Renaissance Europe The 1500’s and Beyond • The Italian algebraists and their work on solutions to cubic and quartic polynomials is probably the most exciting story from the 1500’s. Even if some of it was made up. • But, there are a few things we should talk about before moving on to the (mathematically) exciting periods of the 1600’s and 1700’s. The Abacists vs Traditionalists • Dear Professor: You keep on using that word. I do not think it means what you think it means. • Abacists were abacists in the sense of Liber Abaci, that is, they used the Islamic tradition of decimal numbers and algorithms, and eschewed counting boards in favor of pencil and paper. The Abacists vs Traditionalists • The Abacists won, but it took a while. • “But what if you were stranded on a desert island without paper? Then you’d wish you’d studied how to use a counting board……” Rafael Bombelli • Rafael Bombelli (1526‐ 1572) • His family had been out of favor with local leaders. • Lived in Bologna and was tutored by an engineer/architect. • Worked as a hydraulic engineer, draining wetlands. Rafael Bombelli • While waiting for a certain project to recommence, decided to write an algebra book. • Published in three volumes (two volumes of geometry that were to follow were not published) Rafael Bombelli • Wrote down rules for negative numbers: • Plus times plus makes plus Minus times minus makes plus Plus times minus makes minus Minus times plus makes minus Plus 8 times plus 8 makes plus 64 Minus 5 times minus 6 makes plus 30 Minus 4 times plus 5 makes minus 20 Plus 5 times minus 4 makes minus 20 Rafael Bombelli • Wrote down rules for computing with complex numbers: • Plus of minus times plus of minus makes minus [+√‐n . +√‐n = ‐n] Plus of minus times minus of minus makes plus [+√‐n . ‐√‐n = +n] Minus of minus times plus of minus makes plus [‐√‐n . +√‐n = +n] Minus of minus times minus of minus makes minus [‐√‐n . ‐√‐n = ‐n] Rafael Bombelli • Notation: Modern Bombelli Printed notation 1 5x 5 5x2 2 5 46 Rq 4pRq6 3 2 0 121 Rc 2pRq 0m121 François Viète François Viète (1540 – 1603) was a French lawyer who worked for kings Henri III and Henri IV as a cryptanalyst (a breaker of secret codes). François was born a French Protestant (Huguenot), and when Henry of Navarre, also a Protestant, came to power, it alarmed Catholics both inside and outside of France. François Viète • Philip II of Spain rather fancied his own daughter as successor to Henry III (who, by the way, was flamboyantly gay and unlikely to produce a lawful heir) rather than Henry of Navarre. Philip exchanged many letters with members of the French court in support of his daughter, even after Henry of Navarre assumed the throne as Henry IV. François Viète • These letters, written in code, were given to Viète to decode. He was so successful at this that Philip denounced him for being in league with the Devil. François Viète • During a period of exile from Henry III’s court (this was a very unsettled time in French history) he found the time to write several treatises that are collectively known as The Analytic Art, in which he effectively reformulated the study of algebra by replacing the search for solutions with a detailed study of the structure of equations. François Viète • Viète used upper case vowels (A, E, I, O, U, Y) to represent unknowns and upper case consonants to represent given constants. His symbolism was not complete, in the sense that he still used words to indicate powers – A2 is A quadratum, B3 would be B cubus, and C4 is C quadrato‐quadratum. He did at times use abbreviations such as A quad or C quad‐ quad. His rules for combining powers had to be given verbally. François Viète • Our equation ଶ , in Viète’s symbolism, would be: • B in A Quadratum, + D plano in A, aequari Z solido. • Here, he uses B, D and Z as known (but unspecified) numbers, and A as the unknown. The word “in” represents multiplication. The “+” sign was used for addition, as was “‐“ for subtraction. There was no “=” but instead some proper conjugation of the Latin verb aequare. François Viète • The words plano and solido demonstrate that Viète was strongly influenced by Greek concepts, since they are there to guarantee that the Z has “cubic” units, and the D has “square” units (resulting in “cubic” units when multiplied by A), thus guaranteeing that quantities of the same kind (cubic) are being added. Viète • On the other hand, he accepted larger powers in equations; he even solved a whopper of a 45th degree equation posed by Adriaan van Roomen. • Viète’s symbolism was to be further refined and superseded by that of René Descartes, who only a few years later gave us what amounts to our modern system of algebraic symbolism. Viète • Provided an approximation of correct