David Eugene Smith Quotes

Vieta was the first algebraist after Ferrari to make any noteworthy advance in the solution of the biquadratic. He began with the type <math>x^4 + 2gx^2 + bx = c,</math> wrote it as <math>x^4 + 2gx^2 = c - bx,</math> added <math>gx^2 + \frac{1}{4}y^2 + yx^2 + gy</math> to both sides, and then made the right side a square after the manner of Ferrari. This method... requires the solution of a cubic resolvent.
Descartes (1637) next took up the question and succeeded in effecting a simple solution... a method considerably improved (1649) by his commentator Van Schooten. The method was brought to its final form by Simpson (1745).

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.

The excellent work of Tropfke is an example of the tendency to break away from the mere chronological recital of facts.

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 [of Deventer].

The first noteworthy attempt to write an algebra in England was made by , whose Whetstone of witte (1557) was an excellent textbook for its time. The next important contribution was Masterson's incomplete treatise of 1592-1595, but the work was not up to the standard set by Recorde.
The first Italian textbook to bear the title of algebra was Bombelli's work of 1572. By this time elementary algebra was fairly well perfected, and it only remained to develop a good symbolism. ...this was worked out largely by Vieta (c. 1590), Harriot (c. 1610), Oughtred (c. 1628), Descartes (1637), and the British school of Newton's time (c. 1675).
So far as the great body of elementary algebra is concerned, therefore, it was completed in the 17th century.

The fact that arithmetic and geometry took such a notable step forward... was due in no small measure to the introduction of Egyptian papyrus into Greece. This event occurred about 650 B.C., and the invention of printing in the 15th century did not more surely effect a revolution in thought than did this introduction of writing material on the northern shores of the Mediterranean Sea just before the time of Thales.

In 1673 he wrote his great work De Algebra Tractatus; Historicus & Practicus, of which an English edition appeared in 1685. In this there is seen the first serious attempt in England to write on the history of mathematics, and the result shows a wide range of reading of classical literature of the science. This work is also noteworthy because it contains the first of an effort to represent the imaginary number graphically by the method now used. The effort stopped short of success but was an ingenious beginning.

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

He states that the root of <math>x^3 + 6x = 20</math> is<math>x = \sqrt {\sqrt{108} + 10} - \sqrt {\sqrt{108} - 10}.</math>

He... gave thirteen forms of the cubic which have positive roots, these having already been given by Omar Kayyam.

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.

The Arabs contributed nothing new to the theory, but al-Khowârizmî (c. 825) states the usual rules, and the same is true of his successors.

Of the contemporaries of Newton one of the most prominent was John Wallis. ...Wallis was a voluminous writer, and not only are his writings erudite, but they show a genius in mathematics... He was one of the first to recognize the significance of the generalization of exponents to include negative and fractional as well as positive and integral numbers. He recognized also the importance of Cavalieri's method of indivisibles, and employed it in the quadrature of such curves as y=xⁿ, y=x^1/n, and y=x⁰ + x¹ + x² +... He failed in his attempts at the approximate quadrature of the circle by means of series because he was not in possession of the general form of the binomial theorem. He reached the result, however, by another method. He also obtained the equivalent of <math>ds = \!dx \sqrt{1+(\frac{dy}{dx})^2}</math> for the length of an element of a curve, thus connecting the problem of rectification with that of quadrature.

His writings include works on mechanics, sound, astronomy, the tides, the laws of motion, the Torricellian tube, botany, physiology, music, the calendar (in opposition to the Gregorian reform), geology, and the compass,—a range too wide to allow of the greatest success in any of the lines of his activity. He was also an ingenious cryptologist and assisted the government in deciphering diplomatic messages.

With the coming of the Jesuits in the 16th century, and the consequent introduction of Western science, China lost interest in her native algebra...