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The ''Ars Magna'' (''The Great Art'', 1545) is an important
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
-language book on
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
written by
Gerolamo Cardano Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
. It was first published in 1545 under the title ''Artis Magnae, Sive de Regulis Algebraicis Liber Unus'' (''Book number one about The Great Art, or The Rules of Algebra''). There was a second edition in Cardano's lifetime, published in 1570. It is considered one of the three greatest scientific treatises of the early
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ideas ...
, together with Copernicus' ''
De revolutionibus orbium coelestium ''De revolutionibus orbium coelestium'' (English translation: ''On the Revolutions of the Heavenly Spheres'') is the seminal work on the heliocentric theory of the astronomer Nicolaus Copernicus (1473–1543) of the Polish Renaissance. The book, ...
'' and
Vesalius Andreas Vesalius (Latinized from Andries van Wezel) () was a 16th-century anatomist, physician, and author of one of the most influential books on human anatomy, ''De Humani Corporis Fabrica Libri Septem'' (''On the fabric of the human body'' '' ...
' ''
De humani corporis fabrica ''De Humani Corporis Fabrica Libri Septem'' (Latin, lit. "On the fabric of the human body in seven books") is a set of books on human anatomy written by Andreas Vesalius (1514–1564) and published in 1543. It was a major advance in the histor ...
''. The first editions of these three books were published within a two-year span (1543–1545).


History

In 1535
Niccolò Fontana Tartaglia Niccolò Fontana Tartaglia (; 1499/1500 – 13 December 1557) was an Italian mathematician, engineer (designing fortifications), a surveyor (of topography, seeking the best means of defense or offense) and a bookkeeper from the then Republi ...
became famous for having solved cubics of the form ''x''3 + ''ax'' = ''b'' (with ''a'',''b'' > 0). However, he chose to keep his method secret. In 1539, Cardano, then a lecturer in mathematics at the Piatti Foundation in Milan, published his first mathematical book, ''Pratica Arithmeticæ et mensurandi singularis'' (''The Practice of Arithmetic and Simple Mensuration''). That same year, he asked Tartaglia to explain to him his method for solving cubic equations. After some reluctance, Tartaglia did so, but he asked Cardano not to share the information until he published it. Cardano submerged himself in mathematics during the next several years working on how to extend Tartaglia's formula to other types of cubics. Furthermore, his student
Lodovico Ferrari Lodovico de Ferrari (2 February 1522 – 5 October 1565) was an Italian mathematician. Biography Born in Bologna, Lodovico's grandfather, Bartolomeo Ferrari, was forced out of Milan to Bologna. Lodovico settled in Bologna, and he began his ...
found a way of solving quartic equations, but Ferrari's method depended upon Tartaglia's, since it involved the use of an auxiliary cubic equation. Then Cardano became aware of the fact that
Scipione del Ferro Scipione del Ferro (6 February 1465 – 5 November 1526) was an Italian mathematician who first discovered a method to solve the depressed cubic equation. Life Scipione del Ferro was born in Bologna, in northern Italy, to Floriano and Filip ...
had discovered Tartaglia's formula before Tartaglia himself, a discovery that prompted him to publish these results.


Contents

The book, which is divided into forty chapters, contains the first published algebraic solution to
cubic Cubic may refer to: Science and mathematics * Cube (algebra), "cubic" measurement * Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex ** Cubic crystal system, a crystal system w ...
and
quartic equation In mathematics, a quartic equation is one which can be expressed as a ''quartic function'' equaling zero. The general form of a quartic equation is :ax^4+bx^3+cx^2+dx+e=0 \, where ''a'' ≠ 0. The quartic is the highest order polynomi ...
s. Cardano acknowledges that Tartaglia gave him the formula for solving a type of cubic equations and that the same formula had been discovered by Scipione del Ferro. He also acknowledges that it was Ferrari who found a way of solving quartic equations. Since at the time
negative numbers In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed ma ...
were not generally acknowledged, knowing how to solve cubics of the form ''x''3 + ''ax'' = ''b'' did not mean knowing how to solve cubics of the form ''x''3 = ''ax'' + ''b'' (with ''a'',''b'' > 0), for instance. Besides, Cardano also explains how to reduce equations of the form ''x''3 + ''ax''2 + ''bx'' + ''c'' = 0 to cubic equations without a quadratic term, but, again, he has to consider several cases. In all, Cardano was driven to the study of thirteen different types of cubic equations (chapters XI–XXIII). In ''Ars Magna'' the concept of
multiple root In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root. The notion of multipl ...
appears for the first time (chapter I). The first example that Cardano provides of a polynomial equation with multiple roots is ''x''3 = 12''x'' + 16, of which −2 is a double root. ''Ars Magna'' also contains the first occurrence of complex numbers (chapter XXXVII). The problem mentioned by Cardano which leads to square roots of negative numbers is: find two numbers whose sum is equal to 10 and whose product is equal to 40. The answer is 5 + √−15 and 5 − √−15. Cardano called this "sophistic," because he saw no physical meaning to it, but boldly wrote "nevertheless we will operate" and formally calculated that their product does indeed equal 40. Cardano then says that this answer is "as subtle as it is useless". It is a common misconception that Cardano introduced complex numbers in solving cubic equations. Since (in modern notation) Cardano's formula for a root of the polynomial ''x''3 + ''px'' + ''q''  is :\sqrt \sqrt square roots of negative numbers appear naturally in this context. However, ''q''2/4 + ''p''3/27 never happens to be negative in the specific cases in which Cardano applies the formula.This does not mean that no cubic equation occurs in ''Ars Magna'' for which ''q''2/4 + ''p''3/27 < 0. For instance, chapter I contains the equation ''x''3 + 9 = 12''x'', for which ''q''2/4 + ''p''3/27 = −175/4. However, Cardano never applies his formula in those cases.


Notes


Bibliography

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External links


.pdf of ''Ars Magna''
(in
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
)
Cardano's biography
{{Authority control Mathematics books History of mathematics 1545 books 1545 in science 16th-century Latin books