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stoutfellow) wrote2007-03-07 10:26 am
stoutfellow) wrote2007-03-07 10:26 am
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Ramble, Part 16: "Slowly I Turn..."
Talking of Galileo brings us into the early 17th century. I want to drop back for a moment into the late 16th, in order to talk about François Viète. Viète was the first of a wave of important French mathematicians over the course of a century or so, a list headlined by Pascal, Fermat, and Descartes, but also including such lesser lights as Girard Desargues, Marin Mersenne, and Gilles de Roberval. He was probably the greatest mathematician, and certainly the greatest algebraist, of his era. Two of his achievements are of special importance, under the cut.
I've mentioned before Diophantus' use of syncopated algebra. It was Viète who made the step forward to true symbolic algebra. Though his notation is not quite what later came to be used, it's still a great improvement. Diophantus, limited by the Greek convention of using letters as numerals, had to introduce new symbols to represent a variable and its square and cube. Viète, not so restricted, proposed the use of letters to represent numbers - more specifically, he suggested the use of vowels (voyelles) to represent variables and consonants (consonnes) to represent constants. (That the mnemonic works as well in English as in French is not a coincidence, but one wonders whether speakers of other languages would find it as apt.) This opened the door to equations involving several variables; Diophantus had been forced to deal with such problems by immediately reducing them to single-variable status. In addition to this, Viète introduced a new way of dealing with powers of numbers, suffixing "quad" and "cub" to the variables to represent their squares and cubes. Unfortunately, he still was bound by the requirements of homogeneity; he found it necessary to label certain of his constants with the suffixes "plano" and "solido" for this purpose. Thus, for example, he would write
Another interesting achievement was an early use of recursion, in an algorithm for approximating the zeros of polynomials. For example, suppose one desires a zero of A3+A-3. Begin with an initial estimate, say A=1. Now set A=1+E and expand, obtaining E3+3E2+4E-1. Since E is small, higher powers of E are even smaller; discard them, leaving 4E-1. Set this to 0; we get E=1/4. Our new estimate is thus A=5/4. Repeat the process, setting A=5/4+E; after a time, one obtains a very good approximation of the zero. Viète would not have used this technique on so simple a problem, since equations of third and fourth degree could be solved exactly, but he did use it to estimate solutions of sixth-degree equations.
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Brownie points, by the way, to anyone who can explain the post title.
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I've mentioned before Diophantus' use of syncopated algebra. It was Viète who made the step forward to true symbolic algebra. Though his notation is not quite what later came to be used, it's still a great improvement. Diophantus, limited by the Greek convention of using letters as numerals, had to introduce new symbols to represent a variable and its square and cube. Viète, not so restricted, proposed the use of letters to represent numbers - more specifically, he suggested the use of vowels (voyelles) to represent variables and consonants (consonnes) to represent constants. (That the mnemonic works as well in English as in French is not a coincidence, but one wonders whether speakers of other languages would find it as apt.) This opened the door to equations involving several variables; Diophantus had been forced to deal with such problems by immediately reducing them to single-variable status. In addition to this, Viète introduced a new way of dealing with powers of numbers, suffixing "quad" and "cub" to the variables to represent their squares and cubes. Unfortunately, he still was bound by the requirements of homogeneity; he found it necessary to label certain of his constants with the suffixes "plano" and "solido" for this purpose. Thus, for example, he would write
B 5 in A quad - C plano 2 in A + A cub aequatur D solidowhere a later mathematician would write
5BA2-2CA+A3=D.(A is conceived of as a length; for all quantities to have the same dimensions, C must be an area and D a volume.) Despite its limitations, Viète's notation is a stride forward, and was used by, e.g., Fermat.
Another interesting achievement was an early use of recursion, in an algorithm for approximating the zeros of polynomials. For example, suppose one desires a zero of A3+A-3. Begin with an initial estimate, say A=1. Now set A=1+E and expand, obtaining E3+3E2+4E-1. Since E is small, higher powers of E are even smaller; discard them, leaving 4E-1. Set this to 0; we get E=1/4. Our new estimate is thus A=5/4. Repeat the process, setting A=5/4+E; after a time, one obtains a very good approximation of the zero. Viète would not have used this technique on so simple a problem, since equations of third and fourth degree could be solved exactly, but he did use it to estimate solutions of sixth-degree equations.
The alert may notice that this technique is essentially Newton's Method as applied to polynomials; if f(x) is a polynomial, Viète's technique approximates f(x+e) as f(x)+f'(x)e. Fermat, for one, was to take advantage of this idea later.
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Brownie points, by the way, to anyone who can explain the post title.
Ramble Contents
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