Historians consider the Renaissance one of the outstanding periods of genius in world history. The Renaissance began in Italy about the 14th century, and reached its height in the 15th century. In the 16th and 17th centuries it spread to much of the rest of Europe. The word renaissance, which means "rebirth" in French, refers to the rediscovery by scholars, called humanists, of texts containing the achievements of the ancient Greeks and Romans. In fact, however, the Renaissance was a period of discovery in many fields - of new scientific laws, new forms of art and literature, new religious and political ideas, and new lands, including the Americas. Nevertheless, the recovery and the fuller appreciation of the writings, arts, and attitudes of the ancient Greeks was perhaps the most distinctive feature of the period. New ideas of grace, harmony, and beauty were inspired by the sculpture and other artistic remains of classical Greece and Rome. (Compton's Interactive Encyclopedia)
In 1453 the fall of the Byzantine Empire was marked by the capture of Constantiople by the Turks, who had already overrun the rest of the Empire, including the Balkans; to cap this achievement they returned south, to capture the capital city of Constantinople. This event is said to have contributed greatly to the progress of learning in the West, according to Compton's Interactive Encyclopedia. Many Greeks entered Italy and brought with them manuscripts of Greek literature. This contributed to the process of modernity by reviving classic learning of ancient art and science. (This interpretation is disputed by many historians, among them George Sarton.) Men began once again to exercise their minds, so that they became less servile, and their ideas became clearer and stronger. During this period, men abandoned the medieval features of indistinctiveness of thought and instead began to cultivate Pure Mathematics and Astronomy.
The 16th century was a period of increased intellectual activity. Germany was the leading country in new scientific thought; the Germans brought great productivity in the sciences, mainly due to the country's commercial prosperity. Specifically mathematical activity was largely centered in the Italian cities, and in the central European cities of Nuremberg, Vienna, and Prague. The first great Renaissance contributions to the mathematical sciences were made in Italy and Germany. In Germany advances were made in astronomy and trigonometry, while brilliant contributions to algebra were made in Italy. (Cajori) Among the various Renaissance scholars, two significant mathematicians are worth special mention because of their important contributions to the mathematical field of algebra: the French mathematician of the 16th century, Francois Viéte; and the Italian mathematician Girolamo Cardano.
François Viéte is known as " [t]he greatest French mathematician of the sixteenth century... ", (Eves, p.277) and the " [f]ather of modern algebra", (Catholic Encyclopedia). Perhaps Viéte's importance to the mathematical society is because his mathematical investigations are characteristic of the "new" science at the turn of the 16th century. Viéte was the first to formalize mathematics; which allowed for a simplification of algebraic manipulations. In my opinion, without Viéte's contribution of symbolic algebra mathematicians would be significantly handicapped today. Viéte was born in 1540 in France, the son of Etienne Viéte, an attorney in Fontenay and a notary in Le Busseau. His mother was a first cousin of Barnabé Brisson, President of the Parlement de Paris. Evidence places the Viéte family among the most distinguished in Fontenay. By the age of 20 at the latest, Viéte had already earned a degree in law at the University of Poitiers, in 1560. He soon returned to Fontenay to take rank with the leading barristers at the province, and at age 24 he became a legal advisor of the Huguenot Antoinette d'Aubeterre; who remained his life long confidante. On the side he tutored Catherine, the daughter of Antoinette. Throughout Viéte's lifetime, he gained the acquaintance of highly respectable clients and friends such as the Huguenots, Coligny, Condés, the Queen of Navarre, Henry Navarre, and Francois de Rohan; all significant political figures. At times, Viéte's association with such clients hindered his career. For instance, Viéte was a Huguenot sympathizer at the time of their religious persecution. "Thus Viéte was a victim of political enemies who succeeded in having him banished from court in 1584 to 1589" (Calinger)
His recognized political career began in 1573 when Viéte went to Paris and was appointed by Charles IX to the parliament of city. During the 6 years that Viéte held a position in that parliament, he was also a personal assistant to the King himself. In 1589 he worked for the French state as a parliamentary councilor. The in 1590, during the war with Spain, Viéte assisted Henry IV of France as an interpreter of intercepted Spanish codes.
