Alex's Adventures in Numberland - Alex Bellos [85]
Extracting the cubic formula from Tartaglia became an obsession of Cardano’s. He eventually thought up a ruse – he invited Tartaglia to Milan on the pretext of setting up an introduction to a potential benefactor, the governor of Lombardy. Tartaglia accepted the offer, but once he arrived, discovered that the governor was out of town. Instead, he encountered only Cardano. Worn out from Cardano’s incessant pestering, Tartaglia relented, telling Cardano that if he could keep the formula to himself, he would reveal it to him. But when he passed over the information to Cardano, the crafty Tartaglia wrote the solution in a deliberately abstruse way: as a bizarre poem of 25 lines.
Despite this impediment, the multi-talented Cardano deciphered the method, and he almost kept his promise. He told the solution to only one person, his personal secretary, a young boy named Lodovico Ferrari. This turned out to be problematic, not because Ferrari was indiscreet, but because he improved on Tartaglia’s method to find a way to solve quartic equations. These are equations that require the power of x4. For example, 5x4 – 2x3 – 8x2 + 6x + 3 = 0. A quartic may arise when multiplying one quadratic with another.
Cardano was in a fix – he couldn’t publish Ferrari’s discovery without betraying Tartaglia’s word, but neither could he deny Ferrari the public acclamation that he deserved. Cardano, however, managed to find a clever way out. It turned out that Antonio Fiore, the man who lost the cubic duel against Tartaglia, did in fact know how to solve the cubic, and he had learned the method from an older mathematician, Scipione del Ferro, who had told Fiore from his deathbed. Cardano discovered this after approaching del Ferro’s family and going through the late mathematician’s unpublished notes. Cardano thus felt morally justified in publishing the result, crediting del Ferro as the original inventor, and Tartaglia as the reinventor. The method was included in Cardano’s Ars Magna, the most important book on algebra of the sixteenth century.
Tartaglia never forgave Cardano, and died an angry and bitter man. Cardano, however, lived until he was nearly 75. He died on 21 September 1576, the date he had predicted when casting his horoscope years before. Some maths historians claim he was in perfect health and drank poison just to ensure his prediction would come true.
Rather than just looking at equations with higher and higher powers of x, we can also increase complexity by adding a second unknown number, y. The school algebra favourite that is known as simultaneous equations is usually the task of solving two equations that each have two variables. For example:
y = x
y = 3x – 2
To solve the two equations, we substitute the value of the variable in one equation with the value from the other. In this case, since y = x, then:
x = 3x – 2
Which reduces to 2 x = 2
So x = 1, and y = 1
It’s also possible to understand any equation in two variables visually. Draw a horizontal line and a vertical line that intersect. Define the horizontal line as the x-axis, and the vertical line as the y-axis. The axes intersect at 0. The position of any point in the plane can be determined by referencing a point on both axes. The position (a,b) is defined as the intersection of a vertical line through a on the x-axis and a horizontal line through b on the y-axis.
For any equation in x and y, the points where (x,y) has values for x and y that satisfy the equation describe a line on a graph. For example, the points (0,0), (1,1), (2,2) and (3,3) all satisfy our first equation above, y = x. If we mark, or plot, these points on a graph, it becomes clear that the equation y = x generates a straight line, as in the figure below. Likewise, we can draw the second equation, y = 3x – 2. By assigning x a value and then working out what y is, we can establish that the points (0,–2), (1,1), (2,4) and (3,7) are on the line described