François Viète

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François Viète

François Viète (1540 – 1603), Seigneur de la Bigotière, was a French mathematician, also known as Franciscus Vieta, Francois Vieta or Francois Viete, whose new algebra was an important step towards modern algebra, with innovations such as the use of letters as parameters in equations. He was a lawyer serving as a privy councillor to kings of France, Henry III and Henry IV.


In artem analyticem Isagoge (1591)[edit]

from Frédéric Louis Ritter's French Tr. Introduction à l'art Analytique (1868)

  • On symbolic use of equalities and proportions. Chapter II.
    The analytical method accepts as proven the most famous [ as known from Euclid ] symbolic use of equalities and proportions that are found in items such as:
    1. The whole is equal to the sum of its parts.
    2. Quantities being equal to the same quantity have equality between themselves. [a = c & b = c => a = b]
    3. If equal quantities are added to equal quantities the resulting sums are equal.
    4. If equals are subtracted from equal quantities the remains are equal.
    5. If equal equal amounts are multiplied by equal amounts the products are equal.
    6. If equal amounts are divided by equal amounts, the quotients are equal.
    7. If the quantities are in direct proportion so also are they are in inverse and alternate proportion. [a:b::c:d=>b:a::d:c & a:c::b:d]
    8. If the quantities in the same proportion are added likewise to amounts in the same proportion, the sums are in proportion. [a:b::c:d => (a+c):(b+d)::c:d]
    9.If the quantities in the same proportion are subtracted likewise from amounts in the same proportion, the differences are in proportion. [a:b::c:d => (a-c):(b-d)::c:d]
    10. If proportional quantities are multiplied by proportional quantities the products are in proportion. [a:b::c:d & e:f::g:h => ae:bf::cg:dh]
    11. If proportional quantities are divided by proportional quantities the quotients are in proportion. [a:b::c:d & e:f::g:h => a/e:b/f::c/g:d/h]
    12. A common multiplier or divisor does not change an equality nor a proportion. [a:b::ka:kb & a:b::(a/k):(b/k)]
    13. The product of different parts of the same number is equal to the product of the sum of these parts by the same number. [ka + kb = k(a+b)]
    14. The result of successive multiplications or divisions of a magnitude by several others is the same regardless of the sequential order of quantities multiplied times or divided into that magnitude.
    But the masterful symbolic use of equalities and proportions which the analyst may apply any time is the following:
    15. If we have three or four magnitudes and the product of the extremes is equal to the product means, they are in proportion.[ad=bc => a:b::c:d OR ac=b2 => a:b::b:c]
    And conversely
    10. If we have three or four magnitudes and the first is to the second as the second or the third is to the last, the product of the extremes is equal to that of means. [a:b::c:d => ad=bc OR a:b::b:c => ac=b2]
    We can call a proportion the establishment of an equality [equation] and an equality [equation] the resolution of a proportion.
    • Note: translation utilized Google translate with reference to English translation in Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (1968) Appendix

Quotes about Viète[edit]

Introduction to the Literature of Europe in the Fifteenth, Sixteenth, and Seventeenth Centuries (1866)[edit]

Henry Hallam, 'Vol. 1

  • Ars Magna, published in 1545... contains many valuable discoveries; but that which has been most celebrated is the rule for the solution of cubic equations, generally known by Cardan's name, though he had obtained it from a man of equal genius in algebraic science, Nicolas Tartaglia. ...Cossali has ingeniously attempted to trace the process by which Tartaglia arrived at this discovery; one which, when compared with the other leading rules of algebra, where the invention... has generally lain much nearer the surface, seems an astonishing effort of sagacity. Even Harriott's beautiful generalization of the composition of equations was prepared by what Cardan and Vieta had done before, or might have been suggested by observation in the less complex cases.
    Cardan, though not entitled to the honor of this discovery, nor even equal, perhaps, in mathematical genius to Tartaglia, made a great epoch in the science of algebra; and according to Cossali and Hutton, has a claim to much that Montucla has unfairly or carelessly attributed to his favorite, Vieta.
  • Cossali has given the larger part of a quarto volume to the algebra of Cardan; his object being to establish the priority of the Italian's claim to most of the discoveries ascribed by Montucla to others, and especially to Vieta. Cardan knew how to transform a complete cubic equation into one wanting the second term; one of the flowers which Montucla has placed on the head of Vieta; and this he explains so fully, that Cossali charges the French historian of mathematics with having never read the Ars Magna.