Along with his notable achievements in the political arena, Viéte also made a mark in the field of mathematics. Viéte wrote numerous works on trigonometry, algebra, and geometry such as the "Canon Mathematicus Seu Ad Triangula" in 1579, the "Artem Analyticem Isogoge" in 1591, the "Supplementum Geometriae" in 1593, "De Numerosa Potestatum Resolutione" in 1600, and "De Aequationum Recognitione Et Emendatione". His most famous work was "In Artem" because it contributed greatly to the development of symbolic algebra. Symbolic algebra is one of three types of algebra. This is the type of algebra that we use today. In symbolic algebra which all forms and operations are represented by a fully developed algebraic symbolism, i.e.: x2+ 13x + 9. In my opinion, it allows one to express his mathematical ideas in a formalized way, and it is much more convenient to read and understand. For the first time in mathematical history, Viéte had introduced the new notion of using vowels systematically to represent known quantities, and consonants for unknown ones. The main significance of this notation is the use of letters instead of numbers in the theory of equations.
Like many other Renaissance mathematicians, Viéte was a student of the ancient Greek literature produced by great mathematicians such as Euclid, Pappus, and Diophantus. He did not recognize at the time what the "analytic art" or new algebra meant for the field of algebra, which will be later discussed. Viéte developed this algebraic notation with the intention of a simple means to the end. He said "Behold, the art which I present is new, but in truth so old, so spoiled and defiled by the barbarians, that I consider it necessary, in order to introduce an entirely new form into it, to think out and publish a new vocabulary, having gotten rid of all its pseudo-technical terms, lest it should retain its filth and continue to stink in the old way ..." (Klein, p.153) His main goal was to understand the ancient Greek ideas of Euclid, Pappus, and Diophantus, rather than the later ideas of the Hindus and Arabs. Viéte was particularly interested in the notions of analysis and synthesis, thus according to Viéte his new algebra was merely " a rediscovery of ancient Greek mathematical methods which would lead to a true mathematics, basic to the search for a universal science, a search later continued by Descartes and Leibniz" (Struik). He thought that his new algebra was simply a new tool for understanding the ancient Greek notions of analysis and synthesis. He believed he was merely a preserver of the ancient Greek sciences, not neccessarily an innovator of modern algebra.
In "In Artem", Viéte refers to the ancient Greek sources. He took the ideas of Euclid, Pappus, and Diophantus and wanted to explain them further so that others could have a better understanding of the their works. Thus, he constructed a calculation with "species" rather than with numerical calculations, to make things more clear and organized. Without his systematic algebraic notation, he would not be able to efficiently explain those ideas of analysis and synthesis. (Struik) Viéte wanted to clarify the ideas of analysis and synthesis. "Analysis is the assumption of that which is sought and the arrival, by means of its consequences, at something admitted to be true"(Heath, "Euclid's Elements") While synthesis is " assumption of that which is admitted and the arrival, by means of its consequences, at something admitted to be true" (Heath, "Euclid's Elements") An example of an analysis is as follows:
An example of a synthesis is the opposite line of attack:
In the first chapter of "In Artem", Viéte discusses a type of analysis which he refers to as "rhetic". He claims that the zetetic art applies logic not the numbers but species. The zetetic art is important because it is successful in comparing magnitudes with one another in equations. In the proceeding two chapters, he uses the ideas of Euclid to establish his foundation for his algebraic notation. He takes postulates from Euclid's Elements such as "the whole is equal to the sum of its parts" (Appendix to Jacob Klien, p.322) and "things that are equal to the same thing are equal among themselves"(Appendix to Jacob Klein, p.324) He also refers to the laws of homogeneity such as " homogenous quantities must be compared with homogeneous quantities." (Appendix to Jacob Klein, p.324) In the first couple of chapters, Viéte was emphasized to pay attention to units (i.e.: inches or gallons) when using his algebraic notation, you cannot just throw all types of units into one equation and expect to understand it. Basically he is laying out the rules of using his notation, to avoid confusion and misapplication of his innovation.
