Analysis, Synthesis, the Infinite, and Numbers

Chapter 15 Getting Our Bearings

• Where are we in history? • We’ve finished the middle ages, and have embarked on the Early Modern period (about 1500 – 1800 or so). • In fact, we’ve finished the 1500’s as well. Not Europe

Place Dates Names China Ancient &Medieval The Nine Chapters, and roots, Sea Island indeterminate , Mathematical systems of linear Manual, others equations and linear congruences, polynomial equations India Ancient &Medieval , Square and cube roots, Bramagupta, indeterminate equations, others linear congruences, . Islamic Empire Medieval Al Khwarizmi, , solutions of , al cubics by conics, ’s Tusi, others 5th postulate, Hindu‐ Arabic numeration, Europe

Place Dates Names Mathematics Europe Medieval Leonardo of Pisa, bar Islamic Methods introduced to Hiyya, ibn Ezra, ben Europe Gerson Europe Renaissance / Pacioli, Cardano, Solution of the cubic, Early Modern Targaglia, Ferrari, del systematic algebra & theory of Ferro, Bombelli, Viète, solving equations, Stevin, Regiomontanus, trigonometry, decimal Copernicus, Kepler, fractions, logarithms, Brahe, Napier, others heliocentric astronomy. Getting Our Bearings

• Where are we in history? • We’ve finished the middle ages, and have embarked on the Early Modern period (about 1500 – 1800 or so). • In fact, we’ve finished the 1500’s as well. • We are now entering the 1600’s. Next up

Place Dates Names Mathematics Europe 1600’s Fermat, Descartes, Analytic , theory of equations, , Liebniz, , normals, , max/min, Barrow, Pascal, systematic Oughtred, Harriot, Wallis Europe 1700’s Bernoulli, Further development of the calculus: Bernoulli, differential equations, brachistochrone Bernoulli, …. problem, tautochrone problem, catenary Maclaurin, problem, calculus of variations, multi‐ L’Hopital, dimensional calculus, transcendental Euler functions (logs, exp, trig), complex numbers, multi‐dimensional calculus, partial differential equations, calculus texts, the foundations of calculus, theory of equations, . Galileo and Cavalieri

• For us, the major mathematical interest in his work is his treatment of the infinite, and we’ll discuss that later when we talk about Cantor. • We will also discuss Cavalieri’s work when we talk about the development of calculus prior to Newton and Leibniz.

• 1601‐1665 • Trained as a lawyer at the University of . • Appointed as a judge in Toulouse in 1638. • Married and had five children. Pierre de Fermat

• Considered math his hobby, and never really published any of his works. • Like many others, he was interested in “restoring” lost works of ancient Greek mathematics. • His work is known mainly because of his correspondences with other notable , such as Mersenne (the “walking scientific journal of .”) Pierre de Fermat

• Made contributions to – Number theory – – – Analysis (calculus) • Often didn’t provide proofs. Didn’t like to “polish” his work. This annoyed some of his contemporaries. He was often correct, but not always. Pierre de Fermat

• His correspondence with Pascal signals the birth of modern , and the first major developments in this area since Cardano. • His best work was in number theory, inspired by his reading of ’ . We’ll read about how Euler filled in a great number of the proofs which Fermat overlooked. Pierre de Fermat

• Of course, the most notorious of Fermat’s non‐proofs was of the statement that the has no non‐zero integer solutions for x, y, and z when . (He had already proved the case for n=3) • Known as “Fermat’s Last Theorem” • Fermat wrote, in the margins of his copy of Arithmetica, “I have discovered a truly remarkable proof which this margin is too small to contain.” • Mathematicians searched for a proof ever since this note was discovered and published by his son. “Fermat’s Last Theorem”

• It is pretty well accepted that Fermat was in error about having a proof. I believe the direct quote from historian of mathematics Victor Katz was “Liar, Liar, Pants on Fire.”* • In 1993 English claimed to have a proof, although there were found to be some gaps in it. By 1994 he had closed these gaps and his proof is now accepted as correct. But it is a proof that relies on hundreds of pages and over three centuries of previous work from other mathematicians, and is nothing like anything Fermat could have envisioned.

