Horner’s method of approximation anticipated by Ruffini

Abstract

Not all of the types of symmetry enumerated in this table are available as types of crystal symmetry, for the law of rational indices limits the acceptable axes of symmetry to those of the orders 2, 3, 4, 6. With this limitation the table furnishes the 32 types of crystal symmetry, 11 from each of the first two columns and 10 from the third.

Associated Polynomials and Uniform Methods for the Solution of Linear Problems

“…ASSOCIATED POLYNOMIALS AND UNIFORM METHODS 299 polynomial equations, which depends on evaluating the derivatives of a polynomial by repeated synthetic division (see 9), Cajori [3] points out that the same results were obtained by Ruffini in 1804. Cajori recommends naming this the Rutfini-Horner method.…”

“…Divided differences. It is well known that repeated synthetic division of polynomial P by a fixed number u0 results in the calculation of j 0, 1, n. This fact lies at the heart of the Ruffini-Horner method for solving algebraic equations [3], [9]. This method depends on calculating from the coefficients of a given polynomial the coefficients of a second polynomial all of whose zeros are less than those of the given polynomial by an amount u0.…”

Horner versus Holdred: An Episode in the History of Root Computation

“…Holdred seemed to misunderstand slightly the Viète method since he used, in place of the trial divisor (14), the sum of the coefficients (15). However, both expressions are often dominated by b n−1 , so both will often yield the same trial digit.…”

“…Here, he rightly criticized Holdred for expressing the trial divisor in the Viète procedure as a sum of coefficients (see (15)), and for the incompleteness of Holdred's proof. He also stated that with respect to the method in Holdred's supplement "Mr. Holdred has no claim to style himself the original inventor of it, as I was the first to give him any hint of it .…”

“…Florian Cajori [15] pointed out that the method was anticipated by Paolo Ruffini [68; 69], in Italy. Long before then related techniques were known to Chinese and Arabic writers; see for example, Martin Nordgaard [60,[54][55], Paul Luckey [48; 49], Adolf Youschkevitch [45, 41-48, 242-248; 81, 76-80, 169], Youschkevitch and Boris Rosenfeld [82], Carl Boyer [10,[268][269][270], Ulrich Libbrecht [46,, Roshdi Rashed [65], and Lennart Berggren [7,[48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63].…”