François Viète (Mathematician)

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 nθ 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 Vietas suggestion that cubics can be solved trigonometrically.

— François Viète art

Cossali has given the larger part of a quarto volume to the algebra of Cardan; his object being to establish the priority of the Italians 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.

— François Viète art

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.

— François Viète time

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.

— François Viète change

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.

— François Viète love

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.

— François Viète war

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:—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.

— François Viète art

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 Cardans irreducible case in cubics. He applied the equation 2 cos 1⁄3 φ3 - 3 2 cos 1⁄3 cos φ = 2 cos φ to the solution of x3 - 3 a2x = a2b, when a½ b, by placing x = 2 a cos 1⁄3 φ, and determining φ from b = 2a cos φ.

— François Viète art

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 - z2÷z 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.

— François Viète live

In Vietas 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.

— François Viète knowledge

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.

— François Viète art

Vietas formalism differed considerably from that of to-day. The equation a3 + 3a2b + 3ab2 + b3 = a + b3 was written by him a cubus + b in a quadr. 3 + a in b quadr. 3 + b cubo æqualia a+b cubo.

— François Viète day

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.

— François Viète

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.

— François Viète time

He used capital vowels for the unknown quantities and capital consonants for the known, thus being able to express several unknowns and several knowns.

— François Viète being

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 equationhe placedfor . He then hadHe now letwhence and this gave and

— François Viète work

Although Cardan reduced his particular equations to forms lacking a term in , it was Vieta who began with the general formand made the substitutionthus reducing the equation to the formHe then made the substitution or which led to the forma sextic which he solved as a quadratic.

— François Viète art

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.

— François Viète action

Vietas 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.

— François Viète work

Share your thoughts on François Viète quotes with the community: