During the past two weeks, we studied the works of several of the Greek Mathematicians
By the end of the third century B.C., the Golden Age of Greek mathematics was ending. In 146 B.C. Ptolemy VII, over took Egypt and banished from scientists and scholars who were not loyal to him. Many exiled Alexandrian scholars fled to more remote areas. The last two pre-Christian centuries saw the growth of Roman control. Despite the military machines of Archimedes, Syracuse fell to the Romans in 212 B.C., Carthage in 202 B.C. The Romans conquered Greece in 146 B.C., and Mesopotamia by 64 B.C. After Julius Caesar's assassination in 44 B.C., his grandnephew Augustus Caesar ruled the western Roman empire and also become ruler of the Eastern empire when Mark Antony was defeated. Upon the suicides of Mark Antony and Cleopatra in 30 B.C., Augustus Caesar took over Egypt. This began a period of time of prosperity and relative peace in Alexandria and Egypt which lasted until about 180 A.D. Eventually exploitation by the Romans and the cultural mix of Greeks, Christians, Jews, and Egyptians led to unruly mobs with brawls and bloodshed in the city. Diophantus ( 250 A.D. -- ) and Pappus brought brought a brief return of Greek mathematical glory to Alexandria.
Christianity began as a sect within Palestinian Judaism, and then spread throughout the Roman world. Initially the movement was tolerated by the Roman state, but in the second and third centuries B.C. became the center of blame for internal catastrophes and invasions, and suffered periodic persecutions. During the fourth century B.C. the Roman emperor Constantine was converted to Christianity and under Emperor Theodosius's rule, it became the official religion of the entire empire.
The great days of Greek mathematical reasoning were over. Scholars turned their energies to theological issues. Faith, not scientific inquiry, directed the Church doctrine. Physical science and mathematics were ridiculed, since the Bible was the source of all knowledge. Greek learning became associated with paganism, and libraries and temples were looted and holdings were destroyed.
The empire was frequently threatened with both internal civil wars and external threats. In 330 the empire was divided into an eastern and western half, and in the fifth century the western Roman half was overrun by invading Germanic peoples.
The Eastern (Byzantine) Empire remained independent and isolated for
nearly one thousand more years, and here Greek learning was kept alive,
although dormant. They actively preserved and copied the ancient
Greek works, thereby enabling the later Renaissance.
The new faith Islam based upon the teachings of Mohammed united the desert Arabian tribes, and their power spread quickly throughout the Mediterranean. Damascus and Jerusalem fell to the Arabs in 635 and 637 respectively. They advanced west as far as Spain and France and east through Syria, Persia, and northern India. Only Christian Europe remained out of their control.
Originally ruling from Damascus, they built a new capital on the Tigris River, Baghdad with a population of 800,000 by 762. Arab scholar set up the House of Wisdom which acquired and translated Greek manuscripts, placing them in a library for their use. Thus what might have been lost forever was saved in Arabic translations and enhanced with their refinements.
A good article on the influence of Arabic mathematics especially on the development of algebra can be found at Arabic mathematics at the Mac-Tutor History of Mathematics Site.
The most well-known Arab mathematician of that period was Mohammed ibn Musa Al-Khowarizmi who was associated with the House of Wisdom (it is presumed). His major works consisted of a book on arithmetic and one on algebra. His work " Book of Addition and Subtraction According to the Hindu calculation" contains the earliest Arabic use of the Hindu decimal system of numerals. (Only a Latin translation of the work survives today). Its influence is the reason these base 10 numerals are misnamed Arabic instead of Hindu. An excellent article on his life and works is found at the Mac-Tutor site: Al-Khwarizmi.
In the above article you can an explanation of Al-Khwarizmi solved quadratic equations using the example equation x + 10x2 = 39.
A square and 10 roots are equal to 39
units. The question therefore in this type
of equation is about as follows: what is the square which combined with ten of its
roots will give a sum total of 39? The manner of solving this type of equation is to
take one-half of the roots just mentioned. Now the roots in the problem before us
are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you
add to 39 giving 64. Having taken then the square root of this which is 8, subtract
from it half the roots, 5 leaving 3. The number three therefore represents one root
of this square, which itself, of course is 9. Nine therefore gives the square.
The method of proof he used was a geometric completing of the square.
Exercise: Use the method indicated in the article at
Mac-Tutor site: Al-Khwarizmi.on
x2 + 5x = 36. Do it both algebraically and geometrically (provide the drawing).
No notable mathematicians in Latin civilization were produced from
the decline until the 12th century. The first glimmer of light
was produced by Leonardo of Pisa ( circa 1170 to after 1240),
commonly known today by the name Fibonacci (although he did not use this name). His father's name was Bonacci and Fibonacci is the shortening of filio Bonacci. Read the biographic article at the MacTutor History of Mathematics web site: Fibonacci .
Exercise: After reading the above MacTutor History of
Mathematics article on Fibonacci, write a short summary of his life and
discuss one of his mathematical results (other than the Fibonacci
numbers) in some detail (within a web page). (The result
can be from this article or another that your find). Place a link
to your article on your history web page.