In chapter 4, Viéte describes the four rules for calculation of species. Rule One is "to add a magnitude to a magnitude" (Appendix to Jacob Klein, p.329). In this rule, Viéte states that one can only add homogenous magnitudes; quantities with the same units. For example, according to Viéte's rules for species you cannot add apples with oranges, you can only add apples with apples and oranges with oranges. Viéte uses this rule to make sure that the addition sign is used correctly and not just arbitrarily. The Second Rule is " to subtract a magnitude from a magnitude". This rule is similar to the first rule, except one is subtracting. The 1st and 2nd rules are not appropriate for numbers today. Numbers are abstract entities, adding 10+5=15 is mathematically correct regardless of what units are being added The third and fourth rule, respectively, "To multiply a magnitude by a magnitude: Let there be 2 magnitudes A and B. It is required to multiply the one by the other. Since, then, a magnitude is to be multiplied by a magnitude, they will by their multiplication produce a magnitude heterogeneous in relation to each of them" (Appendix to Jacob Klein, p.332-338) and "To divide a magnitude by a magnitude" (Appendix to Jacob Klein, p.332-338). These two rules basically say that when a magnitude is multiplied (divided) by another magnitude, the solution will always be heterogeneous (different in kind) to the original magnitudes that the process is performing on. Apparently Viéte's claim is logically correct. Multiplying any 2 kinds will always give a new kind that is different form the original two. For example, a side multiplied by a side is a plane (not another side).
Viéte's "In Artem" dealt with variables (species), operators, and units. His laws introduce variables (species) to a certain power, the law of homogeneity, and the idea of denoting unknowns by vowels. Without his accomplishment, people would not be able to work easily with equations. In my opinion, if Viéte had not constructed such a formal original algebraic notation system, the level of mathematical development possibly could have reached a standstill. During the times of the Renaissance, it was hard for people to conceptualize mathematical problems and ideas. Unknowingly, Viéte set up the building blocks for people to use later. Apparently he was successful in creating a language that allowed mankind to express their ideas, logic, and problems in a systematic and organized way. His new algebra was something that could stand on its own, it wasn't merely a tool for understanding analysis and synthesis, instead it was a whole new way for people to look at mathematical problems and notions. Thus as Viéte would say "[We will] leave no problem unsolved."
In contrast with Viéte's respectable reputation and dignified family background, the controversial and enigmatic Italian mathematician Girolamo Cardano led a stormy and unstable life with many ups and downs, twists and turns. "Cardano's character was an enigma to many of his contemporaries...He is a man who has been praised and vilified; by some he has been called a genius, by others a poseur, some have presented him as a benefactor to mankind, others frankly believed him to be an evil spirit, indeed a monster." (Ore, p25)
Girolamo Cardano was born in 1501 in Milan, in northern Italy. He was the illegitimate child of Fazio Cardano and Chiara Micheria. Fazio was a man of universal interests; he was a lawyer in Milan but was also deeply involved in the medical sciences and was an expert in mathematics. In fact, even the great Leonardo da Vinci had consulted with Fazio many times in respect to geometric questions. At a late age Fazio met Girolamo's mother Chiara, who was much younger than Fazio. Chiara did not come from high society and thus was not socially acceptable in Milan. She was a young widow with three children from her past marriage when Fazio met her. Eventually, Fazio married Girolamo's mother.
While growing up, Cardano was an assistant to his father much like a servant. Cardano became anxious to receive an education, despite his father's wishes for him to merely receive home schooling. After a long struggle with his father, Cardano eventually received his father's consent allowing him to attend his father's old university in Pavia to study medicine. However, after only one year war had broken out between Spain and France; thus Cardano had to transfer schools to the University of Padua. At Padua, Cardano showed some of his remarkable abilities. He was a brilliant student and often he would hold his own in disputes with the members of the faculty.
