A.30,000
B.C. - 2001 B.C.
circa
30,000 B.C.:
Paleolithic peoples in Europe etch markings on bones to represent numbers.
circa
5,000 B.C.: The Egyptians use a
decimal number system, a precursor to modern number systems which are also
based on the number 10. The Ancient Egyptians also made use of a
multiplication system that relied on successive doublings and additions in
order to find the products of relatively large numbers. For example, 176
x 313 might be calculated by first finding the double of 313 (313 x 2 = 626),
then finding the double of that number (313 x 4 = 1252), the double of that
number (313 x 8 = 2,504) and so on (313 x 16 = 5,008; 313 x 32 = 10,016; 313 x
64 = 20,032; 313 x 128 = 40,064....). Thus, using these known products
produced by doublings, and knowing that 128 + 32 + 16 = 176, then you add the
known products of 40,064 + 10,016 + 5,008, to acheive the final answer of 176 x
313 = 55,088.
B.2000
B.C. - 501 B.C.
circa 1850 B.C.: The Babylonians possess knowledge of what will
later be known as "The Pythagorean Theorem," an equation that relates
the sides of right triangles whereby the sum of the squares of the two "legs"
(the shortest sides) of the right triangle equal the square of the hypotenuse.
circa
569 B.C.: Pythagoras is born in Samos, Ionia. After traveling
abroad for the sake of learning, Pythagoras founded a philosophical and
religious school in Southern Italy which, among other tenets, believed that all
of nature (reality) consisted of numbers, or the relationship between
numbers. Thus his order of Pythagoreans went on to contribute many
important ideas to the discipline of mathematics, not least among them the
Pythagorean Theorem (cf. 1850 B.C.
above).
C.500
B.C. - 1 A.D.
circa
425 B.C.: Although it had
apparently been known for some time, Theodorus of Cyrene is the first person in recorded history to show that
some square roots produce irrational numbers, that is, they cannot be expressed
as a fraction using integers, and their decimal equivalent neither terminates
nor repeats itself.
287
-- 212 B.C.: The life of Archimedes. Famous in the ancient world for his
machines, many used in the defense of Syracuse against the Romans, Archimedes
claim to fame in posterity focuses more on his pure mathematics, especially in
the field of geometry. Archimedes discovered relationships between a
sphere and a circumscribed cylinder, specifically between their volume and
surface area. He made several more innovative discoveries, and
is considered by many to be one of the greatest mathematicians of all
time. His method of exhaustion -- that is, of finding an area by approximating
it to the area of a series of polygons -- is often considered to be
the beginnings of modern integration mathematics.
D.2
A.D. - 500 A.D.
circa
200 -- 284: The
life of Diophantus. Diophantus, often considered the "father of
algebra," is most famous for his work, Arithmetica. In this work, Diophantus introduces
algebraic equations and how to solve them, as well as other findings in the
theory of numbers. An interesting thing about his algebra, though, is
that he only considered equations with positive rational solutions, that
is, he considered "absurd" such equations that would produce
negative or irrational numbers. For example, as he understood it, in the
equation: 3x + 15 = 6, how could a solution come to equal -3 apples?
circa
220 -- circa 280: The life of Liu Hui. Something to take into consideration
about Chinese mathematicans is that, in general, mathematics seems to have been
taken as something of a lesser art, and that most of its practicioners
contributed to its body of knowledge more or less anonymously, so that the
biographies of many Chinese mathematicians of the past remain very much
unknown. Thus one of the greatest works on mathematics in antiquity, the
Chinese text Nine Chapters on the Mathematical Art, is individually authorless, and instead
reflects the work of many anonymous mathematicians contributing to this one
work, whose personal names have been lost to the dark of ages. And so,
although we know little of the life of Liu Hui, we have record of his
commentary on the Nine Chapters,
and yet know little of the man but what we can imagine from his commentary
on this central document. In this commentary, Liu Hui expresses a
different, more exact and provable way of doing mathematics, and also that he
is at least beginning to understand some of the fundamental concepts of
differential and integral calculus. He found a uniquely original way to
find a closer valuation for pi, using what he knew as the Gougu theorem (to
posterity as the Pythagorean Theorem); which he also utilized, and expanded, to
apply to any number of practical problems dealing with the height and distance
of any numer of topographical objects. Indeed, the brilliant originality,
in both conceptual understanding and writing style, of this man has not been
lost on many historians, ranking him among the greatest mathematicians of all
time.
