A radical life EVARISTE GALOIS The long road to Galois CHAPTER 1 Babylon How many miles to Babylon? Three score miles and ten. Can I get there by candle-light? Yes, and back again. If your heels are nimble and light, You may get there by candle-light.[ Babylon was the capital of Babylonia, an ancient kingdom occupying the area of modern Iraq. Babylonian algebra B.M. Tablet 13901-front (From: The Babylonian Quadratic Equation, by A.E. Berryman, Math. Gazette, 40 (1956), 185-192) B.M. Tablet 13901-back Time Passes . Centuries and then millennia pass. In those years empires rose and fell. The Greeks invented mathematics as we know it, and in Alexandria produced the first scientific revolution. The Roman empire forgot almost all that the Greeks had done in math and science. Germanic tribes put an end to the Roman empire, Arabic tribes invaded Europe, and the Ottoman empire began forming in the east. Time passes . Wars, and wars, and wars. Christians against Arabs. Christians against Turks. Christians against Christians. The Roman empire crumbled, the Holy Roman Germanic empire appeared. Nations as we know them today started to form. And we approach the year 1500, and the Renaissance; but I want to mention two events preceding it. Al-Khwarismi (~790-850) Abu Ja'far Muhammad ibn Musa Al-Khwarizmi was born during the reign of the most famous of all Caliphs of the Arabic empire with capital in Baghdad: Harun al Rashid; the one mentioned in the 1001 Nights. He wrote a book that was to become very influential Hisab al-jabr w'al-muqabala in which he studies quadratic (and linear) equations. It seems that ``al-jabr’’ means ``completion’’ and refers to removing negative terms. It is the origin of the word ``algebra.’’ ``al-muqabala’’ means balancing, and it refers to reducing positive terms if they appear on both sides of the equation. Al-Khwarismi (~790-850) Al-Khwarismi divides equations into six groups, then shows how to solve equations in each group. 1. Squares equal to roots. 2. Squares equal to numbers. 3. Roots equal to numbers. 4. Squares and roots equal to numbers; e.g. x2 + 10 x = 39. 5. Squares and numbers equal to roots; e.g. x2 + 21 = 10 x. 6. Roots and numbers equal to squares; e.g. 3 x + 4 = x2. Life was hard before the invention of decent mathematical notation! Al-Khwarismi (~790-850) His book was widely read also by European mathematicians, and they began to talk of doing things in the Al-Khwarismi way or, as it came to be known, by an algorithm. What about the cubic equation? . The Babylonians had solved some simple cubic equations. But the first serious attempt was perhaps due to a mathematician who is as famous as a mathematician as he is as a poet. Omar Khayyam (1048-1131). A great Persian mathematician working in the Seljuk (Turkish) empire. Solved cubic equations numerically by intersection of conic sections. Stated that some of these equations could not be solved using only straightedge and compass, a result proved some 750 years later. The following web pages discuss his method: Omar Khayyam and a Geometric Solution of the Cubic Omar Khayyam and the Cubic Equation Important general fact . For all of the mathematicians trying to solve cubic equations, negative numbers and 0 were mysterious not well understood concepts. For us, the cubic equation is x3 + ax2 + bx + c =0. For us there is little difference, if any, between the equations x3 + ax = b and x3 = ax + b. For Scipione dal Ferro, Tartaglia et al, the difference was essential because they could only understand the equation if a, b were positive numbers. And, of course, the equation was never written in the form x3 + ax + b = 0; setting to 0 just didn't make any sense. As the year 1500 approaches… . Only second degree equations were known to be solvable by radicals. And then … . But first, a few word from your real numbers sponsor. Solving an equation by radicals . There are, of course many ways of solving equations. Once one knows there are roots, it is only a matter of time to find them. At the heart of what it means to solve an equation is the question of the meaning of numbers. What is a number? . The question is (sort of) easy to answer if the number is an integer, or even a rational number. But what really is the square root of 2. The number π? If Greek mathematicians had had digital computers… and had developed fully the atomic theory of matter, there might not have been a need for irrational numbers. Chances are nature is discrete, and we only needed irrational numbers because we could not deal with zillions of particles at once; we had to come up with a