While Cardano attended the institution, his father passed away. As a source of income Cardano resorted to gambling. His knowledge and understanding of probability is thought to have assisted him in his numerous successes in games. However, his gambling became a fault that lasted throughout his life and robbed Cardano of valuable time, money , and his reputation. At the age of 25, Cardano finished his studies at Padua, thus he applied for admission to the College of Physicians in Milan. His application was denied due to his reputation for aggressiveness and critical opinions. Thus, Cardano decided to become a country doctor in the little village of Sacco.
In Sacco, he married Lucia Bandarini, the daughter of the local innkeeper. Realizing that he could not support a family in Sacco, he made efforts to relocate. He tried again for admission to College of Physicians in Milan, and once again his admission was denied. His luck became worse when he had to move back to Milan into a poorhouse with his family. Once he was back in Milan, Cardano's luck changed. Milan's nobles were deeply interested in scientific inquiry. Due to the respectable reputation of his late father Fazio's name, Cardano was appointed to his father's position as a public lecturer at the Piatti Foundation, a reputable medical institution in Milan. At the Piatti Foundation he was not only a lecturer but he was occasionally allowed to treat patients. During this time he had discovered many cures, resulting in an enhancement of his reputation as a physician among his colleagues.
Finally in 1539, after several unsuccessful attempts for admission to the College of Physicians in Milan, Cardano gained admission into the College. Within a few years, Cardano became the most prominent physician in Milan. "The pope and Europe's royal and imperial heads with their princely families were convinced that no physician could better safeguard their health than Cardano." (Ore, p.13) In addition to his step to success, Cardano began publishing mathematical books such as "The Practice of Arithmetic and Simple Measuration". At this point in time Cardano was at the height of his fame, as a practicing doctor no other could compare, and his books were read everywhere by intellectuals.
Cardano's wife Lucia died in 1546, but Cardano apparently was not effected by this event. He was too busy writing best sellers, and indulging in his scientific studies. He was at the height of his fame. He was treated like royalty and he was the world's leading scientist.
Cardano's fortune once again took a turn for the worst. His eldest son Giambatista poisoned his wife with a cake mixed with arsenic. Cardano supported his son in every way; he hired the best lawyers available, wrote petitions, and appealed for clemency through all his friends and influential patients. Despite his strongest efforts, Giambatista was sentenced to death. This tragic event continually haunted Cardano even until his own death.
In 1570, Cardano himself was put into jail on the charge of heresy (according to one tradition). In some of Cardano's writings, there were statements that could be construed as being impious. It is said that he cast the horoscope of Jesus Christ, and wrote in praise of the Emperor Nero, known in his day as the tormentor of martyrs. In a work entitled "Subtility" he wrote a dialogue about the pros and cons of Christianity. However, because of his powerful and influential friends Cardano was only kept in jail for three months. Upon his release Cardano was forbidden to be a professor and was denied the right to lecture publicly, but worst of all he was denied the right to publish any further books. Cardano decided to travel to Rome, and the reception in Rome was favorable. He was immediately invited by the College of Physicians of Rome as a member, and was granted a pension by the Pope.
Gerlamo Cardano died on September 20 1576. Cardano left his entire estate to his grandson Fazio. His books and manuscripts were left to his friends. His lifetime accomplishments are especially highly regarded in mathematical circles. "...[T]he significance of his contributions to mathematics suggest that he ranks with Copernicus (astronomy), Versalius (anatomy), and Paracelus (alchemy and medicine) as one of the founders of that revolution. He displayed a mastery of calculation and a confidence in dealing with algebraic equations in his work." (Calinger, p233)
"Artis Magne" was Cardano's most widely recognized work, because it contained the first published account of general solutions for cubic and quartic equations by means of radical expression. In addition, this was the first book that paid some attention to computations involving square roots of negative numbers. The notion of imaginary roots was left incomplete until Raphael Bombelli laid out the foundation of a more detailed account of imaginary entities. "The `Artis Magne' has always been highly praised as a milestone in the history of mathematics...Nowadays, the `Artis Magne' would be characterized as a text on algebraic equations. To Cardano's contemporaries it was a breakthrough in the field of mathematics, exhibiting publicly for the first time the principles for solving both cubic and biquadratic equations, giving the roots by expressions formed by radicals, in a manner similar to the method which had been known for equations of the second degree since the Greeks or even the Babylonians."(The Great Art, Forward p.7) Before and during the Renaissance era there was a great deal of interest in the solution of cubic, thus the book and its contents was a great victory in the mathematical field.
