Behold the Ages of Math...

Like most of our knowledge of ancient times, our knowledge of the oldest  math has faded with the millennia, although we can guess as to its nature.  Before  arithmetic, before numbers themselves, there was counting.  A Shepard needed to know that they ended the day with as many animals as when they had set out.

In order to solve this problem, in truly ancient times it was common to mark numbers with pebbles or tallies, even fingers.  These simple, even crude systems were enough to ensure that two numbers were equal without ever defining the concept of number.

However, even in the most ancient times recorded math was already highly developed.  Although the "Pythagorean Theorem" was named for the Greek philosopher Pythagoras, it was already well known to the Chinese, the Hindu, and the Babylonians.  The ancient Egyptians, with their highly developed architecture had an equally evolved geometric system.  Yet our knowledge of these times remains broken and fragmented.  Any conjectures about earlier math will, for now, remain merely conjectures.

(Slightly Less) Ancient Chinese Math (1000 B.C.E.- 1650 C.E)

Although separate from most Western theories of math at the time, the Chinese had a highly sophisticated number theory.  A full thousand years before European civilization the Chinese had approximated pi to sixteen digits.  They had even developed a method for solving systems of equations that was related to modern linear algebra.

Important Works:  Arithmetic in Nine Sections.  A collection of 246 problems dating from the Han dynasty, including a rough estimation for arc length that presents a value of 3 for pi.

Chou-Pei.  Another text of an age with the Arithmetic in Nine Sections.  It contains the beginnings of elementary number theory.

Important People:  Chu Shi-kie.  Understood the properties of what is commonly referred to as "Pascal's Triangle."  He also created a method of advanced matrix manipulation.

The Greeks were some of the best mathematicians of the ancient world.  Their geometry continues to impress and challenge math students of all ages.  They came up with  ingenious solutions to some very famous problems.  Their meticulous nature and careful proofs greatly advanced the science of math, and they successfully gathered many of the discoveries of other cultures.  Their use of the compass and straightedge in geometry continues to influence high school classes today.

Important Works:  Elements.  This is Euclid's masterpiece that set the foundation for much of geometry until the modern age.  Its name suggests that its focus was on an elementary level of geometry, implying that it represented only the beginning of the author's true mastery of the art.

Important People:  Pythagoras. The man whose name was bestowed upon the famous Pythagorean Theorem, was both a philosopher and a mathematician.  Although he was not the first to discover the basic properties of right triangles, he founded a branch of philosophy and mathematics that effected Greek society long after his death.  They believed that the basic properties of number theory (whole numbers, perfect numbers, ratios, etc.) reflected the true nature of the universe.

Euclid.  One of the greatest geometers ever.  Euclidian geometry bears his name, and he is responsible for compiling its fundamental list of postulates.

Archimedes.  Perhaps the greatest mathematician of the time period.  He expanded on the Greek art of geometry, working out many of his problems in three dimensions.

Hindu Math (200 B.C.E. - 1250 C.E.)

The caste system restricted the study of mathematics to the priesthood during this time, but many important discoveries were made in the region that is currently called India.  Unlike the more rigorous Greeks, the Hindus favored calculation over the more artful study of geometry.  As in many civilizations, the majority of the most famous Hindu mathematicians were astronomers by profession, and because of this very few works were published whose focus was truly mathematics.

Important Works:  Lilavati.  Bhaskara's work is the source of much of our knowledge of Hindu arithmetic.  It highlights their system for writing equations.  From this work, we know that the Hindu's had already discovered irrational and negative numbers, understood some of the more basic properties of quadratic equations.

Important People:  Brahmagupta.  The most prominent Hindu mathematician of the seventh century his work in the field of astronomy gives us insight into the advanced state of mathematics in the region during this time.  Bhaskara.  Rose to fame around the year 1150 C.E.  Despite writing almost half a millenium after Brahmagupta the work shows only marginal advances in the field of math.

Srinivasa Ramanujan.  In 1913, this astounding mathematician met the eminent British theorist G. H. Hardy.  Ramanujan possessed amazing intuitive prowess and made a significant contribution to modern number theory.

The Arabs have been an important group throughout the history of mathematics.  They adopted and adapted the Hindu system of numerals and developed the system used worldwide today.  During the Dark Ages of Western Europe, the Arabs preserved many of the discoveries made by Greek scholars.  Although they lacked the rigorous proofs of the Greeks the Arabs also made many discoveries of their own in the field of Algebra.

Important Works:  Fakhri.  Written by al-Karki, it is a very disciplined and professional study of algebra.

Rubaiyat.  Written by Omar Kayyam, it is very highly regarded in intellectually circles in the present day.

Important People:  Omar Kayyam was one of the first mathematicians to solve cubic equations.  He used a geometric solution that he also used to find roots of special quartic equations.

European Math (450 - 1500)

The less said about European mathematics during this time period the better.  The Dark Ages had set in, and many of the advances developed and spread during the Classical period were lost and forgotten in the region.  Almost all discoveries were simply rediscoveries made from older Greek texts or communication with other cultures.  Little original work was accomplished during this stagnant period.

Development was much faster and thorough from the Renaissance on.  Through stronger nations, more formalized relations, and better communication across cultures math began to awaken from its slumber.  Calculus was developed, and most of the fundamental parts of modern math were developed first in this time.

Important Works:  Principia.  Newton's most famous publication.  It marked the beginning of a new age in physics and applied math.  It illustrated the true genius of Newton and paved the way for many of his most important discoveries.

Important People:  Renee Descartes.  His most famous work involves analytic geometry.  By using Cartesian coordinates Descartes finally succeeded in advancing geometry beyond the discoveries of the ancient Greeks.

Fermat.  In the seventeenth century, Pierre de Fermat revolutionized number theory.  Fermat's famous last theorem was written in the margins of a book, proof omitted.  It proved to be one of the most difficult problems for mathematicians, and it took over three hundred years before a solution could be found using the most powerful tools of modern math.

Isaac Newton.  He astounded his peers with his brilliance.  His discoveries, along with the notation derived by Leibnitz led to the birth of modern calculus.

At this point, the advances and discoveries race ahead too fast for justice to be done to them here.  Calculus thrived and flourished, giving birth to a whole new species of applied math that remains important in the present day.  The Industrial Revolution created a new need for the unification of math and science.  As always, math advanced.  Matrix theory in its present form was created leading to a new set of important discoveries.  The modern standards of proof were implemented, and communication between mathematicians increased leading to the sharing of ideas and discoveries.

Important Works:  Proceedings of the St. Petersburg Academy.  This is a journal rather than a single work, and its most important period was during the eighteenth century when it had the honor of publishing Euler's manuscripts.  Euler, besides contributing to the theory of math in general, is respected for standardizing the notation used by mathematicians in the present day.

The metric system.  A very important advancement, although not a published work.  The metric system was the standardization and organization of measurement that marked the success of the scientific method and the general advancement of scientific procedure.  Disquisitiones arithmeticae.  The masterwork of Charles Gauss represents yet another great leap in number theory.

Important People:  To mention a person here by name is to disregard the dozens of other great mathematicians who could not be named.  Truly, as much has happened to the field of math in the last three hundred years as had happened in all the time before.  Among the greatest of the great are Charles Gauss, an extraordinary scholar of the nineteenth century whose potential as a child prodigy was truly realized as an adult, and Albert Einstein the revolutionary physicist.  Among these great names were new tools, more powerful than any before.  Modern mathematicians are aided tremendously by computer, calculator, and communication.  Each year it becomes more difficult to keep abreast of the truly amazing discoveries.  Each year, more names are added to the list of great mathematicians.