A History of Mathematics (1893)[edit]

Florian Cajori

  • Rhaeticus was not a ready calculator only... Up to his time, the trigonometric functions had been considered always with relation to the arc; he was the first to construct the right triangle and to make them depend directly upon its angles. It was from the right triangle that Rhæticus got his idea of calculating the hypotenuse; i.e., he was the first to plan a table of secants. Good work in trigonometry was done also by Vieta and Romanus.
  • Cardan applied the Hindoo rule of "false position" (called by him regula aurea) to the cubic, but this mode of approximating was exceedingly rough. An incomparably better method was invented by Franciscus Vieta... whose transcendent genius enriched mathematics with several important innovations... For this process, Vieta was greatly admired by his contemporaries. It was employed by Harriot, Oughtred, Pell, and others. Its principle is identical with the main principle involved in the methods of approximation of Newton and Horner. The only change lies in the arrangement of the work. This alteration was made to afford facility and security in the process of evolution of the root.
  • Vieta [was] the most eminent French mathematician of the sixteenth century.
  • He was employed throughout life in the service of the state, under Henry III and Henry IV. He was, therefore, not a mathematician by profession, but his love for the science was so great that he remained in his chamber studying, sometimes several days in succession, without eating and sleeping more than was necessary to sustain himself. So great devotion to abstract science is the more remarkable because he lived at a time of incessant political and religious turmoil.
  • During the war against Spain, Vieta rendered service to Henry IV by deciphering intercepted letters written in a species of cipher, and addressed by the Spanish Court to their governor of Netherlands. The Spaniards attributed the discovery of the key to magic.
  • An ambassador from Netherlands once told Henry IV that France did not possess a single geometer capable of solving a problem propounded to geometers by a Belgian mathematician, Adrianus Romanus. It was the solution of the equation of the forty fifth degree:—
    45y - 3795y^3 + 95634y^3 -\ldots+945y^{41} - 45y^{43} + y^{45} = C
    ...Vieta, who, having already pursued similar investigations, saw at once that this awe-inspiring problem was simply the equation by which C = 2 sin φ was expressed in terms of y = 2 sin 1⁄45 φ that since 45 = 3·3·5, it was necessary only to divide an angle once into 5 equal parts, and then twice into 3,—a division which could be effected by corresponding equations of the fifth and third degrees. Brilliant was the discovery by Vieta of 23 roots to this equation, instead of only one. The reason why he did not find 45 solutions, is that the remaining ones involve negative sines, which were unintelligible to him.
  • Detailed investigations on the famous old problem of the section of an angle into an odd number of equal parts, led Vieta to the discovery of a trigonometrical solution of Cardan's irreducible case in cubics. He applied the equation (2 cos 1⁄3 φ)3 - 3 (2 cos 1⁄3 cos φ) = 2 cos φ to the solution of x3 - 3 a2x = a2b, when a > ½ b, by placing x = 2 a cos 1⁄3 φ, and determining φ from b = 2a cos φ.
  • The main principle employed by him in the solution of equations is that of reduction. He solves the quadratic by making a suitable substitution which will remove the term containing x to the first degree. Like Cardan, he reduces the general expression of the cubic to the form x3 + mx + n = 0; then assuming x = (1⁄3 a - z2z and substituting, he gets z6 - bz3 - 1⁄27 a3 = 0. Putting z3 = y, he has a quadratic. In the solution of bi-quadratics, Vieta still remains true to his principle of reduction. This gives him the well-known cubic resolvent. He thus adheres throughout to his favourite principle, and thereby introduces into algebra a uniformity of method which claims our lively admiration.
  • In Vieta's algebra we discover a partial knowledge of the relations existing between the coefficients and the roots of an equation. He shows that if the coefficient of the second term in an equation of the second degree is minus the sum of two numbers whose product is the third term, then the two numbers are roots of the equation. Vieta rejected all except positive roots; hence it was impossible for him to fully perceive the relations in question.
  • The most epoch making innovation in algebra due to Vieta is the denoting of general or indefinite quantities by letters of the alphabet. To be sure, Regiomontanus and Stifel in Germany, and Cardan in Italy, used letters before him, but Vieta extended the idea and first made it an essential part of algebra. The new algebra was called by him logistica speciosa in distinction to the old logistica numerosa.
  • Vieta's formalism differed considerably from that of to-day. The equation a3 + 3a2b + 3ab2 + b3 = (a + b)3 was written by him "a cubus + b in a quadr. 3 + a in b quadr. 3 + b cubo æqualia a+b cubo."
  • In numerical equations the unknown quantity was denoted by N, its square by Q, and its cube by C. Thus the equation x3 - 8 x2 + 16 x = 40 was written 1 C - 8 Q - 16 N œqual. 40.
  • Exponents and our symbol (=) for equality were not yet in use; but... Vieta employed the Maltese cross (+) as the short-hand symbol for addition, and the (-) for subtraction. These two characters had not been in general use before the time of Vieta.