*Just kidding. He really cited 2 Nephi 9:34 René Descartes

• 1596 (Tours, France) – 1650 (Stolkholm, Sweden) • Educated at the Jesuit college of La Flèche in Anjou from 1604 until 1612, studying classics, logic and traditional Aristotelian philosophy. • While in the school his health was poor and he was granted permission to remain in bed until 11 o'clock in the morning, a custom he maintained until the year of his death. René Descartes

• “Philosophy affords the • Began to see mathesis means of discoursing universalis as the with an appearance of underpinnings of all truth on all matters, scientific explanation. and commands the • He devoted some study admiration of the more to this in order to later simple.” tackle the other sciences. René Descartes

• Earned a law degree (which he largely ignored), travelled, joined the army. • Eventually (about 1628) he settled down in Holland. • Wrote a book Le Monde on Natural History (science/) but decided not to publish it after hearing what happened to Galileo. René Descartes

• In 1637 published Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences. – ‘The Method’ of arriving at truth. – ‘‘I think, therefore I am.’’ • Three appendices to this work were La Dioptrique, Les Météores, and La Géométrie. • La Géométrie is what mainly concerns us. René Descartes

• In La Géométrie Descartes developed his analytic geometry as an example of how his method could arrive at certain knowledge. • La Dioptrique contained a theory of as instantaneous movement through an elastic medium. We experience light like we experience resistance that travels from the of action to our hand through a walking stick. • Based on his earlier, unpublished work (Le Monde), and his vortex theory. René Descartes

• He assumed that the universe is filled with matter which, due to some initial motion, had settled down into a system of “vortices” which carry the sun, the stars, the planets and comets in their paths. • Despite the problems with the vortex theory it was championed in France for nearly one hundred years even after Newton showed it was impossible as a dynamical system. René Descartes

• In Le Dioptrique and Les Meteores, “only tried to convince people that my method is better than the ordinary one. I have proved this in my geometry, for in the beginning I have solved a question which, according to Pappus, could not have been solved by any of the ancient geometers.” ‐‐ Letter to Mersenne. • He misread Pappus, by the way. It had been solved. René Descartes

• In 1641 he published a work called Meditationes, in which he explained at some his views on philosophy as sketched out in the Discourse. • In 1644 he issued the Principia Philosophiae, the greater part of which was devoted to physical science, especially the laws of motion and the theory of vortices. René Descartes

• In 1647 he received a pension from the French court in honor of his discoveries. • In 1649 he went to Sweden on the invitation of the Queen in order that she might learn some mathematics from him. She liked to draw her lines at about 5 a.m. and so for the first time, Descartes had to give up his habit of lying in bed until almost noon. • He died a few months later of inflammation of the lungs / pneumonia, brought on by the climate of Sweden and having to arise that early. The Birth of Analytic Geometry

• Born in the 1630’s of two fathers, Pierre de Fermat and René Descartes. • Although their approaches were somewhat different, they both attacked classic problems from , particularly the problems of Apollonius. • Both were searching, in a sense, for the “lost” analysis of the ancient Greeks. (Pappus’ work was lost, and The Method had not been found.) One of Fermat’s Problems

• If, from any number of given points, straight lines are drawn to a point, and if the sum of the of the lines is equal to a given area, the point lies on a of a given position. (Apollonius, via Pappus) • Notice the generality here –perfectly set up for variables and hence for algebra, but the Greeks didn’t have that as an analytic tool. Fermat’s Solution

• Used GH, GL, and GK B P D

• Also used GA, HB, LD, and KC C • Discovered A and O

• Thus, his analysis allowed him toG H M L K find the center O of the circle. Fermat’s Methods B

E

Locus AB Variable segment DE

A D C Fixed straight line AC If, when solving a geometry problem algebraically, you end up with an equation in two unknowns, the resulting solution is a locus, whose points are determined by the motion of one endpoint of a variable , the other endpoint of which moves along a fixed straight line. Fermat’s Methods

• Note that Fermat used one axis, which the “variable line segment” moved along at a fixed and varied in length to define the locus of points. • The fixed angle didn’t have to be 90 degrees. • Notice also that this describes a “mechanical ,” one that is drawn by the endpoint of a moving segment. Fermat’s Methods

• Fermat proved that whenever the relationship among segment lengths gave rise to second degree equations in two variables, the loci defined were conics. He carefully showed that they met the definitions in Apollonius’ Conics. One of Descartes’ Problems