to 10 decimals by using Archimedes’ method. • Wrote what seems to be the first infinite product by noting that Some Astronomy Nicolaus Copernicus • Mikołaj Kopernik, 19 February 1473 –24 May 1543 • Contributed to mathematics, medicine, astronomy, law, economics. Spoke Latin, German, Polish, Greek, and Italian. Nicolaus Copernicus • Refused to participate in reform of the Julian calendar because he knew the inaccuracies of the Ptolemaic system and figured it was impossible to patch up a calendar based on a geocentric model any longer. Nicolaus Copernicus • Most important contribution was De revolutionibus orbium celestium (On the Revolutions of the Heavenly Spheres) • Heliocentric theory Nicolaus Copernicus • If there should chance to be any mathematicians who, ignorant in mathematics yet pretending to skill in that science, should dare, upon the authority of some passage of Scripture wrested to their purpose, to condemn and censure my hypothesis, I value them not, and scorn their inconsiderate judgment. ‐‐ De Revolutionibus Coelestibus Tycho Brahe • Tycho Brahe (1546 – 1601), son of Danish nobleman. • Mainly a very careful and astute observer of the heavens. • Observed a supernova, and comets, and thus disproved the “unchanging celestial spheres” theory of Ptolemaic and Aristotelian philosophy. Tycho Brahe • Proposed a “geo‐ heliocentric” model of the solar system, now known as the Tychonic system. In such a system, the Sun annually circles a central Earth (regarded as essentially different from the planets), while the five planets orbit the Sun. Tycho Brahe • Became very famous and quite rich. • It is said that at one time he owned about 1% of the total wealth of Denmark. • “Married” a commoner and had 8 children. Tycho Brahe • Pretty interesting story altogether. • Lost his nose in a duel in his early twenties. Replaced it with a gold‐ silver alloy nose, which he kept stuck on with paste from a snuff‐box. Tycho Brahe • During a period of excess, kept a dwarf named Jepp (whom he thought to be clairvoyant) as a sort of court jester. • Also had a live moose that drank too much beer at a party and fell down the stairs to his death. Tycho Brahe • At least one book claims, due to extremely toxic levels of mercury found in hairs from his moustache, that he was murdered by his apprentice, Johannes Kepler. Johannes Kepler • Other than being an accused murderer (400+ years after the fact), Johannes Kepler finally unraveled the mathematical laws that governed the motion of the planets. Johannes Kepler • Kepler (1571‐1630) was born in Germany, the son of a mercenary soldier who didn’t return home after leaving for a war when Johannes was 5. • A profoundly religious man throughout his life. Johannes Kepler • Published Astronomia Nova in 1609. • Probably the most important scientific publication prior to Newton’s Principia MathematicaI. Johannes Kepler • Kepler noticed that even with the Copernican heliocentric system, the predictions of planetary motions weren’t much better than with the old Ptolemaic system. • Because the planets orbited the center of the Earth’s orbit, and not the sun. • Orbits were circular. Johannes Kepler • First two laws of planetary motion: 1. The radius vector sweeps out equal areas in equal times. 2. The planets move in elliptical orbits with the Sun at one focus. Johannes Kepler • Third Law appeared in Harmonice Mundi (1619): • “It is absolutely certain and exact that the ratio which exists between the periodic times of any two planets is precisely the ratio of the 3/2‐th power of the mean distances [of the planet to the sun].” Johannes Kepler – Other Firsts • 1604: • 1620: – Inverse square law of – First textbook on photometry heliocentric astronomy – First correct explanation of how the human eye • 1629: works. – First predictions of • 1611: transits in Mercury and – Invention of the Venus. His prediction for astronomical telescope Mercury was out by only with convex objective 6 hours. They occur and eyepiece. about 13 –14 times per century. Johannes Kepler • Brahe’s observations combined with Kepler’s laws led to the Rudolphine Tables giving a factor 20 improvement in the prediction of planetary positions. And just for fun: • After his first wife died, he remarried in order to have someone to look after his children. At the wedding celebrations he noticed that the volumes of wine barrels were estimated by means of a rod slipped in diagonally through the bung‐hole, and he began to wonder how that could work. The result was a study of the volumes of solids of revolution (New Stereometry of wine barrels, 1615) in which Kepler, based on the work of Archimedes, used a resolution into 'indivisibles'. And just for not‐so‐much‐fun: • “His mother was accused of witchcraft.