**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.

**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).

**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.

**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.

**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.

**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.

**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.

**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.

**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.

**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).