continuous model and invent calculus to be able to handle it. Raffaelo Sanzio: The School of Athens ~ 1510 In the future, we might be able to dispense with continuous models. If Greek mathematicians had had digital computers… This purely discrete mathematics will be to current mathematics what a music based on only two notes would be to Mozart’s music. But… Personally, I prefer to think that while our world is probably discrete, it is based on a continuous blueprint, and we want to study the blueprint more than the somewhat imperfect construction based on it. Raffaelo Sanzio: The School of Athens ~ 1510 If Greek mathematicians had had digital computers… But, lacking computers, and not quite sure about atoms, the Greeks invented a continuous mathematics and discovered irrational numbers. They were baffled. Raffaelo Sanzio: The School of Athens ~ 1510 Eudoxus of Cnidus In one of the most brilliant ``tours de force’’ in mathematics, Eudoxus (408 BCE-355 BCE) solved the problem of the irrationals. A semi modern interpretation is that in many ways it is meaningless to ask, for real numbers, is a = b? By Eudoxus, a =b simply means that both a < b and b < a are false. This idea lies behind the finding of formulas for areas and volumes of curved figures by Euclid and Archimedes; it is essential to the notion of convergence. The point is… . That one has to be very explicit by what one means by having a formula to solve algebraic equations. To solve an equation by radicals means to have a formula in which the solutions are expressed as a function of the coefficients; the process of going from the coefficients of the equation to the solution should involve only the usual arithmetic operations (+, -, ×, /), and extraction of roots, and should involve only a finite number of steps. And now to: The Italian Connection THE CUBIC CHRONICLES Luca Pacioli (1445-1509) In Summa de arithmetica, geometria, proportioni et proportionalità, published 1494 in Venice, he summarizes all that was known on equations. He discusses quartic equations stating that the equation that in modern notation is written as x4 = a + bx2 can be solved as a quadratic equation but x4 + ax2 = b and x4 +a = bx2 are impossible at the present state of science; ditto the cubic equation. Great friend of Leonardo da Vinci, briefly a colleague of Scipione dal Ferro. And Then….. THE FIRST ADVANCE OF EUROPEAN MATHEMATICS SINCE THE TIME OF THE GREEKS: Scipione dal Ferro figures out how to solve the depressed cubic equation (ca. 1515) The depressed cubic equation x3 px q Says cubi: I am depressed because I am missing my quadratic term. Scipione dal Ferro (1465- 1526). There may not be any Professor at Bologna. Around 1515 figured reliable portrait of out how to solve the equation x3 + px = q by Scipione dal Ferro on the radicals. Kept his work a complete secret until web. One that pops up is just before his death, then revealed it to his really Tartaglia. student Antonio Fior. Antonio Fior (1506-?). Very little information No portrait of seems to be available about Fior. His main claim Antonio Fior seems to fame seems to be his challenging Tartaglia to to be available. a public equation ``solvathon,’’ and losing the challenge. The two great rivals Nicolo of Brescia who adopted the name Tartaglia (Stutterer) (1499-1557). Hearing a rumor that cubic equations had been solved, figured out how to solve equations of the form x3 +mx2 = n, and made it public. This made Fior think that Tartaglia would not know how to deal with the equations del Ferro knew how to solve and he challenged Tartaglia to a public duel. But Tartaglia figured out what to do with del Ferro’s equation and won the contest. Girolamo Cardano (1501-1576). Mathematician, physician, gambler (which led him to study probability), a genius and a celebrity in his day. Hearing of Tartaglia’s triumph over Fior convinced Tartaglia to reveal the secret of the cubic, swearing solemnly not to publish before Tartaglia had done so. Then he published first. But there are some possible excuses for this behavior. Cardano’s oath I swear to you, by God's holy Gospels, and as a true man of honour, not only never to publish your discoveries, if you teach me them, but I also promise you, and I pledge my faith as a true Christian, to note them down in code, so that after my death no one will be able to understand them. And then, in Ars Magna published in 1545, Cardano revealed the formula to the world. Completing the Picture Lodovico Ferrari (1522-1565). A protégé of Cardano, he discovered how to solve the quartic equation, by reducing it to a cubic.