Cardano’s Solution to the Cubic

Daniel Blanchette, Christian Landry

1300

BC

Babylonians Solved quadratics of the form

x2 + p = qx

Chinese are solving quadratic by completing the square

200 BC

Brahmagupta solves quadratic for positive roots only

628

AD

Yang Hui solves quadratic with negative roots using Pascal’s Triangle

1261

Johannes Guttenberg invents the printing press

1450

- Ottoman empire captures Constantinople - Fall of Byzantine empire- End of 100 year war

1453

- End of Thirteen year war. - Royal Prussia annexed to Poland.

1466

- Unification of Spain- Wedding of Isabela 1 of Castile to Ferdinand Aragon

1469

- Spanish Inquisition begins

1481

- John Widmann uses the + and – signs for the first time in a printed book

1489

- Christopher Columbus discovers the Americas

1492

- Lucia Pacioli writes summa de Arithmetica

1494

- Michealangelo begins work on the Statue of David

1501

-Nilakantha Somayaji writes Tantra sangraha

1501

- Da Vinci begins to paint the Mona Lisa

1503

- Coppernicius says the earth resolves around the sun

1512

- Aztec Empire falls- Montezuma II is killed

1521

- Rome is sacked- End of Italian Renaissance

1527

- England renounces the papacy- King Henry VIII head of Church of England

1531

- Jacques Cartier claims Quebec for France

1534

Scipiones Del Ferro

-Feburary 6th 1465 – November 5th 1526- First to solve depressed cubic- Not much known about mathematics

Niccolo Fontana Tartaglia

- ? 1499 – December 13th 1557- Engineer, Mathematician- Translated Euclid, Archimedes- Studied path of Cannonball- Devised a method to obtain binomial Coefficients- Volume of tetrahedron given lengths- Solved cubic, did not publish.

Gerolamo Cardano

-September 24th 1501 - September 21st 1576- Mathematician, Inventor, Physician, Lawyer- First to describe Typhoid Fever- First to acknowledge imaginary numbers- First to write systematically about Probability- First to publish solution to cubic, quartic- Introduced Binomial Coefficients and theorem

Cardan Grille

Gimbals

Cardan Shaft (Drive Shaft)

Combination Locks

A polynomial with an odd order must have at least one solution that is real. Why?

Questions/ Comments?

Where have you seen the use of auxiliary equations before?

How did Yang Hui obtain negative roots?

Gauss wrote “Oh..Great Archimedes, How come you did not make this discovery? If you had done that, the world would have been thousand of years more advanced”

Do you agree with Gauss?, If yes, why is it that the author of this book did not cover more about Brahmagupta?