He was born at Fontenay-le-Comte, in Poitou, and is believed to have been brought up as a Roman Catholic; but there is no doubt that he was a Huguenot for several years. On the completion of his studies in law at Poitiers Vieta began his career as an advocate in his native town. He left in about 1567, and later became a councillor of the parlement of Brittany, at Rennes. The religious troubles drove him out, and Henri, duc de Rohan, a well-known leader of the Huguenots, took him under his special protection, recommending him in 1580 as a "maître des requetes" (master of requests). Henry of Navarre, at Rohan's instigation, addressed two letters to King Henry III of France on March 3 and April 26 1585, in an attempt to obtain Vieta's restoration to his former office; he failed. After Henry of Navarre became King of France, Vieta was given the position of councillor of the parlement at Tours (1589). He afterwards became a royal privy councillor, and remained so till his death, which took place suddenly at Paris in February 1603. THe cause of his death is unknown. Alexander Anderson, the editor of his scientific writings, speaks of a "praeceps et immaturum autoris fatum."
While at Tours, Vieta discovered the key to a Spanish cipher, consisting of more than 500 characters, and this meant that all despatches in that language which fell into the hands of the French could be easily read. However, his fame now rests entirely on his achievements in mathematics. Being wealthy, he printed numerous papers at his own expense, in which he wrote on various branches of the science, and sent them to scholars in almost every country of Europe. Evidence of his character is found in the fact that he entertained as a guest, for a whole month, a scientific adversary, Adriaan van Roomen, and then paid the expenses of his journey home. Vieta's writings became very quickly known; but, when Franciscus van Schooten issued a general edition of his works in 1646, some were lost.
Vieta's writings lacked discipline. In devising technical terms derived from the Greek he seems to have aimed at making them as unintelligible as possible. None of them has held its ground, and even his proposal to denote unknown quantities by the vowels A, E, I, O, U, Y--the consonants B, C, etc., being reserved for general known quantities--was not taken up. In this denotation he followed, perhaps, some older contemporaries, such as Petrus Ramus, who designated the points in geometrical figures by vowels, making use of consonants, R, S, T, etc., only when these were exhausted. Vieta is sometimes called the father of modern algebra. This does not mean that nobody before him had ever thought of choosing symbols different from numerals, such as the letters of the alphabet, to denote the quantities of arithmetic, but that he made the custom popular. All that is wanting in his writings, especially in his Isagoge in artem analyticam (1591), in order to make them look like a modern school algebra, is the sign of equality--an absence which is more striking because Robert Recorde had made use of the present symbol for this purpose since 1557 and Guilielmus Xylander had employed vertical parallel lines since 1575.
On the other hand, Vieta was well skilled in most modern artifices, aiming at the simplification of equations by the substitution of new quantities having a certain connexion with the primitive unknown quantities. Another of his works, Recensio canonica effectionum geometricarum, bears a modern stamp, being what was later called an algebraic geometry-- a collection of precepts how to construct algebraic expressions with the use of rule and compass only. While these writings were generally intelligible, and therefore of the greatest didactic importance, the principle of homogeneity, first enunciated by Vieta, was so far in advance of his times that most readers seem to have passed it over without adverting to its value. That principle had been made use of by the Greek authors of the classic age; but of later mathematicians only Hero, Diophantus, etc., ventured to regard lines and surfaces as mere numbers that could be joined to give a new number, their sum.
The study of such sums, found in the works of Diophantus, may have prompted Vieta to lay down the principle that quantities occurring in an equation ought to be homogeneous, all of them lines, or surfaces, or solids, or supersolids--an equation between mere numbers being inadmissible. During the centuries that have elapsed between Vieta's day and the present, several changes of opinion have taken place on this subject. Modern mathematicians like to make homogeneous such equations as are not so from the beginning, in order to get values of a symmetrical shape. Vieta himself did not see that far; nevertheless he indirectly suggested the thought. He also conceived methods for the general resolution of equations of the second, third and fourth degrees different from those of Scipione dal Ferro and Lodovico Ferrari, with which he must have been acquainted. He devised an approximate numerical solution of equations of the second and third degrees, wherein Leonardo of Pisa must have preceded him, but by a method which is completely lost.
He knew the connexion existing between the positive roots of an equation (which, by the way, were alone thought of as roots) and the coefficients of the different powers of the unknown quantity. He found out the formula for deriving the sine of a multiple angle, knowing that of the simple angle with due regard to the periodicity of sines. This formula must have been known to Vieta in 1593. In that year Adriaan van Roomen gave out as a problem to all mathematicians an equation of the 45th degree, which, being recognized by Vieta as depending on the equation between sin and sin 4>/45, was resolved by him at once, all the twenty-three positive roots of which the said equation was capable being given at the same time (see trigonometry), Such was the first encounter of the two scholars. A second took place when Vieta pointed to Apollonius's problem of taction as not yet being mastered, and Adriaan van Roomen gave a solution by the hyperbola. Vieta, however, did not accept it, as there existed a solution by means of the rule and the compass only, which he published himself in his Apollonius Callus (1600). In this paper Vieta made use of the centre of similitude of two circles. Lastly he gave an infinite product for the number x (see squaring the circle).
Vieta's collected works were issued under the title of Opera Mathematica by F van Schooten at Leiden in 1646.