Neither the solutions to the cubic and quartic equations were his own discoveries. The former was first discovered by del Ferro and rediscovered by Tartagalia and the latter by Cardano's disciple and son-in-law Lodovico Ferrari. In 1535 Tartagalia was challenged to a problem solving match with Fior. Fior gave Tartaglaia problems concerned with cubic equations. On February 12, 1535 Tartagalia discovered a method of solving cubic equations. For reasons unknown, Tartagalia did not want publish the method yet; so all mathematicians from around the world strived to construct the method yet all failed. Tartagalia tried to keep his method a secret but Cardano persistently begged him to show him the method. Cardano made a solemn promise to Tartaglia that he would not publish the method until Trataglia had himself published it : "I swear to you by the Sacred Gospel, on my faith as a gentlemen, not only never to publish your discoveries, if you tell them to me, but I also promise and pledge my faith as a true Christian to put them down in cipher so that after my death no one shall be able to understand them." (p.77, Ore)
Not surprisingly, Cardano took advantage of his knowledge of the cubic equation's solution and immediately started working on the proof of Tartagalia's rule. Cardano knew that there was more that could be done with this knowledge of the cubic solution. "After I was in possession of this rule and had found the proof for it, I [Cardano] understood that here many other things could be discovered, and with my confidence thus already increased I found such results, partly by myself and in part through the work of Lodovico Ferrari, my former pupil." (Ore, p. 85-86) He and his student Ferrari, made efforts to solve bi-quadratic equations. Although Cardano was able to solve an equation such as 32 = x4 + 2x3 + 2x + 1 by a process employed by Diophantus and the Hindus, he failed to find a general solution. His brilliant pupil Ferrari made the discovery of the general solution of bi-quadratic equations. Cardano was so elated by these two discoveries he had in his own possession, he knew no better way to crown his achievements than by writing Artis Magnae with the rules for solving cubics and quadratic equations. Thus Cardano broke his solemn vow to Tartagalia and went through with his own desires for praise and recognition.
Therefore "Ars Magnae" appeared in 1545 and won immediate acclaim by all prominent mathematicians. "From the mathematical publications of the second half of the 16th century one can see that the book exerted a direct and profound influence upon the rise of European mathematics (p.84, Ore) This was a immense achievement because for centuries mathematicians were in search for a solution of cubics, and many believed that it could not be found. Starting from as early as the Babylonians, mathematicians were concerned with equations of higher degree. Some claim that the Babylonians were the first to solve quadratic equations. They developed an algorithm for solving problems that would lead to a quadratic equation. They used the method of completing the square. A limitation of the Babylonians was that all their answers were positive quantities because the answer was a length. About a century later, Euclid constructed a geometrical way of solving quadratic equations, which later became useful method in the cubic solution found in Cardano's "Artis Magnae". However, Euclid like the Babylonians had no notion of equation or coefficients. He merely worked with geometric entities. Therefore the discovery of solution to cubic and quartics was an accomplishment long awaited.
Since the Renaissance era did not have an efficient algebraic notation available, Cardano had to list a multitude of equation types. Because of his notational difficulties, he bases most of his proofs on geometrical arguments, using the ancient Greek mathematician Euclid's style of reasoning. Nowadays, modern mathematicians would not find a need for such methods, but at that time "Euclid's Elements" held the best position for logical thought. Euclid solved second degree equations by constructing squares, thus Cardano expressed the same logic by using cubes to solve cubic equations of the type x3+ px = q.
