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 My father, Ferris W. Borden 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 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.' ••••• 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, 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 Archimedes Kepler Cavilieri 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 Leibniz 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- 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, for any £, Refer to the following Chapter on Archimedes for an example of the use of exhaustion. Although very similar to the argument employed in proving the existence of a limit in differential and integral calculus, the Greek method dealt with continuous magnitude. The following are examples of the use of this "method of exhaustion": 1) the areas of circles are to each other as the squares on their diameters. "Let the areas of the circles be A and a and let their diameters be 0 and d respectively. If the proportion a: A = d~: o:::: is not true, then let a': A = d;;;': 0 2 where a' is the area of another· circle either greater· or smaller than a. If a' is smaller than a, then in the circle a we can inscribe a polygon of area p such that p is greater than a' and smaller than a. This follows from the principle of exhaustion that if from a magnitude we take more than its half, and from the difference more than its half, and so on, the difference can be made less than any assignable magnitude. If P is the area of a similar polygon inscribed in the circle of area A, then we know that p: P = d~: 0 2 = a': A. But, since p > a' then P > A, which is absurd, since the polygon is inscribed in the circle. In a similar manner it can be shown that the supposition a' > a likewise leads to a contradiction, and the truth of the proposition is therefore established."[3J By an argument based on "reductio ad absurdum" it can be shown that a ratio greater or less than ~ is inconsistent with the principle that the difference could be made as small as desired. [ 4-J 5 2> an approximation of 1:/2 Consider the following ladder of whole numbers: ~ 1 1 2 3 5 7 12 17 29 41 70 99 etc where 1+1=2, 2+1=3, 2+3=5, 5+2=7,· 5+7=12, 12+5=17 and so on .. Then y2 -2X2 = ± 1 Each rung of the ladder consists of 2 numbers, x and y, whose ratio approaches nearer and nearer 1:/2 as the relative error of ±1 becomes as small as we wish. As the positive and negative one alternate between rungs of the ladder, the limiting ratio x: y, is "pinched" between the two extremes. They approximate from both sides the desired 1:/2. Thus the groundwork had been set for Archimedes to proceed with his quadrature of the parabolic section using incommensurable polygons. 6 References: (1) Boyer, Carl B. History of the Calculus and its Conceptual Development. New York: Dover, 1949, p. 34. [ 2) Heath, T. L. ~History of Greek Mathematics. New York: Dover, 1981. [ 3] Heath, T. L. (4) Newman, James R. World of Mathematics. New York: Simon & Schuster, 1956, pp.. 90-100. p • Chapter 2 ARCHIMEDES Archimedes was the greatest mathematicican and physicist of antiquity. Leibniz praised him by saying that those who knew his works marveled les·s at the discoveries of the greatest modern scholars. Pliny called him the "god of mathematics". Many of Archimedes writings have survived, but little is known of his life. According to Plutarch, Archimedes was related to King Hiero of Syracuse. It appears that he studied in Alexandria under the tutelage of Euclid's pupils. After returning to Syracuse, he occasionally worked for the king. From Vitruvius we learn that when Hiero was "exalted" to royal power, he determined to set a golden crown in a shrine as an offering to the immortal gods. Hiero gave a contractor the exact amount of gold but it was rumored that the contractor removed some of this gold and replaced it with an equal weight in silver. Hiero delivered the problem to Archimedes who is said to have found the answer while bathing, observing the displacement of water by his body. He then experimented and observed that the crown displaced more water than a mass, of equal weight, of gold thereby disclosing the contractor's fraud. Tzetzes, the Byzantine grammarian, relates several other Archimedian accomplishments. In Syracuse, Archimedes set fire to beseiging ships with the aid of appropriately placed "burning mirrors". Archimedes is also said to have used the principle of the lever in order to hurl boulders and raise ships against the 7 8 Q • walls. Of this he boasted, "Give me a place to stand on, and I can move the earth."[ 1l In the Sand Reckoner, Archimedes suggested an elaborate scheme of numeration, arranging the numbers in octads or eighth powers of ten. In this work he recognized that aman=aM+n, a law that is the basis of our present operations with logarithms. n Elsewhere he dealt with the summation L n2~ the first example of 1 the systematic treatment of higher series of any kind, the solution to the cubic equations of the form X3 ± ax2 ± bzc = 0, squaring a parabola, the value of fi, and many mensuration formulas. Plutarch writes that Archimedes disdained the practical as "sordid and ignoble" even though he invented many practical weapons, the water screw, and at one time even constructed a complex model planetarium. Plutarch claims that Archimedes was slain at age 75, by a Roman soldier sent by the general Marcell us, merely to bring Archimedes to him. One extremely interesting antecedent to the eventual discovery of the calculus was Archimedes' "Quadrature of a Parabola". Using the properties of a parabola and a summation, he found the area enclosed by a parabola and a chord. 9 fl The points D, C and E are points where the tangent of the parabola are parallel to the chords AC, AB, and BC respectively. Using the properties of the parabola Archimedes demonstrated that each of the triangles ADC and BEC has !til. ~ equal iQ. one-eighth of ·the ~ of A ABC. Euclidean geometry. > Archimedes then constructed four new triangles with bases AD, DC, CE, and EB each having en area equal to one-eighth of the area of triangle ADC. Continuing in this manner, he found that if we call the area of triangle ABC, ~ , then the area, P, of the parabolic section determined by chord AB satisfies: 10 P >AH2 A for any n Therefore P >A +[ 2 A<~ ·JoY> l +[ 22 A Thus becoming the geometric series: P >A <1 + ~ + ~~ +~ +..• + ~.... > Using S = a-ar"• 1 a=l, r= 1. 1-r 4 P>~-~~)JA By a similar method he proves P < ~ -~(~)jA Archimedes proceeds to prove P = ~A by the "double reductio ad absurdum" argument that if the polygons exhausted the parabolic segment, then its area could be neither greater than nor less than ~~ . Inasmuch as Archimedes did not invoke the limit concept, P = ~ 6. was not considered the sum of an infinite series where~(~).., approaches zero in the limit. Rather, in his method of exhaustion he shows ~ (,U;r may be made as small as we wish. In order to be able to define~~ as a sum of the infinite series, it would have been necessary to develop the general concept of a real number. For Greek mathematicians, there was always a gap between the real (finite> and the ideal The following is a proof, using calculus, where AB is any chord: .b, £ ~<-- h ---;> G 12 Every parabola may be expressed as y = ax~ by using rotation end translation. Equation Parabola: Slope at any point: y'= 2ex Let C = {x 3 , ax3 ~) Slope AB = ex,~ - ax.2 ~ = a X3 = x, + x., which is the midpoint of the !t coordinates 2 Area of trazezoid ABGE = -!·2h = D. 2 2 2 = (x 7 ~x,) ~x 1 +ax 2 -2a~, ;xj ] = a By similar reasoning: Area ACBD = -2 =§.. :. Area A CBD = f/g Area A ABC Q.E.D. A synopsis of Archimedes' proof: [3) Archimedes uses twenty-one propositions in order to prove the previous theorem. It is extremely cumbersome, lacking any analytical methods. 0 fl· 14 r · The first five propositions use similar figures to prove basic proportions involving tangents and chords at any A, B, and C. The primary conclusion is that 0 and Q are the midpoints of AB and BC respectively. Propositions VI through XV use a series of levers and suspensions around the center of gravity in order to prove various relations between triangles and rectangular areas. In proposition XVI, Archimedes' final result of these previous propositions, he proves: Area of parabolic segment q R1 R2 ••• RnQ =~ ~ EqQ QE tangent to the parabolic segment at Q and qE parallel to the axis of the parabola. 15 Pr-oposition XVII proves that the area of any segment of a parabola is four-thirds that of the triangle which has the same base as the segment and equal height. The base is a straight line bounding the segment and the height is the greatest perpendicular drawn from the curve to the base of the segment. The vertex will be the point from which the greatest perpendicular is drawn. p Area segment = ~ Area A qPQ 16 Propositions XVIII through XXI proceed to prove that with any base and vertex, at the point where the tangent is parallel to the base: Area A AEC = Area A BDC =~Area A ABC By the method of exhaustion and double reductio ad absurdum Archimedes reasoned that for any parabolic segment ACB, the area can be neither less than nor greater than ~Area A ABC. [ 31 17 References: [1J Burton, Daid M. History of Mathematics. Boston: Allyn & Bacon, 1985, p. 226. [2] Calinger, Ronald. Classics QL Mathematics. Oak Park, Illinois: Moore, 1982, pp. 123-24-, 132, -33. [ 3] Heath, T. L., ed. The Works QL Archimedes. New York: Dover, 1897, [4-J Newman, James R. World of Mathematics. New York: Simon & Schuster, 1956, pp. 179-87. [5] Smith, D. E. History of Mathematics, Vol I. New York: Dover, 1958, pp. 111-115. Chapter 3 GALILEO <1561 - 1642) Galilee was born at Pisa, Italy, on the day of Michelangelo's death, Feb. 18, 1564, and died in the year of Newton's birth, Jan. 8, 1642. He was the son of a Florentine nobleman whose estate had become impoverished. Galilee's father decided his son should study medicine, which decision saved the great mind from serving a