A Historical Survey of Algebraic Methods of Approximating the Roots of Numerical Higher Equations Up to the Year 1819 |
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Page 2
... illustrations and the details of explanation as far as possible from the original sources . The writer has had unusual opportunities for studying such source materials in four notable libraries , the Columbia University Library , the ...
... illustrations and the details of explanation as far as possible from the original sources . The writer has had unusual opportunities for studying such source materials in four notable libraries , the Columbia University Library , the ...
Page 10
... illustrations of his formula : 10 Günther , Quadratische Irrationaliteten der Alten , pp . 45-46 ; Cantor , I ( 1908 ) , pp . 633 , 641 ; Hankel , Zur Geschichte der Mathematik im Alterthum und Mittelalter ( hereafter referred to as ...
... illustrations of his formula : 10 Günther , Quadratische Irrationaliteten der Alten , pp . 45-46 ; Cantor , I ( 1908 ) , pp . 633 , 641 ; Hankel , Zur Geschichte der Mathematik im Alterthum und Mittelalter ( hereafter referred to as ...
Page 19
... illustration we may take his solution of the equation x2 + x = 39 ១០ . 81 In this case we see that x = 5 is too small , while x = Hence the first mean is 5 + 6 II = - I + I 2 6 is too large . I By substitution this proves 6 - I too ...
... illustration we may take his solution of the equation x2 + x = 39 ១០ . 81 In this case we see that x = 5 is too small , while x = Hence the first mean is 5 + 6 II = - I + I 2 6 is too large . I By substitution this proves 6 - I too ...
Page 22
... illustrations.5 == 300x +33915024 ; 13 < 300. I if x = Suppose x3 I , then + 33915024 ; if x = 103 < 300 . 10 IO , then 100 , then 1003 < 300 : 100 + 33915024 ; 1000 , then 10003 > 300. 1000 + 33915024 . + 33925024 ; if x = if x = Hence ...
... illustrations.5 == 300x +33915024 ; 13 < 300. I if x = Suppose x3 I , then + 33915024 ; if x = 103 < 300 . 10 IO , then 100 , then 1003 < 300 : 100 + 33915024 ; 1000 , then 10003 > 300. 1000 + 33915024 . + 33925024 ; if x = if x = Hence ...
Page 28
... illustration . 2. With Vieta the fundamental operation for finding the roots of pure as well as affected equations is the expansion of the powers of a binomial.4 3. For each approximation a , Vieta takes the residual K −ƒ ( a ) as the ...
... illustration . 2. With Vieta the fundamental operation for finding the roots of pure as well as affected equations is the expansion of the powers of a binomial.4 3. For each approximation a , Vieta takes the residual K −ƒ ( a ) as the ...
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Common terms and phrases
3a²e a₁ affected equations algebra Analyse Générale approximating roots Arabs arrangement b₁ b₂ Bürgi c+ ggg calculus canonical form Cantor Cardan Ch'in Chiu-shao Chuquet complete equations cube roots DeLagny derived digits divisor double false position Eductio lateris singularis equa equation x3 finding roots finding the roots formula fractions gives Halley Halley's Hankel Harriot Heron Heron of Alexandria Hindu method Horner's method illustration irrational roots John Wallis Lagrange large numbers Leonardo of Pisa Math Mathematics Mean Numbers method of approximating method of double method of exhaustion method of finding method of solving Newton's method Nouveaux Élémens numerical equations numerical higher equations Oughtred Pitiscus Plane coefficient problem Raphson Regula Aurea Resolvend Rolle's roots of numbers Ruffini solution of numerical Stevin Stevin's rule substitution Subtrahend Taylor's Theon theory of equations Thomas Harriot tion Va³ Vieta Vieta's method Wallis William Oughtred
Popular passages
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.Oó~ = 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.