p • CALIFORNIA STATE UNIVERSITY, NORTHRIDGE THE DEVELOPMENT OF CALCULUS E~DOXUS TO NEWTON A thesis submitted in partial satisfaction of the requirements for the degree of Master of Science in Mathematics by Judith Borden Stevens May 1988 The thesis of Judith Borden Stevens is approved: Barnabas Hughes California State University, Northridge ii DEDICATION This thesis is dedicated to my family. Far from being a professional manuscript, it still took many hours and sacrifices: My mother, Helen Adams Borden <deceased 1973)' who persistently, but lovingly, encouraged me to reach for goals often beyond my physical or intellectual abilities. My father, Ferris W. Borden <deceased 1984-), who always supported my endeavors whatever they might be. Dad was my mentor and inspiration though I could never fill his shoes. My step-mother, Virginia B. Borden, who has carried on my father's support and become my very good friend. My sister Barbara Borden Sanders, who is the truer intellectual of the two of us but always believes I can achieve more than I think I can. My husband's parents, Jean and Wes Stevens, who have been second parents to me since I was nineteen years old. My children Mark B. Stevens, Terry S. Garibay, and Matthew J. Stevens, who managed to put up with their mot her through her "nervous breakdowns" and discouragements. Many is the time, in my career, that they have had to sacrifice because of my lack of availability and time. My husband, Jay D. Stevens, who has exhibited the· "patience of Job" while watching me pursue this goal. Si nee I began my Master' s program we have dealt with surgery, my father's death, execution of a complex will, weddings for all three children, and the births of four beautiful grandchildren <Ashley Brooke Stevens, Allison Gloria iii. ACKNOWLEDGMENTS I would like to extend my most heartfelt thank yous to Dr. John W. Blattner, Dr. Barnabas Hughes, and Dr. Muriel Wright without whose patience this manuscript would never have reached completion. If their dedicated efforts appeared unappreciated, it was due only to my frustration and insecurity. I would also like to thank my typist Mrs. Lois Knutson. I realize she has spent innumerable hours making this thesis legible. I do not believe I could have managed .without her hard work, willing character, and efficiency. The Saugus High School Mathematics department, as well as other special friends on the faculty, deserves a big thank you for putting up with me. Last, I wish to thank Kathie Davidson with whom I have shared this endeavor for the past few years. Without her encouragement and friendship I do not believe I would have found the willpower to remain in the program. v TABLE OF CONTENTS Page DEDICATION. I I ••• I I. I I.' ••••• <I. I. I ••• ' I I. I •• I I •• I I I I •••• I I ••• I... iii ACKNOWLEDGMENTS ................ , . v ABSTRACT .. , •.•. ..•............•.. , , ..... , •.•......•... , ... ·..... vii Chapter 1. EUDOXUS £xhaustive Techniques..................... 1 2. ARCHIMEDES Quadrature of the Parabola and the method of exhaust ion. 7 3. GAL I LEO Motion as a function of time; triangle area as the sum of very narrow lines.... 18 4. KEPLER Idea of the mathematically infinite, maximum-minimum problems .. ,............. 38 5. CAVALIERI Indivisibles for lines, planes and solids. 42 6. FERMAT Areas, tangents and maximum-minimum: using ~ Apollonius adequality ... ,....... .. 48 7. WALLIS Arithmetic analysis; integral as a sum found by a "limiting" process on an infinite arithmetic series ......... ,.... 56 8. BARROW Areas and tangents by geometric methods, inverse operations ... , ................. 62 9. NEWTON Quadrature and tangents in analytic terms, binomial theorem, <atb)n making infinite algebraic series possible, inverse relationships of fluxions and fluents ... 66 10. LEIBNIZ Symbolic interpretation of the differential and the integral, inverse relationship between differential and integral ...... 76 11. NEWTON-LEIBNIZ CONTROVERSY 83 12. CONCLUSION... 86 BIBLIOGRAPHY... 87 vi ABSTRACT THE DEVELOPMENT OF CALCULUS FROM EUDOXUS TO NEWTON by Judith Borden Stevens Master of Science in Mathematics The purpose of this thesis is to show that the discovery of the calculus is a result of a long train of mathematical thought. The mathematicians of antiquity recognized that many geometric problems could not be solved with only the use of Euclidean mathematics. They needed methods for the mensuration of areas under curved figures. Thus, began the slow process of development of the calculus. In order to outline the bas1c aspects of this development, I have selected ten major contributors: 1> Eudoxus <exhaustive techniques), 2 > Archimedes <quadrature of a parabola>, 3> Galilee <motion as a function of time and the