life as a cloth merchant. While at the University of Pisa, Galilee became infatuated with the world of science and geometry and finally persuaded his father to allow him to devote his studies to the sciences. In 1589 he became a professor of mathematics at Pisa, an extremely low-paying and low-esteemed position in those times. It was here that he began his experimental work in physicsj but, because of local controversies, he resigned his chair and began his professor-ship of mathematics at Padua. Here, Galilee carried on his greatest scient i fie work, including construct ion of the telescope, invention of the modern type microscope, theories on motion, and invention of the proportional compasses. [Galilee's proportional compasses have been referred to as one of his major inventions, arousing some curiosity as to their use. This device consisted of a pair of compasses connected by a screw in a pivot point which slides in order to divide each leg in equal proportions: 18 19 a b = c d Proportional compasses were primarily designed in order to enlarge a diagram by a given scale factor. c D CD = 2AB for any eJ 20 By 1614- Galilee found himself involved in disputes over his astronomical views and about statements in the Bible. His acceptance of the Copernican view that the sun was the center of the universe Galilee was reprimanded by the Pope. He was admonished to abstain altogether from teaching or defending this doctrine on penalty of imprisonment. Galilee was silenced until 1623, when the new Pope, Urban III, allowed him to write about Copernicus's theory, provided he did not represent it as reality. He completed his manuscript by 1630 and published it as the "Dialogue" in 1632. His enemies managed to convince the Pope that the teachings were potentially dangerous. Galilee was put under house arrest for life and his "Dialogue" was placed on the Index of Prohibited Books until 1822. Galilee returned to his most important project, "Discourses and Mathematical Demonstrations Concerning Two New Sciences", a treatise analyzing projectile motion and gravitational acceleration. Because the Inquisition would allow no work of Galilee's to be published in Italy, the manuscript had to be smuggled out of the country and printed in Holland. .21 Galilee's major contributions to the discovery of the calculus were his theories on motion as a function of time and on infini tesimals. At first Galilee supposed that the speed of a falling body was proportional to the distance fallen through. When this hypothesis proved unsatisfactory through experimentation, he changed it to read that speed was proportional to the time of the fall. Because free falling bodies attained a velocity beyond the capacity of available measuring implements, Galilee approached the problem of verification by experimenting with models where effects of gravity were ''diluted". He demonstrated that a body moving down an inclined plane of given height attains a velocity independent of the angle of slope, and that its terminal speed is the same as if it had fallen through the same vertical height. Although he probably did not corroborate the following principle by experiment, Galilee, convinced that he was right, claimed that all bodies, heavy or light, are subject to the same acceleration. In other words, there existed some universal constant k, such that velocity is equal to k multiplied by time and that acceleration will equal k. It was not until after his death, when the air pump was invented, that a experimental verification was given by permitting bodies to fall in a vacuum. Experimenting with a pendulum, Galilee obtained further evidence to sustain the principle of "persistence of motion". A swinging bob is analogous to a ball on an inclined plane, which ignoring friction will roll up another inclined plane to the same height as the starting point. This law of "persistence of motion" led to Newton's first law of motion, "the law of inertia": "Every body 22 persists in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed on it."[ 3] Another of Galileo' s important discoveries resulted from his study of the path of projectiles. He showed that the trajectory of the projectile could be resolved into two simultaneous motions: one horizontal, with the velocity The following are some excerpts from Dialogues Concerning Two New Sciences by Galilee Galilei: [3) 23 Nat ur·ally Accelerated Mot ion B "Imagine this page to represent a vertical wall, with a nail driven into it, and from the nail let there be suspended a lead bullet of one or two ounces by means of a fine vertical thread, AB, which hangs about two finger-breadths in front of the wall. Now bring the thread AB with the attached ball into position AC and set it freej fi~st it will be observed to descend along the arc CBD, to pass point B and travel along arc BD, until it almost reaches the horizontal CD, a slight shortage caused by the resistance of air. and the string; from this we may rightly infer that the fall in its descent through arc CB acquired a momentum on reaching B, which was just sufficient to carry it through a similar arc BD, to the same 24 height. Having repeated this experiment many times, let us now drive a nail in the wall close to the perpendicular AB say at E or F, so that it projects out some five or six finger-breadths in order that the thread again carrying the bullet through arc CB may stike up on the nail E when the bullet reaches B, and thus compel it to traverse the arc BG, described about E as center. From this we can see what can be done by the same momentum which previously starting at the same point B carried the same body through arc BD to the horizontal CD. Now, gentlemen, you will observe with pleasure that the ball swings to the point G in the horizontal and you would see the same thing happen if the obstacle were placed at some lower point, say at F, about which the ball would describe the arc BI, the r·ise of the ball always terminating exact 1 y on line CD. But when the nail is placed so low that the remainder of the thread below it will not reach to the height CD then the thread leaps over the nails and twists itself about it." ... "Let us then, for the present, take this as a postulate, the absolute truth of which will be established when we find the inferences from it correspond to and agree perfectly with experiment. .. "[ 3] Galilee then proceeds with a series of Theorems and Corollaries involving motion as a function of time, setting the scene for Newton, who furthered these studies and eventually discovered the inverse nature of the velocity and position functions. I have supplied some of the proofs for the following excerpts: [ 3J 25 THEOREM l PROPOSITION l "The time in which any space is traversed by a body starting from rest and uniformly accelerated is equal to the time in which that same space would be traversed by the same body moving at a uniform speed whose value is the mean of the highest speed end the speed just before acceleration began." THEOREM ~ PROPOSITION ll "The spaces described by a body falling from rest with a uniformly accelerated motion are to each other as the squares of the time-intervals employed in traversing these distances." COROLLARY l "Evidently then the ratio of the distances is the square of the ratio of the final velocities, that is, the lines EP and DO, since they are to each other as AE to AD ... " H c .r 26 Let time be represented by AB with AD and AE any two time-intervals. Let HI represent the distance through which the body starting at rest at H, falls with uniform acceleration. If HL represents the space traversed during the time interval AD, and HM that covered during the interval AE, then the space MH stands to space LH in a ratio which is the square of the ratio of the time AE to the time AD. COROLLARY il. "It follows that, starting from any initial point, if we take any two distances, traversed in any time-intervals whatsoever, those time intervals bear to one another the same ratio as one of the distances to the mean proportional of the two distances." THEOREM III, PROPOSITION III "If one and the same body, starting from rest, falls along an inclined plane and also along a vertical path, each having the same height, the times of descent will be to each other as the lengths of the inclined plane and the vertical." COROLLARY ill "Hence we may infer that the times of descent along planes having different inclinations, but the same vertical height stand to one another in the same ratio as the lengths of the planes." 27 Q ' THEOREM lY, PROPOSITION IV "The times of descent along planes of th~ same length but of different inclinations are to each other in the inverse ratio of the square roots of their heights." time along BA = ~ time along BC V1flf Let BA = BC; BE = height BA; BD = height BC Let BI be the mean proportional to BD and BE /} ie. BD = BI or then BD = y'BD • BE BI BE It has been shown that the time of fall along BA is to that along vertical BE as BA is to BE .. time along BA = -IBA · time along BE = BE time along BE BE time along BD BI time along BD = BD = BI time along BC BC BS ... tim"' along BA = BA But BA = BC and BA = BC = BD time along BC BS BS BS BI due to similar triangles ... time along BA = BD = YBD time along BC Bl V"8E Q.E.D. 28 THEOREM ~ PROPOSITION ~ "The times of descent along planes of different lengths, slopes and heights beer to one another a ratio which is equal to the product of the ratio of the lengths by the square root of the inverse ratio of their heights." 29 To Prove If planes AB and AC have different lengths, inclinations, and heights then time along AC time along AB Let AL be the mean proportional to heights AG and AD then .. AF is a mean proportional between AC and AE ie. AF = J/ AC,AE Now since time along AC = AF ... time along AC = AF time along AB AB Now AF = AL = AG Q. E. D. 30 THEOREM YL PROPOSITION VI "If from the highest or lowest point in a vertical circle there be drawn any inclined planes meeting the circumference, the times of descent along these chords are each equal to the other." A H F Let HG be horizontal F is the "lowest" point and the point of tangency AF is a diameter A the "highest" point B and C are any points on the circle, AB and AC are "inclined planes" To Prove: time along AB = time along AC Draw BD and CE perpendicular to AF '31 Make AI a mean proportional between AE and AD 1. AI = VAE•AD 1. Mean Proportional 2. AE = AI 2. Mean Proportional AI AD 3. t& = 4-. 