circa
250: The
Mayan civilization utilizes a base-20 number system, probably originating from
the fact that humans have a total of 20 fingers and toes.
circa
370 -- 415: The life of Hypatia of Alexandria. The daughter of an
Alexandrian mathematician and philosopher, Hypatia is known as the first major
female figure to contribute to the development of mathematics. (It is
unknown if any of the female Pythagoreans -- who tended to remain individually
anonymous and secretive -- contributed substantially to that school's
advancements.) While she is not believed to have developed anything
original in mathematics, she was well renowned in education
circles for her mastery of, and commentaries on, past and present
knowledge.
E.501
A.D. - 1000 A.D.
598
-- 670: The
life of Brahmagupta, a mathematician and astronomer from India. Perhaps
the most distinguishing mathematical contribution of Brahmagupta's primary
work, Brahmasphutasiddhanta (or, The Opening of the Universe), is the
understanding he shows for the concepts of zero and negative numbers in
arithmetic. He calls negative numbers "debts," and positive
numbers "fortunes," and relates such arithmetical rules as "a
debt minus zero is a debt," "a fortune minus zero is a fortune,"
"a debt subtracted from zero is a fortune," and other such rules that
clearly show his greater understanding in this area in comparison to his
contemporaries. He also espoused a method of multiplication that utilized
the place-value system -- much as we do in Europe and America today.
Brahmagupta also developed ideas toward computing square roots, algebraic
notation, and solving quadratic and indeterminate equations. And
yet, despite these advances he made in mathematics, Brahmagupta's
written works deal primarily with topics in the field
of astronomy, for which he has also gained notoriety.
circa
780 -- 850: The
life of Abu Ja'far Muhammad ibn Musa al-Khwarizmi. Al-Khwarizmi gained his
fame by elucidating an easy to understand form of algebra that was intended for
practical uses. In his treatise, he showed, using both solutions and
geometric methods, how to reduce, balance, and solve equations. Although
many have bestowed the "father of algebra" of title on Diophantus
(cf. circa
200 -- 284), some
actually argue for al-Khwarizmi's claim to that rank, and in fact, the term
"algebra" itself stems from a term in the title of his most famous
and important work.
F.1001
A.D. - 1500 A.D.
953
-- 1029: The
life of Abu Bekr ibn Muhammad ibn al-Husayn al-Karaji. Building on the
algebraic ideas and methods of Diophantus and al-Khwarizmi, al-Karaji receives credit from many
historians for seperating algebra from geometrical explanations, instead using
arithmetical operations, which is of the essence in the algebra of our
day. Dealing with monomials, al-Karaji was able to define the product of
any two terms without requiring a geometrical proof. Al-Karaji, who lived
in Baghdad while writing on mathematics, contributed to the
development of other previous mathematical knowledge, especially that of
Diophantus. However, later in life, he moved to other, wilder countries,
and devoted himself to more practical endeavors, such as the drilling of wells,
the measuring and weighing of buildings, etc.
circa
1135 -- 1213: The
life of Sharaf al-Din al-Muzaffer al-Tusi. Al-Tusi was famous in
his day for travelling throughout the Middle East as a teacher of
mathematics. It has been said that some would travel great distances to be
his pupil. He settled in Baghdad later in life, where he wrote down
his own contributions to mathematics. The original treatise of
al-Tusi is no longer extant; although we have general knowledge of its contents
in the form of briefer summaries and commentaries. Al-Tusi departs from
the school of algebra delineated by al-Karaji by focusing on cubic equations as a way of
studying curves. His method was unique, first by dividing equations into
several different types, then by examining some of these types he is able
to explore equation parameters by utilizing the derivative of a
function, perhaps the first person in history to use this method.
G.1501
A.D. - 1800 A.D.
