16th Century Math

The Sixteenth-Century Mathematics of Italy : Commercial Mathematics


Characteristic of The 16th Century Mathematics.
Arrangement of The Symbols
Cubic and Quartic Equations


Characteristic of The Sixteenth-Century Mathematics

Mathematics in 1400's-1500's in spite of the Renaissance revival was not developed after seventeenth century or Greek.     The only thing focused was a solving of an equation of the third and fourth degree and symbolizing algebra of France.     Though mathematics had a small change in Renaissance, it had powerful energy. However, the most important meaning is to make a modern mathematics start and to estabish a tradition of European mathematics.
In summarzing the mathematical achievements of the sixteenth century, We can say that symbolic algebra was well started, computation with the Hindu-Ariabic numerals became standardized, decimal fractions were developed, the cubic and quartic equations were solved and the theory of equations generally advanced, negative numbers were becoming accepted trigonometry was perfected and systematized, and some excellent tables were computed. The stage was set for the remarkable strides of the next century.


Arrangement of The Symbols

Renaissant algebra started with necessity for commerce and arrangement of algebraic symbols.

Plus(+) and Minus(-) : These symbols appeared in a book about arithmetic written by John Widmann - Called father of arithmetic - for the first time in 1489.
At first, these symbols expressed 'surplus', and 'insufficiency' but later it meant 'addition'and 'subtraction'
The symbol of minus (-) was in the book but the plus symbol(+) was not.   symbol, (+) was originated from Latin, 'et'(means 'or'), whereas we can't know the origin of symbol of minus(-).

Radical symbol(،î) : Heinrich Schreiber Professor of Wien University used (+) and (-) to express addition and subtraction each in his book in 1521.
His disciple, Christoff Rudolff used the radical symbol(،î) including (+), (-), in his bool about algebra in 1525.
He used simple the radical symbol (،î) as (¥) which might be from the first letter of root.

Equal Symbol(=) : This symbol appeared for the first time in known the first English algebraic book written by Robert Recorde (ca. 1510 ~ 1558).  He said the reason why he adopted this symbol.
"There is no other symbol than parallel lines(=) which means equality".

Division Symbol(،ہ): Swiss mathematician Johann Heinrich Rahn used this symbol for the first time in his book published in Zurich in 1659.

Decimal Symbol : Simon Stevin (1546~1620), a former technician, introduced this symbol for the first time.

Inequality Symbol(>,<) : These two Symbols were shown in a book published 10 years after English mathematician Thomas Harriot(1560~1621).
After a century from his death, Pierre Bouguer started to use the symbols of ،أ and ،آ.

Symbols of Multiplication(،؟) and Difference(،­) : These symbols appeared in (1631) writtem by English mathematician William Oughtred(1574~1660).


 symbol of Letters : French mathematician Francois Viete (1540~1603) used letters to distinguish 'the known quantity' from 'the unknown'.
He used consonants - as b,c,d,،¦ - for 'the known quantity' and vowels - as,a,e,i,o and u - for 'the unknown' each.
But today, we use the fore part letters of alphabet - as, a,b,c,،¦ - for 'the known quantity' and hind parts for 'the unknown'
This system started Rene Descartes (1596~1650).
The Introduction of these many mathematical symbols was closely related to the development of printing.


Cubic and Quartic Equations

Probably the most spectacular mathematical achievement of the sixteenth century was the discovery, by Italian mathematicians, of the algebraic solution of cubic and quartic equations.  The story of this discovery, when told in its most colorful version, rivals any page ever written by Benvenuto Cellini.  Briefly told, the facts seem to be these.  About 1515, Scipione del Ferro (1465-1526), a professor of mathematics at the University of Bologna, solved algebraically the cubic equation x3 + mx = n, probably basing his work on earlier Arabic sources.  He did not publish his result but revealed the secret to his pupil Antonio Fior.   Now about 1535, Nicolo Fontana of Brescia, commonly referred to as Tartaglia (the stammerer) because of a childhood injury that affected his speech, claimed to have discovered an algebraic solution of the cubic equation x3 + px2 = n.  Believing this claim was a bluff, Fior challinged Targaglia to a public contest of solving cubic equations, whereupon the latter exerted himself and only a few days before the contest found an algebraic solution for cubics lacking a quadratic term.  Entering the contest equipped to solve two types of cubic equations, whereas Fior could solve but one type, Tartaglia triumphed completely.  Later Girolamo Cardano, an unprincipled genius who taught mathematics and practiced medicine in Milan, upon giving a solemn pledge of secredy, wheedled the key to the cubic form Tartaglia.  In 1545, Cardano published his Ars magna, a great Latin treatise on algebra, at Neuremberg, Germany, and in it appeared Tartaglia's solution of the cubic.  Tartaglia's vehement protests were met by Ludovico Ferrari, Cardano's most capable pupil, who argued that Cardano had received his information from del Ferro through a third party and accused Tartaglia of plagiarism from the same source.  There ensued an acrimonious dispute from which Tartaglia was perhaps lucky to escape alive.
Since the actors in the above drama seem not always to have had the highest regard for truth, one finds a number of variations in the details of the plot.

It was not long after the cubic had been solved that an algebraic solution was discovered for the general quartic (or biquadratic) equation.  In 1540, the Italian mathematician Zuanne de Tonini da Coi proposed a problem to Cardano that led to quartic equation.  Although Cardano was unable to solvce the equation, his pupil Ferrari succeeded, and Cardano had the pleasure of publishing this solution also in his Ars magna.

The repersentative mathematics of the 16th century is algebra originated in Arabia but it developed in Europe because commerce and calculation throve in there.  Italian merchants and bankers, especially, needed now to calculate accurately.
Astronomy contribute to the development of mathematics and 'mathematician' meant 'astronomer' for some time.
Nicolas Corpernicus (1473~1543), Polander, was the most distinguished astronomer who contributed so much to the development of mathematics.
His theory about universe brought the improvement of trigonometry.  He himself wrote a thesis on trigonometry.

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