6. MOTION OF PROJECTILES THEOREM L PROPOSITION l "A projectile which is carried by a uniform horizontal motion compounded with a naturally accelerated vertical motion describes a path which is a semi-parabola." The proof of this theorem requires two theorems from Apollonius and a definition of a parabola: L Definition: Given a right circular cone erected on circle ibkc with apex l. The section of this cone made by a plane parallel to lk is called §.. "arabola. 33 The base be cuts at right angles the diameter ik of cir·cle ibkc and axis ad is parallel to lk. From any point f dr·aw fe parallel to bd then: Proposition (1) Proof: Through e pass plane gh parallel to circle ibkc producing a new circle whose diameter is gh. Because of properties of perpendicular chords of circles id dk = .. (bd) 2 = ad (ef)2 ae Q.E.D. 34 Proposition 2> "Draw a parabola with axis ac, be perpendicular to the base. Choose d such that ad = ac. Then bd will be. tangent to the parabola at b." b c THEOREM 1.L PROPOSITION ll "When the motion of a body is the resultant of two uniform motions, one horizontal, the other perpendicular, the square of the resultant momentum is equal to the sum of the squares of the two component momenta." 35 Q • Theorem III. Proposition III Let motion take place along the line ab, starting from r·est at a, and in this line choose any point c. Let ac represent the time r·equired for the body to fall through the space ac; let.ac also represent the velocity at c acquired by a fall through the distance ac. Let as be a mean proportional between ac and ab. Then the velocity at b is to that at c as the length as is to the length ac. c s b It still remained for Galileo to discover a method of measuring momentum which results from uniform horizontal motion compounded with the vertical motion of a freely falling body starting from rest. The momentum is obtained by placing the square of the resultant equal to the sum of the squares of the two components. 36 The geometrical demonstrations of Gelileo were ell based upon the supposition that the area under a velocity-time curve represented the distance covered. Since he did not possess the concept of limit he said that the moments, small increments of distance, were represented by the lines of the triangle and rectangle and that these geometric figures were actually made up of these lines. Area A ABC = distance . l3 He discussed the infinite, the infinitesimal, and the continuum. He concluded that infinity and indivisibility were incomprehensible but existed. The attributes 'larger', 'smaller' and 'equal', having no place in comparing infinite quanti ties with each other or finites, were admitted as Galileo focused upon the infinite as multiplicity or aggregation. He noticed that 'part' of an infinite set could be put into one to one correspondence with the entire set. For instance, the even numbers can be put into one to one correspondence with the natural numbers. He maintained that magnitudes are made up of 37 indivisibles but could never actually pictur·e the tr·ansition fr·om finite to infinite. References: [1] Burton, David M. The History of Mathematics. Boston: Allyn & Bacon, 1985, pp. 330-34. [ 21 Mach, Ernest. The Science of Mechanics. Chicago: Open Court, 1960, PP· 62, 92, 102, 106, 158. [ 3] Newman, James R. World of Mathematics. New York: Simon & Schuster, 1956, PP· 726-70. Chapter 4 JOHANNES KEPLER (1551 - 1630) Johannes Kepler was born in Weil, Germany, to an impoverished family. His father was a mercenary and a drunkard, while his mother was a violent, ignorant woman actually accused of witchcraft. Born crippled and with impaired vision, he was often unable to at tend school. However, with the patronage of the Duke of WUrt temberg, he was able to enroll at the University of TUbingen where he studied under Michael Mastlin who taught him the controversial works of Copernicus. Kepler espoused the sun-centered theory. Consequently he had to abandon his goal of entering the Lutheran ministry, whose philosophy was incompatible with the Copernican theory. He accepted a position as a teacher, but he lost all of his students with lectures beyond their comprehension. Prevented from obtaining the position of his choice, because of political pr·essures and pressures from the Catholic Church, Kepler took a position as an assistant to Tycho Brahe in his astronomical research. He continued Brahe' s work after the letter's death in 1601. Kepler's three laws, which established the first correct principles of planetary mechanics, overturned Aristotelian cosmology: 1) The earth moves in an elliptical orbit. 2> The line drawn from the sun to a planet sw~eps over equal areas in equal times. 3) The square of the period of the planet is proportional to the cube of its mean distance from the sun. 38 39 The astounding result of these three laws is that together with Newton's second law, force is proportional to acceler-ation, one can prove the "Law of Gravitation" F = G m,m2 , where G is a universal r:2 constant for all planets. Although Kepler is best known for his three great laws, his scientific studies led him to many achievements in pure mathematics. He was the first to clearly present the principle of continuity around their. diameters, tangents, or chords. Kepler regarded infinitely small arcs as straight lines, infinitely narrow plane figures as lines and infinitely thin bodies as planes. Kepler was not free from mistakes, often lapsing into the language of indivisibles, chords of plane figures and plane sections of solids. Apparently he did not distinguish clearly between proofs by means of the method of exhaustion, by ideas of limits, by infinitesimal elements or by indivisibles. Further development was necessary and Kepler' had to be satisfied with conclusions of merely probable correctness. In some of his conclusions he missed the correct solution altogether. The inability to express the idea of limit and continuity reappears even with Newton and Leibniz. Leibniz fell back on his so-called "law of continuity" for justification of his differential calculus, and Newton cloaked his lack of definition with the words instantaneous velocity, or fluxion. One of Kepler's major contributions to the development of the calculus r·esul ted from the pt-oblem of determining the first proportions for wine casks. In his Nova Stereometria Doliorum Vinariorum crude procedures for finding volumes of solids. This brought him to the consideration of many maximum - minimum problems. Among other things, he showed that of all parallelipipeds with square bases, which can be inscribed in a sphere, the cube has the greatest volume. Also, of all right circular cylinders having the same diagonal, the one with the most volume has the diameter- and altitude in the ratio 12 to 1. By making chads on these and other solids, Kepler· noted that as the 41 maximum volume was approached, the change in the volume, for a given chanse in dimensions, became smaller. Several centuries earlier, Oresme observed that the rate of change was least at the maximum point. Although Oresme' s is the point of view we now use with the concept of the derivative, Newton and Leibniz used Kepler's increments and decrements rather than rates of change, and the differential became the primary concept. References: [1] Boyer, Carl B. The History of Calculus and its Conceptual Development. New York: Dover, 1949, pp. 110-17. [ 2] Burton, David M. The History of Mathematics. Boston: Allyn & Bacon, 1985, pp. 337-42. [ 3] Newman, James R. World of Mathematics. New York: Simon & Schuster, 1956, pp. 42, 126-27, 152. [ 4] Smith, D. E. History of Mathematics. New York: Dover, 1958, p. 416. Chapter 5 BONAVENTURA CAVALIERI <1598-1647) Cavalieri was born in Milan. He was a pupil of Galilee and later a Jesuat professor of mathematics at Bologna, which position he held from 1629 until death.t2l He wrote on conics, trigonometry, optics, astronomy, and astrology. He was one of the first to recognize the great value of logarithms. His greatest contribution, however, was his principle of indivisibles, printed in 1635. Having studied Kepler's Stereometria, Cavalieri explained his "method of indivisibles" without making a clear definition of indivisibles. His procedure appears to regard a line as composed of an infinite number of points, a surface as en infinite number of line segments and a solid as an infinite number of plane sections. These were his indivisibles. Cavalieri denied any inspiration from Kepler for this method. His indivisibles led to much criticism as, even with continual subdivisions, the elements must be lines, areas and volumes. Cavalieri was probably aware of this and merely used his "indivisibles" as a calculating device. His method for the mensuration of plane figures was to find the ratio of a required area to a known area. He divided each figure into equally spaced parallel lines "indefinitely" close together. These lines were summed for each figure and the ratio of their sums ascertained when the number of lines became indefinitely great. This ratio was assumed to be equivalent to the ratio of the areas. He used a similar method for the mensuration of solid bodies. Cavalieri did not follow his master, Galilee, in speculations as to the nature of the infinite. It did not appear in his conclusions because 42 43 his center of attention was on the correspondence of the indivisibles of the two figures rather than the total number of indivisibles within the area or volume. Although he encountered opposition to this point of view, he probably introduced some of the most valuable ideas necessary for the invention of the analytical geometry. This invention, combined with the infinitesimals, led to the development of the differential and integral calculus. The proposition for the mensuration of solid figures, still known as "Cavilieri's Theorem", is as follows: "If two solids have equal altitudes, and if sections made by planes parallel to the bases and at equal distances from them are always in a given ratio, then the volumes of the solids are also in this ratio." (ll New proofs for old theorems followed using Cevilieri 's new methods. One of these propositions states that if a parallelogram ABCO is divided by a diagonal then the area of the parallelog·r·am is double the ar·ea of either triangle. Area ABCO = twice area triangle ABO or twice area triangle COB Proof: Let DE = BC Drew CF parallel to BC and EH parallel to AD Then HE = CF 44 Therefore all the lines of triangle ABD are equal to all those of triangle CDB. Consequently the areas of triangles ABD and CDB are equal and the sum of the lines of the parallelogram ABCD is double the sum of the lines of either triangle. From this theorem, Cavilieri went on to prove, by a similar but more involved argument, that: 1. The sum of the squares of the lines in the parallelogram is three times the sum of the squares of the lines in each of the constituent triangles. 