1718
-- 1799: The life of Maria Gaetana Agnesi. The daughter of an affluent merchantman,
and the eldest of the twenty-one children begot by her father (via three
wives), Maria Gaetana Agnesi's main contribution to mathematics consisted of
compiling a clearly-explained and comprehensive text on differential
calculus. In the opinion of all, she succeeded at this much-needed task,
and for her effort was offered a chair in mathematics at the University of
Bologna. Apparently she never accepted this post, however, and instead
devoted her energies, for the remainder of her life, to charitable and
religious ends.
H.1801
A.D. - 1900 A.D.
19th
Century:
The popularization of a multiplication method in India -- popularized because
it used less paper -- probably based on the arithmetic methods of ancient
Indian mathematics. This computational method worked as
follows, considering the problem of 216 x 452: 1) Multiply the last two digits
(6 x 2 = 12), write "2" and remember to carry the 1; 2) Multiply each last digit by
the middle digit of the other number (6 x 5 = 30; 2 x 1 = 2) and sum the
results along with the carryover number (30 + 2 + 1 = 33), write down
"3" (to the left of the "2") and remember to carry the
other 3; 3) Multiply the two middle digits
(1 x 5 = 5) and each first digit with each last digit (2 x 2 = 4; 4 x 6 = 24)
and sum the results along with the carryover number (5 + 4 + 24 + 3 = 36),
write down "6" to the left of the "3") and carry the 3; 4) Multiply each first digit with
each middle digit (2 x 5 = 10; 4 x 1 = 4), sum the results along with the
carryover number (10 + 4 + 3 = 17), write down "7" (to the left of
the "6") and carry the 1; 5) multiply each first digit (2 x 4 = 8) and add
the carryover number (8 + 1 = 9) and write "9" to the left of the
"7," and you will have written down the product of 216 x 452, which
is 97,632.
1776
-- 1831: The
life of Marie-Sophie Germain. The forces of society in 18th century
France (and elsewhere) formed nearly impregnable defenses against the inclusion
of women in many male-dominated activities, and this was especially true of the
academic life. However, Marie-Sophie Germain proved obstinate in
her passion for learning, particularly for mathematics, and
eventually her father caved and would support her for the remainder of her life
as she pursued her passion. Because of her gender, Germain was forced, by
and large, to educate herself in the ways of higher
mathematics, and the little correspondence she did initiate with
other mathematicians she would sign with a male pseudonym. Thanks to
her father's patronage, she remained undeterred by these obstacles
and worked on a theory of elasticity despite the paucity of knowledge
in the field of physics relavent to this problem, and despite her lack of
formal education and opportunity. She also spent time writing papers on number
theory and the curvature of surfaces, and contributed significant developments
to the proof of Fermat's Last Theorem in which she also added a theorem that
would eventually become known as Germain's Theorem. And yet, as testament
to the prejudices of the day, her death certificate did not list her as
mathematician, philosopher or scientist -- but merely as "property
holder."
1850
-- 1891: The life of Sofia Vasilyevna
Kovalevskaya. Raised in a
Russian family of nobility, Sofia Kovalevskaya first became interested in
mathematics by listening to her uncle's reverential discourses on the
subject. Her love for mathematics intensified with age, and she exhibited
a natural capacity for understanding difficult concepts, often surreptitiously
since her father forbid from taking up formal studies. There existed
formidable social barriers at the time in Russia, so Kovalevskaya was forced
into a nominal marriage so that she could be afforded a better opportunity to
realize her dreams of a higher education. Eventually she moved to Berlin
and immediately impressed her professors, whose lectures she attended
unofficially since women were not allowed to matriculate as a student.
After receiving her doctorate in recognition of three noteworthy papers published
on Partial differential equations, Kovalevskaya faced impregnable prejudice as
a woman and was unable to obtain an academic position despite several
recommendations by noted mathematicians of the day. Her patience paid
off, however, and in 1889 she became the third woman to hold a chair at a
European University, and the first mathematician. She went on to
contribute to the fields of analysis and other areas, at the
same time gaining a distinguised place in European society
for her mathematical prowess. She died in 1891 of influenza, in the prime
of her powers.
I.1901
A.D. - 2000 A.D.
