California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 1994 Arguably the scheme that conquered the infinite Timothy Michael Curran Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Science and Mathematics Education Commons Recommended Citation Curran, Timothy Michael, "Arguably the scheme that conquered the infinite" (1994). Theses Digitization Project. 877. https://scholarworks.lib.csusb.edu/etd-project/877 This Project is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. ARGUABLY THE SCHEME THAT CONQUERED THE INTINITE A Project Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment ofthe Requirements for the Degree Master ofArts in Mathematics by Timothy Michael Curran June 1994 ARGUABLY THE SCHEME THAT CONQUERED THE INFINITE A Project Presented to the Faculty of California State University, San Bernardino by Timothy Michael Curran June 1994 Approved by: ^ 9 Dr. Robejrt Stein^ Project Committee Chair, Mathematics Date Dr. Gary Grimns Project Co tee ember This project involves an in-depth look at some ofthe historical developments that preceded the formal invention ofcalculus. It focuses chronologically on the events that led to Newton's discovery ofthe binomial series. The primary purpose ofthis prGject is to The discovery-based activities contain exercises on Pascal's Triangle, Alhazen's method for acquiring formulas for 1^+2^ +3* John Wallis' characteristic ratio ofindex Wallis'famous representation of7t as an infinite product, and Newton's discovery ofthe binomial series. m TABLE OF CONTENTS Abstract............................ ... .iii CHAPTERONE.................................^. 1 Introduction........ 1 APPENDIX A; The Activities.. I I Set I: A Cryptographic Warm-up....... 13 SetTI; Alhazen's Method for Calculating 1^+2*+3*+...-Hw*" for any positive integer k. .22 Set III: Wallis' Characteristic Ratio ofIndex A:.. ....;. 42 Set IV: Wallis' Representation ofk as an Infinite Product.... 59 Set V: Newton's Discovery ofthe Binomial Series 75 BIBLIOGRAPHY..... 85 IV Chapter1 Mathematics, hot unlike any other subject ofstudy, has in its historical background certain developments that stand out among all Others as critical turriing points. This paper involves one such development, namely, Wallis'interpolation scheme which leads to his famous representation of7t as an infmite product. In fact, Wallis himselfCoined the tCrm "interpolation"(originating from words meaning "to polish in between")in his most celebrated •work Arithmetica Infinitorum, published in 1656. The new method of experimentation continuously eniployed by Wallis in its pages involves interpolatiph and induction, leading to multitudes ofgeneralizations, Wallis' use ofinduction was the first of its kind and it led to an abundant harvest offi"esh revelations in mathematics. Arguably, the primary reason for such an explosion ofnewly discovered outcomes in mathematics following the work in by Wallis lies in the fact that, fi"ohi ancient times, until the middle ofthe seventeenth century, mathematicians avoided working with the infinite due to the horrific methodology required. Wallis prepared the way to modern- day approaches to infinite processes. He admitted both the concepts ofinfinite series and limit theory into rnathematical reasoning. The following paragraphs act as a brief chronology ofsignificant events in mathematicswhich ultimately directed Wallis to his profound investigations. The earliest writteh inquiries into the^infinite process date back to abouCSOO BC; with the paradoxes ofZeno. IfZeno were to be reincarnated into today's world, it is likely that he would believe that he couldjump from an airplane without a parachute and, even more frightening, without the fear ofhitting the ground. After all, there are an infinite number of"half-way points" he would have to bypass and, naturally, that would take an infinite amount oftime. At about the same time Aristotle wrote ofZeno's paradoxes, the Pythagoreans discovered the incommensurable(segments that lacked a common measure, so that the ratio oftheir lengths was what we now call irrational). This escalated interest in the idea, ofthe infinite process and led to the next stage in the cultivation ofinfinite theories. In about 450 BC Hippocrates squared the area under a curve. He ascertained that the area ofthe shaded region below was equivalent to the area ofsquare AMCD. Tliis problem was the first ofits kind. It showed that the area under a curve could be squared. Consequently, it sparked interest in one ofthe most tantalizing problems encountered in mathematics, that of"squaring the circle." Shortly