• Let three, four, or more lines be given in position; required the locus of the point from which the same number of lines may be drawn to meet them one to each, at given , such that, in the case of three lines, the of the first two lines may have a fixed relation to the square of the third; or, in the case of four lines, that the rectangle of the first and second may have a fixed relation to rectangle of the third and fourth, and so on. (Apollonius, via Pappus) One of Descartes’ Problems

• Pappus knew that in the case of three or four lines the locus of the point was a . But he didn’t know what the locus was when there were more than 4 lines. • Descartes set out to prove the general case. Descartes’ Solution

T

S

R

E G A x B F y H

D C

X=AB and y=BC; then all other lengths can be represented in terms of these two lengths. For example, the angles in ∆ are known. If ⁄ , then BR = bx. and CR = y+bx. If ⁄ , then CD = cy + bcx. And so on. And on. Descartes’ Methods

• All lines were referred to two principle lines (what we would call axes). • Showed the results that Pappus already knew • Then went on to solve the five‐line problem. • The solution was given by a third‐degree polynomial in x and y. • These and other higher‐degree polynomials also led Descartes into a study of solutions of polynomials. Implications

• Not just the use of algebra to solve geometric problems (although that was important) • Perhaps the greatest implication was the ability to see both curve and equation as two correlated aspects of the same problem, and the complete generality of the methods of solution this gave rise to. Fermat vs Descartes

• Both used one axis instead of two, and neither insisted that the lines measuring the other component had to be at right angles to the axis. • Both used the conics as their chief examples, and both were able to create from equations of degree higher than two. Fermat vs Descartes

• Fermat started with the equation, then showed how it determined a curve. • For Descartes, the curves were primary, and he was able to come up with the necessary equations to describe them and solve problems. Fermat vs Descartes

• Fermat never published his results, although they were circulated widely in manuscript form. • Descartes published his in French (instead of the more typical Latin) and left too many gaps to be filled in. It took others’ commentaries to help his work achieve the recognition it deserved, over 20 years after first appearing. • One historian commented that Descartes wrote La Géométrie to brag, not to explain. Fermat VERUS Descartes

• Take this for what it’s worth, but it’s a good story. Fermat VERUS Descartes

• Jean de Beaugrand published Geostatics in 1636 and Descartes issued a brutal criticism of it. • Upon the appearance of Dioptrics, Beaugrand got an advance copy and started a campaign against it. He sent it to various people he knew, including one M. Fermat. Fermat VERUS Descartes

• Fermat, unaware that there was any political battle brewing, issued what he thought to be a fair, scholarly, scientific critique in a letter to Mersenne. He offered to help M. Descartes in his search for truth. • Mersenne delayed sending it on for a while, but eventually did. Fermat VERUS Descartes

• What Fermat also didn’t know was that Dioptrics was based on Descartes’ previous work, Le Monde, which he elected not to publish. Most of Fermat’s criticisms focused on that were much more fully explained in the original work. • Descartes correctly assumed Fermat had misunderstood things because he didn’t have the full context of his earlier work. Fermat VERUS Descartes

• But then, Fermat got ahold of an early copy of Géométrie. He felt is was seriously lacking in some areas, including a treatment of (which Fermat had worked on himself). • Fermat sent his work on maxima and minima, and his work in analytic geometry to Mersenne, and eventually Descartes saw it – just before official publication of Discourse. Fermat VERUS Descartes

• Descartes did not react well to this or any other criticism. He felt he had nothing to learn from his contemporaries in mathematics, and described them variously as “two or three flies,” “less than a rational animal,” “a little dog,” and “extremely contemptible.” These were the non‐ scatological descriptions. Fermat VERUS Descartes

• In a letter to Mersenne, he said that none of his critics had been able to achieve anything of which the ancient geometers were ignorant. • That included “M. vostre Conseiller De Maximus et Minimus,” i.e., Fermat. Fermat VERUS Descartes

• Fermat’s analytic geometry was particularly annoying to Descartes because he (Fermat) had essentially accomplished what Descartes claimed could only be done because of his Method. Thus the whole foundation of his philosophical and academic mission in life was being questioned. Fermat VERUS Descartes

