This is the quadratic formula, as it is taught to most of us in school:


x_1,2=(-b/2a) ± (1/2a)(b²-4ac
)^1/2



gives the solution to a generic quadratic equation of the

form:

ax² + bx + c = 0.

The development, or derivation, of a mathematical idea, is usually as
logical, deducible and rectilinear as possible. This brings the common
notion about, that its historical development is similarly as
continuous, logical and rectilinear: One mathematician picking up an
idea where another mathematician left it.

Using the quadratic formula as an example, it will be shown that the
historical development of mathematics is not at all rectilinear.
Instead, one will often find parallel developments, interconnections
and confluences, which - to complicate this stuff even further - are
also interrelated with social, cultural, political and religious
matters.

The so-called quadratic formula has been derived in the course of a few millennia to its current form, which is taught to most of us in school. This Entry will strictly concentrate on the historical
development of the quadratic formula. Some mathematical background
may be of use to fully understand the described development, the maths
used in this Entry will be kept at a necessary minimum.

The Original Problem 2000 (or so) BCE

Egyptian, Chinese and Babylonian engineers were really smart people,
they knew how the area of a square scales with the length of its
side. They knew that one can store nine times more bales of hay if
the side of the square loft is tripled. They also found out how to
calculate the area of more complex designs like rectangles and
T-shapes and so on. However, they didn't know how to calculate the
sides of the shapes, the length of the sides, starting from a given
area - which often was what their clients really needed. And so, this
is the original problem: A certain shape¹ must be scaled with a total area, and in the end what
one needs is lengths of the sides, or walls to make a working floor
plan.

1500 BCE The Beginnings - Egypt

The first aspect that finally led to the quadratic equation is that
it is connected to a very pragmatic problem, which in its turn
demanded a 'quick and dirty' solution. We have to note, in this
context, that the Egyptian mathematics did not know equations and
numbers like we do nowadays, it is instead descriptive, rhetoric and
sometimes very hard to follow. It is known that the Egyptian wise-men
(engineers, scribes and priests) were aware of this shortcoming. But
they came up with a way to circumvent this problem: Instead of
learning an operation, or a formula that allows one to calculate the
sides from the area, they calculated the area for all possible sides
and shapes of squares and rectangles and made a look-up table. This
method works much like we learn the multiplication tables by heart in
school instead of doing the operation proper.

So, if someone wanted a loft with a certain shape and a certain
capacity to store bales of papyrus, the engineer would go to his table
and find the most fitting design. The engineers did not have time to
calculate all shapes and sides to make their own table. Instead, the
table they used was a reproduction of a master look-up table. The
copyists did not know if the stuff they were copying made sense or
not as they didn't know anything about maths. So, obviously,
sometimes errors crept in, and copies of the copies were known to be
less trustworthy². These tables
still exist, and it is possible to see where errors crept in during
the copying of the documents.

400 BCE The next step - Babylon and China

The Egyptian method worked fine, but a more general solution, without the need for tables seemed desirable. That's where the Babylonian geeks come into play. Babylonian maths had a big advantage over the one used in Egypt, namely they used a number-system that is pretty much like the one we use today, albeit on a hexagesimal basis, or base-60. Addition and multiplication were a lot easier to perform with this system, so the engineers around 1000 BCE could always double-check the values in their tables. By 400 BCE they found a more general method called 'completing the square' to solve generic problems involving areas. There are no indications that these people used a specific mathematical procedure to find out the solutions, so probably some educated guessing was involved. Around the same time, or a bit later, this method also appears in Chinese documents. The Chinese, like the Egyptians, also did not use a numeric system, but a double checking of simple mathematical operations was made astonishingly easy by the widespread use of the abacus.

300 BCE Geometry - Hellenistic Mediterranean Area

The first attempts to find a more general formula to solve quaratic equations can be tracked back to geometry (and trigonometry) top-bananas Pythagoras (500 BCE in Croton, now Italy) and Euclid (300 BCE in Alexandria, now Egypt), who used a strictly geometric approach, and found a general procedure to solve the quadratic equation. Pythagoras noted that the ratio between the area of a square and the respective length of the side, the square root, were not always integer, but he refused to allow for proportions other than rational. Euclid went even further and found out that this proportion might also not be rational. He concluded that irrational numbers exist.

Euclid's opus Elements covered more or less all the mathematics needed for technical applications from a theoretical point of view. However, it didn't use the same notation with formulas and numbers like we use nowadays. For that reason it was not possible to calculate the square root of any number by hand, in order to obtain a good aproximation for the exact value of the root, which is what the architects and engineers were after. The fact that all, theoretically relevant at least, maths seemed to be complete³ but otherwise useless, the many wars occurring in Europe, and also the early Middle Ages turned the mathematical world in Europe silent until the 13th Century. In this period mathematics also suffers a big shift, going from a pragmatic science to a more mystical, philosophical discipline.

700 CE All Numbers - India

Hindu mathematics used the decimal system (the one we use) at least since 600 CE. One of the most important influences on Hindu mathematics was that it was widely used in commerce. The average Hindu merchant was pretty fast in simple maths. If someone had a debt the numbers would be negative, if someone had a credit the numbers would be positive. Also, if someone had neither credit, nor debt, the numbers would add up to 'zero'. Zero is an important number in the history of mathematics, and it's relatively late appearance, is due to the fact that many cultures had difficulty to conceive 'nothing'. The concept of 'nothing', like in 'shunya', the void, or the concept of 'equilibrium' was already anchored in Hindu culture.

Around 700 CE the general solution for the quadratic equation, this time using numbers, was devised by a Hindu mathematician called Brahmagupta, who among other things used irrational numbers, and who recognized two roots in the solution. The final, complete, solution as we know it today came around 1100 CE, by another Hindu mathematician called Baskhara⁴. Baskhara was the first to recognize that any positive number has two square roots.

820 CE Powerful Islamic Science - Persia

Around 820 CE, near Baghdad, Mohhamad bin Musa Al-Khwarismi, a famous Islamic mathematician⁵, and who knew Hindu mathematics, also derived the quadratic equation. The algebra used by him was entirely rhetorical, and he rejected negative solutions. This particular derivation of the quadratic formula was brought to Europe by Jewish mathematician/astronomer Abraham bar Hiyya (whose Latinised name is Savasorda) who lived in Barcelona around 1100.

1500 CE Renaissance - Europe

With the Renaissance in Europe, the academic attention came back to original mathematical problems. By 1545 Girolamo Cardano, who was a typical Renaissance scientist (ie. interested in alchemy, occultism and suchlike), and one of the best algebraists of his time, compiled the works related to the quadratic equations, that is, he blended Al-Khwarismi's solution with the Euclidean geometry. He was possibly not the first or only one, but the most famous. In his, mainly rhetorical, works he allows for the existence of complex, or imaginary numbers, that is roots of negative numbers. In the end of the 16th Century the mathematical notation and symbolism was introduced by amateur-mathematician François Viète, in France. René Descartes in 1637 published La Géométrie, modern Mathematics was born, and the quadratic formula has adopted the form we know today.