area of a triangle as the sum of very narrow lines>, 4) Kepler <the idea of the mathematically infinite, maximum and minimum), 5 > Cavilieri <theory of indivisibles for lines, planes and solids), 6) Fermat <areas, tangents, maximum, minimum, and Apollonius adequality>, 7> Wallis vii (integral expressed as a sum found by a "limiting" process on an infinite arithmetic series), 8) Barrow (areas and tangents found by geometric methods), 9) Newton <quadrature and tangents in analytic terms, binomial theorem, fluxions and fluents and their inverse relationship>, and 10) Leibniz <symbolic intepretation of the differential and integral>. While pursuing the history of the calculus, I have included brief historical sketches Qf each mathematician for the interest of the reader. I end my paper by discussing the controversy between Leibniz and Newton including their claims of original discovery of the fundamental theorem of integral calculus. vi·i:i· Q • Chapter 1 EUDOXUS of CNIDOS- <c 408- c 355 B.C.) At one time a pupil of Plato, Eudoxus of Cnidos achieved fame in astronomy, geometry, medicine and law. Through various historical accounts, it is believed that he brought the theory of motions of the planets from Egypt to Greece, made separate spheres for the stars, sun, moon and planets, and found the diameter of the sun to be nine times that of the earth. Eudoxus developed a theory of proportion which did not require two of the terms in the proportion to be whole numbers but allowed all of the terms in the proportion to be geometric entities. Euclid states Eudoxus' definition as follows: "Magnitudes are said to be in the same ratio the first to the second and the third to the fourth, when any equimul t i ples whatever be taken of the first and third and any equimultiples whatever of the second and fourth, the former equimul t iples alike exceed, are alike equal to or alike fall short of the latter equimultiples respectively taken in corresponding orde~"[3) a:b = c:d if either i) ae > bf and ce > df or 11> ae = bf and ce = df or iii> ae < bf and ce < df for any integers ~ and L An example for ii> is simple: 1 2 1:2 = 2:4 if either case i) 1 = £ when e>2f and 2e)4f 2 4 or case :J.i) 1. = £ when e = 2f and 2e = 4f 2 4 or case iii> 1. = g_ when e<2f and 2e<4f 2 4 Therefore any case will prove 1. = £ 2 4 However the cases i) and iii) are necessary for incommensurables such as J2 as case ii) will never hold. For example: suppose we wish to prove J2 * 2 1 5 Then we must show that there exists an e and f such that both these cases do not work. let e = 29 and f = 41 then J2(29) > 1<41) as J2<29> ~ 41.01 but (7) <29> < <5) <41) therefore case i) does not hold. Similarly case iii> will not hold and J2 * 2 1 5 In modern terms, this theorem states that if two numbers are unequal then there exists a rational number between these two numbers: J2 > !1. > ?... 29 5 This theory involves only geometric quantities and integral multiples of them so that no general definition of discrete number, rational or irrational, is necessary. Eudoxus did not, as we do, regard the ratio of two incommensurable quantities as a number; 3 nevertheless, Eudoxus' definition does allow for the ordinal idea involved in our present conception of the real number. It must be understood that the Euclidean Greek geometers had no general definitions of length, area and volume. However, it was reasonable to discuss the ratio of the areas of two circles ~s that of squares constructed on the diameters of the circles. The proof of the correctness of this proportion requires, in this case, the comparison of circles and squares. Using his theory of proportion, Eudoxus developed a rigorous argument for dealing with problems involving two dissimilar heterogeneous or incommensurable quantities such as the circle and the square. The procedure which Eudoxus proposed is now known as the method of exhaustion. This axiom states that given two unequal magnitudes (zero was not a number or a magnitude for the Greeks), "if from the greater there be subtracted a magnitude greater than its half, and·from that which is left a magnitude greater its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out."[ 2, 4] This axiom is known as the axiom of Archimedes; however, Archimedes himself ascribed it to Eudoxus. By continuing the process indicated in this axiom one could now produce a magnitude as small as desired. Eudoxus' "method" guarantees that for any E, no matter how small, from another magnitude, say x, we can continually "cut away" a value greater than one-half of the remainder and eventually produce a magnitude less than that of E, <x-2/3x)-2/3 <x-2/3x)... < E 4 for any £, Refer to the following Chapter on Archimedes for an example of the use of exhaustion.