Ars Magna (Gerolamo Cardano)

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The title page of the Ars Magna. The full title is Artis Magnæ, Sive de Regulis Algebraicis Liber Unus (Book number one about The Great Art, or The Rules of Algebra).

The Ars Magna (Latin: "The Great Art") is an important book on Algebra written by Gerolamo Cardano. It was first published in 1545 under the title Artis Magnæ, 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^[1] one of the three greatest scientific treatises of the early Renaissance, together with Copernicus' De revolutionibus orbium coelestium and Vesalius' De humani corporis fabrica. It is interesting to note that the first editions of these three books were published within a two year span (1543–1545).

1 History
2 Contents
3 Notes
4 Bibliography
5 External links

[edit] History

In 1535, Tartaglia became famous for having solved cubics of the form x³ + 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 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 into mathematics during the next several years working on how to extend Tartaglia's formula to other types of cubics. Furthermore, his student Lodovico Ferrari 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 become aware of the fact that Scipione del Ferro had discovered Tartaglia's formula before Tartaglia himself, a discovery that prompted him to publish these results.

[edit] Contents

The book, which is divided into forty chapters, contains the first published solution to cubic and quartic equations. Cardano acknowledges that Tartaglia gave him the formula for solving a type of cubic equations and that the same formula had been discovered by Scipiano del Ferro. He also acknowledges that it was Ferrari who found a way of solving quartic equations.

Since at the time negative numbers were not generally acknowledge, knowing how to solve cubics of the form x³ + ax = b did not mean knowing how to solve cubics of the form x³ = ax + b (with a,b > 0), for instance. Besides, Cardano, also explains how to reduce equations of the form x³ + ax² + 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 appears for the first time (chapter I). The first example that Cardano provides of a polynomial equation with multiple roots is x³ = 12x + 16, of which −2 is a double root.

Ars Magna also contains the first occurrence of complex numbers (chapter XXXVII). Since (in modern notation) Cardano's formula for a root of the polynomial x³ + px + q is

$\sqrt[3]{-\frac q2+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}+\sqrt[3]{-\frac q2-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}},$

square roots of negative numbers appear naturally in this context. However, q²/4 + p³/27 never happens to be negative in the specific cases in which Cardano applies the formula.^[2] 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. Then Cardano says that this answer is “as subtle as it is useless”.

[edit] Notes

^ See, for instance, the foreword that Oystein Ore wrote for the English translation of the book, mentioned at the bibliography.
^ This does not mean that no cubic equation occurs in Ars Magna for which q²/4 + p³/27 < 0. For instance, chapter I contains the equation x³ + 9 = 12x, for which q²/4 + p³/27 = −175/4. However, Cardano never applies his formula in those cases.

[edit] Bibliography

Calinger, Ronald (1999), A contextual history of Mathematics, Prentice-Hall, ISBN 0-02-318285-7
Cardano, Gerolamo (1545), Ars magna or The Rules of Algebra, Dover (published 1993), ISBN 0-486-67811-3
Gindikin, Simon (1988), Tales of physicists and mathematicians, Birkhäuser, ISBN 3-7643-3317-0