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
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.
Babylonian Algebra

1800 - 1600 B.C.
 Solved
quadratic equations
 Very little use of symbols
 Rhetorical 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
Alternate Words Used For Unknowns
 Latin


Italian
German
A Journey Through Time
x  5x  7 x  x  6
3
2
x3  5 x 2  7 x  x  6
 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)
x  5x  7 x  x  6
3
2
 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
,
x  5x  7 x  x  6
3
2
16th century Germany
+ , - ,
Cristoff Rudolff, Coss, 1525
Michael Stifel, Arithmetica Integra, 1544
“The unknown” was represented differently for
each power.
x  5x  7 x  x  6
3
2
 Nicholas Chuquet, French Physician in 1484, denoted
his unknowns with exponents by using superscripts.
 54 denoted 5x4
.
3.5.
denoted
3
5
 13.m.52.p.71. montent R2.11.p.60.
 Rafael Bombelli, 1572
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.


Thomas Harriot,1620’s


5a3 + 7e2
James Hume, 1636


5aaa + 7ee
Pierre Herigone, 1634


53+72 would no longer be sufficient if we wanted to say 5A3+7E2
5aiii + 7eii
Rene Descartes, 1637

5a3 + 7e2
x  5x  7 x  x  6
3
2
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.
Abstract Algebra

19th Century


Gibbs (American, 1839-1903)


Developed vectors in three dimensional space.
Cayley (British, 1821-1895)


British mathematicians looked at vectors, matrices,
transformations, etc.
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.gapsystem.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>