51 The Cossic Art Writing Algebra with Symbols

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



.



13.m.52.p.71. montent R2.11.p.60.

3.5. denoted

3

5

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, < />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,
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