thereafter, the Sophist Antiphon proposed that there existed a regular polygon, with a sufficient number ofsides, whose area matched that ofa circle having a radius equal to the length ofthe polygon's apothem. Already and quite understandably. we see the confusion perpetrated by processes involving the infinite. The Pythagoreans' discovery ofthe existence ofincommensurable geometric magnitudes(lengths, areas, volumes)ultimately led to the next stage in the development of the theory ofthe infinite process, the writing ofthe 13 books ofthe Elements by Euclid in about 300 BC. Incommensurable geometric magnitudes forced a thorough reexamination ofthe foundations ofmathematics and this task was undertaken by Euclid. He created a "continuity axiom" that allowed the Greeks to deal with geometric magnitudes that could not be "measured by numbers." An important application ofthe continuity axiom involved the Greeks "method ofexhaustion." The method ofexhaustion was utilized by the Greeks as they attempted to calculate the area ofa curvilinear figure. This application involved filling up, or e>diausting, the curvilinear figure by means ofa sequence ofpolygons. It was devised, apparently by Eudoxus,to provide a geometric approach to acquiring certain limits. This was an early form of the modem-day means used which is informally called "taking the limit." The method ofexhaustion was logically clear, but was very cumbersome,for it, in practice, hinged on the idea that the difference between the area ofthe curvilinear figure and the last polygon in the sequence could be made as small as desired by making the sequence ofpolygons sufficiently large. In many instances, this led to difficult geometry problems. The classical era ofGreek mathematics probably reached its climax in the third century BC. with the squaring ofthe parabola by Archimedes. In the Preface to his Treatise on the Quadrature ofthe Parabola, Archimedes writes: "Many mathematicians have endeavored to square the circle, the ellipse, or the segment ofa circle, ofan ellipse, or ofa hyperbola. No one, however,seems to have thought ofattempting the quadrature ofthe segment ofa parabola, which is precisely the one that can be carried out." This was a significant event because since Hippocrates'squaring ofa particular crescent in 450 BC., many ofthe greatest minds over the course ofthe next 190 years tried in vain to square other curvilinear figures. Essentially, Archimedes'work constituted the very beginning of the calculus, and indeed one ofhis proofs(he gave two)anticipated methods developed in the theory ofintegration, nearly 2000 years later. Archimedes'quadrature ofthe parabola b was equivalent, in modem terms,to evaluation of J{px^ ¥q)dx.Generalization ofthis a result to other functions involved the function concept and tools ofalgebra and analytic geometry. Nearly 1900 years later a discovery related to Archimedes quadrature ofthe parabola was made by Cavalieri. In about 1630, he calculated the area under the curves j for A: = 3,4,5,...,9. Cavalieri's work was based on formulas developed by the Arab mathematician Alhazen in about 1000 AD.for the sums ofthe first n cubes and fourth powers. Alhazen's brilliant acquisition ofa means for obtaining these formulas helped bridge the gap between early Greek mathematics and the explosion ofideas in the seventeenth century, including this one by Cavalieri, that simplified further work involving infinite processes. Another bridge was the translation, in the thirteenth century, of Aristotle's Physics. Aristotle's work by no means represented the best scientific thinking ofhis day. However, its translation had a profound influence on European thinking. He explored the nature of the infinite along with the existence ofindivisibles or infmitesimals. He summoned scholars "to discuss the infinite and to inquire whether there is such a thing or not, and,if there is, what it is"{ Book III, Ch.4 }. Because ofthis summoning,thirteenth century philosophers and mathematicians became fascinated with the mysteries ofthe infinite. A sample ofthe problems solved at this timefollows; 1. Ifa point moves throughout the first halfofa certain time interval with a constant velocity, throughout the next quarter ofthe interval at double the initial velocity, throughout the following eighth at triple the initial velocity, and so on ad infinitum; then the average velocity during the whole time interval will be double the initial velocity. 2. Ifan aliquot part(one Ath)should be taken from some quantity(a), and from the first remainder such a part is taken, and from the second remainder