In 1564 Viète took a position in the service of Antoinette d'Aubeterre. He was employed to supervise the education of Antoinette's daughter Catherine, who would later become Catherine of Parthenay (Parthenay is about half-way between Fontenay-le-Comte and Poitiers). Catherine's father died in 1566 and Antoinette d'Aubeterre moved with her daughter to La Rochelle. Viète moved to La Rochelle with his employer and her daughter.
This was a period of great political and religious unrest in France. Charles IX had become king of France in 1560 and shortly after, in 1562, the French Wars of Religion began. It is a gross over-simplification to say that these wars were between Protestants and Roman Catholics but fighting between the various factions would continue on and off until almost the end of the century. In 1570 Viète left La Rochelle and moved to Paris. Although he was never employed as a professional scientist or mathematician, Viète was already working on topics in mathematics and astronomy and his first published mathematical work appeared in Paris in 1571. While Viète was in Paris, Charles IX authorised the massacre of the Huguenots, who were an increasingly powerful group of French Protestants, on 23 August 1572. This must have been an extremely difficult time for Viète for, although not active in the Protestant cause, he was a Huguenot himself. Charles did not live very long after this event, the massacre apparently haunting him for the rest of his life. However, on 24 October 1573 Charles appointed Viète to the government of Brittany which was based at Rennes.
Viète moved to Rennes to take up his position of counsellor there. He remained at Rennes until March 1580 when he returned to Paris. Charles IX had died on 30 May 1574 and, on Charles' death Henry III became king. Henry made concessions to the Protestant Huguenots in 1576 and the Roman Catholics formed the Holy League to look after their own interests by military actions. In this tense atmosphere Viète was appointed by Henry III as royal privy counsellor on 25 March 1580, and he was attached to the parliament in Paris.
In 1584 the Holy League was strengthened when Henry III's brother died and the Protestant Henry of Navarre became heir to the throne. Fearing that the Protestants might gain control in France, the Holy League fought more vigorously for the Roman Catholic cause. The royal court contained factions with different political aims and in 1584 Viète's position as a known Huguenot became untenable and he was banished by his political enemies from the court. Leaving Paris, Viète went to Beauvoir-sur-Mer, on the coast about 130 km northwest of his home town of Fontenay-le-Comte. During the five years that he spent at Beauvoir-sur-Mer, Viète was able to devote himself entirely to his mathematical studies. In many ways Viète's enemies did mathematics a favour, for it was during this period without formal duties that Viète's most important mathematics was done.
In 1587 Henry of Navarre defeated the army of Henry III. A rising of the people of Paris, a Holy League stronghold, on 12 May 1588, caused the king to flee to Chartres. At this stage Henry III sent for Viète and, in April 1589, brought him back into his parliament which was now set up at Tours. Henry III was reconciled with Henry of Navarre (since it suited them to combine forces) and together they tried to retake Paris in 1589. Henry III was, however, assassinated by a Jacobin friar on 1 August of that year.
Philip II of Spain, a champion of the Roman Catholic Counter-Reformation, supported the Holy League by sending money and troops to France. After the murder of Henry III, Philip claimed the throne of France for his daughter, Isabella Clara Eugenia. A letter to Philip dated 28 October 1589 written in code fell into the hands of Henry of Navarre who was to become the next king, Henry IV.
Following the assassination of Henry III, Viète worked for Henry IV. He was now in a sounder position, as a Protestant supporter of a Protestant King. Viète was certainly well known for his mathematical abilities by this time and, as one of the Henry IV's most loyal supporters, it was natural for Henry to turn to Viète to decode messages being sent to his enemy Philip II of Spain. It took Viète some time to crack the complicated code. At first he was only able to decode parts of the message and forwarded parts to Henry IV, but eventually Viète sent him the fully decoded message on 15 March 1590. However :-
... when Philip, assuming that the cipher could not be broken, discovered that the French were aware of his military plans, he complained to the Pope that black magic was being employed against his country.Although Viète was never a professional mathematician, he did lecture on mathematics. For instance in 1592 he lectured at Tours and discussed recent claims that the circle could be squared, an angle trisected, and the cube doubled using only ruler and compass. He showed in these lectures that the "proofs" which had been published earlier in the year were fallacious.
In 1592 Henry IV did not control Paris, and he was still opposed by the Holy League in France who were supported by Spain. Henry converted back to Roman Catholicism in July 1593, perhaps for political rather than religious reasons. Viète followed the example of his king and also converted to Roman Catholicism. Henry's conversion was certainly effective, for resistance against him lessened and he took Paris on 22 March 1594. Henry declared war on Philip II of Spain in January 1595 and continued to wipe out resistance by the League and its Spanish allies.
During the period referred to in the previous paragraph, Viète had again come to the King's rescue by solving a mathematical problem. In 1593 Roomen had proposed a problem which involved solving an equation of degree 45. The ambassador from the Netherlands made comments to Henry IV on the poor quality of French mathematicians saying that no Frenchman could solve Roomen's problem. Henry put the problem to Viète who solved it by realising that there was an underlying trigonometric relation. As a result of this a friendship grew up between Viète and Roomen. Viète proposed the problem of drawing a circle to touch 3 given circles to Roomen (the Apollonian Problem) and Roomen solved it using hyperbolas, publishing the result in 1596. Viète himself, published his answer to Roomen's problem in 1595, stating in the introduction :-
I, who do not profess to be a mathematician, but who, whenever there is leisure, delight in mathematical studies ...Viète continued to serve Henry IV in Paris until 1597 when he went back to his home town of Fontenay-le-Comte. Two years later he was back in Paris, again in the service of Henry IV, but he was dismissed by Henry on 14 December 1602. He died almost exactly one year later.
