1. The Cossic Art Writing 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
The development of Algebra notation progressed in 3 stages. The first was the rhetorical or written/verbal stage.
The second was the syncopated stage. This was when they would use abbreviated words in their expressions.
And finally the third stage is what we now see and use today.
No matter how an algebra equation is written, there is some unknown we are trying to find which must be solved from what we know.

3. Egyptian Algebra 1850 B.C, Rhind Papyrus 300 B.C., Cairo 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.
The Rhind Papyrus was actually written in 1650BC, but it’s thought that the findings represented math from around 1850.
One of the reasons Egyptian Algebra didn’t evolve because of their way of dealing with fractions.

4. Babylonian Algebra 1800 - 1600 B.C. Solved quadratic equations
Very little use of symbols
Rhetorical teachings, taught through examples.
They solved quadratic equations even though they didn’t recognize negative numbers and roots. They did, however, find approximations to problems that had no exact rational solution.

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.
Arithmetica was the book he wrote.

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
The first real progress we made in algebra was in the 9th century by Al-Khwarizmi
Wrote al jabr w’al muqabala which is known to be the first book written on algebra.
He used shai to represent an unknown quantity and later when the book was translated into latin this word became res which means “thing”.

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

9. A Journey Through Time
Mathematicians tried to come up with an easier way to express unknowns and for it to be used as a universal language that would clarify ideas. Here’s an example of a typical equation you might see today. We will show you how this equation evolved through the development of algebra.

10. 1202, Leonardo of Pisa 13th & 14th centuries
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)
In 1202, Leonardo of Pisa wrote the equation rhetorically like we talked about before.

11. co means “cosa” –the unknown quantity
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
“v” or Universale stood for the grouping of terms and was placed after the square root R.
In the the late 15th century mathematicians started using symbols
Ce and cu were used by italians as the sqaure and the cube.
With this notation we were unable to write more than one unknown in an expression.
However, the Hindus at this same time were using colors to represent different unknowns so they could represent more than one in an equation. This had started as early as the 7th century.

12. “The unknown” was represented differently for each power.
16th century Germany
+ , - ,
Cristoff Rudolff, Coss, 1525
Michael Stifel, Arithmetica Integra, 1544
“The unknown” was represented differently for each power.
Different powers had different symbols for each power.
Its first power was called “the root”.
The second “the square”
The third “cubus”
Any higher powers, would have been written as a combination of these three powers… so if you wanted to write the 6th power, you would use the second power and 3thrd power. The fourth power was 2x2. Higher powers that would have been too long to write out, introduced new symbols.. ( I tried to find what these might have been , but I couldn’t find anything on the subject.))
Show on board, the symbols for the powers… and then the new equation that would have been written.

13. Nicholas Chuquet, French Physician in 1484, denoted his unknowns with exponents by using superscripts.
54 denoted 5x4
denoted
13.m.52.p.71. montent R2.11.p.60.
Rafael Bombelli, 1572
Chuquet was ahead of his time by using 0, especially as an exponent.
But like others of his time, he only used one unknown.
His work wasn’t published at the time he wrote it so it wasn’t ‘til about the 16th century that we started actually using this system of exponents. Finally Rafael Bombelli started using this in 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.
Using this process, of both letters for constants and unknowns he could write general forms of equations and not rely on specific examples, in which different numbers chosen might affect the solution.
He was the first to really use letters, as a major part of algebra.

15. Now, mathematicians were able to write equations with more than one unknown.
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
While Viete was coming up with his way of expressing unknowns, there were others who were coming out with their own ways to express letters in algebra.
Thomas Harriot in the 1620’s would write as many a’s or e’s as he wanted to express, so like in our above example, he would write 3 a’s for the third power, and 2 e’s for the 2nd power.
Then in 1634 Pierre Herigone expressed unknowns the same way Viete might have except he used lowercase and didn’t superscript the exponents.
Then, James Hume, would use small Roman Numerals to express exponents.
In 1637, the notation Rene Descartes used, is what we now use today. Harriots and Herigones might have been easier to typeset but the clarity of writing it in this way was why his notation stuck.

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.
He also used a sign that looked like this………… for equality.
Therefore, in our equation that we have been looking at… Descarte’s would have looked very similar to our own. ( show equation on board)
Eventually Descartes introduced the equation ax + by = c , which is still used to describe the equation of a line today.
Mathematicians of his time didn’t treat exponents as independent of geometric dimension. His notation itself led to the introduction of reciprocals, roots of powers, and limits of roots of powers, and even complex exponents.

17. Abstract Algebra 19th Century Gibbs (American, 1839-1903)
British mathematicians looked at vectors, matrices, transformations, etc.
Gibbs (American, )
Developed vectors in three dimensional space.
Cayley (British, )
Developed an algebra of matrices
Galois (French, )
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

19. Timeline
1850 B.C, Egyptians solved problems equivalent to a linear equation with one unknown.
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, ); Developed vectors in three dimensional space.
Cayley (British, ) ; introduced matrices.
Galois (French, ) ; introduced concept of a group

21. References
Christoff Rudolff, <
Berlinghoff, William P., and Fernando Q. Gouvêa. Math Through the Ages: a Gentle History for Teachers and Others. Farmington: Oxton House, pp
The History of Algebra, <