2. The volume of a cone is one-third that of the circumscribed cylinder. 3. The area of a parabolic segment is two-thirds the area of the circumscribed rectangle.[ 1l Theorem one claims that the volume of a pyramid is equal to one- third the volume of a prism with the same base. 8 A c.. I ' ...... l I ' I I ...... II - ...- - - D 45 The volume of Pyramid C-DEF, with base A DEF and height CF, is equal to the volume of pyramid E-ABC with base ~ ABC and height BE. Similarly, the volume of pyramid C-ADE, with base A ADE and height h, is equal to the volume of pyramid C-ABE with base A ABE and height h. But since pyramid C-ABE is the same as E-ABC we have ell 3 pyramids of equal volume and therefore equal to one-third the volume of the prism. The other two theorems follow directly from this proposition. The above were known to Archimedes but the value of the use of indivisibles for Cevilieri's proofs had great significance. For example: "The sum of the cubes of the lines of a parallelogram is four times the sum of the cubes of the lines of one of the constituent triangles."[lJ In order to prove this well known t_heorem, Cavilieri employed a special case of the binomial theorem. 46 Let AD = c, GH = a, HE = b then a+b = c for all the lines of the parallelogram. <1> Lc3 = 2La3 + 6La2 b due to symmetry. Now Lc3 = CLC2 as c is constant therefore LC3 = cLc2 = CL {atb):2 = CL82 + 2cLab + CLb2 But earlier Cavilieri showed that 2:a2 = Lb 2 =12:c 2 Therefore l:c~ = 2c2:a2 + 2c2:ab = ~ c2:c 2 + 2c2:ab -:3 = 3 2 = ~ l:c~ + 4La 2 b ;:3 Therefore 4La2 b = 1 2:c 3 and 2:a 2 b = ,1. 2:c 3 12- Substituting into (1) l:c::;; = 2La3 + ~,Lc::;' or l:c'"' = 4La3 Q.E.D. Cevilieri realized that these results could be generalized for all values of n. They are, in a sense, equivalent to what we would now a express by the notation I xndx = an+l 0 n+l Archimedes had come to this conclusion for n = 1, 2 and 3 while Arab mathematicians proved it for n = 4. However, Cavilieri presented the first public statement of the generalized theorem using infinitesimal methods. Later it would be extended to include negative, rational and 47 irrational values of n. The problem remained a geometrical theorem concerning ratio, Cavilieri having no vision of a new analysis such as the calculus. He regarded this method of "indivisibles" as a device in order to avoid the method of exhaustion. He also showed a lack of emphasis on the algebraic and arithmetical elements. Cavilieri regarded area and volume as intuitively clear, geometrical concepts and was preoccupied with their ratios rather than any single numerical value which might be associated with any single area or volume. References: £1] Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York: Dover, 1949, pp. 113-19. £2] Eves, Howard. An Introduction to the History of Mathematics. New York: Holt, Rinehart & Winston, 1953. [3] Newman, James R. <1956) World of Mathematics. New York: Simon & Schuster, 1956, pp. 41-2. [4] Smith, D. E. History of Mathematics Vol. I and II. New York: Dover, 1958, pp. 362-63. Chapter 6 PIERRE de FERMAT Born in Beaumont de Lomagne, Pierre de Fermat the "Prince of Amateurs", was perhaps the last great mathematician to. pursue the subject as a sideline to his career. By profession, Fermat was a modest, retiring lawyer and magistrate of the parliament of Toulouse. The results of his studies are chiefly known through his letters to men such as Pascal, Mersenne and Descartes. Volumes have been written on his advancement of the number theory and analytical geometry. The son of a prosperous leather merchant, Fermat attended the University of Toulouse and the University of Orleans where he received a law degree in 1631. He probably received little education in mathematics, but by 1636 had set down a theory now known as analytic geometry which remained in manuscript form until 1679. He derived his pleasure from mathematical research and wrote only one manuscript during his lifetime, published in book form after his death, entitled "Introduction to Plane and Surface Loci", 1660, concealing his identity under the initials M. P. E. A. S. Upon an offer from Roberval to edit and publish his works in 1637, he is quoted as ·saying, "Whatever of my works is judged worthy of publication, I do not want my name to appear there."[2l Fortunately, he did carry on a voluminous correspondence with friends. Many of his theorems have been found in the margins of his copy of the Sachet edition of Diophantus' Arithmetic. Many of the steps to his proofs were missing due to lack of space and time. Recently, as reported in the Los 48 49 Angeles Times, 3/8/88, and Time magazine "Solving the Puzzle" Volume 131, number 12, 3/21/88, Yoiche Miyaoka, of Japan, claims to have found a proof of Fermat's famous "Last Theorem": "It is impossible to write a cube as a sum of two cubes, a fourth power as a sum of two fourth powers and, in general, any power beyond the second as a sum of two similar powers. For this I have discovered· a truly wonderful proof, but the margin is too small to contain it!"[ 2l At this time, scientists and mathematicians are awaiting a first draft of Miyaoka's manuscript which will enable them to do a precise study of his work and determine if the 321 year old theorem has finally been proved. Although the majority of Fermat's work is in the area of number theory, his development of the analytic geometry gave him a method to represent a curve by an equation, whereby he discovered many properties of the curve. He applied the idea of infinitesimals to finding quadrature, solving maxima and minima problems, and drawing of tangents. Fermat's method of determining maximum and minimum values first appeared in an article in 1638 [1). According to a letter written to Roberval, Fermat claimed he was using the method as far back as 1629. It is substantially that which is applied today to elementary problems. In the expression whose values are to be investigated, he substituted for the independent variable, say x, the new value Desired is the value of x which makes x2 (a-x> a relative maximum or minimum depending on a < 0 or a > 0 x2 2ax - 3x2 + E = 0 Putting E = 0, 2ax - 3x2 = 0 and x 1!:!3.· , or x 0 is obtained as the value of x = .:3 = One must remember that Fermat was not using any concept of function or symbols representing variables. Therefore, although our modern notation suggests the concept of limit as E approaches zero, Fermat's vanishing E is in the sense that it actually is zero. Developing an example without inserting E might be as follows: 2 2 Let ax - x = C where C is any number less than a maximum. Then there are at least two roots, say x, and but when C is a maximum these roots are equal. Therefore: 2 2ax, = 3x1 and x 1 = ~ or x 1 = 0 Naturally this is the same result achieved by setting the derivative 51 equal to zero. Fermat was actually visualizing this value of x as the point at which the curve has a horizontal tangent. His method did not provide any criterion for distinguishing between maxima, minima and points of inflection where the tangent is horizontal. Fermat provided early solutions to the problem, in the differential calculus, of finding a tangent to a curve touching at a given point. Problem: Given parabola BON with vertex D, axis DC and a point B on it, draw BE tangent to parabola BON at B and meeting the axis at E. 52 Taking any point, I, in straight line BE and from it drawing ordinate OI and from B ordinate BC, the ratio of CD to DI will be greater than the ratio of the square on BC to the square on OI because 0 is outside the parabola. ~ > DI Due to similar triangles, BC squared is to OI squared as CE squared is to IE squared. Therefore, the ratio of CD to DI will be greater than that of the square on CE to the square on IE. Since point B is given, the applied line BC is given, hence point C is given as well as CD. Therefore, let CD be equal to d, CE be a, and CI be e. 2 2 2 Then d has a greater ratio to d - e than a to a + e - 2ae _g_ adequal to a 2 ~ __d__ adequal to --=a-_2 ______d-e 2 de - 2dae adequal to a 2 e or de2 + a2 e adequal to 2dae de + a2 adequal to 2ad therefore a 2 = 2ad and a= 2d [3,4lt t[Adequality - "nearest as possible to equality". Fermat used the Diophantine technique of false assumption. He assumed two fractions were equal in order to determine under which conditions they were equal. Fermat "adequat ed" members of an equation. He then simply divided all terms of the adequality ·by a factor until at least one term no longer contained it. The technique was finite, involving 53 neither infinitesimals nor approaches to a limit. Obviously controversy arose over the apparent division by zero.] Fermat's method of tangents was