In addition to Cardano's magnificent display of the cubic solution in "Ars Magnae", he is also recognized and distinguished from other mathematicians for acknowledging the use of imaginary or complex numbers to get real solutions. ".[H]e calculated formally with imaginary numbers but did not accept them." (Calinger, p.234) Unlike other mathematicians before him, he does not avoid or brush imaginary numbers aside. He found something strange when he applied his formula to certain cubics. He found negative numbers under the radical sign. Nobody before him had learned how to take the square root a negative number. In "Ars Magnae", Cardano does give a calculation with complex number, however he did not really understand his own calculation. " In his `Ars Magnae', he takes notice of negative roots of an equation, calling them fictitious, while the positive roots are called real. He paid some attention to computations involving the square root of negative numbers, but failed to recognize imaginary roots. Cardano observed the difficulty in the irreducible case (case when a cubic equation has roots that are expressed by the difference of 2 cube roots of complex imaginary numbers) in the cubic, which, like the quadrature of the circle, has since 'so much tormented the perverse ingenuity of mathematicians'. But he did not understand its nature" (p.135, Cajori). Cardano calculated formally with negative roots but he did not really accept them. Clearly, Cardano used imaginary numbers when it was convinient for solving cubic equtaions, but he did not understand what they actually were. Since he did not understand them, he does not further discuss him in his work.
Perhaps the main obstacle in Cardano's understanding of imaginary roots is that in his search for a solution to cubic equations his "cause was to preserve the purity of the Euclidean tradition" ( Tanner) Cardano solved the cubic equations with geometrical conceptualization (Euclidean logic) , thus the notion of negative roots ( or magnitudes) was not easy to understand of visualize. Who would ever think of a magnitude, such as a side of a square to be negative in length? It is difficult for people to conceptualize this notion of negative roots now, and thus it must have been even more difficult for people of the time of the Renaissance. "Cardano and his age were not less conversant with abstractions and intangible reality..." (Tanner, p.177) In "The Great Arte", Cardano indirectly admits that he does not have a full understanding of complex numbers and that he merely uses them for calculation. ".[S]ince such a remainder is negative, you have to imagine -15-that is the difference between AD and 4AB.namely 5+ 25-40 and 5- 25- 40, or 5+-15 and 5-15. Putting aside the mental tortures involved, multiply 5+-15 and 5-15, making 25-(-15) which is +15. Hence the product is 40" (Cardano, p219) Cardano uses the word "imagine" to indicate that he is not comfortable enough with complex numbers to explain exactly what they are, but for the reader to assume that they exist so that one can perform the necessary computations for cubic equations. Thus a reason for Cardano's blindness in understanding negative roots was that "... Cardano's mind still ran on his geometrical arguments...of Euclid's Elements Book 2. Prop.7" (Tanner, p.169) . "Cardano's other cause, in defence of Euclid, is not to be underestimated, despite its negative and retrograde appearance. It underlines a hiatus that mathematics meets at every breakthrough, and is most familiar today in its 'applied' aspects. In the step from static to dynamic theory, when inapplicable rules for matter at rest are used to derive properties of matter of motion, in their classical treatment. It was also the basis of many ancient paradoxes, which admit as arguments against such a step would have halted advance at the start, but whose serious consideration is a source of unexpected insights even today." (Tanner, p.178) His knowledge of Euclidean methods was both a blessing and a curse on his studies of cubics. At the same time it allowed him solve certain cubics, but also handicapped him in understanding the true notion of imaginary roots which could have allowed him to advance even further in his mathematical accomplishments.
Cardano and Viéte's individual contributions were major stepping stones toward further mathematical discoveries. Cardano provided a tool for solving equations of the 3rd and 4th degrees. He began the crusade for understanding the notion of imaginary roots and complex numbers, which led to higher worlds of mathematical thought and abstraction. His contributions ignited European mathematical development. Viéte's new systematic algebra opened doors for further mathematical development. He created a tool which assisted other mathematicians to engage in detailed mathematical discoveries. He allowed people to express their thoughts mathematically with a higher degree of detail and precision. Both these accomplished Renaissance mathematicians contributed to the foundation of elementary algebra. Without Viéte and Cardano, algebra would still be as underdeveloped as it before and during the Renaissance period.