N-th Order Linear Differential Equations 191imaginary. In 1748 , Euler introduced the use of i for A and in 1832, Gauss
gave the name complex numbers to the quantities a+ bi.
Prior to 1545 , mathematicians recognized only positive real numbers as
roots of polynomials. Therefore, to solve a polynomial equation before 1545
meant to find the positive real roots. The linear equation ax = b where a and
b were both positive could be solved both algebraically and geometrically by
early civilizations. The ancient Babylonians ( c. 2000 B .C.) knew how to solve
the quadratic equation ax^2 + bx + c = 0 algebraically by both the method of
completing the square and by substituting into the quadratic formula. They
could also solve certain special cubic equations. The Persian poet, astronomer,
and mathematician Omar Khayyam (c. 1044-1123) was able to geometricallysolve every type of cubic equation for its positive roots. It was not until almost
five hundred years later tha t the algebraic solution of the cubic equation was
accomplished. Shortly thereafter the algebraic solution of the quartic equation
was achieved also.
About 1515 , Scipio del Ferro discovered the algebraic solution of the cubicequation of the form x^3 +bx = c. He did not publish his result but revealed his
secret to his pupil Antonio Maria Fior (Florido). In 1530, Zuanne de Tonini
d a Coi sent the following two problems to Niccolo Fontana to be solved:x^3 + 3x^2 = 5 and x^3 + 6x^2 + 8x = 1000.
Fontana was also known as Tartaglia (the stammerer) because of a saber
wound he received when he was only thirteen years old at the hands of the
French in the 1512 massacre at Brescia. In 1535 , Tartagli a claimed to havealgebraically solved the cubic equation of the form x^3 + ax^2 = c. Florido, who
b eli eved that Tartagli a was merely boasting, challenged Tartagli a to a public
contest. Each contestant was to submit to the other the same number of cubic
equations to be solved within a given period of time. Tartaglia accepted the
challenge. He knew he could defeat his opponent ifhe submitted only problems
of the type which he could solve, namely, x^3 + ax^2 = c, and if he could also
solve problems of the type which Florido co uld solve, namely, x^3 + bx = c.
Tartagli a exerted himself and a few days before the scheduled contest he
discovered how to so lve problems of the type Florido could solve. Knowing
how to so lve both types of cubic equations, whereas his opponent only knew
how to solve one type of cubic equ ation, Tartagli a triumphed completely.
However , Tartaglia did not publish his method of solution. In 1539, Girolamo
Cardano (Cardan), an unscrupulous man who practiced medicine in Milan,
wrote Tartaglia requesting a meeting. At the meeting Tartaglia, upon h aving
pledged Cardan to secrecy, revealed his method of solution- at first in cryptic
verse and later in full detail. Cardan admits receiving t he solution from
Tartagli a but denies receiving any explanation of the method. In 1545 , Cardan
published his Ars Magna in which he reduced the general cubic equation
x^3 + px^2 + qx = r to the form x^3 +bx = c and then solved the latter equation.
Tartaglia protested vehemently. But Cardan was ably defended by one of