encyclopedia
The ages
The Mathmen,
which
Britannica,
of Mathematics
page
vol 1,
17
of Mathematics
,);f; .
f p r,i-P,:-
\.._21/He was one of
He soon
by Tales,
0
- --
and like Thales
for free!23
started
~-
on 1 yon
the
e r ie s
vol 1, page
82
J-- /'
/,"//
5
known
'facts'.
1) The sum of the angles
angles (1800).
of a triangle
equals
two right
2) The sum of the exterior angles
four right angles (3600) .
of a triangle
3) The sum of the interior a~g~Jf
right angles, where n=the~umber
a polygon
of sides.
..----'"
equals
equals
2n-4
4) The sum of the exterior angles of any polygon equals
four right angles (3600), regardless
of the number of
sides.
5) Three reqular polygons - a triangle, a square, ana a
hexagon - fill the sRace about a point on a plane.
There
is some question
ever.
about
the validity
Not all of the 'facts'
Non - Euclidean
describes
geometry,
Einstienian
his 'facts'
he assumed
which
space~s
',~
.~
of these
to prove
is consistant
based
proofs,
how-
~
~L"
them are valed.
and actually
on the assumption
bette~.
that
~
'~J
are false(..---/Thisdiscussion .is essentially
the same)c'\)
as that of the parellel postulate25
which I shall discuss much
later
in thes paper.
Any discussion
theories
of Pythagoras
must
include
his remarkable
of numbers.
(ypythagOras
believed that all things - physical and
mental, all nature and all ideas - are built on a
pattern of integers.
Fractions he did not consider
numbers.
They were only rotios, relations between
numbers.
Having discovered
the figurate numbers,
24 George
Gamow,
One, Two, Three,
Infinity,
page 103
25 Pythagoras's
proposition
number one and Euclids'
parellel postulate can each be proven from the
other .
IN
~./
"
Pythagoeas thought that he proved that geometry itself was formaed on some sort of number pattern.
And
it followed 'logically', he asserted, that all material things in nature were formed in a si~~ler manner and that -numbers rule the universe.~j
Pythagoras
of number
numbers
discovered
prime and composite
that he discovered
are numbers
for example,
ered the perfect
and 8128.
numbers
3,550,336,
highest
of which
which
1500 years
before
Today
is over 1300 digits
are the sums of each others
factors
discovered
17 are known,
numbers
are the only friendly
numbers
num-
They are numbers
=
i. e. 284
of 284).
1.
2 +
of 220),
The above
that Pythagoras
discovered!
It was 2000 years
before
Euler alone
60 pairs and today over 400 pairs ore known.
Still another
found
group of numbers
One is the ancestor
are female.
of numbers,
Odd numbers
26 Leon Perry,
87 Ibid
another
496
the
'Friendly'
4 + 5 + 10 + 11 +20 + 22 + 44 + 55 + 110 (the factors
220 = 1 + 2 + 4 + 71 + 142 (the factors
discov-
the next perfect
long!27
of Pythagoras.
Six,
Pythagoras
6 and 28, his students
type
Perfect
the sum of their factors.
was discovared.
bers are also an invention
Another
numbers.
one plus two plusthree.
It was another
number,
are 'perfect'
that equal
equals
numbers.
pair was discoveree!
are male and female
not a number.
are male.
The Mathmen,
numbers~
Even numbers
Five is the marriage
page 32
Leonard
number
-7
because
it is the sum of the first
female
number
angular
They
(2 + 3~.
numbers,
corespond
tagDns
square
numbers
numbers,
pentoginal
of dots
Triangular
=
6, 6 + 4
to triangles
Square
cetera
triples
c will
which
3 on a side,
to squares
1.
squares,
pen-
This corresponds
of 2 wide,
3 wide,
is most
of the sides
4 wide
3 then 6 then 7 et
are similar.
