Syncopated Algebra
275 - 1600


The change from rhetorical to syncopated to symbolic was a "transition." The year associated with syncopated algebra is 275 because Diophantus was probably the first to take steps toward an algebraic notation. These steps were mostly stenographic abbreviations, to help with printing. He had abbreviations for each of the following: the unknown, powers of it up to the sixth, subraction sign, equal sign, and reciprocals.


200 - 284 -- Diophantus of Alexandria

He tried to solve indeterminate problems of the second degree (but he was not the first). The famous "cattle problem" was before his time (apparently sent from Archimedes to Eratosthenes). Diophantus didn't restrict the solutions to integers, he tried rational solutions.



An example of syncopated algebra would be to write

as "unknown cubed 1, unknown squared 13, unknown 5."



One of the best sources of ancient Greek algebra problems is the Palatine Anthology. These were assembled about 500. An example of one of the problems would be the following:

Demochares has lived a fourth of his life as a boy, a fifth as a youth, a third as a man, and has spent 13 years in his dotage. How old is he?

The 3 Graces were carrying baskets of apples, and in each was the same number. The 9 Muses met them and asked each for apples and they gave the same number to each Muse and the 9 and the 3 each had the same number. Tell me how many they gave and how they all had the same number. (The problem is indeterminant. Find the smallest permissible solution.)

Bear in mind the anthology contained 46 number problems not unlike the above and there were no algebraic symbols to solve them!!


The beginning of mathematical tradition in China is the textbook Nine Chapters on the Mathematical Art. It is the oldest textbook on arithmetic, but the date and origin is unknown. (Here's an example of some work in the text.) We do know that it is a collection of many efforts over several centuries. Original copies were destroyed in the famous Burning of the Books of 213 B.C. The text today is a commentary on the original Nine Chapters, by Liu Hui in 263. He has theoretical verifications of the problems and added a lot of his own contributions.

The book is the first to show a systematic method for solving simultaneous linear equations.

Check out the link to the history of Chinese mathematics

The Hindus syncopated their algebra with the works of Brahmagupta (c. 628) and Bhaskara (c. 1150). They are the first known to have tried to solve ax + by = c. The Indians had a far superior system of mathematics than the Greeks during the time period from 400 to 1200 (except in geometry).


476 - 550 -- Aryabhata

Aryabhata was the first to solve the Diophantine equation

Aryabhata studied the summation of arithmetic and geometric series, made a table of sines of angles in the first quadrant, and tried to solve quadratic and linear indeterminate equations.



598 - 670 -- Brahmagupta

Brahmagupta decided to let these equation (Diophantine equations) have integral solutions only.


 

1114 - 1185 -- Bhaskara

He affirmed the existence and validity of negative and positive roots. Europeans did not claim this until the 16th or 17th century. He completed the square in his book Sridhara and called it the "Hindu Rule."

He predicted (in his role as astrologer) for his daughter Lilavati the lucky day and hour for her marriage. She was waiting for the exact hour by watching a water clock (in which time could be told according to the amount of water in the device). As she leaned over the clock to peer closely, a pearl from her wedding headdress fell into the water and stopped the time. She never married as a result.


1170 - 1250 -- Leonardo Pisano Fibonacci

In 1202, Fibonacci's Liber abaci is published, with syncopated algebra. He included Hindu-Arabic numerals and prompted the acceptance of this system in Europe.


In 1494 the first printed edition of the Suma by friar Luca Pacioli appeared. The work is compiled from many sources and although it has superior notation than Fibonacci's Liber abaci, it contains little that is not found there. This book's algebra is syncopated.



Origins of Algebra

Go to timeline of rhetorical algebra

Go to timeline of symbolic algebra


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