History of Mathematics (1925) Vol. 2[edit]

David Eugene Smith

  • Vieta (c. 1590) rejected the name "algebra" as having no significance in the European languages, and proposed to use the word "analysis," and it is probably to his influence that the popularity of this term in connection with higher algebra is due.
  • Vieta: 1QC - 15QQ + 85C - 225Q + 274N, aequator 120. Modern form:
    x^6 - 15x^4 + 85x^3 - 225x^2 + 274x = 120
  • He used capital vowels for the unknown quantities and capital consonants for the known, thus being able to express several unknowns and several knowns.
  • 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 had
    u^2 + (2z + a)u +(z^2 + az + b) = 0.
    He now let 2z + a = 0, whence z = -\frac{1}{2}a,and this gave
    u^2 - \frac{1}{4}(a^2 - 4b) = 0.
    u = \pm \frac{1}{2} \sqrt{a^2 - 4b}.
    x = u + z = -\frac{1}{2}a \pm \sqrt{a^2 - 4b}.
  • Although Cardan reduced his particular equations to forms lacking a term in x^2, it was Vieta who began with the general form
    x^3 + px^2 + qx + r = 0
    and made the substitution x = y -\frac{1}{3}p, thus reducing the equation to the form
    y^3 + 3by = 2c.
    He then made the substitution
    z^3 + yz = b,   or   y = \frac{b - z^2}{z},
    which led to the form
    z^6 + 2cz^2 = b^2,
    a sextic which he solved as a quadratic.
  • Vieta was the first algebraist after Ferrari to make any noteworthy advance in the solution of the biquadratic. He began with the type x^4 + 2gx^2 + bx = c, wrote it as x^4 + 2gx^2 = c - bx, added gx^2 + \frac{1}{4}y^2 + yx^2 + gy 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).

The Development of Mathematics (1940)[edit]

Eric Temple Bell

  • Letters had been used before Vieta to denote numbers, but he introduced the practice for both given and unknown numbers as a general procedure. He thus fully recognized that algebra is on a higher level of abstraction than arithmetic. This advance in generality was one of the most important steps ever taken in mathematics. The complete divorce of algebra and arithmetic was consummated only in the nineteenth century, when the postulational method freed the symbols of algebra from any necessary arithmetical connotation.
  • Improving on the devices of his European predecessors, Vieta gave a uniform method for the numerical solution of algebraic equations. was essentially the same as Newton's (1669)... Although Vieta's method has been displaced by others... The method applies to transcendental equations as readily as to algebraic when combined with expansions to a few terms by Taylor's or Maclaurin's series.
  • An algebraic equation of degree 45 which Vieta attacked in reply to a challenge indicates the quality of his work in trigonometry. Consistently seeking the generality underlying particulars, Vieta had found how to express sin (n a positive integer) as a polynomial in sin θ, cos θ. He saw at once that the formidable equation of his rival had manufactured from an equivalent of dividing the circumference of the unit circle into 45 equal parts. ...More important than this spectacular feat was Vieta's suggestion that cubics can be solved trigonometrically.
  • Vieta's principle advance in trigonometry was his systematic application of algebra. ...he worked freely with all six of the usual functions, and... obtained many of the fundamental identities algebraically. With Vieta, elementary (non-analytic) trigonometry was practically completed except on the computational side. All computation was greatly simplified early in the seventeenth century by the invention of logarithms.

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