Pythagorean
famous for. They are
a2 + b2 = c2, and a, b, and
the condition
be the lengths
et cetera.
4 on a side et cetera.
by adding
The other
Pythagoras
fulfill
in triangles,
15 et cetera.
can be obtained
are what
integers
=
10, 10 +5
to the number
numbers
into tri-
I~
correspond
which
are devided
numbers
of 2 on a side,
numbers
et cetera;
=
and the first
are obtained by ading
.~~
to the number 1 i. e. 1 + 2 = 3,
2 then 3 then 4 et cetera
3 + 3
number
Figurate
to patterns
et cetera.
male
of a right
triangle.
Pythag-
oras developed
a formula to figure out such numbers:
n2 + (~(n2 _ 1))2 = (~(n2 + 1))2
where
n is an odd integer.
II
One day Pythagoras
eble fact:
fractions!"28
fo~nd
there
were
integer
numbers
He was making
the hypotenuse
nor fraction
wha-, was to him an incred-
which
were
an icosolese
to have a length
for.
of
He, however,
irrational.
It was Hippasus,
proved
«to
be irrational.
Vb et
cetera
28 Leon
'\
discovered
right
J2:
could
integers
triangle
which
Later
Theodorus
The Mathmen,
page 40
proved
nor
and
he found
not prove
one of Pythagorases
to be i rra tiona 1 .
Perry,
neither
it was
students,
~
no
V5,
who
liThe famous
bution
Greek
to mathematics
he trained
his school,
the hexahedron
hedron
possible
Eudoxus.
theory
of concentric
to enter
thousand
(8 sides),
circles,
which
of planets
with
Another
work
years.
of irrational
which
become
the predecesor
IIHe (Aristotle)
was universal,
31 Leon Perry,
to fol-
actually
meant
The Mathmefl,
of Greek
of Plato.
(4 sides),
the dodecaare the only
mathematics
was an elaborate
the earth
was
He did a large
(fractions).3ID
His
explanation
in the center,
lasted
of his was the development
numbers.32
He also did work
to modern
integral
Encyclopedia
calculusl33
advanced
Britannica,
number
theory.
page 42
32 "Mathematics,
Historyll, Encyclopedia
vol ume 11, page 64!iY';U
33 Michal Moffatt,
page 80
of the
had no special field.
His knowand he wrote about everything.
29 IIMathematics,
Historyll,
volume 11, page 645
30 Arithmatic
(20 sides)
of proportions
system
ledge
In
The only
the tetrahedron
the scene
a geometric
would
mathematicians
and music.
that
a student
on the theory
as to the movement
famous
in which
(shapes).
He was originally
of work
a school
for the students
the 6ctahedron
polyhedrons
amount
many
is to prove
and the icosahedron
The next figure
contri-
of this science.,,29
'roads'
arithmatic30,
(6 sides),
regular
no great
he did was to show the uniqueness
that
(12 sides),
of so many
he had four
thing
figures;
made
But he created
the creaters
geometry,
very significant
Plato
the works
among
Academy,
astronomy,
Platonic
himself.
and directed
that he is numbered
low:
philosopher
Thw ages of Mathematics
Britannica,
vol 1,
of
He wrote on logic, physics, metaphysics,
astronomy,
meteorology,
botany, zoology, embryology,
medicin~,
ethics, psychology,
politics,
economics,
and literature.
His treaties were used as textbooks
in his
school.
But they were more than that.
For they formed
an encyclopedia
of everything
known in his time.
This
encyclopedia
had no peer of rival for 2000 years until
the French Encyclopedia
in A. D. 1751 - 1765. 34
11
Aristotle
which
rivaled
ebrated
the great
for gibing
his great
ematics
great
who 1 ived from
384 to 322 B. C. , set up a school,
Academy!34
us the key to the mastery
book on logic,
because
IIAristotle
is that
he thought
key.1I35
it was complete,
Lyceum,
has long been celof reasoning.
