History of the Triangle History of the Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted .The so called 'Pascal' triangle was known in China as early as 1261. In '1261 the triangle appears to a depth of six in Yang Hui and to a depth of eight in Zhu Shijiei in 1303. Yang Hui attributes the triangle to Jia Xian, who lived in the eleventh century'. They used it as we do, as a means of generating the binomial coefficients. It wasn't until the eleventh century that a method for solving quadratic and cubic equations was recorded, although they seemed to have existed since the first millennium. At this time Jia Xian 'generalised the square and cube root procedures to higher roots by using the array of numbers known today as the Pascal triangle and also extended and improved the method into one useable for solving polynomial equations of any degree'. There are some proofs that this number triangle was familiar to the Arab astronomer, poet and mathematician Omar Khayyam as early as the XI century. Most probably the number triangle came to Europe from China through Arabia. The Chinese representation of the binomial coefficients, often equally called Pascal`s Triangle being found in his work published for the first time after his death ( in 1665 ) and dealing with figurate numbers, is found for the first time on the title page of the European Arithmetic written by Appianus, in 1527. Know More About :- Multiplication of Whole Numbers
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Blaise Pascal was not the first man in Europe to study the binomial coefficients, and never claimed to be such; indeed, both Blaise Pascal and his father Etienne had been in correspondence with Father Marin Mersenne, who published a book with a table of binomial coefficients in 1636. Many authors discussed the ideas with respect to expansions of binomials, answers to combinatorial problems and figurate numbers, numbers relating to figures such as triangles, squares, tetrahedra and pyramids. In 1407 an edition of Jordanus' de Arithmetica contains the following table. 1523: Nicolo Tartaglia first publishes the generalization of the figurate numbers. Some 30 years later, in his General Treatise, he publishes the Triangle in table form. Tartaglia is the first mathematician to publish a general formula for solving cubic equations. His name in Italian means "stammerer". This cruel nickname was given to him after severe facial wounds he suffered at the age of twelve when attacked by a soldier invading his hometown of Brescia nearly killed him; these wounds left him able to speak only with difficulty for the rest of his life. 1539: Gerolamo Cardano, the Italian algebraist, correctly determines that the number of ways to take 2 or more things from a set of n things is 2n-n-1. 1544: The German mathematician Michael Stifel publishes the extended Figurate Triangle in the figure shown below. Stifel gives credit to Cardano's work published five years earlier. 1591: François ViÊte gives names to the first few columns of the Triangle in Latin; "numeri trianguli", "pyramidales", "triangulo-trianguli", "triangulo-pyramidales" These names are also used in the next century by Pierre de Fermat, who was Pascal's main correspondent in solving the Problem of Points, and William Oughtred, a British mathematician who influences many of his countrymen who come after him. 1631: William Oughtred publishes his Clavis Mathematicae, which influences his student John Wallis and is later owned in a 3rd edition printing by Isaac Newton; both Wallis and Newton are instrumental in the work that connects the binomial coefficients to the new field of calculus later in this century. 1633: The lifetime work of Henry Briggs entitled Trigonometria Britannica is published two years after his death by his friend Henry Gellibrand; he has a chapter on the figurate numbers, which he refers to as "the calcuator of many uses". 1636: Father Marin Mersenne publishes his Harmonicorum Libri XII; Mersenne in his life meets with both Blaise Pascal and his father Etienne, and there is little doubt both of them read the book and saw this table.
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