51 The Cossic ArtWriting Algebra with Symbols

The Cossic ArtWriting Algebra with Symbols Nate Jones & Chelsea Landis...  Solved quadratic equations  Very little use of symbols  Rhetorical teachings, taught through examples...  N

The Cossic Art

Writing Algebra with Symbols

Nate Jones & Chelsea Landis

 The English called the study “ the Cossic Art” which

means “the Art of Things”.

 Algebraists were called cossists, and algebra the cossic

art, for many years

 3 stages of Algebra

 1 Rhetorical Stage

 2 Syncopated Stage

 3 Symbolic Stage

Babylonian Algebra

 1800 - 1600 B.C.

 Solved quadratic equations

 Very little use of symbols

 Rhetorical teachings, taught through examples.

Diophantine Algebra

rational roots he only gave one as the solution There was no structure to his work

solved a different way.

Introduction to Algebra

muqabala”

 It is asked, therefore, what thing together with 10 of its roots or what is the same, ten times the root obtained from it, yields 39.

 X + 10 √x = 39

 X2 + 10x = 39

A Journey Through Time

such as R for “square root” (Leonardo)

 Late 15th century, Luca Pacioli

 cu.m.5.ce.p.7.co. -Rv.co.p.6.

 co means “cosa” –the unknown quantity

 ce & cu “censo” and “cubo” – square and the cube

 R denoted square root, v “universale”

 - denoted equals

3 5 2 7 6

x − x + x = x +

 16th century Germany

 + , - ,

 Cristoff Rudolff, Coss, 1525

 Michael Stifel, Arithmetica Integra, 1544

each power.

3 5 2 7 6

x − x + x = x +

 Nicholas Chuquet, French Physician in 1484, denoted

his unknowns with exponents by using superscripts

Major Breakthrough in the 16th Century!!

 Francois Viete - Lawyer, Mathematician, advisor to King Henri IV of France

 Focused on algebraic equations in his mathematical writings.

 Introduced letters for both constants and unknowns.

 In his own words, “ In order that this work may be assisted by some art, let the given magnitudes be

distinguished from the undetermined unknowns by a constant, everlasting and very clear symbol, as, for instance, by designating the unknown magnitude by means of the letter A or some other vowel…and the given magnitudes by means of the letters B,G,D or other consonants.

 Now, mathematicians were able to write equations with more than one unknown.

 53+72 would no longer be sufficient if we wanted to say 5A3+7E2

More on Descartes…

unknowns.

alphabet for constants

root sign to indicate the expressions length.

x − x + x = x +

 1850 B.C, Egyptians solved problems equivalent to a linear equation with one unknown

 1800 - 1600 B.C Babylonians solved quadratic equations

 300 B.C, Egyptians solved problems equivalent to a system of two second degree equations in two unknowns

 9th Century- Al-Khwarizmi wrote “aljabr w’al muqabala (source of the word algebra)

 1202, Leonard of Pisa, used rhetorical writing to express algebra equations

 13th & 14th centuries, introduced R for square root, along with other

occasional abbreviations

 15th century, Luca Pacioli introduced symbolic algebra

 Nicholas Chuquet, 1484, introduced exponents written as superscripts

 Cristoff Rudolff wrote Coss, 1525

Timeline Continued…

 16th century Germany, “The unknown” was represented differently for each power

 Rafael Bombelli, 1572, started reusing Chuquet’s system of exponents

 Francois Viete, 16th century, Introduced letters for both constants and

unknowns

 Thomas Harriot,1620’s, Pierre Herigone, 1634, James Hume, 1636, Rene Descartes, 1637; introduced different ways of writing equations with

exponents and unknowns

 Gibbs (American, 1839-1903); Developed vectors in three dimensional space

 Cayley (British, 1821-1895) ; introduced matrices

 Galois (French, 1811-1832) ; introduced concept of a group

 The History of Algebra,

<http://www.ucs.louisiana.edu/~sxw8045/history.htm>