believed to be an application of this method of maxima but he never explained what quantity he was maximizing. A running argument fermented between the followers of Descartes and Fermat which was primarily due to the fact that Fermat believed all his work was from the employment of the method of maxima and minima. Descartes felt that although Fermat's results were correct, his methods were wrong. [3] Fermat used similar methods with the problem of determining the center of gravity of a segment of a paraboloid. Let the center of gravity, 0, be a units from the vertex. On decreasing the altitude h of the segment by E, the center of gravity is changed. Fermat knew that the distance between the centers of gravity of the two segments are proportional to the altitudes and that the volumes of the segments are to each other as the squares of the altitudes. He therefore set up an adequali ty, involving a, h, and E, allowing E to "vanish", obtaining the result that a = <2/3)h. The determination of the center of gravity was not a new \ result, having been calculated by Archimedes nineteen centuries \ earlier. What was significant was that Fermat determined the center by means of methods equivalent to differential calculus rather than a summation similar to our integral calculus. Recognizing the similar results, I therefore find it rather surprising that Fermat did not recognize the inverse nature of the summation and tangent problems. Fermat did do a great amount of work on procedures for the determination of quedratures. As we have seen, many mathematicians 54 a had shown in various forms our modern integral Jxndx = antl. Fermat 0 n+l established a &eneral result for positive integral values of n and possessed a proof for rational fractional values as well. In finding the area under y = xp/q• from 0 to x, he took points on the axis with abscissas x. ex. e2 x.. . where e < 1. The areas of the rectangles at successive ordinates will form an infinite geometric progression. He found the sum of these rectangles to be: ~ r./ -JL :1 x-v· L!- e.~ Now each rectangle had to be infinitely small. In order to do this, he let e = E"" Then the sum becomes: Now let e approach 1. Therefore so does E and the sum becomes ~- r X t- which is the area under the curve. pi-"$- Fermat's Treatise .Q!l. Quadrature simply contains a method of resolving specfic problems. He found answers to specific problems he posed. He did not seek some general method. If he had, he might be known as the father of calculus. 55 References: [ 1] Boyer, Carl B. The History of the Calculus and Its Conceptual Development. New York: Dover, 1949, pp. 120-300. [ 21 Burton, David M. The History of Mathematics. Boston: Allyn & Bacon, 1985, pp. 484-94. [3] Mahoney, Michael Sean. The Mathematical Career of Pierre de Fermat. Princeton, New Jersey: Princeton University Press, 1973. £41 Newman, James R. World 2L Mathematics. New York: Simon & Schuster, 1956, pp. 53, 132. [5] Smith, D. E. History of Mathematics, Vol I & II. New York: Dover: 1958, pp. 377-78. Chapter 7 JOHN WALLIS <1616 -1703) Born in Ashford, Kent, John Wallis was one of the most prominent of Newton's contemporaries. He studied theology at Emmanuel College in Cambridge, receiving his M. A. in 1640, in which year he was ordained. In 1649 he was elected to the Savilian professorship of geometry at Oxford, a position which he held until death. He was awarded the degree of doctor of divinity in 1653 and became Chaplain to Charles II in 1660. He was one of the founders of the Royal Society. Much of Wallis's work has been overshadowed by that of Newton. Synthesizing the methods of Cavilieri with analytic geometry and more advanced algebraic analysis was one of Wallis' chief contributions. He was one of the first to recognize the significance of the generalization of exponents to include negative and fractional as well as the positive integral numbers. He employed Cavalieri's notion of indivisibles in the quadrature of such curves as y = xn, He divided the curve, by parallel ordinates, into strips which approximated parallelograms, . found the sum of them, and determined a value that the sum approached as the number of strips was indefinitely increased. These methods led him to attempt the quadrature of a circle y = <1 - x2 )m letting m = 1/2. Not in possession of the binomial theorem, he formed the series corresponding to m = 0, 1, 2, 3, and tried to find, by interpolation, the expression which would correspond to m = 112. 56 57 By an indirect method he succeeded in expressing n in the form of an infinite product: K = 2x2x4x4x6x6. . . = lim 24 ',(n!) 4 2 lx3x3x5x5x7 ... n~~ [ <2n)!l 2 <2n+l) A synopsis of the proof in modern language is as follows: Integration by parts and mathematical induction tell us: n/2 I2n = I sin2"xdx = K rlx3x... <2n-1) 0 2 l_2x4x ... 2n n/2 2 1 and I 2 n ... 1 - I sin "+ xdx = 2x4 ... {2n) 0 3x5 ... <2n+l> for n € N Since on the interval K. 1 ~ I--· 0 ± 1 ~ J ..,n*? = 2 I2n l2n fi -·2 = 2n+1 2n+2 which approaches 1 . lim I.-..c~l = 1 and .. lim nrf=j n~oo I2n n~oo 2 I2n K, = lim 2 n~oo n -.2 = lim 2x4 ... 2n 2x4 ... <2n) n~~ u3x5 ... <2n+ili 1) o-1x3 ... <2n-1)J = lim 2x2x4x4x6x6 ... <2n) <2n) n~~ lx3x3x5x5 ... <2n-1) <2n) <2n+l) ~ K 2 [ 4) 58 1 Although Wallis did not evaluate J <1-x~> dx, he 0 1 constructed a table for J (1-x2 )n for certain positive integral 0 values of n and with difficult analysis he arrived at the expression: !. = 1 = 3x3x5x5x7x7 .. . 1t 2x4x4x6x6x8 .. . Wallis also obtained the equivalent of our modern formula ds = V1+~):c dx for the length of an element of a curve, thus connecting the problem of rectification with that of quadrature. He recognized that the analytic method was to replace the synthetic, when he defined a conic as a cur·ve of second degr·ee rather than a section of a cone. In his book Opera Mathematics, Vol II, 1693, earlier published as De Algebra Tractatus, 1673, Wallis describes the mannet- in which he made the transition from the geometry of lines to the arithmetic of numbers. He proved that the area of a triangle is the product of the base and one-half the altitude. Using infinitely many parallelograms he found each of their altitudes to be Wallis claimed that it could be verified by inscribed and circumscribed figures. Modern mathematics has eliminated his infinitely small magnitudesi nevertheless, his direct arithmetical analysis rather than the methods of exhaustion, promoted the development of the integral calculus. I Ow l Although discovered before, Wallis's demonstration of the a equivalent of our modern theorem Jxndx = an+l 0 n+l is of particular interest. He first observed the following relationships: 0+1 = l, 0+1+2 = 1 0+1+2+3 = _1 1+1 2 2+2+2 -~2 3+3+3+3 2 Wallis concluded that the result would be the same for an infinite number. 60 He then noted: 0+1 - l +- l . 0+1+4 l + l . 0+ 1+4+9 - l +- l etc. 1+1 - 3 6 J 4+4-+4 - 3 12 I 9+9+9+9 - 3 18 Therefore the greater the number of terms the more closely the number approximates one-third. Thus, according to Wallis, we can eliminate the difference for an infinite number of terms and as a consequence the ratio for an infinite number of terms will be 1/3. He then proceeded to show, by analogy, that the ratios for the third, fourth, fifth and higher powers will be 1/4, 1/5, 1/6 and so on. He then justified the rule for all powers, rational or irrational except -1. CWe now define Jx- 1 dx = lnlxi+C) Wallis used "interpol at ion", his concept of the continuous curve, and induction. The basis for the concept of the definite integral had now been laid by both Fermat and Wallis. However, Fermat did not fully explain the nature of his symbol E and Wallis confused his work with the infinitesimal. Comparing and using small rectangles as lines and writing ;}; = 0 lead to the conception of the integral as a total entity rather than the limit of a sum. As a consequence, no one before Barrow fully realized the significance of the inverse relationship of tangents and quadratures. References: [1] Boyer, Car·l B. The History of the Calculus and its Conceptual Development. New York: Dover, 1949, pp. 170-73. [ 2J Burton, David M. The History of Mathematics. Boston: Allyn & Bacon, 1985, pp. 368-70, 400. [3J Newman, James R. World of Mathematics. New York: Simon & Schuster, 1956, p. 140. 61 [ 4) Olmsted, John M. Real Variables. New York: Appleton- Century-Crafts, 1956, pp. 526-27. [ 5) Smith, D. E. History of Mathematics. New York: 1958, pp. 406-09. Chapter 8 ISAAC BARROW <1630 - 1677> Isaac Barrow was born in London, England. He received his / degree from Trinity College in 1648. In 1662, after extensive travel, he was elected professor of Geometry at Gresham College and in 1664 the first Lucasian professor. In 1670 Barrow resigned the chair in favor of Isaac Newton, devoting his at tent ion to theology. Subsequently he became chaplain to Charles II, master of Trinity College and Vice Chancellor of the University. He ranked as one of the best Greek scholars, one of the leading theologians of England, and a profound student of physics and astronomy. Barrow was probably the first to recognize the genius of Newton when serving as his professor at Cambridge. In his work, he made use of the "differential triangle" which is still the basis of initial work in differentiation. The following is an example of Barrow's method of constructing tangents to curves. Here we see the differential triangle and a method of drawing tangents which resembles differentiation even more than that of Fermat. 62 63 Imagine the circle to be a polygon with many sides. Let one be PQ with PQ intersecting the x axis at T. Let P be T From similar triangles: RT - !t --RP a Since Q is on the circle B> Cx+e) 2 + Subtracting equation A from equation B we have 2ex + e 2 + 2ay + a2 = 0 1. 2ex + e2 + m + a2 =0 2a 2a 2a 2a 2. !Ut + e2 + y+ !L.=,O a 2a 2 3. + + c+ v= o 4.. since RP = y and RT = !t RP a 5. :. !t = RT a y 6. ~+~) t Q+~) = 0 7. ~+~)= -~+~) 8. RT = -y(yt %) (x+%) But e and a are infinitesimal increments and therefore can be neglected. 