He didn1t
write
Organon,
on Math-
but he was still
a~~;/
mathematician.
, ~~.
r:~(J
/~s
yet, as a mathematician,
Euc1ids
fame is
~~;f
not tcIue to his own research.
Few of the theorems in
I
~).P
his textbooks
are of his own.
What Euclid did, and
what made him great, was to take all the knowledge
accumulated
in mathematics
to his time and codify it
into a single work.
In doing so, he evolved, as a
starting point, a series of axioms and postulates that'
~
~
we rea d m ira b 1e for the ir brev i ty and e 1ega nc e .II3
l
t
I
6y
"Euc 1ids El ements
G r e e k \</ r i tin g son
in 300 B. C. su perseded
mat hem at ic s .II37
a 11 preced i ng
~
<-
'
'-,/)1;1I'
~
~;- if
'W
r ~}, /
IIWe kn.ow his (Euclid) Elements,
whose influence
has
.1)£
not been equaled in the history of science.
For twentY,.!l
centuries,
the great mathematicians
of Greece, Egypt,
.,v
Persia, Arabia, and India got their first stimulus from)
it.
Each pupil copied the manuscripts
in order to have ~
one of his own.
'
The fir s t pr in ted ed it ion 0 fEu c 1 ida p pea red jus t
ten years before Col umbus found the New Worl d. One by
~
one, there followed more than 1000 other editions-in
more copies, in more languages,
t~an any other book
~,
wit h the ex c e p t ion 0 f the Bib 1e /yJ 8
UV
fAvfi
;11:.. J ~
r./\
"\'
l
A
)
~
f 1~
'\\~
34 Leon
Perry,
The Mathmen,
page
Asimov,
37 IIMathematics
page 395
38 Leon
Perry,
ll
,
Asimov
51
on Numbers,
Encyclopedia
The Mathmen,
53
<
fi~If:
page
Americana,
page
I
I
lvi)
vytl
r1~)
35 Ibid
36 Isaac
W ~
134
volume
17,
~
'
Very little is known about the background
however
, known that he taught
Museum.
in an Egyptian
The library of Alexandria,
to have had 600,000 papyrus
~Archimedes,
scientist
Archimedes
Despite
ancy.
He also worked
calculus.
--
on r~g
decimals
a challenge
8 equations
and invented
can be mathematically
with
of digits
long; The law of the lever, concerning
said "give my another
~
one" in reference
the
basic
The sand reckoner,
which
expressed;
to a rival, Apollonius,
with an answer billions
bod~es,
feat of his
the laws of boy-
variables
levers; On floating
he was
He invented
and levers and formulated
that any number
cattle problem,
years later."39
A remarkable
Some of his major works are:
demonstrates
was the greatest
his many inventions,
of differential
screw, pulleys,
is said
times, and hes equal did
two thousand
in pure mathematics.
water
limits.
of ancient
went to Museum.
was solving problems
called
served Museum,
the son of an astronomer,
Isaac Newton,
more interested
which
university
rolls!
and mathematician
not arise until
It is,
of Euclid.
it has 8
of trillions
the behavior
set down laws of boyancy.
earth to stand on and I shall lift this
to the powers
weapons
of levers and pulleys.
including
catapult
He also
remarkable
10 pound
stones, cranes which could lift and throw ships, and
to hurl ~ ton and
ships miles away!
39 Isaac Asimov,
of
He once
designed
lenses to ignite
The
Asimov
on Numbers,
page 173
J,
Starting
developed.
in the 17th century
Algebra
a new wave of mathematical
was f~lY
developed('nd,
Some of the greatest
men~e~
lived within
during
Also new light fell upon previous
this period.
~ew fields
thought
invented.
k<~Lc"
200 years of each other
questions,
which were unanswered.
One of the largest unsolved
postulate.
century.