Some of Viète's first work was directed towards the production of a major work on mathematical astronomy Ad harmonicon coeleste. It was a work which was never published but four manuscripts, one of them an autograph, have survived and were rediscovered by Libri. The contents of these manuscripts are described in  where it is stated that Viète was interested purely in the geometry of the planetary theories of both Ptolemy and Copernicus, and did not consider the question of whether the theories represented the actual physical reality. Perhaps rather surprisingly Viète came to the conclusion that Copernicus's theory was not valid geometrically.
Although the Ad harmonicon coeleste was never published, Viète did begin publishing the Canon Mathematicus in 1571 which was intended as a mathematical introduction to the astronomy treatise. The Canon Mathematicus covers trigonometry; it contains trigonometric tables, it also gives the mathematics behind the construction of the tables, and it details how to solve both plane and spherical triangles. It is interesting that in the second part of the Canon Mathematicus Viète :-
... wrote decimal fractions with the fractional part printed in smaller type than the integral and separated from the latter by a vertical line.Viète introduced the first systematic algebraic notation in his book In artem analyticam isagoge published at Tours in 1591. The title of the work may seem puzzling, for it means "Introduction to the analytic art" which hardly makes it sound like an algebra book. However, Viète did not find Arabic mathematics to his liking and based his work on the Italian mathematicians such as Cardan, and the work of ancient Greek mathematicians. One would have to say, however, that had Viète had a better understanding of Arabic mathematics he might have discovered that many of the ideas he produced were already known to earlier Arabic mathematicians.
In his treatise In artem analyticam isagoge Viète demonstrated the value of symbols introducing letters to represent unknowns. He suggested using letters as symbols for quantities, both known and unknown. He used vowels for the unknowns and consonants for known quantities. The convention where letters near the beginning of the alphabet represent known quantities while letters near the end represent unknown quantities was introduced later by Descartes in La Gèometrie. This convention is used today, often without people realising that a convention is being used at all. (If I asked for a solution to ax = b nobody asks: "For which quantity do I solve the equation ?")
Viète made many improvements in the theory of equations. However, if we are to be strictly accurate we should say that he did not solve equations as such but rather he solved problems of proportionals which he states quite explicitly is the same thing as solving equations. However, he was restricted by a condition of homogeneity of dimension. The problem is that if we ask for a solution of x3 + x = 1 then we ask for the solution to a problem which does not make sense geometrically. For x3 is a cube while x is a line and clearly it makes no sense to add a three dimensional object to a one dimensional object. Viète therefore looked for solutions of equations such as A3 + B2A = B2Z where, using his convention, A was unknown and B and Z were knowns. The dimensions here are "correct" each term being of dimension 3. Viète wrote in the In artem analyticam isagoge (see  or ):-
The first and permanent law of equalities or proportions which, because it is conceived from homogeneous quantities is called the law of homogeneous quantities, is this: homogeneous quantities must be compared with homogeneous quantities.He presented methods for solving equations of second, third and fourth degree. He knew the connection between the positive roots of equations and the coefficients of the different powers of the unknown quantity. Perhaps it is worth noting that the word "coefficient" is actually due to Viète. When Viète applied numerical methods to solve equations as he did in De numerosa potestatum he gave methods which were similar to those given by earlier Arabic mathematicians. For example his methods are compared with those of Sharaf al-Din al-Tusi in the paper  and . In the first the author argues that although the methods appear to be similar at first sight, there are many important differences. He deduces that the work of Viète is not based on that of Sharaf al-Din al-Tusi. In , however, Rashed argues that the methods of Sharaf al-Din al-Tusi and of Viète are very close indeed.
Viète also wrote books on trigonometry and geometry such as Supplementum geometriae (1593). He gave geometrical solutions to doubling a cube and trisecting an angle in this book.
In 1593 Viète published a second book, which in many ways was motivated by his lecture course at Tours in the previous year (which we mentioned above), examining various problems such as doubling the cube, trisecting an angle and the construction the tangent at any point on an Archimedian spiral. Also, in this book, he calculated π to 10 places using a polygon of 6 × 216 = 393216 sides. He also represented π as an infinite product which, as far as is known, is the earliest infinite representation of π.
Finally we should mention that Viète is often called "the father of algebra". As the author of  argues this, on the one hand, is unfair on the many fine algebraists who preceded Viète. On the other hand it is unfair to Viète since his contributions were of much wider mathematical importance.
It would also be interesting to know how much Viète's ideas were influenced by those of Harriot. In  a quotation from a book about Harriot written in 1900 by H Stevens is given:-
... it appears that Harriot's system of analytics or algebra was based on that of his friend and correspondent François Viète as Viète's was avowedly based on that of the ancients. ... Full credit was given by Harriot and his friends to the distinguished French mathematician.Although this seems to make Harriot's dependence on Viète clear, one would have to say that the two men give very similar approaches to solving equations algebraically, yet Harriot shows deeper understanding than does Viète. I [EFR] feel that one must allow the possibility that ideas flowed in both directions and that Viète's algebra must have benefited from his correspondence with Harriot.
Article by: J J O'Connor and E F Robertson
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