9. RT becomes -y2 /x Now the tangent at P will form a right angle with OP. Therefore if PT is the desired tangent, from similar triangles: 10. OR = RP RP RT 11. RT = We now know that Barrow was exactly correct. 65 Neither Barrow nor Fermat justified the neglect of the terms a, e and E. According to my references, Barrow appears to have discovered the fundamental inverse relationships between differential and integral calculus, but because he did not develop enough analytic representation such as our symbols y and x, he was unable to make use of them. Although he reduced inverse tangent problems to quadrature he never expressed them in terms of the antiderivative and never expressed his tangent method in an algorithmic form. Had he done so he would have been reknowned as the "inventor of the calculus". When he resigned from the Lucasian chair, Barrow turned his lectures over to Newton and John Collins to prepare for public at ion. In 1670, when Barrow's Geometrical Lectures were published, evidence shows that Newton was already in possession of his methods. About Bar·row and his pr·edecessor·s comes Newton's quote, "If I have seen farther than others, it is because I have stood on the shoulders of giants." [2] References: [1) Boyer, Carl B. The History Qi the Calculus and its Conceptual Development. New York: Dover, 1949, pp.. 164, 182-87. [2) Burton, David~ The History of Mathematics. Boston: Allyn & Bacon, pp. 363-65, 383. [ 3] Newman, James R. <1956>. World of Mathematics. New York: Simon & Schuster, 1956, pp. 53-55. [ 4-l Smith, D. E. History of Mathematics. New York: Dover, 1958, pp. 396-98. Chapter 9 ISAAC NEWTON (1642 - 1727> Isaac Newton was born at Woolsthorpe, in Lincolnshire, on Christmas day, 1642, during the tumultuous era of the Cromwell rebellion. His father, who died before his famous son was born, was a yeoman farmer, as were his ancestors. Newton was a small, feeble child,. extremely inattentive to his studies, ranking lowest in his class. He was skilled in working with his hands, constructing windmills and water clocks, painting and drawing. There was nothing to suggest exceptional genius. As he grew older, Newton's health improved and he became more competitive in his academic skills. At the age of eighteen, he was accepted to the famous and influential Trinity College, in Cambridge. He had little money and became what was then called a Sijar, a student who paid his way through college by doing odd jobs and waiting on his tutor. Little is known about Newton's first two years at Trinity College. He had always been a loner of very sober tastes and rigid morals and probably found his more wealthy and rowdy classmates displeasing. When he was twenty-one, he came into contact with Isaac Barrow, the Lucasian Professor of Mathematics at Cambridge, so named after Henry Lucas who originally provided the money to found the professorship. In 1665 Newton received his B. A. degree with no particular distinction. However, under Barrow's tutelage and encouragement, he had developed a strong interest i~ mathematical physics and optics. When the Great Plague of 1665 became severe, 66 67 il ' closing the University, Newton left the city and returned to his birthplace at Woolsthorpe for eighteen months, in order to pursue these interests. Unlike most men, Newton never showed any wish to publish his works. As a consequence, to this day, historians are sorting out evidence as to the dates of mariy of Newton's discoveries. It is believed that this quiet period of time at Woolsthorpe was one of his most fruitful. As an old man he claimed "All this was in the two plague years of 1665 and 1666, for in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since." [ 1 J It was here that he began his studies on the calculus, described by him as the "method of fluxions", and used this theory in finding the tangent and the radius of curvature at any point on a curve. He worked on the theory of light; began his great investigations into the theory of gravity, and applied the method of fluxions to the study of equations. In 1716 he wrote, "I invented the method of series and fluxions in the year 1665, improved them in the year 1666 and I still have in my custody several mathematical papers written in 1664, 1665, 1666, some of which happen to be dated. "[6] He did not publish these results or tell anybody what he was doing until 1669 when he gave Isaac Barrow a written account of some of his work which was then handed among a few prominent mathematicians. It was not published until over thirty years later. This lack of published material was the cause of the unfortunate dis pute between Newton and Leibniz from 1684 until 1716. The conflict over the originality of ideas on calculus is discussed in Chapter 11. 68 In 1667 Newton returned to Cambridge and was soon elected to a professorship at Trinity College. Shortly after, Isaac Barrow resigned his position as Lucasian Professor in favor of the twenty six year old. During these years, Isaac Newton extended his work on optics, his first reflecting telescope, end the fundamental laws of gravitational attraction end planetary motion. In 1672 Newton was elected a fellow of the Royal Society. The stage was now set for the appearance of what is held by most men as the greatest scientific book ever written, The Mathematical Principles of Natural Philosophy "nat ural philosophy", which we call physical science. ) In 1689 Newton was elected a Member of Parliament by the University. At this time, possibly from exhaustion because of quarrels with Leibniz, Newton became very restless and indifferent to science. He set about seeking an administrative position but received several rejections. He became melancholy and dejected, believing his friends had failed and deceived hi~ He accused his friends of persecuting him and was clearly on the edge of a nervous breakdown. By the end of 1693 he appeared to have recovered and by 1696 had finally secured the title of Warden of the Mint, in 1699 Master of the Mint. In 1703 Newton was elected President of the Royal Society and r·emained so until his death in 1727. During this period he revised his Principia twice, and published his work Opticks. Queen Anne, in 1705, conferred the title of knight on Isaac Newton in order to honor 69 him for his services to science. From that date, he was a national figure, as such an honor had never been conferred for achievements in pure science. Until 1716, in addition to his work at the Mint and with the Royal Society, Newton devoted much of his time to his controversy with Leibniz. For years previous to Leibniz' death in 1716, the latter troubled Newton and embitter-ed his life. Three books, written long before, were published during this period of Newton's life: De Analysi, Arithmetica Universalis and Opticks. In February, 1727, Newton went to London to preside at a meeting of the Royal Society. He became ill and never recovered, dying March 20th. Great honors were paid to him at his funeral. His body lay in state, like that of a sovereign, in a chamber adjoining Westminster Abbey. He was buried in the Abbey itself. No Englishman before or after has received such extraordinay respect on his death. It appears that during the two plague years at Woolsthorpe, while studying and pursuing answers to the questions surrounding elliptical orbits, Isaac Newton, quite by accident, found that "fluxions" and "fluents" were inversely related. ln modern terms: On b n r a, bl, J f{x)dx =lim 2: f where F' He was convinced that the laws that governed celestial motion were the same mechanical laws which governed the motions of bodies on earth even though popular opinion was that they had mysterious laws of their own. Although a deeply religious man, he did not believe that religion or mystery had anything to do with planetary mot ion. Since he believed that the moon was subject to the same laws as a 70 moving body on earth, his problem was to find what force kept it from "flying away" as any body moving in a circular path should do. Newton deduced that the mutual pull of the earth and the moon depends on the inverse square of their distance from one another. He also supposed that the force was directly proportional to the masses. Thus it was not necessary to know the mass of the moon because both the centrifugal force, the force pulling away from the center, and the gravitational force were equal, in order to keep the moon moving at a constant speed. From these conjectures arose Newton's three universal laws of motion, published in the Principia. 1) Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by impressed forces. 2> The change of motion (rete of change of momentum) is proportional to the impressed force and takes place in the direction of the straight line in which the force is impressed. 