It was unproved
In 1733 Girolamo
Saccheri
known to each other independently
that no controdiction
unsuccesfully
or indirectly.
in 1829 and the hungarian
geometry. ,,39 They followed
. J
was Euc1lds parellel
by the other axioms up to the 17th
it by 'reductio ad absurdum'
Ivanovich
problems
discovered
similar
tried to prove
"The 6).ssian Nikolai
Johann Bolyai
in 1832, un-
a non - Euclidean
lines as Saccheri
but asserted
could be found.
"It was during the 17th century that John Napier
revealed his invention of logarythems.
That Galileo
Galilei founded the mathematics of dynamics.
That
Johannes Kepler induced his laws of planetary motion.
That Gerald Desargues and Blaise Pascal formulated
projective geometry.
That Pierre de Fermat laid the
foundations of modern number theory.
And that Pascal,
Fermat, and Christiaan Huygens made distinguished contributions to the theory of probibility.,,40
"The development of analysis in the 17th century
by the mathematicians Pierre de Fermat, Rene Descartes,
and Isaac Newton soon left behind clawsical methods
and problems, and an enormous wealth of new descoveries
revealed an interaction between theoretical mathematics
and all branches of physics and astronomy."4l
39 "Mathematics",
page 396
Encyclopedia
Americana,
volume
17,
40 Ibid
41 "Mathematics, History",
volume-II, page 648
42 Ibid
Encyclopedia
Britannica,
_-?
12.
The 19th century
had an enormous
were contridictions
quanity
of new methods
which
of old beliefs.
''')In
1843, after years of cogitation, the Irish mathematician William Bowan Hamilton was led to invent his
quarternion algebra in which the commutative law of
multiplication does not hold.G42
""-
,'yIn1844 the german mathematician Hermann Gunther
~Grassman
published the first edition of his remarkable
Ausdehnungslehre,
in which he developed classes of
algebras of much greater generalarity than Hamiltons
quarternion algebra.
By weakoning or deleting various
of the laws of common algebra, or by replacing some
of the laws by others that are also consistant with
the remaining ones, an eno.rmous variety of algebraic
structures can be created.~43
Hamilton
and Grassman
mathematician
opened
the world
to abstract
can use any set of consistant
axioms he choses.
"There has never been a man like Newton,
be one like him.
Not Einstein,
Plank, not anybody
in addition
to formulating
ential and integral
problems
developed
which
not Archimedes,
else measured
and there never will
not Galileo,
up to near his stature.,,44
the laws of gravity
calculus.
algebra.
He developed
invented
systems
42 "Mathematics",
page 400
to solve many
system of limits.
Encyclopedia
Americana,
volume
43 Ibid
44 Petr Beckmann,
A History
of
?f,
page
137
Newton
deffer-
could not be solved until he solved them.
an excellent
not
17
Newton
BIBLIOGRAPHY
v
Asimov, Isaac, Asimov on Numbers, Hew York,
Pocket Books, 1978
Beckmann, Petr, A History of ~
The Golem Press, 1971
, Boulder,
Cook, Peter Do , The Ages of Mathematics vol I!) New York,
Doubleday and Company, 1977
Gamow, George, One, Tw0 Three, Infinity, New York,
The Viking Press, 19i1
Huff, Darrell, and Irving Geia, How to Take _a Chance,
New York, W. W. Norton & Company, 1959
IIMathematics," Encyclopedia Americana, 1979,
volume 17, pp. 431-434
"Mathematics, HistoIlT", Encyclopedia Britannica~ 1979,
volume 11, ppo 639-670
Menninger, Karl, A cultural history of Numbers,
The Me I. T. Press, 1970
.
Cambridge,
Moffatt, Michal, The Ages of Mathematics vol I , H ew York
Doubleda~r & Company, 1977
Perry, Leon, The Mathmen, New York, McGraw-Hill Book
Company, 19$
Waerden, B. L. Van Der, Science Awakening , New York,
Oxford University Press, 1961