David Eugene Smith: Equation

“Although Cardan reduced his particular equations to forms lacking a term in x^2, it was Vieta who began with the general formx^3 + px^2 + qx + r = 0and made the substitution x = y -\frac{1}{3}p, thus reducing the equation to the formy^3 + 3by = 2c.He then made the substitutionz^3 + yz = b, or y = \frac{b - z^2}{z},which led to the formz^6 + 2cz^2 = b^2,a sextic which he solved as a quadratic.”

David Eugene Smith

Source: History of Mathematics (1925) Vol.2, p.465

“[Zuanne de Tonini] da Coi… impuned Tartaglia to publish his method, but the latter declined to do so. In 1539 Cardan wrote to Tartaglia, and a meeting was arranged at which, Tartaglia says, having pledged Cardan to secrecy, he revealed the method in cryptic verse and later with a full explanation. Cardan admits that he received the solution from Tartaglia, but… without any explanation. At any rate, the two cubics x^3 + ax^2 = c and x^3 + bx = c could now be solved. The reduction of the general cubic x^3 + ax^2 + bx = c to the second of these forms does not seem to have been considered by Tartaglia at the time of the controversy. When Cardan published his Ars Magna however, he transformed the types x^3 = ax^2 + c and x^3 + ax^2 = c by substituting x = y + \frac{1}{3}a and x = y - \frac{1}{3}a respectively, and transformed the type x^3 + c = ax^2 by the substitution x = \sqrt[3]{c^2/y}, thus freeing the equations of the term x^2. This completed the general solution, and he applied the method to the complete cubic in his later problems.”

David Eugene Smith

Source: History of Mathematics (1925) Vol.2, p.461

Problem , Time

“Cardan's originality in the matter seems to have been shown chiefly in four respects. First, he reduced the general equation to the type x^3 + bx = c; second, in a letter written August 4, 1539, he discussed the question of the irreducible case; third, he had the idea of the number of roots to be expected in the cubic; and, fourth, he made a beginning in the theory of symmetric functions. …With respect to the irreducible case… we have the cube root of a complex number, thus reaching an expression that is irreducible even though all three values of x turn out to be real. With respect to the number of roots to be expected in the cubic… before this time only two roots were ever found, negative roots being rejected. As to the question of symmetric functions, he stated that the sum of the roots is minus the coefficient of x2</sup”

David Eugene Smith

Source: History of Mathematics (1925) Vol.2, pp.461-464

Idea , Question , Time

“It is difficult to say when algebra as a science began in China. Problems which we should solve by equations appear in works as early as the Nine Sections (K'iu-ch'ang Suan-shu) and so may have been known by the year 1000 B. C. In Liu Hui's commentary on this work (c. 250) there are problems of pursuit, the Rule of False Position… and an arrangement of terms in a kind of determinant notation. The rules given by Liu Hui form a kind of rhetorical algebra.
The work of Sun-tzï contains various problems which would today be considered algebraic. These include questions involving indeterminate equations. …Sun-tzï solved such problems by analysis and was content with a single result…
The Chinese certainly knew how to solve quadratics as early as the 1st century B. C., and rules given even as early as the K'iu-ch'ang Suan-shu… involve the solution of such equations.
Liu Hui (c. 250) gave various rules which would now be stated as algebraic formulas and seems to have deduced these from other rules in much the same way as we should…
By the 7th century the cubic equation had begun to attract attention, as is evident from the Ch'i-ku Suan-king of Wang Hs'iao-t'ung (c. 625).
The culmination of Chinese is found in the 13th century. …numerical higher equations attracted the special attention of scholars like Ch'in Kiu-shao (c.1250), Li Yeh (c. 1250), and Chu-Shï-kié (c. 1300), the result being the perfecting of an ancient method which resembles the one later developed by W. G. Horner”

David Eugene Smith

1819
Source: History of Mathematics (1925) Vol.2, Ch. 6: Algebra

Way , Question , Problem , Appearance

“It is difficult to say who it is who first recognized the advantage of always equating to zero in the study of the general equation. It may very likely have been Napier, for he wrote his De Arte Logistica before 1594, and in this there is evidence that he understood the advantage of this procedure. Bürgi also recognized the value of making the second member zero, Harriot may have done the same, and the influence of Descartes was such that the usage became fairly general.”