3) To every action there is always an opposed and equal reaction. Previous to this time, the Ancients and Kepler· believed that to keep a body moving at a constant speed required a force in that direction acting upon it rather than the body continuing at that speed, in that direction, unless another force acted upon it. The indirect consequence of these suppositions was Newton's discovery that the path of a body could be determined from its' velocity and its velocity from its acceleration. Newton's approach to fluxions, as seen by Philip Jourdain, is 71 best understood using our symbolism.. [8] Newton stated that finding velocity of a particle at a given instant and finding the tangent to a curve at a given point are synonymous. We are actually finding the "differential quotient" of a function. Consider a curve described by moving points. If this motion is unifor~ the number measuring any increment of the distance divided by the number measuring the corresponding increment of time gives the value for the measure of average velocity during that time. However, the smaller the increment of time, t, the more nearly the increment of the curve corresponds to a straight line segment. Thus we note the increment of t by "At" and the corresponding increment of distances, by " 1!:::. s" and the average velocity in this element of mot ion by A s At This was defined, by Newton, as a quotient of "infinitely small" increments. For uniform motion where s = at+b, a and b some constants, ~ s = a. Now if we consider some uniformly accelerated ~t motion and denote the velocity by v, then~= h, where h represents At the acceleration. Newton then decomposed the velocity of the moving point into two others parallel to the coordinate axes of x and y. These velocities were called the "fluxions" of x and y, and the velocity of the point was the fluxion of the arc. Reciprocally the arc was the fluent of the velocity with which it was described. From the given equation of the curve one could determine the relations between the fluxions "time". ( 10 3 In order to better explain Newton's approach to the mathematical problems of fluxions and fl uents the following is a synopsis of some sections taken from the Princi~ia. De Alsebra Tractatus. and Opticks. Principia, Book I, Section I, Lemma II - 1687 [4] In this lemma, . Newton explains his concept of the Eudoxus - Archimedes "Principle of Exhaustion" . ..s A B c D F X. 73 Newton inscribed and circumscribed rectangles in a curved figur·e and stated that if the widths of these rectangles are "diminished" and their number approaches infinity, then the sum of the area of the inscribed rectangles will equal the sum of the areas of the circumscribed rectangles. Principia, Book I, Section I, Lemma XI, Scholium [4] This scholium explains Newton's nascent and evanescent quantities as the limits of ratios. He states that, by the "ultimate ratio" of evanescent quantities, he means the ratio of the quantities not befcr·e nor· after· they vanish, but at the instant at which they vanish. He then proceeds to define "ultimate velocity" as that "ultimate ratio." We would now call this: the instantaneous velocity. Newton, at this point, has introduced the derivative which in modern language would be written v = lim t-10 where v is instantaneous velocity, s is the position function and t is the time. Principia, Book II, Section II, Lemma II [4] This lemma describes Newton's "fluxions" as the limits of ratios as their "moments" diminish. In other words, he has replaced the "ultimate ratio" of evanescent quantities with the word "fluxion" Newton clearly shows his knowledge of the product rule for derivatives. Naming a the fluxion of A, b the fluxion of B and so on, he states that the moment, or derivative, of ABC is aBC+AbC+ABc. Newton's flowing quantities, "fluents" are curves "generated" by fluxions. 74 (l • De Quadrature Curvarum, 1704- £4-l Although not published until 1704, Newton claims in this book that he discovered his "method of fluxions" in 1665. Here Newton clearly states that he has found a method of determining quantities from the velocities of the motions or increments with which they are gener·ated and that ~hese velocities are, in fact, the fluxions of the curve. He begins by describing in his laborious fashion, the fluxion of the quantity xn. "In the same time that the quantity x, by flowing, becomes x+O, the quantity x•, will become And the augments 0 and nox•""- 1 + (n2 -n)oox"-2 +... 2 are~o each other as one and nx"- 1 + (n2 -n)ox"-2 +... 2 Now let these augments vanish and their ultimate ratio will be one 1 to nx'-.- ." He proceeds to state that with this new "method of fluxions" one no longe~ needs to resort to sums of infinitely small figures in order to determine the fluents, thereby eliminating the necessity of specific formulas for the quadrature "of various curves. Thus the final breakthrough had occurred. Isaac Newton had discovered the inverse relationship between differentiation and integration together with methods of finding one given the other. \ The world now had a method of finding position, given an acceleration function, work, given the force function, area, given the equation of the curve and many other applications. This theorem is now known as the "first fundamental theorem of 75 integral calculus.": if f References: [1] Andrade, da C., E. N. Sir Isaac Newton. Garden City, New York: Doubleday, 1954, pp. 1-140. [2] Boyer, Carl B. The History Q[ the Calculus and its Conceptual Development. New York: Dover, 1949, pp. 191-203. [ 3] Burton, David M. The History of Mathematics. Boston: Allyn & Bacon, 1985, pp. 325-569. [ 4] Caj ori, Florian. !:. History· of the Conceptions of Limits ! Fluxions in Gr·eat Britain. Chicago: Open Court, 1919, pp. 1-36. [ 5] Cohen, I. Bet·nat'd. Introduction to Newton's Principia. Cambridge, Massachusetts: . Harvard University Press, 1971. [6] De Morgan, Augustus. Essays on the Life and Works of Newton. Chicago: Open Court, 1914, pp. 1-65, 84. [7] Herival, John. The Background to Newton's Principia. England: Oxford at Clarendon, 1965, pp. 2-93. [ 8] Newman, James R. Wor·ld of Mathematics. New York: Simon & Schuster·, 1956, pp. 58-62, 142-45, 254-93. [ 9] Smith, D. E. History of Mathematics. New York: Dover, 1958, pp. 398-404. Chapter 10 GOTTFRIED WILHELM LEIBNIZ <1646 -1716) Gottfried Leibniz, born in Leipzig June 21, 1646, was the only first class pure mathematician produced by Germany in the 17th century. His father, a jurist and professor of moral philosophy, died when Leibniz was six years old. Though left with no direction as far as studies, given access to his father's library, he turned to the world of books. He taught himself Latin and became acquainted with all the classical writers. In 1661 Leibniz became a student at the University of Leipzig. Outstripping all his contemporaries, he soon became something of a child prodigy. He paid little attention to the traditional lectures on Euclid's Elements, much to his regret later. After graduation at the age of seventeen, he began to concentrate on legal studies, earning his master's degree the following year. He acquired a teaching position at Leipzig and he proceeded to write Ars Combinatoria in 1666. Although it contained little new mathematical material, nevertheless, this treatise did establish a new mathematics-like "language of reasoning". It also suggested that a calculus of reasoning could be devised that would provide an automatic method of solution for ·all problems which could be expressed in his scientific language. In 1666 Leibniz left Leipzig when he was refused a degree of doctorate of law on the pretense that he was too young. He received his doctorate at Altdorf, refused the offer of a professorship, and took a post in the service of the archbishop-elect of Mainz. 76 77 Leibniz remained in this diplomatic career until 1676, at which time he moved to Hannover to become librarian to the Duke of Brunswick. The leisure of his diplomatic office allowed Leibniz the opportunity to pursue his career in mathematics. In 1672 he was sent to Paris to present his plan to Louis XIV for France to seize Egypt from the Turks, thereby reducing France's aggressive policy toward Holland and Germany. Although the proposal failed, Leibniz' stay in Paris was extended until 1676, putting him in contact with the great intellectuals of the continent. Among these was Huygens, who recognized the mathematical brilliance of Lei bnh and undertook to fill in what he lacked in training. At this time Leibniz found a rule for determining the sum of certain converging infinite series. He also worked on the invention of a calculating machine which performed all four arithmetic operations. In 1673 Leibniz visited London in order to encourage help in peace negotiations. While there, he made the acquaintance of Henry Oldenburg, permanent Secretary of the Royal Society. At this time, Leibniz was unanimously elected into the society. Although negotiations failed, Leibniz again made the necessary contacts in order to continue his mathematical studies. Returning to Paris in 1673, Leibniz began to develop the principal f~atures and notation for his version of the calculus. His first symbols were understandably poor, but from them he developed our modern notation. In a manuscript dated Oct. 29, 1675, never published, Leibniz made his symbolic connection of the direct and inverse tangent problems. <"To find the locus of the function 78 provided that the locus which determines the subtangent is known">: [ 4] The following is a glossary of Leibniz' original terms: omn = sum ,1. = dy horizontal bars = parenthesis a=dx - THEOREM 1. L omn.R.. = whatever ~ may be 2 THEOREM £ omn Xi. = x omnL - omn omn{ In the middle of the paper Leibniz replaces omn with the symbol J Thus the theorems become: ~ = JJi& and Jx.l = xJl - JJ£a.. 2 a.. Leibniz was not yet using the differential under the integral sign, but it may be found in a manuscript soon after. His two theorems would then become our well known: the difference of the y' s. Hence one equation can be transformed All integrals were understood to be definite, but no special notation for the limits was used. It was noticed by Leibniz that the operation J raised the dimension by one degree. If Jy = z then conversely y = z/d and again Leibniz emphasizes that the d - operation lowers the dimension by one degree. Immediately afterward he presented some examples which show that he placed a constant factor before the integral sign and that he treated the sum of integrals as the integral of a sum. His final remark concerned his 'transmutation': "let the tangent at the point P Q E· then the area generated "ordinatu~' from PQ will be equal to the triangle <112>xy, as can be deduced from the resulting integral X J Edy + ydj dx " 0 2dx [ 4-l A little later in the same year, Leibniz turned to curves with a given subnormal. From ~~.!/& he finds a 2 x =-f.- , not noticing the missing constant of integration due to his integral curve always starting at the origin. During this calculation Leibniz changes his notation, replacing~ by the equivalent dx. By November 1676, Leibniz was able to state the rule for integral and fractional values of n. 80 p • Leibniz' work on the calculus was based upon the "characteristic triangle" used by Barrow and Pascal. For a curve 'I = f !J =f(!