A Historical Survey of Algebraic Methods of Approximating the Roots of Numerical Higher Equations Up to the Year 1819

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Teachers college, Columbia university, 1922 - Algebra - 64 pages
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Page 44 - An attempt towards the improvement of the method of approximating, in the extraction of the roots of equations in numbers," he shows the identity of the coefficients in his series and the coefficients in Halley's General Analytical Speculum.
Page 55 - ... y— 38,205,440,000=0, and this way may be taken to be the meaning of the following table, which Ch'in gives in connection with his explanation of the problem: Barring the omission of zeros, which we must attribute to a copyist's error, this result coincides with the correct one given just above. But we still have too meager information as to the details of the work for us to be able to affirm confidently that Horner's method was known to the Chinese in the 13th century; we can only say that...
Page 54 - Yoshio Mikami, The Development of Mathematics in China and Japan (New York: Chelsea Publishing Company . 1913), 324.
Page 58 - Newton's method of exhaustion, based on equations whose terms constitute an infinite series, was conceived as an aid to his work in the calculus. The place of this method as a constituent part of algebra is largely due to its modification and systematic development by Raphson. Its technique was improved by DeLagny, Halley, Taylor, and Simpson and its scientific basis was clarified and strengthened by Lagrange, Fourier, Budan, and Sturm.
Page 58 - Archimides; a specific process was described by Heron of Alexandria. In algorithmic work double false position was used by the Arabs and Leonardo of Pisa. By the Renaissance writers these processes were extended to the solutions of numerical higher equations. Though eclipsed by the methods of exhaustion invented by Vieta, Newton, and Horner, methods of double false position have never become obsolete.
Page 59 - Rolle's Method of Cascades and the method of recurring series developed by Bernoulli and Euler have been used only slightly. The methods invented by Collins, Fontaine, and some of those used by DeLagny were never taken up into the mathematical activities of the world.
Page 57 - Horner published his method in 1819, and it soon became widely used in England and later in the United States, and to a less degree in Germany, Austria, and Italy.
Page 63 - ... 18. A sphere of yellow pine 1 foot in diameter floating in water sinks to a depth x given by 2 x3 - 3 ж2 + 0.O~ = 0.
Page 58 - The method of averages was probably used by Archimedes; a definite exposition of it is given by Heron of Alexandria. Its applicability was restricted; but it was often incorporated into other methods, as in Chuquet's Rule of Mean Numbers.
Page 29 - J. Wallis, A Treatise of Algebra, Both Historical and Practical, London, 1685, p.

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