1882
-- 1935: The life of Emmy Amalie Noether. The daughter of a distinguished
mathematician, Max Noether, Emmy early on decided to forgo a life of teaching
languages at a girls' school to study mathematics at university despite the
formidable obstacles for women that such a path entailed. In time she
gained a reputation as an innovative mathematical thinker, and in 1919 she
finally overcame the ban on her gender and was granted permission to be
included, officially, on the Faculty at the University of Gottingen. Her
work in the theory of invariants laid some of the pre-conceptual groundwork for
Einstein's general theory of relativity, and Hilbert's related work on field
equations for gravitation (cf. 1900 and 1915 below); Einstein also complimented
Noether for her "penetrating mathematical thinking." With the
rise of the Nazi party in Germany, Noether, who was Jewish, was forced from her
post, and she came to the United States to teach and lecture, especially at
Bryn Mawr College.
NOTE ON WOMEN IN
MATHEMATICS: By reading the biographies of the women considered in
this timeline, especially from Marie-Sophie Germain to Emmy Amalie
Noether -- that is, from the 18th centruty to the the 20th -- it is
possible to see the slow degrees in which society afforded opportunities
to women, and not, perhaps, only in the area of mathematics. One can see
that the common thread that compelled these women to blind themselves to their
social oppression, and to continue forward with their passion, was their
undeniable love for learning and thinking about mathematics. And though
none of them, individually, was able to completely break the barriers
surrounding them and thus provide themselves the full opportunity
they deserved, they nonetheless persevered and in doing so contributed
piecemeal, and not unsignificantly, toward the road to ultimate equality which,
it can and should be argued, has not been fully constructed to this day.
1900: In 1900, at the Second
International Congress of Mathematicians, David Hilbert gave a speech entitled, "The Problems of
Mathematics," declaring the great vitality of mathematics in relation to
its unsolved problems. He went on to mark 23 problems for the coming
century including the continuum hypothesis, the well ordering of the reals, the
transcendance of powers of algebraic numbers, Goldbach's conjecture, the
extension of Dirichlet's principle, the Riemann hypthesis, and many more.
As the century progressed, many of the problems were solved, and each time it
became a major event in the world of mathematics. Hilbert went on to
contribute many important ideas across a wide range of mathematical branches,
showing a genius for synthesizing such branches and explaining their
interconnectiveness. Also, he nearly trumped Einstein in discovering the
correct field equations for general relativity (cf. 1915 below).
1915: Albert Einstein publishes his Theory of General
Relativity. Before the 20th Century, Newton's law of gravitation,
presented in 1687, held rule as an accurate theory on the force of
gravity. However, as the 19th Century progressed, certain problems arose
concerning this theory, and a new explanation was warranted. After familiarizing
himself with various types of mathematics which he had hitherto adjudged as
"luxuries," Einstein created his General Theory of Relativity,
which presented a picture of gravity as the curvature of space. The
implications of this theory resonated in several fields of study throughout the
20th Century, and helped to make Einstein the household name he is
today.
1970: In this year, Alan Baker receives the Fields Medal at
the International Congress of Mathematicians, in Nice, France, for his work
with Diophantine equations, problems originating from the work of Diophantus (cf. circa 200 above). Baker also made
a significant contribution to Hilbert's seventh problem (cf. 1900 above), which asked whether a to the q power was transcendental when a and q are algebraic. He is
also famous for his many published works on number theory.
1994: Andrew John Wiles provides proof of Fermat's Last Theorem.
This Theorem remained famous because it was the last of the claims that Fermat
made, yet had not sufficiently proved. This equation
had befuddled mathematicians for more than 300 years, and yet, in the
undertaking to solve it, had produced several new developments in
mathematics. Wiles learned of this problem at the age of 10, and from
that point forward felt a personal passion towards finding its solution.
After several years of hard work, he found a solution, only to have a small
error found that nearly compelled him to give up the chase. However,
almost a year later, in 1994, he found a solution which has been accepted as
valid.
J.2001
A.D. - present
2001:
Vladimir Voevodsky, at
the 24th International Congress of Mathematics in Beijing, China, receives the
Fields Medal (sharing it with Laurent Lafforgue) for for making outstanding
advances in algebraic geometry; a field considered by many historians to have
been founded in history by Sharaf al-Din al-Muzaffer al-Tusi (cf. 1135 above).
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