The Cossic Art Writing Algebra with Symbols

1. The Cossic ArtWriting Algebra with Symbols Nate Jones & Chelsea Landis

2. 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

3. Egyptian Algebra • 1850 B.C, Rhind Papyrus • Solved problems equivalent to a linear equation with one unknown. • 300 B.C., Cairo Papyrus • Solved problems equivalent to a system of two second degree equations with two unknowns.

4. Babylonian Algebra • 1800 - 1600 B.C. • Solved quadratic equations • Very little use of symbols • Rhetorical teachings, taught through examples.

5. Greek Algebra • Didn’t add much to Algebra over what Babylonians already had. • Their achievement was applying deductive reasoning and describing general procedures.

6. Diophantine Algebra • 250 A.D. • He accepted only positive rational roots • When a quadratic equation had two positive rational roots he only gave one as the solution. There was no structure to his work. • Had no general methods. • Each of the 185 problems in Arithmetica was solved a different way.

7. Introduction to Algebra • 9th century Al-Khwarizmi wrote “al jabr w’al muqabala” • Shai - unknown quantity • “Quaeritur ergo, quae res…” • 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

8. Alternate Words Used For Unknowns • Latin • Italian • German “Causa” “Cosa” “Coss”

9. A Journey Through Time

10. 1202, Leonardo of Pisa • The cube of seven things less five squares is equal to the root of six more than the thing. • 13th & 14th centuries • Rhetorical mathematics with some exceptions such as R for “square root”. (Leonardo)

11. Late 15th century, Luca Pacioli • cu.m.5.ce.p.7.co.-------Rv.co.p.6. • comeans “cosa” –the unknown quantity • ce&cu “censo” and “cubo” – square and the cube • R denoted square root,v “universale” • ------ denoted equals

12. 16th century Germany • + , - , • Cristoff Rudolff, Coss, 1525 • Michael Stifel, Arithmetica Integra, 1544 • “The unknown” was represented differently for each power.

13. Nicholas Chuquet, French Physician in 1484, denoted his unknowns with exponents by using superscripts. • 54 denoted 5x4 • . 3.5. denoted • 13.m.52.p.71. montent R2.11.p.60. • Rafael Bombelli, 1572

14. 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.

15. 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 • Thomas Harriot,1620’s • 5aaa + 7ee • Pierre Herigone, 1634 • 5a3 + 7e2 • James Hume, 1636 • 5aiii + 7eii • Rene Descartes, 1637 • 5a3 + 7e2

16. More on Descartes… • Used lowercase letters from end of alphabet as unknowns. • Used lowercase letters from beginning of alphabet for constants. • He also introduced the overline bar in the square root sign to indicate the expressions length.

17. Abstract Algebra • 19th Century • British mathematicians looked at vectors, matrices, transformations, etc. • Gibbs (American, 1839-1903) • Developed vectors in three dimensional space. • Cayley (British, 1821-1895) • Developed an algebra of matrices • Galois (French, 1811-1832) • Helped develop the concept of a group (a set of operations with a single operation which satisfies three axioms)

18. "God created the natural numbers. Everything else is the work of man." - Kronecker 1823 - 1891

19. Timeline • 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 • Michael Stifel, Arithmetica Integra, 1544

20. 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

21. References • Christoff Rudolff, <http://www.gap-system.org/~history/Printonly/Rudolff.html> • Berlinghoff, William P., and Fernando Q. Gouvêa. Math Through the Ages: a Gentle History for Teachers and Others. Farmington: Oxton House, 2002. pp 95-100 • The History of Algebra, <http://www.ucs.louisiana.edu/~sxw8045/history.htm>