The heart of these notes is the study of solutions to
cubic and quartic

equations, and this forms one of the most fascinating stories in the

history of mathematics. The discovery of their solutions in the sixteenth

century through the work of several Italian mathematicians was

the most dramatic advance to have taken place in algebra in centuries .

This section provides a mere sketch of the relevant history. For

more information, one can turn to the two volumes by B. L. van der

Waerden, Geometry and Algebra in Ancient Civilizations and A History

of Algebra From Al-Khwarizmi to Emmy Noether, and the references

contained within them. Also worth looking at is one primary source,

Gerolamo Cardano’s Ars Magna of 1545, as well as the book Algebra

in Ancient and Modern Times by V. S. Varadarajan, with its mix of

mathematics and history.

Every ancient civilization developed mathematics to at least some

degree, in part to serve the practical needs of measurement, construction,

and commerce. The work of the ancient Greeks may be the most

familiar, thanks to the codification of much of it by Euclid, but important

contributions were made as well within early Babylonian, Egyptian,

Hindi, and Chinese civilizations. Van derWaerden suggests, given

some of the common features of mathematics across these civilizations,

that a body of mathematical results existed within Central Europe in

the period around 3000 to 2500 BCE, and that this then spread to

Great Britain, the Near East, India, and China.

One particular problem that all these civilizations had addressed

was the solving of quadratic equations. Modern algebraic notation

was not available until many centuries later. As a result, mathematics

was often described through words and examples, sometimes relying

on geometry to express what we now consider to be algebraic concepts.

Through the working out of examples, a general method would be

implicitly laid out.

Let’s look at one example of the Babylonian treatment of quadratic

equations, as described by van der Waerden, using the translation by

the great historian of ancient mathematics and science, Otto Neugebauer,

of Old-Babylonian text BM 13901. A problem is stated as follows:

”I have subtracted (the side of) the square from the area , and it

is 14, 30.” What we are to find is the length of the side of the square.

Babylonians used base 60 in writing numbers, so 14, 30 is to be interpreted

as (14×60)+30, or 870. Thus, in effect, the problem is to solve

the equation

x^{2} − x = 870.

The general quadratic equation in this form would be

x^{2} − ax = b

and the quadratic formula would tell us in this case that one solution

is

Substituting a = 1 and b = 870, we obtain x = 30.

The Babylonian text gives the same recipe in words. It says to take

1, the coefficient of the side of the square, divide it into two parts to

get 1/2, which is written as 0; 30 (recall that this is base 60 and think

of the semi-colon as a decimal point), form the product

0; 30 × 0; 30 = 0; 15

and add 14, 30. The result is 14, 30; 15, which has square root 29; 30.

The final step in the text is to add to 29; 30 the 0; 30 which was multiplied

by itself, obtaining 30 as the side of the square.

The point to be drawn from this, curious notation aside, is that

solving quadratic equations was well understood by Babylonians, and

other civilizations throughout the world, over 2000 years ago.

The mathematics that flowered in ancient Greece is surely the best

known of that developed in ancient civilizations, culminating with the

work of Diophantos of Alexandria on solving equations around 250

C.E. For the next thousand years, the principal developments in algebra

took place within Arab countries. Most notable of the Arabic

mathematicians is Muhammad ben Musa Al-Khwarizmi, who lived

roughly between 780 and 850 near Baghdad. As described by van der

Waerden, Al-Khwarizmi introduced the solution of problems by al-jabr

and al-muqabala. The process of jabr is that of adding equal terms

to both sides of an equation in order to eliminate negative terms. In

contrast, muqabala is the reduction of positive terms by subtracting

equal amounts from both sides of an equation. The combination of the

two words – al-jabr wal-muqabala– is sometimes used more generally to

mean performing algebraic operations, or simply the science of algebra.

Van der Waerden describes an example of Al-Khwarizmi’s. In modern

notation, one wishes to solve the equation

x^{2} = 40x − 4x^{2}.

Using the operation of al-jabr, one adds 4x^{2} to both sides to obtain the

equation

5x^{2} = 40x

or

x^{2} = 8x.

From this, Al-Khwarizmi obtains the answer x = 8.

Al-Khwarizmi treats quadratic equations in his treatise The Compendious

Book on Calculation by al-jabr and al-muqabala.. I refer to

van der Waerden for more details. See also his treatment of the algebraic

work of two later Arabic scholars, Tabit ben Qurra and Omar

Khayyam (the famed poet).

The return of significant mathematical activity within Europe occurred

in Italy, no doubt because its cities were major trade centers

with connections to Arabic ports along the Mediterranean. In the thirteenth

century, Leonardo da Pisa, better known to us under the name

Fibonacci, wrote several mathematical works that were preserved. Of

special note for us is that he provided a treatment in the Liber Abbaci

of linear and quadratic equations, citing the work of Al-Khwarizmi. A

number of Italians continued to work on algebra in subsequent centuries,

but we may jump ahead to Fra Luca Pacioli, who lived from

1445 to 1514. His main work, Summa de arithmetica , geometrica, proportioni

e proportionalit`a, was published in Venice in 1494. Of note is

that in it he writes, with regard to cubic and quartic equations, that

”it has not been possible until now to form general rules .” This was

the setting as the sixteenth century dawned.