<') Leibniz noticed that the characterietic triangle PQR was similar to the triangle PVW, formed by the normal n, the subnormal c;:r , and the ordinate y at the point of contact, and also similar to the triangle PVU formed by the tangent t, subtangent s and the ordinate y. 81 From the similarity of the characteristic triangle PQR and triangle PVW he deduced: dx _ gy or er dx = ydy y- p- Regarding dx and dy as infinitely small and summing up, Lei bniz derived: J r dx = J y dy He supposed the subnormal to be inversely proportional to the ~ ordinate: Gr=~ and found that: # _ eC:x • --..:3 In another application Leibniz used the fact that for sufficiently small values, PR could be considered the same length as ds of the curve. Due to similar triangles, .Jl _ J. ~ .Jl -J. or yds = ndx • PR - dx -'7' ds - dx Summing: J y ds = J ndx which could be used for the formulas to find surface of revolution obtained by rotating the original curve about the x axis.[4,6J Although Leibniz continued to refine the calculus, neither he, Newton nor any immediate successors solved the difficulties which centered about the conception of a "limit." That is, Qx lim A v ~ .. dx Ax-iO Ax where for any number, no matter how small there is a Ax which differs from 0 by less than that number. Leibniz' contribution to modern symbolis~ discovery of many theorems, together with his many other mathematical contributions entitle him to an outstanding place in history, despite the fact that he did not originate the calculus. 82 References: [1] Boyer, Carl R. The History of the Calculus and its Conceptual Development. New York: Dover, 19~9, pp. 178-223. [ 21 Burton, David M. The History of Mathematics. Boston: Allyn & Bacon, 1985, pp. 38~-~05. [3] Hall, A. Rupert. Philosophers at War. London: Cambridge University Press, 1980, pp. 10-259. [41 Hoffman, J. E. Leibniz ~Paris 1672 ~ 1676. London: Cambridge University Press, 1974, pp. 12-307. [51 Mach, Ernest. The Science of Mechanics. Chicago: Open Court, 1960. [6] Newman, James R. World of Mathematics. New York: Simon & Schuster, 1956, pp. 143, 57-61, 152, 275, 286. [7] Smith, D. E. History of Mathematics. New York: Dover, 1956, pp. 417-18. [81 Wolf, A. History of Science. Technolosy and Philosophy in the 16th and 17th Centuries. Boston: Peter Smith, 1968. Chapter 11 THE NEWTON-LEIBNIZ CONTROVERSY Outside his Cambridge lecture room, little was known of Newton's mathematics ·of fluxions. His great Principia, 1687, was couched in the style of Greek geometry. Because Newton used his discoveries only to pursue knowledge of the physical laws of motion, much of his invention of the calculus was hidden in solutions to broader physical problems. Thus arose the controversy between Newton and Leibniz which still exists today. Ther·e is no doubt in most minds that Newton did recognize the reciprocity between integration In De Quadrature Curvarum. 1704, Newton claims that he thought of fluents, fluxions and notation by dots as early as 1665. A manuscript, dated Nov. 13, 1665, gives rules for finding velocities of two or more lines described by bodies, the lines being related to each other by an equation, such as x3 -2ay:=-=tzzx-yyx+zyy-z3 =0. Leibniz, having been in England in 1673, had heard something indefinite as to what Newton had done and desired to know more. In a letter to Oldenburg, secretary of the Royal $ociety, dated June 3, 1676, which Newton wished to be communicated to Leibniz, Newton dwells on the various consequences of the binomial theorem, but says nothing on fluxions. In a second letter dated Oct. 24-th, 1676, he again discusses the binomial theorem, together with the letters of a certain sentence challenging Leibniz to put them together: 83 84 aaaaaa cc d ae eeeeeeeeeeeee f f iiiiiii 11 nnnnnnnnn oooo qqqq rr· ssss ttttttttt vvvvvvvvvvvv x 12v x) Translated, the sentence becomes "given equation any whatsoever, flowing quantities involving, fluxions to find, and vice versa.".[3] On June 21, 1677, Leibniz wrote a letter to Oldenburg stating all conclusions at which he had arrived, including his new symbols. In 1684 Leibniz published his method, whereas Newton's Principia, 1687, still gave nothing more than the most general description of it, and avoided its direct use entirely. By 1695, the mainland had developed this method into a most powerful tool while England paid little attention. Soon the friends of Newton began to claim his rights. Wallis referred only to the "method of fluxions" rather than differential calculus, Fatio de Duilleir published an implied charge of plagiarism on the part Leibniz, and Keill asserted that Leibniz had stolen Newton's method, merely changing names and symbols. After an appeal fr·om Leibniz, the Royal Society appointed a committee of eleven members, to examine the archives and to defend Newton. The committee reported that Leibniz had no method prior· to 1672 when Newton had directed his letters to Collins, later published under the name of Commercium Epistolicum. 1712. The committee provided nQ. evidence that Leibniz had access to those letters. Although Leibniz was to claim condemnation_without a hearing, he received little attention until 1713 at which time he and Newton communicated their claims through a mutual friend, the Abbe Conti. Nothing was resolved. De Morgan claims that Newton demanded to be "absolute monarch" in the world of science and mathematics; therefore, unwilling to admit dual authorship. However, there is strong evidence in "Remarks on the Dispute" published by Leibniz 85 at the end of 1713 in response to Keill's "Letter from London", that he is arguing that Newton had no morel right to the discovery of calculus rather than legal. This document addresses Newton's lack of use of his method but presents no argument for non-existence. Although the controversy still rages, most modern textbooks give independent credit to both men. If, in fact, Leibniz received his fundamental ideas from Newtonian manuscripts and personal correspondence, nevertheless Leibniz' differentials have prevailed, and he is to be credited with our more modern, workable and understandable system. References: [1] Andt~ade, da C., E. N. Sir Isaac Newton. Garden City, New York: Doubleday, 1954, pp. 1-140. (2] Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York: Dover, 1949, pp. 219-23. £3] Burton, David M. The History of Mathematics. Boston: Allyn & Bacon, 1985, pp. 325-405. £41 Cajori, Florian. A History of the Conceptions of Limits ~ Fluxions in Great Britain. Chicago: Open Court, 1919. [51 De Morgan, Augustus. Essays Qrr the Life and Works of Newton. Chicago: Open Court, 1914, pp. 1-65, 84. [6] Hall, A. Rupert. Philosphers at War. London: Cambridge University Press, 1980. [7] Herivel, John. The Background to Newton's Principia. England: Oxford at the Clarendon, 1965, pp. 2-93. [ 10] Newman, James R. World of Mathematics. New York: Simon & Schuster, 1956, pp. 143, 148, 152, 165, 259, 278. Chapter 12 CONCLUSION As the previous chapters show, the discovery of the calculus cannot be ascribed to any single individuaL It evolved from the development of many ideas. Newton's fluxions and Leibniz' differential calculus had been anticipated by Wallis, Fermat and Barrow. Newton and Leibniz were able to generalize on the previous ideas of these men. They, in turn, elaborated on the discoveries of Cavalieri, Kepler and Galilee whose infinitesimal devices, we have seen, were anticipated by Eudoxus and Archimedes. One of the reasons for the twenty-five hundred year interval, during which the ideas leading to calculus evolved, was a hesitancy to separate mathematics from a strictly physical interpretation. Mathematicians emplo~ed infinitesimals and indivisibles, justifying their existance with their consistency with Euclidean geometry. This dependence upon the geometric idea of limits had to be removed . .. Calculus, in the modern sense, had to wait for Augustin-Louis Cauchy <1789-1857> who developed an acceptable arithmetic theory of limits and gave a formal, precision to the . concepts of cent inuit y, differentiability and the definite integral. 86 87 ,, . BIBLIOGRAPHY Andrade, da C., E. N. Sir Isaac Newton. Garden City, New York: Doubleday, 1954. Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York: Dover, 1949. Burton, David M. ~ History Q[ Mathematics. Boston: Allyn and Bacon, 1985. Butts, R. E. & Davis, J. W. The Methodological Heritage Q[ Newton. Toronto: University of Toronto Press, 1970. Cajori, Florian, ~ History of the Conceptions of Limits ~ Fluxions in Great Britain. Chicago: Open Court, 1919. Calinger, Ronald. Classics of Mathematics. Oak Park, Illinois: Moore, 1982. Cohen, I Bernard. Introduction to Newton's 'Principia'. Cambridge, Massachusetts: Harvard University Press, 1971. De Morgan, Augustus. Essays on the Life and Works Q[ Newton. Chicago: Open Court, 1914. Eves, Howard. An Introduction to the History of Mathematics. New York: Holt, Rinehart, & Winston, 1953. Greenstreet, W. J. Isaac Newton 1642 ~ 1727. London: G. Bell & Sons, 1927. Hall, A. Rupert. Philosophers ~ War. London: Cambridge University Press, 1980. Heath, Thomas Little Heat~ ~ L. Herivel, John. The Background to Newton's Principia. England: Oxford at the Clarendon Press, 1965. · Hoffman, J. E. Leibniz in Paris 1672 ~ 1676. London: Cambridge University Press, 1974. Mach, Ernest. The Science Q[ Mechanics. Chicago: Open Court 1960. 88 p • Mahoney, Michael Sean. The Mathematical Career of Pierre de Fermat. Princeton, New Jersey: Princeton University Press, 1973. Newman, James R. World QL Mathematics. New York: Simon & Schuster, 1956. Newton, Sir Isaac. The Mathematical Principles of Natural Philosophy. Olmstead, John M. Real Variables. New York: Appleton- Century-Crafts, 1956. Smith, D. E. History of Mathematics. New York: Dover, 1958. Wolf, A. History QL Science Technology and Philosophy in the 16th and 17th Centuries. Boston: Peter Smith, 1968. time along AE AE and time alon~ AE = AE