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Blaise Pascal wrote Sums of Numerical Powers in 1654. See Pengelley, p. 5 who is citing Pascal for the following promise in his treatise: "Given, starting with the unit, some consecutive numbers, for example 1, 2, 3, 4, one knows, by the methods the Ancients made known to us, how to find the sum of their squares, and also the sum of their cubes; but these methods, applicable only to the second and third degrees, do not extend to higher degrees. In this treatise, I will teach how to calculate not only the sum of squares and of cubes, but also the sum of the fourth powers and those of higher powers up to infinity . . . " David Pengelley describes in his paper that Pascal's binomial expansions "tediously" solve for sums of series. In comparison, Bernoulli numbers, worked out after Pascal, are credited as the more functional pattern that emerges for formulas. When considering more correspondence of 1636 between Pierre de Fermat and Pascal, (also cited by Pengelley,) I think we catch that Fermat and Pascal were on another scent than specifically Bernoulli numbers when they were thinking there are good figurate number methods to solve for the sums of powers:
In his correspondence of 1636, Pierre de Fermat called the problem of finding formulas for sums of powers "what is perhaps the most beautiful problem of all arithmetic", and claimed a recursive solution using figurate numbers, which could then be applied to integrate the "higher parabolas" x^k (Boyer, 1943; Katz, 1998, p. 481ff). The only details Fermat gave were his claims about "figurate numbers" that
"The last number multiplied by the next larger number is double the collateral triangle; the last number multiplied by the triangle of the next larger is three times the collateral pyramid; the last number multiplied by the pyramid of the next larger is four times the collateral triangulo-triangle; and so on indefinitely in this same manner." [Taking the "last number" as x and "the collateral" also as the xth: (x)*((x+1)th of n-PT figurate) = (n)*(xth of (n+1)PT figurate]
How might interrelationships of series in Pascal's Triangle relate to Euler/Pascal formulas that solve for shells, powers, sums of powers, sums-of-sum-of-power, etc.? The Euler/Pascal cube of relationships graph an answer.
Fermat's above quoted figurate observation is evidence of his focus on figurate content. After all, the figurate content of powers make quite visible figurate patterns if the accumulating values of the sieved series that chart powers accumulating from integers are graphed with colored dots.
Doesn't Pascal's promise for his publication Sums of Numerical Powers sound as if it didn't quite pan out?
But beauty is in all of the interrelationships between series: figurate numbers, power series, multiplicative relations, summed relations, order relations, cumulative relations . . .
. . . and the Euler/Pascal relationships are all about powers (or dimensions) as functions of BOTH a power-level rank AS WELL AS a power's accumulation-level. |
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