• Descartes retaliated, offering critiques of some of Fermat’s work, and even suggesting that Fermat was indebted to Descartes for his work in analytic geometry. • This last charge is almost certainly untrue. The general consensus is that Fermat was totally unaware of Descartes’ work. Fermat VERUS Descartes

• In fact, Fermat did seem to have a better approach to the finding of and tangents, and to some aspects of refraction in . • He also had the support of M. Pascal and M. Gilles Personne de Roberval, which particularly galled Descartes. • Descartes called Fermat a “Gascon.” He also implied he came upon his results by lucky guesses (this was easy to believe because of Fermat’s frequent lack of proof). Fermat VERUS Descartes

• The tradition is that throughout their dispute, Descartes was hot‐headed and acid‐penned, while Fermat was more courteous and humble. This came in part from interpreting some events in his life as a judge and lawyer which suggested he tended to shrink from controversy. This latter point may be based on stories spread by his enemies. In any case, he was at least more subtle in his attacks on Descartes than vice‐versa. Fermat VERUS Descartes

• After their various exchanges through the 1630’s, the controversy died down. Descartes had succeeded in suggesting to many that Fermat was not a brilliant mathematician, and his work was not as well received as earlier. • Descartes continued to publish in philosophy, and won acclaim in that field. He died in 1650, honored and celebrated. Fermat VERUS Descartes

• Interesting Post‐Script #1. In 1658, when requested to supply some original letters to/from Descartes for inclusion in a collection, Fermat instead opened the old debate with Descartes’ supporters, even exaggerating some claims that could be easily disproven by someone who had Descartes’ letters. Fermat VERUS Descartes

• He also used politesse to –perhaps –attack Descartes and his work: • The conclusions that can be taken from the fundamental proposition of M. Descartes’ Dioptrique are so beautiful and ought naturally to produce such lovely results throughout every part of the study of refraction that one would wish –not for for the glory of our deceased friend, but more for the argumentation and embellishment of the sciences – that this proposition were genuine and legitimately demonstrated, and all the more as it is from these [conclusions] that one is able to say that multa sunt falsa probabiliora veris (often, falsehoods are more acceptable than truth). . . . Fermat VERUS Descartes

• “Glory” of our deceased friend – Descartes’ own definition of glory was a type of “love that one has for one’s self.” • Falsehood more certain than truth” may be a literary reference to a court case argued by Cicero in 81 BC in which a group of men stole an inheritance from a rightful heir on a false charge. Fermat VERUS Descartes

• Interesting Post‐Script #2: • While Descartes largely abandoned mathematics after the 1630’s Fermat continued to make several important discoveries, although mostly they were unheralded among his peers. Fermat’s Contributions:

• The Principle of Least Time, which influenced studies of refraction and other areas in physics. • Analytic Geometry much closer to our own than was Descartes’. • Important advances in number theory. • Work on maxima and minima that influenced many later mathematicians and laid the groundwork for calculus. • But he did champion Viete’s cumbersome notation to the bitter end, while Descartes’ notation is very nearly what we use today. Descartes’ Contributions to Algebra

• • Descartes chose letters from the end of the alphabet for unknowns, and letters from the beginning for constants. The printer asked if it mattered which among x, y, z was used to denote the unknowns, and when Descartes replied that it didn’t matter, chose x because it occurred less frequently than y or z in French (probably in English, too). We owe to that printer (at least in part) our never‐ending search for x in Algebra classes. Further Development of Analytic Geometry • There were numerous commentaries on Géométrie, which were necessary since Descartes didn’t explain things. • Roberval • Debeaune So When Did Analytic Geometry Begin to Look Like It Does Today? • Newton took a big step forward. • “…one finds for the first time the systematic construction of two axes. . . .There is no hesitation, however, with respect to negative coordinates, and the curves are plotted completely and correctly in all four quadrants.” ‐‐ Carl Boyer, History of Analytic Geometry, speaking of Newton’s Enumeratio, an appendix to Opticks. From Enumeratio The Definitive Formulation

• In the latter part of the 1700’s, Lagrange, Monge, and Lacroix developed something like the analytic geometry we know today, complete with equations of lines and conics, , point‐ forms of lines, the , and so forth. Earlier, Euler had used the ideas of analytic geometry to study surfaces in three .