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