The Cossic ArtWriting Algebra with Symbols

Nate Jones & Chelsea LandisNate 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

Egyptian Algebra

1850 B.C, Rhind PapyrusSolved 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.

Babylonian Algebra

1800 - 1600 B.C.Solved quadratic equationsVery little use of symbolsRhetorical teachings, taught through

examples.

Greek Algebra

Didn’t add much to Algebra over what Babylonians already had.

Their achievement was applying deductive reasoning and describing general procedures.

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.

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

Latin

Italian

German

Alternate Words Used For Unknowns

A Journey Through Time

3 25 7 6x x x x

3 25 7 6x x x x

1202, Leonardo of PisaThe cube of seven things less five squares is

equal to the root of six more than the thing.

13th & 14th centuriesRhetorical mathematics with some exceptions

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 25 7 6x x x x

16th century Germany + , - , Cristoff Rudolff, Coss, 1525 Michael Stifel, Arithmetica Integra, 1544

“The unknown” was represented differently for each power.

3 25 7 6x x x x

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

3 5

3 25 7 6x x x x

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

Thomas Harriot,1620’s 5aaa + 7ee

Pierre Herigone, 1634 5a3 + 7e2

James Hume, 1636 5aiii + 7eii

Rene Descartes, 1637 5a3 + 7e2

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.

3 25 7 6x x x x

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)

1823 - 1891

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

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

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>