David Eugene Smith

History of Mathematics (1925) Vol.2

Art , Study

“The law which asserts that the equation X = 0, complete or incomplete, can have no more real positive roots than it has changes of sign, and no more real negative roots than it has permanences of sign, was apparently known to Cardan; but a satisfactory statement is possibly due to Harriot (died 1621) and certainly to Descartes.”

David Eugene Smith

Source: History of Mathematics (1925) Vol.2, p.469

Law , Change

“The first writer on algebra whose works have come down to us is Ahmes. He has certain problems in linear equations and in series, and these form the essentially new feature in his work. His treatment of the subject is largely rhetorical.”

David Eugene Smith

Source: History of Mathematics (1925) Vol.2, Ch. 6: Algebra

Problem

“There are only four Hindu writers on algebra whose names are particularly noteworthy. These are Āryabhata, whose Āryabhatiyam (c. 510) included problems in series, permutations, and linear and quadratic equations; Brahmagupta, whose Brahmasiddhānta (c. 628) contains a satisfactory rule for solving the quadratic… Mahāvīra, whose Ganita-Sāra Sangraha (c. 850) contains a large number of problems involving series, radicals, and equations; and Bhāskara, whose Bija Ganita (c. 1150)… extends the work through quadratic equations.”

David Eugene Smith

Source: History of Mathematics (1925) Vol.2, Ch. 6: Algebra

Problem

“The problem of the biquadratic equation was laid prominently before Italian mathematicians by Zuanne de Tonini da Coi, who in 1540 proposed the problem, "Divide 10 parts into three parts such that they shall be continued in proportion and that the product of the first two shall be 6." He gave this to Cardan with the statement that it could not be solved, but Cardan denied the assertion, although himself unable to solve it. He gave it to Ferrari, his pupil, and the latter, although then a mere youth, succeeded where the master had failed. …This method soon became known to algebraists through Cardan's Ars Magna, and in 1567 we find it used by Nicolas Petri”

David Eugene Smith

of Deventer
Source: History of Mathematics (1925) Vol.2, pp.467-468

Youth , Problem , Parting

“In the work of Vieta the analytic methods replaced the geometric, and his solutions of the quadratic equation were therefore a distinct advance upon those of his predecessors. For example, to solve the equation x^2 + ax + b = 0 he placed u + z for x. He then hadu^2 + (2z + a)u +(z^2 + az + b) = 0.He now let 2z + a = 0, whence z = -\frac{1}{2}a,and this gaveu^2 - \frac{1}{4}(a^2 - 4b) = 0.
u = \pm \frac{1}{2} \sqrt{a^2 - 4b}.andx = u + z = -\frac{1}{2}a \pm \sqrt{a^2 - 4b}.</center”

David Eugene Smith

Source: History of Mathematics (1925) Vol.2, p.449

“Algebra in the Renaissance period received its first serious consideration in Pacioli's Sūma (1494)… which characterized in a careless way the knowledge… thus far accumulated. By the aid of the crude symbolism then in use it gave a considerable amount of work in equations.
The noteworthy work… and the first to be devoted entirely to the subject, was Rudolff's Coss (1525). This work made no decided advance in the theory, but it improved the symbolism for radicals and made the science better known in Germany. Stiffel's edition of this work (1553-1554) gave the subject still more prominence.
The first epoch-making algebra to appear in print was the Ars Magna of Cardan (1545). The next great work… to appear in print was the General Trattato of Tartaglia…”

David Eugene Smith

Source: History of Mathematics (1925) Vol.2, p. 384; Ch. 6: Algebra

Way , Knowledge , Appearance , Science

“When we speak of the early history of algebra it is necessary to consider… the meaning of the term. If… we mean the science that allows us to solve the equation ax^2 + bx + c = 0, expressed in these symbols, then the history begins in the 17th century; if we remove the restriction as to these particular signs, and allow for other and less convenient symbols, we might properly begin the history in the 3rd century; if we allow for the solution of the above equation by geometric methods, without algebraic symbols of any kind, we might say that algebra begins with the Alexandrian School or a little earlier; and if we say that we should class as algebra any problem that we should now solve with algebra (even though it was as first solved by mere guessing or by some cumbersome arithmetic process), the the science was known about 1800 B. C., and probably still earlier.<”

David Eugene Smith

Source: History of Mathematics (1925) Vol.2, Ch. 6: Algebra, p. 378

History , Problem , Science